diff --git a/-NE2T4oBgHgl3EQfmAfb/content/tmp_files/2301.03995v1.pdf.txt b/-NE2T4oBgHgl3EQfmAfb/content/tmp_files/2301.03995v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..a737e860a73c7ccb366e10658bcfe43e2dcc0443 --- /dev/null +++ b/-NE2T4oBgHgl3EQfmAfb/content/tmp_files/2301.03995v1.pdf.txt @@ -0,0 +1,591 @@ +Efficiently unquenching QCD+QED at O(𝜶) +Tim Harris,𝑎,∗ Vera Gülpers,𝑎 Antonin Portelli𝑎 and James Richings𝑎,𝑏 +𝑎School of Physics and Astronomy, University of Edinburgh, +Edinburgh EH9 3FD, United Kingdom +𝑏EPCC, University of Edinburgh, +EH8 9BT, Edinburgh, United Kingdom +E-mail: tharris@ed.ac.uk +We outline a strategy to efficiently include the electromagnetic interactions of the sea quarks +in QCD+QED. When computing iso-spin breaking corrections to hadronic quantities at leading +order in the electromagnetic coupling, the sea-quark charges result in quark-line disconnected +diagrams which are challenging to compute precisely. An analysis of the variance of stochastic +estimators for the relevant traces of quark propagators helps us to improve the situation for certain +flavour combinations and space-time decompositions. We present preliminary numerical results +for the variances of the corresponding contributions using an ensemble of 𝑁f = 2 + 1 domain-wall +fermions generated by the RBC/UKQCD collaboration. +The 39th International Symposium on Lattice Field Theory (Lattice2022), +8-13 August, 2022 +Bonn, Germany +∗Speaker +© Copyright owned by the author(s) under the terms of the Creative Commons +Attribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). +https://pos.sissa.it/ +arXiv:2301.03995v1 [hep-lat] 10 Jan 2023 + +Efficiently unquenching QCD+QED at O(𝛼) +Tim Harris +1. +Introduction +Several lattice QCD predictions which form important input for precision tests of the Standard +Model have uncertainties at or below the 1% level, for example the HVP contribution to (𝑔 − 2)𝜇, +𝑓𝐾/ 𝑓𝜋, 𝑔A or the Wilson flow scale √𝑡0 to name a few [1, 2]. +However, to further improve +such predictions, QCD with iso-spin symmetry is not a sufficiently accurate effective description +of the low-energy dynamics and QED, which contributes one source of iso-spin breaking due to +the different up- and down-quark electric charges, must be included. Recent efforts have been +successful at including iso-spin breaking corrections, and some of which fully account for the +effects of the sea-quark electric charges [3, 4, 5, 6, 7]. +Nevertheless, many computations of +iso-spin breaking effects still neglect to incorporate these dynamical effects in an approximation +known as electroquenching. As the FLAG report notes in Section 3.1.2 [2], computations using the +electroquenched approximation might feature an uncontrolled systematic error. +In this work we aim to include the effects of the electric charge of the sea quarks in the +perturbative method known as the RM123 approach. This amounts to computing at least two +additional Wick contractions. +In order to sum the vertices in the resulting diagrams over the +lattice volume, some approximations must be used which often introduce additional fluctuations, +for example due to the auxiliary fields of a stochastic estimator. Here we investigate some simple +decompositions which may avoid large contributions to the variance, so that sufficiently precise +results can be obtained to systematically include all sources of iso-spin breaking without incurring +a large computational cost. +2. +Sea-quark effects in the RM123 method +Due to the smallness of the fine-structure constant 𝛼 ∼ 1/137 and the renormalized light- +quark mass difference (𝑚R +u − 𝑚R +d )/Λ ∼ 1%, it is natural to expand physical observables (i.e. in +QCD+QED) in these parameters to compute iso-spin breaking corrections, as was first outlined in +Refs. [8, 9]. In the resulting expansion of an observable 𝑂 +⟨𝑂⟩ = ⟨𝑂⟩ +��� +𝑒=0 + 1 +2𝑒2� 𝜕 +𝜕𝑒 +𝜕 +𝜕𝑒 ⟨𝑂⟩ +� +𝑒=0 + . . . +(1) +the leading corrections in the electric charge 𝑒 = +√ +4𝜋𝛼 are parameterized in terms of the correlation +function +𝜕 +𝜕𝑒 +𝜕 +𝜕𝑒 ⟨𝑂⟩ = (−i)2 +∫ +d4𝑥 +∫ +d4𝑦 ⟨𝐽𝜇(𝑥)𝐴𝜇(𝑥)𝐽𝜈(𝑦)𝐴𝜈(𝑦)𝑂⟩c +(2) +where the electromagnetic current for u, d, s quark flavours is defined +𝐽𝜇 = +∑︁ +𝑓 =u,d,s +𝑄 𝑓 ¯𝜓 𝑓 𝛾𝜇𝜓 𝑓 , +𝑄u = 2 +3, +𝑄d = 𝑄s = −1 +3. +(3) +By choosing the expansion point to be a theory with 𝛼 = 0 and iso-spin symmetry 𝑚u = 𝑚d, +only correlation functions in the 𝑁f = 2 + 1 theory need to be evaluated, which we denote with +𝑒 = 0 in Eq. (1). The precise definition of such a theory using an additional set of renormalization +conditions is necessary to fix the meaning of the leading-order term on the right-hand side (and +2 + +Efficiently unquenching QCD+QED at O(𝛼) +Tim Harris +𝑊1 +𝑂 +𝑊2 +𝑂 +𝑊3 +𝑂 +𝑊4 +𝑂 +Figure 1: Wick contractions which appear at leading order in the expansion of a hadronic observable 𝑂 +in the electromagnetic coupling. Each closed fermion line has contributions from all of the quark flavours +u, d, s, . . . with the appropriate charge factors. +conversely the iso-spin breaking corrections themselves). Otherwise the predictions of QCD+QED +are unambiguously defined, up to its intrinsic accuracy, by fixing 𝑁f quark masses and the QCD +coupling as the electric coupling does not renormalize at this order. In the above, the ellipsis stands +for the mass counterterms which are needed to make physical predictions due to the contribution to +the quark self-energy induced by QED. +After integrating out the fermion and photon fields, the resulting Wick contractions 𝑊𝑖 are +shown in Fig. 1, which contribute to the derivative with respect to the electric charge through the +connected correlation function +𝜕 +𝜕𝑒 +𝜕 +𝜕𝑒 ⟨𝑂⟩ = +4 +∑︁ +𝑖=1 +⟨𝑂𝑊𝑖⟩c. +(4) +The first two subdiagrams, which arise soley from the electric charges of the sea quarks, can be +expressed in terms of a convolution with the photon propagator (in some fixed gauge) 𝐺 𝜇𝜈(𝑥) = +⟨𝐴𝜇(𝑥)𝐴𝜈(0)⟩ +𝑊1,2 = −𝑎8 ∑︁ +𝑥,𝑦 +𝐻𝜇𝜈 +1,2(𝑥, 𝑦)𝐺 𝜇𝜈(𝑥 − 𝑦), +(5) +where 𝐻1,2 are the traces of quark propagators 𝑆 𝑓 (𝑥, 𝑦) = ⟨𝜓 𝑓 (𝑥) ¯𝜓 𝑓 (𝑦)⟩ +𝐻𝜇𝜈 +1 (𝑥, 𝑦) = +∑︁ +𝑓 ,𝑔 +𝑄 𝑓 𝑄𝑔 tr{𝛾𝜇𝑆 𝑓 (𝑥, 𝑥)} tr{𝛾𝜈𝑆𝑔(𝑦, 𝑦)}, +(6) +𝐻𝜇𝜈 +2 (𝑥, 𝑦) = − +∑︁ +𝑓 +𝑄2 +𝑓 tr{𝛾𝜇𝑆 𝑓 (𝑥, 𝑦)𝛾𝜈𝑆 𝑓 (𝑦, 𝑥)}. +(7) +These two diagrams are the main subject of these proceedings, and the techniques advocated for +the first can be effectively reused for the third diagram, 𝑊3. In the following sections we introduce +stochastic estimators only for the quark lines and compute the subdiagrams by convoluting with the +exact photon propagator which avoids introducing additional stochastic fields for the U(1) gauge +potential. The final diagram 𝑊4, which only contributes if the observable 𝑂 depends explicitly +on the (charged) fermion fields, is the only one surviving the electroquenched approximation, and, +can in most cases be computed efficiently provided that the leading-order diagram is already under +control. +3 + +Efficiently unquenching QCD+QED at O(𝛼) +Tim Harris +We note that the variance of the contributions to the connected correlation functions on the +r.h.s. of Eq. (4) crudely factorizes +𝜎2 +𝑂𝑊1,2 ≈ ⟨𝑂⟩2 +c ⟨𝑊1,2⟩2 +c + ⟨𝑂𝑊1,2⟩c +(8) +≈ 𝜎2 +𝑂𝜎2 +𝑊1,2, +(9) +where in the first line we have made the Gaussian approximation, and in the second line we have +assumed that the fluctuations are much larger than the signal ⟨𝑂𝑊1,2⟩c. Thus, in the following +sections we will analyse the variance of individual subdiagrams 𝑊1,2 in order to gain a rough +insight into the fluctuations of the total correction, in a similar fashion to the analysis of Ref. [10]. +In that case, however, the correction to the factorization of the variance is exponentially suppressed +in the separation between the vertices of the subdiagrams. +3. +Quark-line disconnected subdiagram 𝑊1 +We begin by noting that the hadronic part of the diagram factorizes into two traces, +𝐻𝜇𝜈 +1 (𝑥, 𝑦) = 𝑇𝜇(𝑥)𝑇𝜈(𝑦), +(10) +each of which, with the current defined in Eq. (3) and in the 𝑁f = 2 + 1 theory with iso-spin +symmetry, is the difference of the light- and strange-quark propagators +𝑇𝜇(𝑥) = 1 +3 tr{𝛾𝜇[𝑆ud(𝑥, 𝑥) − 𝑆s(𝑥, 𝑥)]}. +(11) +It is convenient to rewrite this difference as a product [10] +𝑆ud − 𝑆s = (𝑚s − 𝑚ud)𝑆ud𝑆s +(12) +which makes the explicit suppression of 𝑇𝜇 in the SU(3)-symmetry breaking parameter 𝑚s − 𝑚ud +explicit. This additionally results in a suppression of the variance of 𝑊1 by (𝑚s − 𝑚ud)4. This +suppression results in a cancellation of a quartic short-distance divergence in the variance of the +contribution of each individual flavour to 𝑊1, explaining this favourable flavour combination. +While the identity in Eq. (12) is easily derived for Wilson-type fermions, here we sketch that +it holds exactly for the domain-wall fermion valence propagator 𝑆 𝑓 = ˜𝐷−1 +𝑓 which (approximately) +satisfies the Ginsparg-Wilson relation [11]. Recalling the definition of ˜𝐷 𝑓 in terms of the 5D +Wilson matrix 𝐷5, 𝑓 (see Ref. [12] for unexplained notation) +˜𝐷−1 +𝑓 = (P−1𝐷−1 +5, 𝑓 𝑅5P)11, +(13) +where the matrix indices indicate the coordinate in the fifth dimension, the result is obtained +immediately from +˜𝐷−1 +ud − ˜𝐷−1 +s += (𝑚s − 𝑚ud)(P𝐷−1 +5,ud𝑅5𝐷−1 +5,s𝑅5)11 +(14) +by noting that the following matrix projects on the physical boundary +(𝑅5)·· = (𝑅5P)·1(P−1)1·. +(15) +4 + +Efficiently unquenching QCD+QED at O(𝛼) +Tim Harris +𝐿/𝑎 +𝑇/𝑎 +𝑚 𝜋 +𝑚 𝜋𝐿 +𝑎 +𝑁cfg +24 +64 +340 MeV +4.9 +0.12 fm +50 +Table 1: The parameters of the C1 ensemble of 𝑁f = 2 + 1 Shamir domain-wall fermions used in the +numerical experiments in this work, see Ref. [17] for details. +The preceding identity is easily demonstrated using the explicit representations +𝑅5 = +��� +� +𝑃+ +𝑃− +��� +� +, +P−1 = +������ +� +𝑃− +𝑃+ +𝑃+ +... +... +𝑃+ +𝑃− +������ +� +, +(16) +where 𝑃± = 1 ± 𝛾5. +Using the identity for the difference, there are two independent estimators for the trace +Θ𝜇(𝑥) = 1 +3 (𝑚s − 𝑚ud) 1 +𝑁s +𝑁s +∑︁ +𝑖=1 +𝜂† +𝑖 (𝑥)𝛾𝜇{𝑆ud𝑆s𝜂𝑖}(𝑥), +(17) +T𝜇(𝑥) = 1 +3 (𝑚s − 𝑚ud) 1 +𝑁s +𝑁s +∑︁ +𝑖=1 +{𝜂† +𝑖 𝑆s}(𝑥)𝛾𝜇{𝑆ud𝜂𝑖}(𝑥), +(18) +where the auxiliary quark fields 𝜂𝑖(𝑥) have zero mean and finite variance. +The properties of +both estimators were investigated in detail in Ref. [10], where it was shown that the contribution +to the variance from the auxiliary fields for the second split-even estimator was in the region of +a factor O(100) smaller than the first standard estimator, which translates into the same factor +reduction in the cost. The split-even estimator has since been used extensively for disconnected +current correlators [13, 14, 15], while in the context of the twisted-mass Wilson formulation similar +one-end trick estimators have often been employed for differences of twisted-mass propagators [16]. +In this work we propose an estimator for the first diagram 𝑊1 using +W1 ≈ +� +𝑎4 ∑︁ +𝑥 +T𝜇(𝑥) +� � +𝑎4 ∑︁ +𝑦 +T𝜈(𝑦)𝐺 𝜇𝜈(𝑥 − 𝑦) +� +(19) +where independent estimators are used for the two traces to avoid incurring a bias with a finite +sample size. The convolution in the second parentheses can be efficiently computed using the +Fast Fourier Transform (FFT). With a minor modification, an estimator using all possible unbiased +combinations of samples can be written at the cost of performing O(𝑁s) FFTs. +The standard +estimator is obtained by replacing both occurances of T𝜇 with Θ𝜇 in Eq. (19). +We performed an analysis of the variance for the standard and split-even estimators for W1 +using the domain-wall ensemble generated by the RBC/UKQCD collaboration whose parameters +are listed in Tab. 1. The photon propagator is computed in the QED𝐿 formulation [18] in the +Feynman gauge. The results for the variances, which are dimensionless numbers, are shown in +Fig. 2. In addition, we plot the variance for the contribution of a single flavour Wu +1 using the +5 + +Efficiently unquenching QCD+QED at O(𝛼) +Tim Harris +10−4 +10−3 +10−2 +10−1 +100 +101 +102 +103 +104 +105 +106 +107 +108 +109 +1 +10 +100 +1000 +σ2 +Ns +Wu +1 +Wuds +1 +(standard) +Wuds +1 +(split-even) +1/N 2 +s +Figure 2: Left: Comparison of the variance versus the number of sources for the 𝑊1 quark-line disconnected +diagram, using a single flavour (red squares), the standard estimator for u, d, s flavours (blue circles) and the +split-even estimator (green triangles). The dashed line shows 1/𝑁2 +s scaling. In this figure, the (local) currents +are not renormalized and the charge factors are not included. +standard estimators for the traces. We note that all the variances are dominated by the fluctuations +of the auxiliary fields for small 𝑁s, and in particular scale like 1/𝑁2 +s in that region. +As expected, the standard estimator including the light-quark and strange-quark contributions +(blue circles) is suppressed with respect to the contribution of a single flavour (red squares). +Furthermore, the variance of the split-even estimator (green triangles) is reduced by a factor of 104 +with respect to the standard one (blue circles). This reduction is commensurate with the reduction +in the variance observed for the disconnected contribution to the current correlator [10], which +suggests the same mechanisms are present here. For 𝑁s ∼ 100, the variance is independent of +the number of auxiliary field samples which indicates that it is dominated by the fluctuations of +the gauge field. In this case no further variance reduction is possible for a fixed number of gauge +configurations. Finally we note that the convolution of the second parentheses of Eq. (19) can be +simply inserted sequentially in any of the diagrams of type 𝑊3. +4. +Quark-line connected subdiagram 𝑊2 +In contrast to the quark-line disconnected subdiagram, there is no cancellation in the variance +in the connected subdiagram 𝑊2 between the light and strange-quark contributions. In this case, +power counting suggests that the variance diverges with the lattice spacing like 𝑎−4 as 𝑎 → 0 and is +expected to be dominated by short-distance contributions. Translation averaging should therefore +be very effective and one way to implement it is to use an all-to-all estimator [19] for the quark +propagator +S 𝑓 (𝑥, 𝑥 + 𝑟) = 1 +𝑁s +𝑁s +∑︁ +𝑖=1 +{𝑆 𝑓 𝜂𝑖}(𝑥)𝜂† +𝑖 (𝑥 + 𝑟), +(20) +6 + +Efficiently unquenching QCD+QED at O(𝛼) +Tim Harris +10−7 +10−6 +10−5 +10−4 +10−3 +10−2 +10−1 +100 +101 +102 +103 +104 +105 +106 +107 +108 +109 +0 +2 +4 +6 +8 +10 +12 +σ2 +|r|/a +H2(r)G(r) +H2, Ns = 1 +¯H2, NX = 1 +H2, Ns = ∞ +10−2 +10−1 +100 +101 +102 +103 +104 +105 +106 +1 +10 +100 +1000 +σ2 +Ninv +W2, R/a = 4 +� +r≤R H2G +� +r>R ¯H2G +R = 0, Ns = ∞ +Figure 3: Left: the variance for the stochastic estimator (red squares) and point source estimator (blue +circles) for the minimum number of inversions required, for the contribution with fixed separation between +the currents |𝑟|. The green triangle indicates the gauge variance for the point 𝑟 = 0. Right: the variance for +the short-distance (red squares) and long-distance (blue circles) for the choice 𝑅/𝑎 = 4, versus the number +of inversions. The green band indicates the gauge variance for the contribution from 𝑟 = 0 only. The dashed +lines indicate the expected leading 𝑁−2 +inv and 𝑁−1 +inv scaling for the short- and long-distance components. +using independent fields for each propagator in the trace +H 𝜇𝜈 +2 +(𝑟) = 𝑎4 ∑︁ +𝑥 +∑︁ +𝑓 +𝑄2 +𝑓 tr{𝛾𝜇S 𝑓 (𝑥, 𝑥 + 𝑟)𝛾𝜈S 𝑓 (𝑥 + 𝑟, 𝑥)}. +(21) +As written, the estimator is feasible to compute for a small number of separations 𝑟 between the +vertices and, although it introduces a (mild) signal-to-noise ratio problem at large 𝑟, should be +efficient at small |𝑟| ≤ 𝑅 given the leading extra contribution vanishes like 𝑁−2 +s , c.f. Sec. 3. +For the remainder |𝑟| > 𝑅, we propose using 𝑁𝑋 randomly selected point sources 𝑋𝑛 [20] +¯𝐻𝜇𝜈 +2 (𝑟) = 𝐿3𝑇 +𝑁𝑋 +𝑁𝑋 +∑︁ +𝑛=1 +𝐻𝜇𝜈 +2 (𝑋𝑛, 𝑋𝑛 + 𝑟) +(22) +so that the total is split between short- and long-distance contributions +W2 = 𝑎4 ∑︁ +|𝑟 |≤𝑅 +H2(𝑟)𝐺 𝜇𝜈(𝑟) + 𝑎4 ∑︁ +𝑟>𝑅 +¯𝐻𝜇𝜈 +2 (𝑟)𝐺 𝜇𝜈(𝑟), +(23) +using the efficient stochastic estimator for the noisy short-distance contribution. Ref. [21] introduced +an importance sampling based on current separations for higher-point correlation functions, whereas +in this case we make the separation based on the expected contributions to the variance. This +approach avoids completely factorizing the trace which would require either O(𝑉) contractions or +O(𝑁2 +s ) FFTs to include the photon line which we deemed unfeasible. +In Fig. 3 (left) we illustrate the variance of each of the terms in Eq. (23) for the sum over a +fixed separation |𝑟| between the currents, for the case 𝑁s = 𝑁𝑋 = 1. As expected, the variance +from the contribution around |𝑟| ∼ 0 dominates both the stochastic (red squares) and point source +estimator (blue circles), and we observe the mild signal-to-noise ratio problem in the stochastic +7 + +Efficiently unquenching QCD+QED at O(𝛼) +Tim Harris +estimator. The green triangle denotes the gauge variance for the case 𝑟 = 0, which is approximately +suppressed by (𝐿3𝑇)/𝑎4 compared to 𝑁𝑋 = 1 indicating translation averaging is very effective for +the short-distance contribution. In the right-hand panel, we see variance of the short- and long- +distance contributions with the choice 𝑅/𝑎 = 4 as a function of the number of inversions (where +𝑁𝑋 = 1 corresponds to 12 inversions). The variance is dominated by the short-distance contribution +(red squares) which however scales favourably like 𝑁−2 +inv, while the long-distance contribution (blue +circles) which scales only like 𝑁−1 +inv is much suppressed. Deviations from the former scaling indicate +that the gauge variance may be reached with just 𝑁inv ∼ 1000, which although is larger than required +for 𝑊1 is still achievable with modern computational resources, and universal for all observables. +5. +Conclusions +In this work we have examined the Wick contractions which arise due to the charge of the +sea quarks in the RM123 method. Such diagrams contribute, in principle, even to observables +constructed from neutral fields and are therefore ubiquitous in the computation of iso-spin breaking +corrections. We have proposed stochastic estimators for the quark lines in such diagrams which +completely avoids the need to sample the Maxwell action stochastically, thus eliminating one +additional source of variance. As for the case of disconnected contributions to current correlators, +we have shown it is beneficial to consider certain flavour combinations which have greatly suppressed +fluctuations. We have shown that the split-even estimators generalize also to domain-wall fermions +and perform well compared with naïve estimators. Thus the frequency-splitting strategy of Ref. [10] +should generalize appropriately for this fermion formulation. In the second topology, however, there +is no cancellation of the short-distance effects in the variance by considering multiple flavours. +In this case, we propose decomposing the diagram into a short-distance part to be estimated +stochastically and a long-distance part estimated using position-space sampling. The variance is +reduced sufficiently so that the gauge variance can be reached with a reasonable computational cost. +Given their short-distance nature, these estimators should also succeed with smaller quark masses, +and furthermore as the diagrams are universal to all iso-spin breaking corrections we anticipate +that these simple decompositions ought to be beneficial in large-scale simulations. In particular we +are developing these methods for refinements of our computations of iso-spin breaking corrections +within the RBC/UKQCD collaboration, for example to meson (leptonic) decay rates [22, 23]. +Acknowledgments +We use the open-source and free software Grid as the data parallel C++ library +for the lattice computations [24]. The authors warmly thank the members of the RBC/UKQCD +collaboration for valuable discussions and the use of ensembles of gauge configurations. T.H., A.P. +and V.G. are supported in part by UK STFC 1039 grant ST/P000630/1. A.P. and V.G. received +funding from the European Research Council (ERC) under the European Union’s Horizon 2020 +research and innovation programme under grant agreement No 757646 and A.P. additionally under +grant agreement No 813942. This work used the DiRAC Extreme Scaling service at the University +of Edinburgh, operated by the Edinburgh Parallel Computing Centre on behalf of the STFC DiRAC +HPC Facility (www.dirac.ac.uk). This equipment was funded by BEIS capital funding via STFC +capital grant ST/R00238X/1 and STFC DiRAC Operations grant ST/R001006/1. DiRAC is part of +the National e-Infrastructure. +8 + +Efficiently unquenching QCD+QED at O(𝛼) +Tim Harris +References +[1] +T. Aoyama et al. In: Phys. Rept. 887 (2020), pp. 1–166. arXiv: 2006.04822 [hep-ph]. +[2] +Y. Aoki et al. In: Eur. Phys. J. C 82.10 (2022), p. 869. arXiv: 2111.09849 [hep-lat]. +[3] +S. Aoki et al. In: Phys. Rev. D 86 (2012), p. 034507. arXiv: 1205.2961 [hep-lat]. +[4] +T. Ishikawa et al. In: Phys. Rev. Lett. 109 (7 Aug. 2012), p. 072002. +[5] +R. Horsley et al. In: J. Phys. G 46 (2019), p. 115004. arXiv: 1904.02304 [hep-lat]. +[6] +L. Bushnaq et al. In: (Sept. 2022). arXiv: 2209.13183 [hep-lat]. +[7] +S. Borsanyi et al. In: Nature 593.7857 (2021), pp. 51–55. arXiv: 2002.12347 [hep-lat]. +[8] +G. M. de Divitiis et al. In: JHEP 04 (2012), p. 124. arXiv: 1110.6294 [hep-lat]. +[9] +G. M. de Divitiis et al. In: Phys. Rev. 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In: PoS LATTICE2015 (2016), p. 023. +9 + diff --git a/-NE2T4oBgHgl3EQfmAfb/content/tmp_files/load_file.txt b/-NE2T4oBgHgl3EQfmAfb/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..6929d31816d1e8eaae56672ef135d833fb82f6b6 --- /dev/null +++ b/-NE2T4oBgHgl3EQfmAfb/content/tmp_files/load_file.txt @@ -0,0 +1,371 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf,len=370 +page_content='Efficiently unquenching QCD+QED at O(𝜶) Tim Harris,𝑎,∗ Vera Gülpers,𝑎 Antonin Portelli𝑎 and James Richings𝑎,𝑏 𝑎School of Physics and Astronomy, University of Edinburgh, Edinburgh EH9 3FD, United Kingdom 𝑏EPCC, University of Edinburgh, EH8 9BT, Edinburgh, United Kingdom E-mail: tharris@ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content='uk We outline a strategy to efficiently include the electromagnetic interactions of the sea quarks in QCD+QED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' When computing iso-spin breaking corrections to hadronic quantities at leading order in the electromagnetic coupling, the sea-quark charges result in quark-line disconnected diagrams which are challenging to compute precisely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' An analysis of the variance of stochastic estimators for the relevant traces of quark propagators helps us to improve the situation for certain flavour combinations and space-time decompositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' We present preliminary numerical results for the variances of the corresponding contributions using an ensemble of 𝑁f = 2 + 1 domain-wall fermions generated by the RBC/UKQCD collaboration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' The 39th International Symposium on Lattice Field Theory (Lattice2022), 8-13 August, 2022 Bonn, Germany ∗Speaker © Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-NoDerivatives 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content='0 International License (CC BY-NC-ND 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content='0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' https://pos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content='sissa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content='it/ arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content='03995v1 [hep-lat] 10 Jan 2023 Efficiently unquenching QCD+QED at O(𝛼) Tim Harris 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' Introduction Several lattice QCD predictions which form important input for precision tests of the Standard Model have uncertainties at or below the 1% level, for example the HVP contribution to (𝑔 − 2)𝜇, 𝑓𝐾/ 𝑓𝜋, 𝑔A or the Wilson flow scale √𝑡0 to name a few [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' However, to further improve such predictions, QCD with iso-spin symmetry is not a sufficiently accurate effective description of the low-energy dynamics and QED, which contributes one source of iso-spin breaking due to the different up- and down-quark electric charges, must be included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' Recent efforts have been successful at including iso-spin breaking corrections, and some of which fully account for the effects of the sea-quark electric charges [3, 4, 5, 6, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' Nevertheless, many computations of iso-spin breaking effects still neglect to incorporate these dynamical effects in an approximation known as electroquenching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' As the FLAG report notes in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content='2 [2], computations using the electroquenched approximation might feature an uncontrolled systematic error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' In this work we aim to include the effects of the electric charge of the sea quarks in the perturbative method known as the RM123 approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' This amounts to computing at least two additional Wick contractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' In order to sum the vertices in the resulting diagrams over the lattice volume, some approximations must be used which often introduce additional fluctuations, for example due to the auxiliary fields of a stochastic estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' Here we investigate some simple decompositions which may avoid large contributions to the variance, so that sufficiently precise results can be obtained to systematically include all sources of iso-spin breaking without incurring a large computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' Sea-quark effects in the RM123 method Due to the smallness of the fine-structure constant 𝛼 ∼ 1/137 and the renormalized light- quark mass difference (𝑚R u − 𝑚R d )/Λ ∼ 1%, it is natural to expand physical observables (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' in QCD+QED) in these parameters to compute iso-spin breaking corrections, as was first outlined in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' [8, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' In the resulting expansion of an observable 𝑂 ⟨𝑂⟩ = ⟨𝑂⟩ ��� 𝑒=0 + 1 2𝑒2� 𝜕 𝜕𝑒 𝜕 𝜕𝑒 ⟨𝑂⟩ � 𝑒=0 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' (1) the leading corrections in the electric charge 𝑒 = √ 4𝜋𝛼 are parameterized in terms of the correlation function 𝜕 𝜕𝑒 𝜕 𝜕𝑒 ⟨𝑂⟩ = (−i)2 ∫ d4𝑥 ∫ d4𝑦 ⟨𝐽𝜇(𝑥)𝐴𝜇(𝑥)𝐽𝜈(𝑦)𝐴𝜈(𝑦)𝑂⟩c (2) where the electromagnetic current for u, d, s quark flavours is defined 𝐽𝜇 = ∑︁ 𝑓 =u,d,s 𝑄 𝑓 ¯𝜓 𝑓 𝛾𝜇𝜓 𝑓 , 𝑄u = 2 3, 𝑄d = 𝑄s = −1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' (3) By choosing the expansion point to be a theory with 𝛼 = 0 and iso-spin symmetry 𝑚u = 𝑚d, only correlation functions in the 𝑁f = 2 + 1 theory need to be evaluated, which we denote with 𝑒 = 0 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' The precise definition of such a theory using an additional set of renormalization conditions is necessary to fix the meaning of the leading-order term on the right-hand side (and 2 Efficiently unquenching QCD+QED at O(𝛼) Tim Harris 𝑊1 𝑂 𝑊2 𝑂 𝑊3 𝑂 𝑊4 𝑂 Figure 1: Wick contractions which appear at leading order in the expansion of a hadronic observable 𝑂 in the electromagnetic coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' Each closed fermion line has contributions from all of the quark flavours u, d, s, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' with the appropriate charge factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' conversely the iso-spin breaking corrections themselves).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' Otherwise the predictions of QCD+QED are unambiguously defined, up to its intrinsic accuracy, by fixing 𝑁f quark masses and the QCD coupling as the electric coupling does not renormalize at this order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' In the above, the ellipsis stands for the mass counterterms which are needed to make physical predictions due to the contribution to the quark self-energy induced by QED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' After integrating out the fermion and photon fields, the resulting Wick contractions 𝑊𝑖 are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' 1, which contribute to the derivative with respect to the electric charge through the connected correlation function 𝜕 𝜕𝑒 𝜕 𝜕𝑒 ⟨𝑂⟩ = 4 ∑︁ 𝑖=1 ⟨𝑂𝑊𝑖⟩c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' (4) The first two subdiagrams, which arise soley from the electric charges of the sea quarks, can be expressed in terms of a convolution with the photon propagator (in some fixed gauge) 𝐺 𝜇𝜈(𝑥) = ⟨𝐴𝜇(𝑥)𝐴𝜈(0)⟩ 𝑊1,2 = −𝑎8 ∑︁ 𝑥,𝑦 𝐻𝜇𝜈 1,2(𝑥, 𝑦)𝐺 𝜇𝜈(𝑥 − 𝑦), (5) where 𝐻1,2 are the traces of quark propagators 𝑆 𝑓 (𝑥, 𝑦) = ⟨𝜓 𝑓 (𝑥) ¯𝜓 𝑓 (𝑦)⟩ 𝐻𝜇𝜈 1 (𝑥, 𝑦) = ∑︁ 𝑓 ,𝑔 𝑄 𝑓 𝑄𝑔 tr{𝛾𝜇𝑆 𝑓 (𝑥, 𝑥)} tr{𝛾𝜈𝑆𝑔(𝑦, 𝑦)}, (6) 𝐻𝜇𝜈 2 (𝑥, 𝑦) = − ∑︁ 𝑓 𝑄2 𝑓 tr{𝛾𝜇𝑆 𝑓 (𝑥, 𝑦)𝛾𝜈𝑆 𝑓 (𝑦, 𝑥)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' (7) These two diagrams are the main subject of these proceedings, and the techniques advocated for the first can be effectively reused for the third diagram, 𝑊3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' In the following sections we introduce stochastic estimators only for the quark lines and compute the subdiagrams by convoluting with the exact photon propagator which avoids introducing additional stochastic fields for the U(1) gauge potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' The final diagram 𝑊4, which only contributes if the observable 𝑂 depends explicitly on the (charged) fermion fields, is the only one surviving the electroquenched approximation, and, can in most cases be computed efficiently provided that the leading-order diagram is already under control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' 3 Efficiently unquenching QCD+QED at O(𝛼) Tim Harris We note that the variance of the contributions to the connected correlation functions on the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' (4) crudely factorizes 𝜎2 𝑂𝑊1,2 ≈ ⟨𝑂⟩2 c ⟨𝑊1,2⟩2 c + ⟨𝑂𝑊1,2⟩c (8) ≈ 𝜎2 𝑂𝜎2 𝑊1,2, (9) where in the first line we have made the Gaussian approximation, and in the second line we have assumed that the fluctuations are much larger than the signal ⟨𝑂𝑊1,2⟩c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' Thus, in the following sections we will analyse the variance of individual subdiagrams 𝑊1,2 in order to gain a rough insight into the fluctuations of the total correction, in a similar fashion to the analysis of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' In that case, however, the correction to the factorization of the variance is exponentially suppressed in the separation between the vertices of the subdiagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' Quark-line disconnected subdiagram 𝑊1 We begin by noting that the hadronic part of the diagram factorizes into two traces, 𝐻𝜇𝜈 1 (𝑥, 𝑦) = 𝑇𝜇(𝑥)𝑇𝜈(𝑦), (10) each of which, with the current defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' (3) and in the 𝑁f = 2 + 1 theory with iso-spin symmetry, is the difference of the light- and strange-quark propagators 𝑇𝜇(𝑥) = 1 3 tr{𝛾𝜇[𝑆ud(𝑥, 𝑥) − 𝑆s(𝑥, 𝑥)]}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' (11) It is convenient to rewrite this difference as a product [10] 𝑆ud − 𝑆s = (𝑚s − 𝑚ud)𝑆ud𝑆s (12) which makes the explicit suppression of 𝑇𝜇 in the SU(3)-symmetry breaking parameter 𝑚s − 𝑚ud explicit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' This additionally results in a suppression of the variance of 𝑊1 by (𝑚s − 𝑚ud)4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' This suppression results in a cancellation of a quartic short-distance divergence in the variance of the contribution of each individual flavour to 𝑊1, explaining this favourable flavour combination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' While the identity in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' (12) is easily derived for Wilson-type fermions, here we sketch that it holds exactly for the domain-wall fermion valence propagator 𝑆 𝑓 = ˜𝐷−1 𝑓 which (approximately) satisfies the Ginsparg-Wilson relation [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' Recalling the definition of ˜𝐷 𝑓 in terms of the 5D Wilson matrix 𝐷5, 𝑓 (see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' [12] for unexplained notation) ˜𝐷−1 𝑓 = (P−1𝐷−1 5, 𝑓 𝑅5P)11, (13) where the matrix indices indicate the coordinate in the fifth dimension, the result is obtained immediately from ˜𝐷−1 ud − ˜𝐷−1 s = (𝑚s − 𝑚ud)(P𝐷−1 5,ud𝑅5𝐷−1 5,s𝑅5)11 (14) by noting that the following matrix projects on the physical boundary (𝑅5)·· = (𝑅5P)·1(P−1)1·.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' (15) 4 Efficiently unquenching QCD+QED at O(𝛼) Tim Harris 𝐿/𝑎 𝑇/𝑎 𝑚 𝜋 𝑚 𝜋𝐿 𝑎 𝑁cfg 24 64 340 MeV 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content='12 fm 50 Table 1: The parameters of the C1 ensemble of 𝑁f = 2 + 1 Shamir domain-wall fermions used in the numerical experiments in this work, see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' [17] for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' The preceding identity is easily demonstrated using the explicit representations 𝑅5 = ��� � 𝑃+ 𝑃− ��� � , P−1 = ������ � 𝑃− 𝑃+ 𝑃+ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' 𝑃+ 𝑃− ������ � , (16) where 𝑃± = 1 ± 𝛾5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' Using the identity for the difference, there are two independent estimators for the trace Θ𝜇(𝑥) = 1 3 (𝑚s − 𝑚ud) 1 𝑁s 𝑁s ∑︁ 𝑖=1 𝜂† 𝑖 (𝑥)𝛾𝜇{𝑆ud𝑆s𝜂𝑖}(𝑥), (17) T𝜇(𝑥) = 1 3 (𝑚s − 𝑚ud) 1 𝑁s 𝑁s ∑︁ 𝑖=1 {𝜂† 𝑖 𝑆s}(𝑥)𝛾𝜇{𝑆ud𝜂𝑖}(𝑥), (18) where the auxiliary quark fields 𝜂𝑖(𝑥) have zero mean and finite variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' The properties of both estimators were investigated in detail in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' [10], where it was shown that the contribution to the variance from the auxiliary fields for the second split-even estimator was in the region of a factor O(100) smaller than the first standard estimator, which translates into the same factor reduction in the cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' The split-even estimator has since been used extensively for disconnected current correlators [13, 14, 15], while in the context of the twisted-mass Wilson formulation similar one-end trick estimators have often been employed for differences of twisted-mass propagators [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' In this work we propose an estimator for the first diagram 𝑊1 using W1 ≈ � 𝑎4 ∑︁ 𝑥 T𝜇(𝑥) � � 𝑎4 ∑︁ 𝑦 T𝜈(𝑦)𝐺 𝜇𝜈(𝑥 − 𝑦) � (19) where independent estimators are used for the two traces to avoid incurring a bias with a finite sample size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' The convolution in the second parentheses can be efficiently computed using the Fast Fourier Transform (FFT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' With a minor modification, an estimator using all possible unbiased combinations of samples can be written at the cost of performing O(𝑁s) FFTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' The standard estimator is obtained by replacing both occurances of T𝜇 with Θ𝜇 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' We performed an analysis of the variance for the standard and split-even estimators for W1 using the domain-wall ensemble generated by the RBC/UKQCD collaboration whose parameters are listed in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' The photon propagator is computed in the QED𝐿 formulation [18] in the Feynman gauge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' The results for the variances, which are dimensionless numbers, are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' In addition,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' we plot the variance for the contribution of a single flavour Wu 1 using the 5 Efficiently unquenching QCD+QED at O(𝛼) Tim Harris 10−4 10−3 10−2 10−1 100 101 102 103 104 105 106 107 108 109 1 10 100 1000 σ2 Ns Wu 1 Wuds 1 (standard) Wuds 1 (split-even) 1/N 2 s Figure 2: Left: Comparison of the variance versus the number of sources for the 𝑊1 quark-line disconnected diagram,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' using a single flavour (red squares),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' the standard estimator for u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' s flavours (blue circles) and the split-even estimator (green triangles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' The dashed line shows 1/𝑁2 s scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' In this figure, the (local) currents are not renormalized and the charge factors are not included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' standard estimators for the traces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' We note that all the variances are dominated by the fluctuations of the auxiliary fields for small 𝑁s, and in particular scale like 1/𝑁2 s in that region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' As expected, the standard estimator including the light-quark and strange-quark contributions (blue circles) is suppressed with respect to the contribution of a single flavour (red squares).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' Furthermore, the variance of the split-even estimator (green triangles) is reduced by a factor of 104 with respect to the standard one (blue circles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' This reduction is commensurate with the reduction in the variance observed for the disconnected contribution to the current correlator [10], which suggests the same mechanisms are present here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' For 𝑁s ∼ 100, the variance is independent of the number of auxiliary field samples which indicates that it is dominated by the fluctuations of the gauge field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' In this case no further variance reduction is possible for a fixed number of gauge configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' Finally we note that the convolution of the second parentheses of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' (19) can be simply inserted sequentially in any of the diagrams of type 𝑊3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' Quark-line connected subdiagram 𝑊2 In contrast to the quark-line disconnected subdiagram, there is no cancellation in the variance in the connected subdiagram 𝑊2 between the light and strange-quark contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' In this case, power counting suggests that the variance diverges with the lattice spacing like 𝑎−4 as 𝑎 → 0 and is expected to be dominated by short-distance contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' Translation averaging should therefore be very effective and one way to implement it is to use an all-to-all estimator [19] for the quark propagator S 𝑓 (𝑥,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' 𝑥 + 𝑟) = 1 𝑁s 𝑁s ∑︁ 𝑖=1 {𝑆 𝑓 𝜂𝑖}(𝑥)𝜂† 𝑖 (𝑥 + 𝑟),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' (20) 6 Efficiently unquenching QCD+QED at O(𝛼) Tim Harris 10−7 10−6 10−5 10−4 10−3 10−2 10−1 100 101 102 103 104 105 106 107 108 109 0 2 4 6 8 10 12 σ2 |r|/a H2(r)G(r) H2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' Ns = 1 ¯H2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' NX = 1 H2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' Ns = ∞ 10−2 10−1 100 101 102 103 104 105 106 1 10 100 1000 σ2 Ninv W2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' R/a = 4 � r≤R H2G � r>R ¯H2G R = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' Ns = ∞ Figure 3: Left: the variance for the stochastic estimator (red squares) and point source estimator (blue circles) for the minimum number of inversions required,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' for the contribution with fixed separation between the currents |𝑟|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' The green triangle indicates the gauge variance for the point 𝑟 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' Right: the variance for the short-distance (red squares) and long-distance (blue circles) for the choice 𝑅/𝑎 = 4, versus the number of inversions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' The green band indicates the gauge variance for the contribution from 𝑟 = 0 only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' The dashed lines indicate the expected leading 𝑁−2 inv and 𝑁−1 inv scaling for the short- and long-distance components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' using independent fields for each propagator in the trace H 𝜇𝜈 2 (𝑟) = 𝑎4 ∑︁ 𝑥 ∑︁ 𝑓 𝑄2 𝑓 tr{𝛾𝜇S 𝑓 (𝑥, 𝑥 + 𝑟)𝛾𝜈S 𝑓 (𝑥 + 𝑟, 𝑥)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' (21) As written, the estimator is feasible to compute for a small number of separations 𝑟 between the vertices and, although it introduces a (mild) signal-to-noise ratio problem at large 𝑟, should be efficient at small |𝑟| ≤ 𝑅 given the leading extra contribution vanishes like 𝑁−2 s , c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' For the remainder |𝑟| > 𝑅, we propose using 𝑁𝑋 randomly selected point sources 𝑋𝑛 [20] ¯𝐻𝜇𝜈 2 (𝑟) = 𝐿3𝑇 𝑁𝑋 𝑁𝑋 ∑︁ 𝑛=1 𝐻𝜇𝜈 2 (𝑋𝑛, 𝑋𝑛 + 𝑟) (22) so that the total is split between short- and long-distance contributions W2 = 𝑎4 ∑︁ |𝑟 |≤𝑅 H2(𝑟)𝐺 𝜇𝜈(𝑟) + 𝑎4 ∑︁ 𝑟>𝑅 ¯𝐻𝜇𝜈 2 (𝑟)𝐺 𝜇𝜈(𝑟), (23) using the efficient stochastic estimator for the noisy short-distance contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' [21] introduced an importance sampling based on current separations for higher-point correlation functions, whereas in this case we make the separation based on the expected contributions to the variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' This approach avoids completely factorizing the trace which would require either O(𝑉) contractions or O(𝑁2 s ) FFTs to include the photon line which we deemed unfeasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' 3 (left) we illustrate the variance of each of the terms in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' (23) for the sum over a fixed separation |𝑟| between the currents, for the case 𝑁s = 𝑁𝑋 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' As expected, the variance from the contribution around |𝑟| ∼ 0 dominates both the stochastic (red squares) and point source estimator (blue circles), and we observe the mild signal-to-noise ratio problem in the stochastic 7 Efficiently unquenching QCD+QED at O(𝛼) Tim Harris estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' The green triangle denotes the gauge variance for the case 𝑟 = 0, which is approximately suppressed by (𝐿3𝑇)/𝑎4 compared to 𝑁𝑋 = 1 indicating translation averaging is very effective for the short-distance contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' In the right-hand panel, we see variance of the short- and long- distance contributions with the choice 𝑅/𝑎 = 4 as a function of the number of inversions (where 𝑁𝑋 = 1 corresponds to 12 inversions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' The variance is dominated by the short-distance contribution (red squares) which however scales favourably like 𝑁−2 inv, while the long-distance contribution (blue circles) which scales only like 𝑁−1 inv is much suppressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' Deviations from the former scaling indicate that the gauge variance may be reached with just 𝑁inv ∼ 1000, which although is larger than required for 𝑊1 is still achievable with modern computational resources, and universal for all observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' Conclusions In this work we have examined the Wick contractions which arise due to the charge of the sea quarks in the RM123 method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' Such diagrams contribute, in principle, even to observables constructed from neutral fields and are therefore ubiquitous in the computation of iso-spin breaking corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' We have proposed stochastic estimators for the quark lines in such diagrams which completely avoids the need to sample the Maxwell action stochastically, thus eliminating one additional source of variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' As for the case of disconnected contributions to current correlators, we have shown it is beneficial to consider certain flavour combinations which have greatly suppressed fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' We have shown that the split-even estimators generalize also to domain-wall fermions and perform well compared with naïve estimators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' Thus the frequency-splitting strategy of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' [10] should generalize appropriately for this fermion formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' In the second topology, however, there is no cancellation of the short-distance effects in the variance by considering multiple flavours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' In this case, we propose decomposing the diagram into a short-distance part to be estimated stochastically and a long-distance part estimated using position-space sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' The variance is reduced sufficiently so that the gauge variance can be reached with a reasonable computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' Given their short-distance nature, these estimators should also succeed with smaller quark masses, and furthermore as the diagrams are universal to all iso-spin breaking corrections we anticipate that these simple decompositions ought to be beneficial in large-scale simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' In particular we are developing these methods for refinements of our computations of iso-spin breaking corrections within the RBC/UKQCD collaboration, for example to meson (leptonic) decay rates [22, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' Acknowledgments We use the open-source and free software Grid as the data parallel C++ library for the lattice computations [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' The authors warmly thank the members of the RBC/UKQCD collaboration for valuable discussions and the use of ensembles of gauge configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=', A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' are supported in part by UK STFC 1039 grant ST/P000630/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme under grant agreement No 757646 and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' additionally under grant agreement No 813942.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content=' This work used the DiRAC Extreme Scaling service at the University of Edinburgh, operated by the Edinburgh Parallel Computing Centre on behalf of the STFC DiRAC HPC Facility (www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content='dirac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfmAfb/content/2301.03995v1.pdf'} +page_content='ac.' metadata={'source': 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+3Institute of Automation Chinese Academy of Sciences +{caopu, soeaver, ldx, ls1995, priv}@bupt.edu.cn zhiwei.liu@nlpr.ia.ac.cn +Abstract +Recently, inversion methods have focused on additional +high-rate information in the generator (e.g., weights or +intermediate features) to refine inversion and editing results +from embedded latent codes. +Although these techniques +gain reasonable improvement in reconstruction, +they +decrease editing capability, especially on complex images +(e.g., containing occlusions, detailed backgrounds, and +artifacts). A vital crux is refining inversion results, avoiding +editing capability degradation. +To tackle this problem, +we introduce Domain-Specific Hybrid Refinement (DHR), +which draws on the advantages and disadvantages of two +mainstream refinement techniques to maintain editing +ability with fidelity improvement. +Specifically, we first +propose Domain-Specific Segmentation to segment images +into two parts: in-domain and out-of-domain parts. The +refinement process aims to maintain the editability for +in-domain areas and improve two domains’ fidelity. +We +refine these two parts by weight modulation and feature +modulation, which we call Hybrid Modulation Refinement. +Our proposed method is compatible with all latent code +embedding methods. +Extension experiments demonstrate +that our approach achieves state-of-the-art in real image +inversion and editing. +Code is available at https: +//github.com/caopulan/Domain-Specific_ +Hybrid_Refinement_Inversion. +1. Introduction +Generative Adversarial Networks (GANs) have shown +promising results in image generation. Synthetic images +are photorealistic with high resolution and are difficult to +distinguish from real images [24, 27, 28, 26, 61]. Mean- +while, image manipulation and controllable generation are +deeply explored thanks to their highly semantic latent space. +Moreover, GANs can represent a high-quality image prior +*corresponding author. +Figure 1. Inversion and editing results of our method. We pre- +serve image details, including background and occlusion, in both +inversion and manipulation processes. +to improving various tasks, such as face parsing [59, 58, +62, 60, 57, 56], style transfer [33, 63], face super-resolution +[53]. +Inversion is built to convert real images into GANs’ la- +tent space. The inverted latent codes are required to recon- +struct given images by pretrained generator, which also em- +arXiv:2301.12141v1 [cs.CV] 28 Jan 2023 + +TO +indul +Inversion +ASA +TO +Smile +SAANN +LUTO +Young +TO +Exposure +LTO +Lipstick +SAANNbeds semantic information to edit or apply in other GAN- +based tasks. Two types of methods generally reach image +embedding. One is training an image encoder to convert +given images to latent codes [49, 41], while another is min- +imizing the discrepancy between given images and recon- +structed images to optimize initial latent codes iteratively +[28]. This process attains the corresponding latent codes +to reconstruct or edit the images. However, latent codes are +low bit-rate [52], and high-rate details of images may not be +reconstructed faithfully. Hence, many works focus more on +refining results by additional high-rate information, e.g., in- +termediate features [36], generator weight [44, 5], recently. +As reconstruction performance increases by refinement +with high-rate information, editing capability is inevitably +decreased, especially on images containing complex parts. +This phenomenon is due to the destruction of pretrained +GAN prior. +High-rate information needs drastic change +to reconstruct complex parts. +We demonstrate this phe- +nomenon in Figure 3. Meanwhile, complex images pre- +vail in the natural world. For example, face accessories, +hats, occlusions, and complex backgrounds usually appear +in face photos. +As there are two mainstream refinement methods, they +show different manipulation impacts. One is weight modu- +lation, in which the generator’s weight is tuned [44, 12] or +predicted [5] by given images. Another is feature modula- +tion [52, 36], in which the input image would also invert to +feature space by encoder or optimization. Generally, weight +modulation can maintain editing capacity better, while fea- +ture modulation breaks it since high-rate level editing is re- +quired [38]. +Based on the above illustration, we further explore the +idea of ”divide and conquer.” Specifically, we divide the +image into in-domain and out-of-domain parts. In-domain +parts imply areas close to generators’ output distribution +and are desired to perform well on both inversion and edit- +ing. +Correspondingly, out-of-domain parts are segments +challenging to inverse or edit and desired to reconstruct +faithfully. Hence, we introduce a hybrid method to han- +dle them. We refine in-domain parts by tuning generator +weight since it can maintain editing capability. For out-of- +domain parts, we straightforwardly invert them by interme- +diate features to keep spatial image details. Notably, our +hybrid refinement method first analyzes and combines fea- +ture and weight modulation for improved GAN inversion, +and achieves extraordinary results as shown in Figure 1. +Extensive experiments are presented to demonstrate the +effects of our Domain-Specific Hybrid Refinement. +We +achieve state-of-the-art and gain significant improvement in +both fidelity and editability. The key contributions of this +work are summarized as follows: +• We analyze the reasons for editing capability degrada- +tion in the refinement process. Based on our analysis, +we introduce in-domain and out-of-domain and pro- +pose Domain-Specific Segmentation to segment im- +ages into these two parts for better inversion. +• We propose Hybrid Modulation Refinement to im- +prove inversion results of in-domain and out-of- +domain parts. We conduct weight modulation on in- +domain part and feature modulation on out-of-domain +part, which can preserve editing capability when refin- +ing the image details. +• We conduct extensive experiments and user studies to +demonstrate the effects of our method. We reach ex- +traordinary performance on real-world image inversion +and editing and achieve state-of-the-art. +2. Related Work +2.1. GAN Inversion +GAN inversion aims to embed real-world images into a +pretrained generator’s latent space, which can be used to +reconstruct and edit input images. Generally, methods can +be divided into two stages. +The first stage aims to attain low-rate latent codes, usu- +ally in Z/W/W + spaces. The latent codes are gained by +an encoder or optimization process. Training an encoder +[49, 43, 54, 17, 8, 41] to predict latent codes is efficient for +inference and is easier to get better trade-offs between fi- +delity and manipulation [49, 41]. Optimizing initial latent +codes by reconstruction discrepancy gains better fidelity. +However, it may cost several minutes per image [28, 8, 1, 2] +and decreases editability during per-image tuning. Due to +low-rate characteristics, latent codes can only reconstruct +coarse information and drop the details from original im- +ages. Meanwhile, there is a trade-off between fidelity and +editability, and many methods introduce additional regular- +ization modules (e.g., latent code discriminator [49] and la- +tent space alignment [41]) to address it. +In the second stage, reconstruction and manipulation re- +sults from latent codes are refined by high-rate information. +Refinement methods are mainly divided into weight mod- +ulation and feature modulation. Weight modulation meth- +ods predict or finetune generator weight to improve fidelity. +Some methods use hypernet [18] to predict weight offsets. +The others tune generator by given images, which attain bet- +ter fidelity but cost much time. Another branch further in- +vert images to latent feature, which we call feature modula- +tion. HFGI [52] proposes a distortion consultation approach +for high-fidelity reconstruction. SAM [36] segments images +into various parts and inverts them into different intermedi- +ate layers by predicting ”invertibility.” All of them only use +one of feature and weight modulation to refine results and +suffer editing capability degradation. In this work, we com- + +generator +manifold +latent space +weight +modulation +𝐺(𝑤; 𝜃∗) +𝐺(𝑤) +𝑤 +generator +manifold +latent space +𝐺(𝑤, 𝑓) +𝐺(𝑤) +𝑤 +image space +image space +feature +modulation +generator +manifold +latent space +feature +modulation +𝐺(𝑤, 𝑓; 𝜃∗) +𝐺(𝑤) +𝑤 +image space +weight +modulation +𝐺(𝑤) +𝐺(𝑤; 𝜃∗) +𝐺(𝑤; 𝜃∗) +𝐺(𝑤, 𝑓; 𝜃∗) +(a) Weight Modulation +(b) Feature Modulation +(c) Hybrid Modulation +Figure 2. Comparison of different refinement mechanisms. We suppose that the refinement results are similar to the given images in +all pipelines. The first row shows two previous mainstream refinement mechanisms, weight, and feature modulation. Weight modulation +changes the generator manifold and feature modulation introduces spatial high-rate information to recover image details. The bottom +part demonstrates our hybrid refinement method, combining these two modulation mechanisms to retain editing capability. We tune the +generator on invertible and editable areas, which causes lower manifold deviation, and the result is shown on G(w; θ∗). To reconstruct +faithfully, we use feature modulation on the other area and attain G(w, f; θ∗). +bine these two aspects by their pros and cons to reach more +promising results. +2.2. GAN-based Manipulation +GANs’ latent spaces encode highly rich semantic infor- +mation, which develop the GAN-based manipulation task. +It aims to edit given images by changing latent codes in +certain directions. +Many works propose multiple meth- +ods to find semantic editing directions in latent spaces. +Some methods obtain the edit vectors of the correspond- +ing attributes by means of supervision with the help of +attribute-labeled datasets [10, 15, 47, 45]. And others ex- +plore the latent space by unsupervised [19, 46, 50, 51] or +self-supervised ways [23, 39] to find more semantic direc- +tions way. +3. Method +3.1. Preliminaries +Inversion is built to bridge real-world images and GANs’ +latent space. +As latent codes are low-rate, which limits +their reconstruction performance, much research has re- +cently focused on additional high-rate information in gen- +eration process, which we call refinement methods. They +can be mainly divided into two categories: weight modula- +tion and feature modulation. We first formulate them and +analyze the causes of editing capacity degradation. +Formulation. +We denote the original generation process +as X = G(w), where G is the generator, w is latent code +which can represent each latent space (e.g., Z/W/W +). +We use encoded latent codes as w in the refinement process. +Weight modulation methods predict [5] or optimize [28] +θ by minimizing reconstruction error, and are denoted as +X += G(w; θ∗). +And feature modulation methods in- +vert images into the intermediate feature, which follows +X = G(w, f). Defining L as the distance of images, we +can illustrate these two refinement processes as follow: +θ∗ = arg min +θ +L(x, G(w; θ)) +(1) +f ∗ = arg min +f +L(x, G(w, f)) +(2) +Impacts on editing capability. Weight and feature modu- +lation impacts image manipulation in different aspects. The + +Easy Sample +Easy Sample +Hard Sample, Occlusion +Hard Sample, Artifact +Figure 3. Impacts on editing capability of weight modulation. +We show the input images, inversion results, and two editing re- +sults (smile and age) from PTI [44]. For those easy samples, edit- +ing results are reasonable. However, editing capability degrades +significantly on hard samples. +schematics are shown in Figure 2. Since the feature modu- +lation mechanism fixes the intermediate feature distribution +at one of layers, the effects of edit vectors applied to previ- +ous layers cannot edit the features of latter layers. Although +many existing works make efforts to maintain the editing ef- +fects, including training with adaptive distortion alignment +[25, 52], their solutions still sacrifice fidelity or editing re- +sults [36]. +Meanwhile, weight modulation shows promising editing +performance but also gains unreasonable results on com- +plex images, as shown in Figure 3. Editing results are more +reasonable on easy samples than on hard samples. +The +main reason for editing capacity degradation is the signif- +icant weight deviation caused by refining complex images, +which we show in Figure 2(a). To reconstruct given images, +the weight modulation mechanism may change the genera- +tor manifold much. Therefore, the highly semantic charac- +teristic of the pretrained generator is broken, which would +decrease the editing capability. +In conclusion, the critical problem of refinement meth- +ods is how to decrease weight deviation with reconstruc- +tion improvement. We next propose our method to answer +this question. +3.2. Overview +In this work, we conduct Domain-Specific Hybrid +Refinement (DHR) to deal with real-world image inversion +and the pipline is shown in Figure 4. Based on the above +analysis, we explore the idea of ”divide and conquer.” +We first propose the concepts of in-domain and out-of- +domain. In-domain implies areas that have a similar distri- +bution with the generator’s output space and are easy to in- +vert, while out-of-domain areas misalign with output space +and are difficult to invert. For example, in face domain, +in-domain areas mainly consist of face and hair, while out- +of-domain areas consist of occlusions, backgrounds, and ar- +tifacts. Meanwhile, in-domain areas are more editable, e.g., +smile, lipstick, and eyes openness. +Hence, we propose a hybrid refinement method, which +segments images into in-domain and out-of-domain areas +and applies weight and feature modulation to improve fi- +delity and preserve editing capability. Our framework is +shown in Figure 4, which consists of three components. +The image Embedding module aims to embed images +into latent codes, which we use an off-the-shelf encoder +(i.e., e4e [49] and LSAP [41]). Given input images X, the +encoder predicts its W + space latent codes, which we de- +note as w = E(X), where E is an encoder. +Domain-Specific Segmentation predicts a binary mask +which indicates in-domain and out-of-domain areas: +m = S(X) +where m ∈ {0, 1}h×w. It segments images into two parts, +which will be used for refinement. +In Hybrid Modulation Refinement, weight modulation is +applied to in-domain areas to recover image details in both +inversion and editing results. Thanks to the low reconstruc- +tion discrepancy of in-domain part, weight deviation would +not be large, and editing capacity would be preserved. For +out-of-domain parts, we use feature modulation to refine +them spatially and not to edit them. Hence, those hard-to- +invert parts would not influence editing ability. We mod- +ulate weight θ and feature f by minimizing reconstruction +error in in-domain and out-of-domain part, respectively: +Xrec = G(w, f, m; θ) +(3) +The difference with vanilla weight and feature modula- +tion can be seen in Figure 2. Based on hybrid ways, genera- +tor manifold would not change a lot, which highly maintains +the editing capability of the original GAN. +3.3. Domain-specific Segmentation +The first challenge is segmenting images into in-domain +and out-of-domain at the pixel-level. +An end-to-end +domain-specific segmentation model is required for a large, +manually annotated dataset. Although previous work [36] + +omFEOTURUUSH +FEATORLNSTPTOAIKO +2,2 +OTomHybrid Modulation Refinement +Domain-Specific Segmentation +θ∗ +Segmentation +𝒘" +𝒇∗ +Input +Encoder +Image Embedding +Figure 4. Overview of our Domain-Specific Hybrid Refinement framework. We use an off-the-shelf image encoding mechanism and +introduce Domain-Specific Segmentation and Hybrid Modulation Refinement. The former segments the input images with two domains: +in-domain and out-of-domain. They are refined by weight modulation and feature modulation in the latter method. +Figure 5. Illustration of Domain-Specific Segmentation module. We use a parsing model and superpixel algorithm with coarse opti- +mization to segment input images into in-domain (white areas) and out-of-domain (black areas) parts. +trains an invertibility prediction model by self-supervision, +results are inaccurate in some complex areas, which we il- +lustrate in our ablation study. In this work, we propose a +Domain-Specific Segmentation module, combining parsing +and superpixel modules to generate domain segments. Our +module is robust for real images and does not require data +annotation. The pipeline is shown in Figure 5. +The parsing model categorizes face components [69] like +eye, mouth, and background. For parsing results mp, we +manually set some categories as out-of-domain and the oth- +ers as in-domain. However, it is not robust on some complex +images, as shown in Figure 5. The parsing result represents +a coarse mask, where complicated paradigms are not seg- +mented well. Therefore, we introduce a superpixel module +with coarse optimization to improve the segmentation re- +sults. +We use a superpixel algorithm [3] for image partition- +ing, shown in the middle route of Figure 5. This step finely +segments images to distinguish each area. We denote each +partition as {mi +s}S +i=1. Categorizing each partition into in- +domain and out-of-domain without manual annotation is a +crucial challenge. We first apply a coarse optimization in W +space, where latent codes initialed by mean values are only +optimized by a few steps. Since in-domain are those easy- +to-invert areas, the coarse inversion result Xcoarse could +reconstruct in-domain areas. We calculate the perceptual +loss L between the coarse reconstruction image and the in- +put image, as shown at the bottom of Figure 5. White area + +Parsing Result +Domain Segment +om: +Face +Parsing +loma +ome +om +Superpixel +K +Input X +Superpixel Result +om +out-of-domain +Coarse +Optimization +in-domain +Xcoarse +LPIPS(X, Xcoarse)m +KPTOAIKO +2,2 +OTmeans higher loss value, while black area means lower loss +value. As can be seen, the loss of the occlusion area is sig- +nificantly higher than the face area. We calculate the aver- +age loss of each partition as follows: +vi = L ⊙ mi +s +||mis|| +and the result {mi +s, vi}S +i=1 is visualized. Then we binarize +superpixel results by adaptive threshold τ and attain ms. +Finally, domain-specific segmentation results are com- +bined by parsing results and superpixel results: +m = mp × ms +Our Domain-Specific Segmentation module can gain fine +segmentation results without data annotation. +3.4. Hybrid Modulation Refinement +Figure 6. Illustration of Hybrid Modulation Refinement mod- +ule. We refine in-domain areas and out-of-domain areas by weight +and feature modulation, respectively. Black lines indicate forward +flow, and orange and blue lines represent gradients. +To faithfully recover image details and maintain editing +capability from original GAN, we introduce a Hybrid Mod- +ulation Refinement module. It consists of two mainstream +refinement aspects: weight modulation and feature modu- +lation. Weight modulation aims to minimize in-domain re- +construction error by tuning the generator’s parameters. In +contrast, feature modulation is applied to out-of-domain ar- +eas by optimizing an intermediate feature of the generator. +The forward and backward processes are shown in Figure 6. +For lth layer of total k stages in generator, the original +generator’s feature is denoted as fl = Gl(w; θ), and an ad- +ditional modulated feature is marked as f, which is initialed +by fl. Fixing the latent codes, the original feature fl is only +relevant to θ. Given segmentation result m, we formulate +the forward process as follows: +f ′ = fl ⊙ m + f ⊙ (1 − m) +(4) +Then f ′ represents the output of the first l layers and gener- +ates the final images, which follow Eq 3. +For backward, we use mean square error L2 and percep- +tual loss Llpips as objectives in the refinement process. To +make weight and feature focus on the corresponding areas, +we update them in a parallel optimization process. Calcu- +lating the reconstruction errors, we backward loss with seg- +mentation result m: +L = L2 + λLlpips +(5) +∇f = ∂ +∂f [L ⊙ (1 − m) +||1 − m|| +] +(6) +∇θ = ∂ +∂θ[L ⊙ (m) +||m|| +] +(7) +where λ is a hyper-parameter. The parallel optimization +mechanism constrains the impact from different domains. +Based on Domain-Specific Segmentation and Hybrid +Modulation Refinement, we segment images into in-domain +and out-of-domain areas and refine them by weight and fea- +ture modulation, which significantly improve fidelity with +editing capability remaining. +4. Experiments +4.1. Experimental Settings +Datasets. +We evaluate all methods on the CelebA-HQ +[24, 35] test set (2,824 images). Encoders and the generator +are trained on FFHQ [27] (70,000 images). +Baselines. We compare our model with previous state-of- +the-art refinement methods, i.e., ReStyle [4], HFGI [52], +SAM [36], and PTI [44]. +We use pSp [43], e4e [49] +and LSAP [41] as encoders. Moreover, the performance +of encoder-based methods is also reported. All of model +weights of encoders and generators come from their official +release. +Metrics. +We evaluate all methods in two respects: inver- +sion and editing. For inversion ability, we conduct MSE, +LPIPS [67], and identity similarity calculated by a face +recognition model [22]. MSE straightforwardly measures +the image distortion, and LPIPS evalutaes the visual dis- +crepancy. Identity similarity further compares the identity +consistency during inversion. Moreover, we perform user +studies to evaluate the perceptual performance of inversion +and editing. +4.2. Main Results +Quantitative results. We first evaluate the reconstruction +ability. Qualitative results are reported in Table 1. We com- +pare our method with four previous refinement methods and +employ two encoders. We conduct experiments with en- +coders and original Wpivot for PTI. As one can see, DHR +achieves the best performance on all metrics. Employed +with e4e, it gains 0.0036 MSE, which is 7.5% e4e, 8.3% + +om +Loss +gradient of weight modulation +gradient of feature modulationFigure 7. Inversion and editing results. The second column is the segmentation results of in-domain and out-of-domain areas. Our method +restores almost all image details. +ReStyle, 17% HFGI, 25% SAM, and 48% PTI. For LPIPS +and identity similarity, it demonstrates similar superiority +and surpasses other methods by a large margin. DHR with +LSAP achieves the best performance of all metrics. +Qualitative results. +We illustrate the inversion and edit- +ing results of DHR in Figure 7. +The second column is +the results from the Domain-Specific Segmentation module, +and the third is our inversion results. Some dedicated ar- +eas are categorized into out-of-domain domain (black area). +We edit them by InterFaceGAN and GANSpace, using four +editing directions, i.e., smile, young, exposure, and lipstick. +All image details are preserved in both inversion and editing +results, such as hairstyle (the first row), earrings (the fifth +row), and hats. Our results are faithful and photorealistic. +We further conduct a qualitative comparison with other +methods, shown in Figure 8. Although inversion results are + +Input +Inversion +Smile +Young +Exposure +LipstickFigure 8. Comparisons with previous methods. We compare the inversion and editing results with PTI [4], HFGI [52], and SAM [36]. +Although HFGI and SAM reach reasonable inversion results, image distortion and details loss occur in editing results. Our method attains +the best fidelity and editing performance. Image details are reserved in both phases, and our results are the most natural. +Method +Encoder +MSE ↓ +LPIPS ↓ +Similarity ↑ +ReStyle [4] +pSp +0.0276 +0.1298 +0.5816 +e4e +0.0429 +0.1904 +0.5062 +HFGI [52] +e4e +0.0210 +0.1172 +0.6816 +LSAP +0.0210 +0.0945 +0.7405 +SAM [36] +e4e +0.0143 +0.1104 +0.5568 +LSAP +0.0117 +0.0939 +0.6184 +PTI [44] +e4e +0.0074 +0.0750 +0.8633 +LSAP +0.0067 +0.0666 +0.8696 +Wpivot +0.0084 +0.0845 +0.8402 +DHR (ours) +e4e +0.0036 +0.0455 +0.8704 +LSAP +0.0035 +0.0436 +0.8780 +Encoder-Only +pSp [43] +0.0351 +0.1628 +0.5591 +e4e [49] +0.0475 +0.1991 +0.4966 +LSAP [41] +0.0397 +0.1766 +0.5305 +Table 1. Fidelity results on face domain. We compare DHR to +three previous refinement methods with two powerful encoders. +The results of these encoders are also presented at the bottom. +reasonable, editing capability degradation occurs in base- +lines, e.g., decorations and occlusion blur. +4.3. User Study +We conduct user studies to demonstrate the performance +of inversion and editing. +Results are shown in Table 2. +We randomly select 50 different images and invert and edit +them by HFGI [52], SAM [36], PTI [44], and our method. +We then ask three users to make a preference for each pair of +images. A higher value implies users prefer our results. As +can be seen, our results are highly preferred by users, which +Method +Inversion +Editing +Smile +Young +Exposure +Lipstick +ReStyle [4] +94% +100% +100% +90% +94% +HFGI [52] +94% +84% +86% +100% +96% +SAM [36] +100% +92% +94% +100% +100% +PTI [44] +96% +100% +100% +84% +88% +Table 2. User Study. We conduct user studies on inversion and +editing tasks. The values in the table indicate the percentage of +images where users prefer our results. Results show our method is +more faithful and photorealistic. +are most all above 90%, compared to previous state-of-the- +art methods. It illustrates that our method decreases image +distortion and attains better photorealism of reconstruction +and manipulation results. +5. Conclusion +In this work, we propose Domain-Specific Hybrid Re- +finement to improve GAN inversion and editing capabil- +ity. Specifically, we analyze the causes of editing ability +degradation in refinement process and introduce ”divide and +conquer” to address it. Our method consists of Domain- +Specific Segmentation and Hybrid Modulation Refinement, +which segments images into in-domain and out-of-domain +parts and refines them by weight and feature modulation, +respectively. Our method attains promising results in both +inversion and editing with considerable improvement. + +Young +Lipstick +Exposure +50 +Smile +Input +Inversion +Inversion +edit +Inversion +edit +Inversion +edit +PTI +HFGI +SAM +DHR (ours)References +[1] Rameen Abdal, Yipeng Qin, and Peter Wonka. +Im- +age2stylegan: How to embed images into the stylegan latent +space? 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Datasetgan: Efficient labeled data factory with +minimal human effort. +In Proceedings of the IEEE/CVF +Conference on Computer Vision and Pattern Recognition, +pages 10145–10155, 2021. +[69] Yinglin Zheng, Hao Yang, Ting Zhang, Jianmin Bao, +Dongdong Chen, Yangyu Huang, Lu Yuan, Dong Chen, +Ming Zeng, and Fang Wen. +General facial representa- +tion learning in a visual-linguistic manner. arXiv preprint +arXiv:2112.03109, 2021. + diff --git a/2dFLT4oBgHgl3EQfqy9t/content/tmp_files/load_file.txt b/2dFLT4oBgHgl3EQfqy9t/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..0c7ffadcf1fc28ae8383e5114f74a90b5e91c53f --- /dev/null +++ b/2dFLT4oBgHgl3EQfqy9t/content/tmp_files/load_file.txt @@ -0,0 +1,578 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf,len=577 +page_content='What Decreases Editing Capability?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' Domain-Specific Hybrid Refinement for Improved GAN Inversion Pu Cao1,2 Lu Yang1 Dongxu Liu1 Zhiwei Liu3 Shan Li1 Qing Song1* 1Beijing University of Posts and Telecommunications 2Metavatar 3Institute of Automation Chinese Academy of Sciences {caopu, soeaver, ldx, ls1995, priv}@bupt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content='cn zhiwei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content='liu@nlpr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content='ia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content='cn Abstract Recently, inversion methods have focused on additional high-rate information in the generator (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=', weights or intermediate features) to refine inversion and editing results from embedded latent codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' Although these techniques gain reasonable improvement in reconstruction, they decrease editing capability, especially on complex images (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=', containing occlusions, detailed backgrounds, and artifacts).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' A vital crux is refining inversion results, avoiding editing capability degradation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' To tackle this problem, we introduce Domain-Specific Hybrid Refinement (DHR), which draws on the advantages and disadvantages of two mainstream refinement techniques to maintain editing ability with fidelity improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' Specifically, we first propose Domain-Specific Segmentation to segment images into two parts: in-domain and out-of-domain parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' The refinement process aims to maintain the editability for in-domain areas and improve two domains’ fidelity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' We refine these two parts by weight modulation and feature modulation, which we call Hybrid Modulation Refinement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' Our proposed method is compatible with all latent code embedding methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' Extension experiments demonstrate that our approach achieves state-of-the-art in real image inversion and editing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' Code is available at https: //github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content='com/caopulan/Domain-Specific_ Hybrid_Refinement_Inversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' Introduction Generative Adversarial Networks (GANs) have shown promising results in image generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' Synthetic images are photorealistic with high resolution and are difficult to distinguish from real images [24, 27, 28, 26, 61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' Mean- while, image manipulation and controllable generation are deeply explored thanks to their highly semantic latent space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' Moreover, GANs can represent a high-quality image prior corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' Inversion and editing results of our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' We pre- serve image details, including background and occlusion, in both inversion and manipulation processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' to improving various tasks, such as face parsing [59, 58, 62, 60, 57, 56], style transfer [33, 63], face super-resolution [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' Inversion is built to convert real images into GANs’ la- tent space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' The inverted latent codes are required to recon- struct given images by pretrained generator, which also em- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content='12141v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content='CV] 28 Jan 2023 TO indul Inversion ASA TO Smile SAANN LUTO Young TO Exposure LTO Lipstick SAANNbeds semantic information to edit or apply in other GAN- based tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' Two types of methods generally reach image embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' One is training an image encoder to convert given images to latent codes [49, 41], while another is min- imizing the discrepancy between given images and recon- structed images to optimize initial latent codes iteratively [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' This process attains the corresponding latent codes to reconstruct or edit the images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' However, latent codes are low bit-rate [52], and high-rate details of images may not be reconstructed faithfully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' Hence, many works focus more on refining results by additional high-rate information, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=', in- termediate features [36], generator weight [44, 5], recently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' As reconstruction performance increases by refinement with high-rate information, editing capability is inevitably decreased, especially on images containing complex parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' This phenomenon is due to the destruction of pretrained GAN prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' High-rate information needs drastic change to reconstruct complex parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' We demonstrate this phe- nomenon in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' Meanwhile, complex images pre- vail in the natural world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' For example, face accessories, hats, occlusions, and complex backgrounds usually appear in face photos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' As there are two mainstream refinement methods, they show different manipulation impacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' One is weight modu- lation, in which the generator’s weight is tuned [44, 12] or predicted [5] by given images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' Another is feature modula- tion [52, 36], in which the input image would also invert to feature space by encoder or optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' Generally, weight modulation can maintain editing capacity better, while fea- ture modulation breaks it since high-rate level editing is re- quired [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' Based on the above illustration, we further explore the idea of ”divide and conquer.” Specifically, we divide the image into in-domain and out-of-domain parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' In-domain parts imply areas close to generators’ output distribution and are desired to perform well on both inversion and edit- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' Correspondingly, out-of-domain parts are segments challenging to inverse or edit and desired to reconstruct faithfully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' Hence, we introduce a hybrid method to han- dle them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' We refine in-domain parts by tuning generator weight since it can maintain editing capability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' For out-of- domain parts, we straightforwardly invert them by interme- diate features to keep spatial image details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' Notably, our hybrid refinement method first analyzes and combines fea- ture and weight modulation for improved GAN inversion, and achieves extraordinary results as shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' Extensive experiments are presented to demonstrate the effects of our Domain-Specific Hybrid Refinement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' We achieve state-of-the-art and gain significant improvement in both fidelity and editability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' The key contributions of this work are summarized as follows: We analyze the reasons for editing capability degrada- tion in the refinement process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' Based on our analysis, we introduce in-domain and out-of-domain and pro- pose Domain-Specific Segmentation to segment im- ages into these two parts for better inversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' We propose Hybrid Modulation Refinement to im- prove inversion results of in-domain and out-of- domain parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' We conduct weight modulation on in- domain part and feature modulation on out-of-domain part, which can preserve editing capability when refin- ing the image details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' We conduct extensive experiments and user studies to demonstrate the effects of our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' We reach ex- traordinary performance on real-world image inversion and editing and achieve state-of-the-art.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' Related Work 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' GAN Inversion GAN inversion aims to embed real-world images into a pretrained generator’s latent space, which can be used to reconstruct and edit input images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' Generally, methods can be divided into two stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' The first stage aims to attain low-rate latent codes, usu- ally in Z/W/W + spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' The latent codes are gained by an encoder or optimization process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' Training an encoder [49, 43, 54, 17, 8, 41] to predict latent codes is efficient for inference and is easier to get better trade-offs between fi- delity and manipulation [49, 41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' Optimizing initial latent codes by reconstruction discrepancy gains better fidelity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' However, it may cost several minutes per image [28, 8, 1, 2] and decreases editability during per-image tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' Due to low-rate characteristics, latent codes can only reconstruct coarse information and drop the details from original im- ages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' Meanwhile, there is a trade-off between fidelity and editability, and many methods introduce additional regular- ization modules (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=', latent code discriminator [49] and la- tent space alignment [41]) to address it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' In the second stage, reconstruction and manipulation re- sults from latent codes are refined by high-rate information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' Refinement methods are mainly divided into weight mod- ulation and feature modulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' Weight modulation meth- ods predict or finetune generator weight to improve fidelity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' Some methods use hypernet [18] to predict weight offsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' The others tune generator by given images, which attain bet- ter fidelity but cost much time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' Another branch further in- vert images to latent feature, which we call feature modula- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' HFGI [52] proposes a distortion consultation approach for high-fidelity reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' SAM [36] segments images into various parts and inverts them into different intermedi- ate layers by predicting ”invertibility.” All of them only use one of feature and weight modulation to refine results and suffer editing capability degradation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' In this work, we com- generator manifold latent space weight modulation 𝐺(𝑤;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' 𝜃∗) 𝐺(𝑤) 𝑤 generator manifold latent space 𝐺(𝑤, 𝑓) 𝐺(𝑤) 𝑤 image space image space feature modulation generator manifold latent space feature modulation 𝐺(𝑤, 𝑓;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' 𝜃∗) 𝐺(𝑤) 𝑤 image space weight modulation 𝐺(𝑤) 𝐺(𝑤;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' 𝜃∗) 𝐺(𝑤;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' 𝜃∗) 𝐺(𝑤, 𝑓;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' 𝜃∗) (a) Weight Modulation (b) Feature Modulation (c) Hybrid Modulation Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' Comparison of different refinement mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' We suppose that the refinement results are similar to the given images in all pipelines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' The first row shows two previous mainstream refinement mechanisms, weight, and feature modulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' Weight modulation changes the generator manifold and feature modulation introduces spatial high-rate information to recover image details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' The bottom part demonstrates our hybrid refinement method, combining these two modulation mechanisms to retain editing capability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' We tune the generator on invertible and editable areas, which causes lower manifold deviation, and the result is shown on G(w;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' θ∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' To reconstruct faithfully, we use feature modulation on the other area and attain G(w, f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' θ∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' bine these two aspects by their pros and cons to reach more promising results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' GAN-based Manipulation GANs’ latent spaces encode highly rich semantic infor- mation, which develop the GAN-based manipulation task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' It aims to edit given images by changing latent codes in certain directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' Many works propose multiple meth- ods to find semantic editing directions in latent spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' Some methods obtain the edit vectors of the correspond- ing attributes by means of supervision with the help of attribute-labeled datasets [10, 15, 47, 45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' And others ex- plore the latent space by unsupervised [19, 46, 50, 51] or self-supervised ways [23, 39] to find more semantic direc- tions way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' Method 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' Preliminaries Inversion is built to bridge real-world images and GANs’ latent space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' As latent codes are low-rate, which limits their reconstruction performance, much research has re- cently focused on additional high-rate information in gen- eration process, which we call refinement methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' They can be mainly divided into two categories: weight modula- tion and feature modulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' We first formulate them and analyze the causes of editing capacity degradation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' Formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' We denote the original generation process as X = G(w), where G is the generator, w is latent code which can represent each latent space (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=', Z/W/W +).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' We use encoded latent codes as w in the refinement process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' Weight modulation methods predict [5] or optimize [28] θ by minimizing reconstruction error, and are denoted as X = G(w;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' θ∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' And feature modulation methods in- vert images into the intermediate feature, which follows X = G(w, f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' Defining L as the distance of images, we can illustrate these two refinement processes as follow: θ∗ = arg min θ L(x, G(w;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' θ)) (1) f ∗ = arg min f L(x, G(w, f)) (2) Impacts on editing capability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' Weight and feature modu- lation impacts image manipulation in different aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' The Easy Sample Easy Sample Hard Sample, Occlusion Hard Sample, Artifact Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' Impacts on editing capability of weight modulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' We show the input images, inversion results, and two editing re- sults (smile and age) from PTI [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' For those easy samples, edit- ing results are reasonable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' However, editing capability degrades significantly on hard samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' schematics are shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' Since the feature modu- lation mechanism fixes the intermediate feature distribution at one of layers, the effects of edit vectors applied to previ- ous layers cannot edit the features of latter layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' Although many existing works make efforts to maintain the editing ef- fects, including training with adaptive distortion alignment [25, 52], their solutions still sacrifice fidelity or editing re- sults [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' Meanwhile, weight modulation shows promising editing performance but also gains unreasonable results on com- plex images, as shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' Editing results are more reasonable on easy samples than on hard samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' The main reason for editing capacity degradation is the signif- icant weight deviation caused by refining complex images, which we show in Figure 2(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' To reconstruct given images, the weight modulation mechanism may change the genera- tor manifold much.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' Therefore, the highly semantic charac- teristic of the pretrained generator is broken, which would decrease the editing capability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' In conclusion, the critical problem of refinement meth- ods is how to decrease weight deviation with reconstruc- tion improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' We next propose our method to answer this question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' Overview In this work, we conduct Domain-Specific Hybrid Refinement (DHR) to deal with real-world image inversion and the pipline is shown in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' Based on the above analysis, we explore the idea of ”divide and conquer.” We first propose the concepts of in-domain and out-of- domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' In-domain implies areas that have a similar distri- bution with the generator’s output space and are easy to in- vert, while out-of-domain areas misalign with output space and are difficult to invert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' For example, in face domain, in-domain areas mainly consist of face and hair, while out- of-domain areas consist of occlusions, backgrounds, and ar- tifacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' Meanwhile, in-domain areas are more editable, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=', smile, lipstick, and eyes openness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' Hence, we propose a hybrid refinement method, which segments images into in-domain and out-of-domain areas and applies weight and feature modulation to improve fi- delity and preserve editing capability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' Our framework is shown in Figure 4, which consists of three components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' The image Embedding module aims to embed images into latent codes, which we use an off-the-shelf encoder (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=', e4e [49] and LSAP [41]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' Given input images X, the encoder predicts its W + space latent codes, which we de- note as w = E(X), where E is an encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' Domain-Specific Segmentation predicts a binary mask which indicates in-domain and out-of-domain areas: m = S(X) where m ∈ {0, 1}h×w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' It segments images into two parts, which will be used for refinement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' In Hybrid Modulation Refinement, weight modulation is applied to in-domain areas to recover image details in both inversion and editing results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' Thanks to the low reconstruc- tion discrepancy of in-domain part, weight deviation would not be large, and editing capacity would be preserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' For out-of-domain parts, we use feature modulation to refine them spatially and not to edit them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' Hence, those hard-to- invert parts would not influence editing ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' We mod- ulate weight θ and feature f by minimizing reconstruction error in in-domain and out-of-domain part, respectively: Xrec = G(w, f, m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' θ) (3) The difference with vanilla weight and feature modula- tion can be seen in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' Based on hybrid ways, genera- tor manifold would not change a lot, which highly maintains the editing capability of the original GAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' Domain-specific Segmentation The first challenge is segmenting images into in-domain and out-of-domain at the pixel-level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' An end-to-end domain-specific segmentation model is required for a large, manually annotated dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' Although previous work [36] omFEOTURUUSH FEATORLNSTPTOAIKO 2,2 OTomHybrid Modulation Refinement Domain-Specific Segmentation θ∗ Segmentation 𝒘" 𝒇∗ Input Encoder Image Embedding Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' Overview of our Domain-Specific Hybrid Refinement framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' We use an off-the-shelf image encoding mechanism and introduce Domain-Specific Segmentation and Hybrid Modulation Refinement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' The former segments the input images with two domains: in-domain and out-of-domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' They are refined by weight modulation and feature modulation in the latter method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' Illustration of Domain-Specific Segmentation module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' We use a parsing model and superpixel algorithm with coarse opti- mization to segment input images into in-domain (white areas) and out-of-domain (black areas) parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' trains an invertibility prediction model by self-supervision, results are inaccurate in some complex areas, which we il- lustrate in our ablation study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' In this work, we propose a Domain-Specific Segmentation module, combining parsing and superpixel modules to generate domain segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' Our module is robust for real images and does not require data annotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' The pipeline is shown in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' The parsing model categorizes face components [69] like eye, mouth, and background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' For parsing results mp, we manually set some categories as out-of-domain and the oth- ers as in-domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' However, it is not robust on some complex images, as shown in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' The parsing result represents a coarse mask, where complicated paradigms are not seg- mented well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' Therefore, we introduce a superpixel module with coarse optimization to improve the segmentation re- sults.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' We use a superpixel algorithm [3] for image partition- ing, shown in the middle route of Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' This step finely segments images to distinguish each area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' We denote each partition as {mi s}S i=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' Categorizing each partition into in- domain and out-of-domain without manual annotation is a crucial challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' We first apply a coarse optimization in W space, where latent codes initialed by mean values are only optimized by a few steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' Since in-domain are those easy- to-invert areas, the coarse inversion result Xcoarse could reconstruct in-domain areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' We calculate the perceptual loss L between the coarse reconstruction image and the in- put image, as shown at the bottom of Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' White area Parsing Result Domain Segment om: Face Parsing loma ome om Superpixel K Input X Superpixel Result om out-of-domain Coarse Optimization in-domain Xcoarse LPIPS(X, Xcoarse)m KPTOAIKO 2,2 OTmeans higher loss value, while black area means lower loss value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' As can be seen, the loss of the occlusion area is sig- nificantly higher than the face area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' We calculate the aver- age loss of each partition as follows: vi = L ⊙ mi s ||mis|| and the result {mi s, vi}S i=1 is visualized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' Then we binarize superpixel results by adaptive threshold τ and attain ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' Finally, domain-specific segmentation results are com- bined by parsing results and superpixel results: m = mp × ms Our Domain-Specific Segmentation module can gain fine segmentation results without data annotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' Hybrid Modulation Refinement Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' Illustration of Hybrid Modulation Refinement mod- ule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' We refine in-domain areas and out-of-domain areas by weight and feature modulation, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' Black lines indicate forward flow, and orange and blue lines represent gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' To faithfully recover image details and maintain editing capability from original GAN, we introduce a Hybrid Mod- ulation Refinement module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' It consists of two mainstream refinement aspects: weight modulation and feature modu- lation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' Weight modulation aims to minimize in-domain re- construction error by tuning the generator’s parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' In contrast, feature modulation is applied to out-of-domain ar- eas by optimizing an intermediate feature of the generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' The forward and backward processes are shown in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' For lth layer of total k stages in generator, the original generator’s feature is denoted as fl = Gl(w;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' θ), and an ad- ditional modulated feature is marked as f, which is initialed by fl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' Fixing the latent codes, the original feature fl is only relevant to θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' Given segmentation result m, we formulate the forward process as follows: f ′ = fl ⊙ m + f ⊙ (1 − m) (4) Then f ′ represents the output of the first l layers and gener- ates the final images, which follow Eq 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' For backward, we use mean square error L2 and percep- tual loss Llpips as objectives in the refinement process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' To make weight and feature focus on the corresponding areas, we update them in a parallel optimization process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' Calcu- lating the reconstruction errors, we backward loss with seg- mentation result m: L = L2 + λLlpips (5) ∇f = ∂ ∂f [L ⊙ (1 − m) ||1 − m|| ] (6) ∇θ = ∂ ∂θ[L ⊙ (m) ||m|| ] (7) where λ is a hyper-parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' The parallel optimization mechanism constrains the impact from different domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' Based on Domain-Specific Segmentation and Hybrid Modulation Refinement, we segment images into in-domain and out-of-domain areas and refine them by weight and fea- ture modulation, which significantly improve fidelity with editing capability remaining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' Experiments 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' Experimental Settings Datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' We evaluate all methods on the CelebA-HQ [24, 35] test set (2,824 images).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' Encoders and the generator are trained on FFHQ [27] (70,000 images).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' Baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' We compare our model with previous state-of- the-art refinement methods, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=', ReStyle [4], HFGI [52], SAM [36], and PTI [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' We use pSp [43], e4e [49] and LSAP [41] as encoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' Moreover, the performance of encoder-based methods is also reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' All of model weights of encoders and generators come from their official release.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' Metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' We evaluate all methods in two respects: inver- sion and editing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' For inversion ability, we conduct MSE, LPIPS [67], and identity similarity calculated by a face recognition model [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' MSE straightforwardly measures the image distortion, and LPIPS evalutaes the visual dis- crepancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' Identity similarity further compares the identity consistency during inversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' Moreover, we perform user studies to evaluate the perceptual performance of inversion and editing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' Main Results Quantitative results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' We first evaluate the reconstruction ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' Qualitative results are reported in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' We com- pare our method with four previous refinement methods and employ two encoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' We conduct experiments with en- coders and original Wpivot for PTI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' As one can see, DHR achieves the best performance on all metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' Employed with e4e, it gains 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content='0036 MSE, which is 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content='5% e4e, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content='3% om Loss gradient of weight modulation gradient of feature modulationFigure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' Inversion and editing results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' The second column is the segmentation results of in-domain and out-of-domain areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' Our method restores almost all image details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' ReStyle, 17% HFGI, 25% SAM, and 48% PTI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' For LPIPS and identity similarity, it demonstrates similar superiority and surpasses other methods by a large margin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' DHR with LSAP achieves the best performance of all metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' Qualitative results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' We illustrate the inversion and edit- ing results of DHR in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' The second column is the results from the Domain-Specific Segmentation module, and the third is our inversion results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' Some dedicated ar- eas are categorized into out-of-domain domain (black area).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' We edit them by InterFaceGAN and GANSpace, using four editing directions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=', smile, young, exposure, and lipstick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' All image details are preserved in both inversion and editing results, such as hairstyle (the first row), earrings (the fifth row), and hats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' Our results are faithful and photorealistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' We further conduct a qualitative comparison with other methods, shown in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' Although inversion results are Input Inversion Smile Young Exposure LipstickFigure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' Comparisons with previous methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' We compare the inversion and editing results with PTI [4], HFGI [52], and SAM [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' Although HFGI and SAM reach reasonable inversion results, image distortion and details loss occur in editing results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' Our method attains the best fidelity and editing performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' Image details are reserved in both phases, and our results are the most natural.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' Method Encoder MSE ↓ LPIPS ↓ Similarity ↑ ReStyle [4] pSp 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content='0276 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content='1298 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content='5816 e4e 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content='0429 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content='1904 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content='5062 HFGI [52] e4e 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content='0210 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content='1172 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content='6816 LSAP 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content='0210 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content='0945 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content='7405 SAM [36] e4e 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content='0143 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content='1104 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content='5568 LSAP 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content='0117 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content='0939 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content='6184 PTI [44] e4e 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content='0074 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content='0750 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content='8633 LSAP 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content='0067 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content='0666 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content='8696 Wpivot 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content='0084 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content='0845 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content='8402 DHR (ours) e4e 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content='0036 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content='0455 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content='8704 LSAP 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content='0035 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content='0436 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content='8780 Encoder-Only pSp [43] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content='0351 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content='1628 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content='5591 e4e [49] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content='0475 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content='1991 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content='4966 LSAP [41] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content='0397 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content='1766 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content='5305 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' Fidelity results on face domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' We compare DHR to three previous refinement methods with two powerful encoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' The results of these encoders are also presented at the bottom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' reasonable, editing capability degradation occurs in base- lines, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=', decorations and occlusion blur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' User Study We conduct user studies to demonstrate the performance of inversion and editing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' Results are shown in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' We randomly select 50 different images and invert and edit them by HFGI [52], SAM [36], PTI [44], and our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' We then ask three users to make a preference for each pair of images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' A higher value implies users prefer our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' As can be seen, our results are highly preferred by users, which Method Inversion Editing Smile Young Exposure Lipstick ReStyle [4] 94% 100% 100% 90% 94% HFGI [52] 94% 84% 86% 100% 96% SAM [36] 100% 92% 94% 100% 100% PTI [44] 96% 100% 100% 84% 88% Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' User Study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' We conduct user studies on inversion and editing tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' The values in the table indicate the percentage of images where users prefer our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' Results show our method is more faithful and photorealistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' are most all above 90%, compared to previous state-of-the- art methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' It illustrates that our method decreases image distortion and attains better photorealism of reconstruction and manipulation results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' Conclusion In this work, we propose Domain-Specific Hybrid Re- finement to improve GAN inversion and editing capabil- ity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' Specifically, we analyze the causes of editing ability degradation in refinement process and introduce ”divide and conquer” to address it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' Our method consists of Domain- Specific Segmentation and Hybrid Modulation Refinement, which segments images into in-domain and out-of-domain parts and refines them by weight and feature modulation, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' Our method attains promising results in both inversion and editing with considerable improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' Young Lipstick Exposure 50 Smile Input Inversion Inversion edit Inversion edit Inversion edit PTI HFGI SAM DHR (ours)References [1] Rameen Abdal, Yipeng Qin, and Peter Wonka.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' Im- age2stylegan: How to embed images into the stylegan latent space?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' In Proceedings of the IEEE/CVF International Con- ference on Computer Vision, pages 4432–4441, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' [2] Rameen Abdal, Yipeng Qin, and Peter Wonka.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' Im- age2stylegan++: How to edit the embedded images?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' In Proceedings of the IEEE/CVF conference on computer vi- sion and pattern recognition, pages 8296–8305, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' [3] Radhakrishna Achanta, Appu Shaji, Kevin Smith, Aurelien Lucchi, Pascal Fua, and Sabine S¨usstrunk.' metadata={'source': 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Wen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' General facial representa- tion learning in a visual-linguistic manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content=' arXiv preprint arXiv:2112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} +page_content='03109, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFLT4oBgHgl3EQfqy9t/content/2301.12141v1.pdf'} diff --git a/2tFQT4oBgHgl3EQf2jaP/content/tmp_files/2301.13424v1.pdf.txt b/2tFQT4oBgHgl3EQf2jaP/content/tmp_files/2301.13424v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..d49bfbf2bb71b14b2cc56b436736598fa07fc61d --- /dev/null +++ b/2tFQT4oBgHgl3EQf2jaP/content/tmp_files/2301.13424v1.pdf.txt @@ -0,0 +1,1616 @@ +INRADIUS OF RANDOM LEMNISCATES +MANJUNATH KRISHNAPUR, ERIK LUNDBERG, AND KOUSHIK RAMACHANDRAN +ABSTRACT. A classically studied geometric property associated to a complex polynomial p is the +inradius (the radius of the largest inscribed disk) of its (filled) lemniscate Λ := {z ∈ C : |p(z)| < 1}. +In this paper, we study the lemniscate inradius when the defining polynomial p is random, namely, +with the zeros of p sampled independently from a compactly supported probability measure µ. If +the negative set of the logarithmic potential Uµ generated by µ is non-empty, then the inradius is +bounded from below by a positive constant with overwhelming probability. Moreover, the inradius +has a determinstic limit if the negative set of Uµ additionally contains the support of µ. +On the other hand, when the zeros are sampled independently and uniformly from the unit circle, +then the inradius converges in distribution to a random variable taking values in (0, 1/2). +We also consider the characteristic polynomial of a Ginibre random matrix whose lemniscate we +show is close to the unit disk with overwhelming probability. +1. INTRODUCTION +Let p(z) be a polynomial of degree n and Λ be its (filled) lemniscate defined by Λ = {z : |p(z)| < +1}. Denote by ρ(Λ) the inradius of Λ. By definition, this is the radius of the largest disk that is +completely contained in Λ. In this paper, we study the inradius of random lemniscates for various +models of random polynomials. +The lemniscate {z : |zn − 1| < 1} has an inradius asymptotically proportional to 1/n. In 1958, P. +Erd¨os, F. Herzog, and G. Piranian posed a number of problems [10] on geometric properties of +polynomial lemniscates. Concerning the inradius, they asked [10, Problem 3] whether the rate of +decay in the example {|zn − 1| = 1} is extremal, that is, whether there exists a positive constant C +such that for any monic polynomial of degree n, all of whose roots lie in the closed unit disk, the +inradius ρ of its lemniscate Λ satisfies ρ ≥ C +n . This question remains open. C. Pommerenke [33] +showed in this context that the inradius satisfies the lower bound ρ ≥ +1 +2e n2 . +Our results, which we state below in Sec. 1.4 of the Introduction, show within probabilistic set- +tings that the typical lemniscate admits a much better lower bound on its inradius. Namely, if the +zeros of p are sampled independently from a compactly supported measure µ whose logarithmic +potential has non-empty negative set, then the inradius of Λ is bounded below by a positive con- +stant with overwhelming probability, see Theorem 1.1 below. Let us provide some insight on this +result and explain why the logarithmic potential of µ plays an important role. First, the lemniscate +Λ can alternatively be described as the sublevel set { 1 +n log |p(z)| < 0} of the discrete logarith- +mic potential 1 +n log |p(z)| = +1 +n +� log |z − zk| where zk are the zeros of p(z). For fixed z the sum +1 +n +� log |z − zk| is a Monte-Carlo approximation for the integral defining the logarithmic potential +Uµ(z) of µ, and, in particular, it converges pointwise, by the law of large numbers, to Uµ(z). With +the use of large deviation estimates, we can further conclude that each z in the negative set Ω− of +Uµ is in Λ with overwhelming probability. The property of holding with overwhelming probabil- +ity survives (by way of a union bound) when taking an intersection of polynomially many such +events. This fact, together with a suitable uniform estimate for the derivative p′(z) (for which we +can use a Bernstein-type inequality), allows for a standard epsilon-net argument showing that an +1 +arXiv:2301.13424v1 [math.PR] 31 Jan 2023 + +2 +MANJUNATH KRISHNAPUR, ERIK LUNDBERG, AND KOUSHIK RAMACHANDRAN +arbitrary compact subset of Ω− is contained in Λ with overwhelming probability. Since Ω− is as- +sumed nonempty, this leads to the desired lower bound on the inradius, see the proof of Theorem +1.1 in Section 3 for details. +Under an additional assumption that the negative set Ω− of the logarithmic potential of µ contains +the support of µ, the inradius converges to the inradius of Ω− almost surely, see Corollary 1.2; in +particular, the inradius has a deterministic limit. +On the other hand, for certain measures µ, the inradius does not have a deterministic limit and +rather converges in distribution to a nondegenerate random variable, see Theorem 1.5 addressing +the case when µ is uniform measure on the unit circle. We also consider the lemniscate associated +to the characteristic polynomial of a random matrix sampled from the Ginibre ensemble, and we +show that the inradius is close to unity (in fact the whole lemniscate is close to the unit disk) with +overwhelming probability, see Theorem 1.6. +See Section 1.4 below for precise statements of these results along with some additional results +giving further insight on the geometry of Λ. +1.1. Previous results on random lemniscates. The current paper fits into a series of recent stud- +ies investigating the geometry and topology of random lemniscates. Let us summarize previous +results in this direction. We note that the lemniscates studied in the results cited below, in contrast +to the filled lemniscates of the current paper, are level sets (as opposed to sublevel sets). +Partly motivated to provide a probabilistic counterpart to the Erd¨os lemniscate problem on the +extremal length of lemniscates [10], [5], [11], [12], the second and third authors in [23] studied the +arclength and topology of a random polynomial lemniscate in the plane. When the polynomial has +i.i.d. Gaussian coefficients, it is shown in [23] that the average length of its lemniscate approaches +a constant. They also showed that with high probability the length is bounded by a function with +arbitrarily slow rate of growth, which means that the length of a lemniscate typically satisfies a +much better estimate than the extremal case. It is also shown in [23] that the number of connected +components of the lemniscate is asymptotically n (the degree of the defining polynomial) with +high probability, and there is at least some fixed positive probability of the existence of a “giant +component”, that is, a component having at least some fixed positive length. Of relevance to the +focus of the current paper, we note that the proof of the existence of the giant component in [23] +shows that for a fixed 0 < r < 1, there is a positive probability that the inradius ρ of the lemniscate +satisfies the lower bound ρ > r. +Inspired by Catanese and Paluszny’s topological classification [7] of generic polynomials (in terms +of the graph of the modulus of the polynomial with equivalence up to diffeomorphism of the do- +main and range), in [9] the second author with M. Epstein and B. Hanin studied the so-called +lemniscate tree associated to a random polynomial of degree n. The lemniscate tree of a poly- +nomial p is a labelled, increasing, binary, nonplane tree that encodes the nesting structure of the +singular components of the level sets of the modulus |p(z)|. When the zeros of p are i.i.d. sam- +pled uniformly at random according to a probability density that is bounded with respect to Haar +measure on the Riemann sphere, it is shown in [9] that the number of branches (nodes with two +children) in the induced lemniscate tree is o(n) with high probability, whereas a lemniscate tree +sampled uniformly at random from the combinatorial class has asymptotically +� +1 − 2 +π +� +n many +branches on average. +In [21], partly motivated by a known result [11], [43]) stating that the maximal length of a rational +lemniscate on the Riemann sphere is 2πn, the second author with A. Lerario studied the geometry +of a random rational lemniscate and showed that the average length on the Riemann sphere is + +INRADIUS OF RANDOM LEMNISCATES +3 +asymptotically π2 +2 +√n. Topological properties (the number of components and their nesting struc- +ture) were also considered in [21], where the number of connected components was shown to +be asymptotically bounded above and below by positive constants times n. Z. Kabluchko and I. +Wigman subsequently established an asymptotic limit law for the number of connected compo- +nents in [16] by adapting a method of F. Nazarov and M. Sodin [28] using an integral geometry +sandwich and ergodic theory applied to a translation-invariant ensemble of planar meromorphic +lemniscates obtained as a scaling limit of the rational lemniscate ensemble. +1.2. Motivation for the study of lemniscates. The study of lemniscates has a long and rich history +with a wide variety of applications. The problem of computing the length of Bernoulli’s lemnis- +cate played a role in the early study of elliptic integrals [1]. Hilbert’s lemniscate theorem and its +generalizations [26] show that lemniscates can be used to approximate rather arbitrary domains, +and this density property contributes to the importance of lemniscates in many of the applications +mentioned below. In some settings, sequences of approximating lemniscates arise naturally for ex- +ample in holomorphic dynamics [25, p. 159], where it is simple to construct a nested sequence of +“Mandelbrot lemniscates” that converges to the Madelbrot set. In the classical inverse problem of +logarithmic potential theory—to recover the shape of a two-dimensional object with uniform mass +density from the logarithmic potential it generates outside itself—uniqueness has been shown to +hold for lemniscate domains [37]. This is perhaps surprising in light of Hilbert’s lemniscate the- +orem and the fact that the inverse potential problem generally suffers from non-uniqueness [40]. +Since lemniscates are real algebraic curves with useful connections to complex analysis, they have +frequently received special attention in studies of real algebraic curves, for instance in the study of +the topology of real algebraic curves [7], [2]. Leminscates such as the Arnoldi lemniscate appear +in applications in numerical analysis [39]. Lemniscates have seen applications in two-dimensional +shape compression, where the “fingerprint” of a shape constructed from conformal welding sim- +plifies to a particularly convenient form—namely the nth root of a Blaschke product—in the case +the two-dimensional shape is assumed to be a degree-n lemniscate [8], [44], [35]. Lemniscates have +appeared in studies of moving boundary problems of fluid dynamics [18], [24], [19]. In the study +of planar harmonic mappings, rational lemniscates arise as the critical sets of harmonic polyno- +mials [17], [22] as well as critical sets of lensing maps arising in the theory of gravitational lensing +[30, Sec. 15.2.2]. Lemniscates also have appeared prominently in the theory and application of +conformal mapping [3], [15], [13]. See also the recent survey [36] which elaborates on some of the +more recent of the above mentioned lines of research. +1.3. Definitions and Notation. Throughout the paper, µ will denote a Borel probability measure +with compact support S ⊂ C. The logarithmic potential of µ is defined by +Uµ(z) = +� +S +log |z − w|dµ(w). +It is well known that Uµ is a subharmonic function in the plane, and harmonic in C \ S. For such +µ, we denote the associated negative and positive sets of its potential by +Ω− = {z ∈ C : Uµ(z) < 0}, Ω+ = {z ∈ C : Uµ(z) > 0}. +It is easy to see that Ω− is a (possibly empty) bounded open set. +Assumptions on the measure. Let µ be a Borel probability measure with compact support S ⊂ C. +We define the following progressively stronger conditions on µ. +(A) For each compact K ⊂ C, +C(K) = sup +z∈K +� +S +(log |z − w|)2 dµ(w) < ∞. + +4 +MANJUNATH KRISHNAPUR, ERIK LUNDBERG, AND KOUSHIK RAMACHANDRAN +(B) There is some C < ∞ and ε > 0 such that for all z ∈ C and all r ≤ 1, we have +µ (B(z, r)) ≤ +C +(log(1/r))2+ε . +(C) There exists δ > 0 such that +sup +z∈C +� +S +dµ(w) +|z − w|δ < ∞. +(D) There is some C < ∞ and ε > 0 such that for all z ∈ C and all r > 0, we have +µ (B(z, r)) ≤ Crε. +1.4. Main results. In all theorems (except Theorem 1.6), we have the following setting: +Setting: µ is a compactly supported probability measure on C with support S. The random +variables Xi are i.i.d. from the distribution µ. We consider the random polynomial pn(z) := +(z − X1) . . . (z − Xn) and its lemniscate Λn := {z +: |pn(z)| < 1}. We write ρn = ρ(Λn) for the +inradius of Λn. +Throughout the paper, w.o.p. means with overwhelming probability, i.e., with probability at least +1 − e−cn for some c > 0. +The theorems below concern the random lemniscate Λn. Observe that Λn consists of all z for which +log |pn(z)| < 0, or what is the same, +1 +n +n +� +k=1 +log |z − Xk| < 0. +By the law of large numbers, the quantity on the left converges to Uµ(z) pointwise. Hence we +may expect the asymptotic behaviour of Λn to be described in terms of Uµ and its positive and +negative sets Ω+, Ω−. The first three theorems make this precise under different conditions on the +underlying measure µ. +Theorem 1.1. Assume that µ satisfies assumption (A). Suppose that Ω− ̸= ∅ and let ρ = ρ(Ω−). Fix +compact sets K ⊂ Ω−, and L ⊂ Ω+ \ S. Then for all large n, +K ⊂ Λn, +w.o.p., +and +L ⊂ Λc +n +w.o.p. +In particular, if ρn denotes the inradius of Λn, then +ρn ≥ a +w.o.p., +∀a ∈ (0, ρ) +Corollary 1.2. In the setting of Theorem 1.1, lim inf ρn ≥ ρ a.s. Further, if S ⊆ Ω−, then ρn → ρ a.s. +Ideally, we would have liked to say that a compact set L ⊆ Ω+ is contained inside Λc +n w.o.p. +However, this is clearly not true if some of the roots fall inside L. Making the stronger assumption +(D) on the measure and further assuming that Uµ is bounded below by a positive number on L, +we show that L is almost entirely contained in Λc +n. +Theorem 1.3. Let µ satisfy assumption (D). Let L be a compact subset of {Uµ ≥ m} for some m > 0. +Then there exists c0 > 0 such that +Λn ∩ L ⊂ +n� +k=1 +B(Xk, e−c0n), +w.o.p. + +INRADIUS OF RANDOM LEMNISCATES +5 +In particular, if Uµ ≥ m everywhere, then the whole lemniscate is small. It suffices to assume that +Uµ ≥ m on the support of µ, by the minimum principle for potentials (Theorem 3.1.4 in [34]). +Corollary 1.4. Suppose µ satisfies assumption (D) and Uµ ≥ m on S. Then there is a c0 > 0 such that +Λn ⊂ �n +k=1 B(Xk, e−c0n) and ρn ≤ ne−c0n w.o.p. +A class of examples illustrating Theorem 1.1 and Theorem 1.3 is given at the end of the section. +What happens when the potential Uµ vanishes on a non-empty open set? In this case log |pn| has +zero mean, and is (approximately) equally likely to be positive or negative. Because of this, one +may expect that the randomness in Λn and ρn persists in the limit and we can at best hope for a +convergence in distribution. The particular case when µ is uniform on the unit circle is dealt with +in the following theorem. +Theorem 1.5. Let µ be the uniform probability measure on S1, the unit circle in the complex plane. Then, +ρn +d→ ρ for some random variable ρ taking values in (0, 1 +2). Further, P{ρ < ε} > 0 and P{ρ > 1 +2 − ε} > 0 +for every ε > 0. +As shown in the proof of Theorem 1.5, the random function log |pn(z)| converges, after appropri- +ate normalization, almost surely to a nondegenerate Gaussian random function on D, and this +convergence underlies the limiting random inradius ρ. We note that similar methods can be used +to study other measures µ for which Uµ vanishes on non-empty open set (such as other instances +where µ is the equilibrium measure of a region with unit capacity), however the case of the uni- +form measure on the circle is rather special, as the resulting random function log |pn(z)| as well as +its limiting Gaussian random function has a deterministic zero at the origin (which is responsible +for the limiting inradius taking values only up to half the radius of D. +Another setting where one can rely on convergence of the defining function log |pn(z)| is in the +case when the polynomial pn has i.i.d. Gaussian coefficients. Actually, the convergence in this +case is more transparent (and does not require additional tools such as Skorokhod’s Theorem) as +pn can already be viewed as the truncation of a power series with i.i.d. coefficients. This case has +a similar outcome as in Theorem 1.5, except the value 1/2 is replaced by 1 due to the absence of a +deterministic zero. +One can ask for results analogous to Theorems 1.1 and 1.3 when the zeros are dependent random +variables. A natural class of examples are determinantal point processes. We consider one special +case here. +The Ginibre ensemble is a random set of n points in C with joint density proportional to +e− �n +k=1 |λk|2 � +j √e, Theorem 1.3 applies to show that Λn is contained in a union of +very small disks. +Let us carry out the computations to verify (3). By rescaling, it is clear that Uµr(z) = log r + +Uµ1(z/r), hence it suffices to consider r = 1. +Uµ1(z) = 1 +π +� +D +log |z − w|dA(w). +For |z| ≥ 1, the integrand is harmonic with respect to w ∈ D, hence Uµ(z) = log |z| by the mean- +value theorem. For |z| < 1, we separate the integral over the two regions where |w| < |z| and +|w| > |z|. Harmonicity of w �→ log |z − w| on {|w| < |z|} and the mean-value property gives +� +|w|<|z| +log |z − w|dA(w) = π|z|2 log |z|. +We switch to polar coordinates w = reiθ for the second integral. +� 1 +|z| +� 2π +0 +log |z − reiθ|rdθdr = +� 1 +|z| +� 2π +0 +log |ze−iθ − r|dθ +� +�� +� +2π log r +rdr += +� 1 +|z| +2πr log rdr += 2π +�1 +4 − |z|2 +2 log |z| + |z|2 +4 +� +, +where we have again used the mean value property (this time over a circle) for harmonic functions +to compute the inside integral in the first line above. Combining these integrals over the two +regions and dividing by π we arrive at (3). + +INRADIUS OF RANDOM LEMNISCATES +7 +FIGURE 1. Lemniscates of degree n = 30, 40, 400, 15 with zeros sampled uniformly +from the disks of radii 0.5, 1, 1.5, 1.7 (order: from top-left to bottom-right). The +dotted circle has radius rc. +-1.5 +-1.0 +-0.5 +0.0 +0.5 +1.0 +1.5 +-1.5 +-1.0 +-0.5 +0.0 +0.5 +1.0 +1.5 +-1.5 +-1.0 +-0.5 +0.0 +0.5 +1.0 +1.5 +-1.5 +-1.0 +-0.5 +0.0 +0.5 +1.0 +1.5 +-1.5 +-1.0 +-0.5 +0.0 +0.5 +1.0 +1.5 +-1.5 +-1.0 +-0.5 +0.0 +0.5 +1.0 +1.5 +FIGURE 2. Lemniscates of degree n = 20, 30, 40 with zeros sampled uniformly +from the unit circle. A unit circle is also plotted for reference in each case. +Outline of the paper. We review some preliminary results in Section 2 that serve as tools in the +proofs the results stated above. We prove Theorem 1.1 and Corollary 1.2 in Section 3, and we +prove Theorem 1.3 and Corollary 1.4 in Section 4. The proof of Theorem 1.5, concerning uniform +measure on the circle, is presented in Section 5, and Theorem 1.6, related to the Ginibre ensemble, +is proved in Section 6. +2. PRELIMINARY RESULTS +We start with two preparatory lemmas which we use repeatedly in the proofs of our theorems. +Lemma 2.1. Let µ be a Borel probability measure with compact support S ⊂ C satisfying Assumption +(A). Whenever K is a non-empty compact subset of Ω− or a compact subset of Ω+ with K ∩ S = ∅, there + +1.0 +1.0 +0.5 +0.5 +0.0 +0.0 +-0.5 +0.5 +-1.0 +1.0 +1.0 +0.5 +0.0 +0.5 +1.0 +1.0 +0.5 +0.0 +0.5 +1.0 +1.0 +1.0F +0.5 +0.5 +0.0 +0.0F +-0.5 +0.5 +.. +1.0 +1.0 +1.0 +0.5 +0.0 +0.5 +1.0 +1.0 +0.5 +0.0 +0.5 +1.08 +MANJUNATH KRISHNAPUR, ERIK LUNDBERG, AND KOUSHIK RAMACHANDRAN +-1.0 +-0.5 +0.0 +0.5 +1.0 +-1.0 +-0.5 +0.0 +0.5 +1.0 +-1.0 +-0.5 +0.0 +0.5 +1.0 +-1.0 +-0.5 +0.0 +0.5 +1.0 +-1.0 +-0.5 +0.0 +0.5 +1.0 +-1.0 +-0.5 +0.0 +0.5 +1.0 +FIGURE 3. Lemniscates of degree n = 20, 30, 40 with i.i.d. Gaussian coefficients +plotted together with a unit circle for reference. +-1.5 +-1.0 +-0.5 +0.0 +0.5 +1.0 +1.5 +-1.5 +-1.0 +-0.5 +0.0 +0.5 +1.0 +1.5 +-1.5 +-1.0 +-0.5 +0.0 +0.5 +1.0 +1.5 +-1.5 +-1.0 +-0.5 +0.0 +0.5 +1.0 +1.5 +-1.5 +-1.0 +-0.5 +0.0 +0.5 +1.0 +1.5 +-1.5 +-1.0 +-0.5 +0.0 +0.5 +1.0 +1.5 +FIGURE 4. Lemniscates of degree n = 20, 30, 40 generated by the characteristic +polynomial of a Ginibre matrix together with a unit circle plotted for reference. +exists a constant c(K) > 0 such that +inf +z∈K +� +S +(log |z − w|)2 dµ(w) ≥ c(K). +Proof. Let µ satisfy assumption (A), and let K ̸= ∅ be compact. Then, by the Cauchy-Schwarz +inequality we have for all z ∈ K +� +S +(log |z − w|)2 dµ(w) ≥ +�� +S +|log |z − w|| dµ(w) +�2 +≥ |Uµ(z)|2 +Thus in order to prove the lemma, it suffices to show that |Uµ(z)|2 is bounded away from zero for +z ∈ K, whenever K ⊂ Ω−, or K ⊂ Ω+ and K ∩ S = ∅. +Suppose first that K ⊂ Ω− is compact. Since subharmonic functions are upper semi-continuous +and hence attain a maximum on any compact set, there exists c1(K) > 0 such that Uµ(z) ≤ +−c1(K), for all z ∈ K. Hence |Uµ(z)|2 ≥ c1(K)2, for z ∈ K. In the other case, let K ⊂ Ω+ be +compact and disjoint from the support S of µ. Notice then that Uµ(z) is positive and harmonic +on K. An application of Harnack’s inequality now gives the existence of the required constant +(depending only on K). This concludes the proof of the lemma. +□ +The second lemma is based on a net argument which allows us to control the size of the modulus +of a polynomial by its values at the points of the net. + +INRADIUS OF RANDOM LEMNISCATES +9 +Lemma 2.2. Let G be a bounded Jordan domain with rectifiable boundary. Let p(z) be a polynomial of +degree n. Then, there exists a constant C = C(G) > 0, and points w1, w2...wCn2 ∈ ∂G such that +(4) +∥p∥∂G ≤ 2 +max +1≤k≤Cn2 |p(wk)| +Proof. The key to the proof is a Bernstein-type inequality (see [32, Thm. 1]) +(5) +|p′(z)| ≤ C1n2M, +where M := ∥p∥∂G, and C1 is a constant that depends only on G. With this estimate in hand, the +proof reduces to the following argument that is well-known but which we nevertheless present in +detail for the reader’s convenience. Let ℓ = ℓ(∂G) denote the length of ∂G. Let N be a positive +integer to be specified later. Divide ∂G into N pieces of equal length, with w0, w1..., wN denoting +the points of subdivision. Let z0 ∈ ∂G be such that M = ∥p∥∂G = |p(z0)|. If z0 is one of the wj, +then the estimate (4) clearly holds. If that is not the case, then z0 lies |z0 −wj| ≤ ℓ +N , for some j with +0 ≤ j ≤ N. We can now write +(6) +M − |p(wj)| ≤ |p(z0) − p(wj)| = +����� +� z0 +wj +p′(t)dt +����� ≤ C1n2M ℓ +N . +Here we have used the Bernstein-type inequality (5) to estimate the size of |p′|. If we now choose +N = 2ℓC1n2, then the estimate (6) becomes +M − |p(wj)| ≤ M +2 +which concludes the proof of the lemma. +□ +We will also need the following concentration inequality (see Section 2.7 of [6]). This result, re- +ferred to as “Bennett’s inequality”, is similar to the well-known Hoeffding inequality, but note +that, instead of being bounded, the random variables are merely assumed to be bounded from +above. +Theorem 2.3 (Bennett’s inequality). Let X1, X2, ..., Xn be independent random variables with finite +variance such that Xi ≤ b for some b > 0 almost surely for all i ≤ n. Let +S = +n +� +i=1 +(Xi − E(Xi)) +and ν = �n +i=1 E(X2 +i ). Then for any t > 0, +P(S > t) ≤ exp +�−ν +b2 h +�bt +ν +�� +, +where h(u) = (1 + u) log(1 + u) − u for u > 0. +3. PROOFS OF THEOREM 1.1 AND COROLLARY 1.2 +Proof of Theorem 1.1. We divide the proof into two steps. +Step 1: Compact subsets of Ω− lie in Λn +By our hypothesis Ω− ̸= ∅. Let K ⊂ Ω− be compact. We wish to show that K ⊂ Λn w.o.p. We +may assume without loss of generality that K = G for some bounded Jordan domain G with + +10 +MANJUNATH KRISHNAPUR, ERIK LUNDBERG, AND KOUSHIK RAMACHANDRAN +rectifiable boundary, since any connected compact is contained such a domain. Recall that Λn = +{z : log |pn(z)| < 0}. Writing +log |pn(z)| = +n +� +k=1 +log |z − Xk| +as a sum of i.i.d. random variables, for z ∈ Ω− we will use a concentration inequality to show that +log |pn(z)| is negative with overwhelming probability. We then use lemma 2.2 to get a uniform +estimate on K and finish the proof. +Fix z0 ∈ K. For k = 1, 2, ..., n, define Yk = log |z0 − Xk|, and let +Z := +n +� +k=1 +(Yk − EYk). +Notice since z0 ∈ Ω− +EYk = Uµ(z0) < 0 +and by the assumption in the statement of the theorem, we also have +σ2 +z0 := EY 2 +k = +� +S +(log |z0 − u|)2dµ(u) < ∞ +Now applying Theorem 2.3 to our problem with b ≥ supz∈K,w∈S log(|z|+|w|), ν = nσ2 +z0, we obtain +P{log |pn(z0)| > − log(2)} = P{log |pn(z0)| − nUµ(z0) > −nUµ(z0) − log(2)} += P{Z > −nUµ(z0) − log(2)} +≤ exp +� +−nσ2 +z0 +b2 h +� −b +σ2z0 +Uµ(z0) − b log(2) +nσ2z0 +�� +. +Since subharmonic functions are upper semi-continuous and hence attain a maximum on any +compact set, we have, Uµ(z) ≤ −M for all z ∈ K and some M > 0. Also, by Lemma 2.1, 0 < +c1(K) ≤ σ2 +z ≤ c2(K) < ∞, for all z ∈ K. This bound together with the fact that h is an increasing +function can now be used in the above estimate to get +(7) +P{log |pn(z0)| > − log(2)} ≤ exp +� +−nσ2 +z0 +b +h +� −b +σ2z0 +Uµ(z0) − b log(2) +nσ2z0 +�� +≤ exp (−cn) +for some constant c = c(K) > 0 depending only on K. Using lemma 2.2 in combination with a +union bound and the estimate (7), we obtain +P{log ∥pn∥K < 0} ≥ P{ +max +1≤k≤Cn2 log |pn(wk,n)| + log(2) < 0} += 1 − P{ +max +1≤k≤Cn2 log |pn(wk,n)| > − log(2)} += 1 − P +� +� +Cn2 +� +k=1 +{log |pn(wk,n)| > − log(2)} +� +� +≥ 1 − Cn2 exp (−cn) +where in the last inequality we used (7). This proves that K ⊂ Λn w.o.p. and concludes the proof +of the first part. + +INRADIUS OF RANDOM LEMNISCATES +11 +Step 2: Compact subsets L of Ω+ \ S are in Λc +n. +Without loss of generality, we may assume that L is a closed disc in Ω+ \ S. Since S is a compact +set disjoint from L, there exists δ > 0 such that the distance d(L, S) = δ. Notice that for all z ∈ L, +we have − log |z − Xi| ≤ − log δ. Now fix z0 ∈ L. An application of Bennett’s inequality to the +random variables − log |z0 − Xi| yields, +(8) +P (− log |pn(z0)| + nUµ(z0) ≥ nUµ(z0) − 1) ≤ exp +� +−nσ2 +z0 +b +h +� b +σ2z0 +Uµ(z0) − +b +nσ2z0 +) +�� +. +The quantities b, h have an analogous meaning as in Step 1. By Lemma 2.1, σ2 +z is bounded below, +and by assumption it is also bounded above, by some positive constants depending only on L. +Furthermore, Lemma 2.1 shows that Uµ(z) ≥ c(L) > 0 for all z ∈ L. Making use of all this in (8), +we can now estimate +P (log |pn(z0)| > 1) = P (log |pn(z0)| − nUµ(z0) > −nUµ(z0) + 1) += 1 − P (log |pn(z0)| − nUµ(z0) ≤ −nUµ(z0) + 1) += 1 − P (− log |pn(z0)| + nUµ(z0) ≥ nUµ(z0) − 1) +≥ 1 − exp +� +−nσ2 +z0 +b +h +�bUµ(z0) +σ2z0 +− +b +nσ2z0 +�� +≥ 1 − exp (−C0(L)n) . +(9) +This estimate shows that individual points of L are in Λc +n with overwhelming probability. To finish +the proof, we once again use a net argument to show that L ⊂ Λc +n w.o.p. We first observe that if +z, w ∈ L, and X is one of the Xk’s, the mean value theorem gives +| log |z − X| − log |w − X|| ≤ |z − w| +δ +, +where we have used that d(L, S) = δ > 0 (and that L is a disk). The triangle inequality then yields +(10) +| log |pn(z)| − log |pn(w)|| ≤ n|z − w| +δ +, +for z, w ∈ L. +Choose a net of n2 equally spaced points w1, w2, ...wn2 on ∂L, and note that any point on ∂L is +within C1/n2 of some point in the net, where C1 is a constant depending on the radius of L. From +(10) we have that +(11) +| log |pn(z)| − log |pn(w)|| ≤ C2 +n , +for z, w ∈ L with |z − w| ≤ C1 +n2 , +where C2 = C1/δ is a constant. +We are now ready to show that for large n, infz∈L log |pn(z)| > 0 w.o.p. Indeed, note that the point +on ∂L where the infimum of log |pn| is attained must be within C1/n2 of some point in the net +{w1, w2..., wn2}. Then by (11), +P +� +inf +L log |pn(z)| > 0 +� +≥ P +� +� +n2 +� +k=1 +{log |pn(wk)| > 1} +� +� . + +12 +MANJUNATH KRISHNAPUR, ERIK LUNDBERG, AND KOUSHIK RAMACHANDRAN +Therefore, we obtain +P +� +inf +L log |pn(z)| > 0 +� +≥ P +� +� +n2 +� +k=1 +{log |pn(wk)| > 1} +� +� += 1 − +n2 +� +k=1 +P (log |pn(wk)| ≤ 1) +≥ 1 − n2 exp(−C0 n). +by the pointwise estimate (9). This concludes the proof of the theorem. +□ +Proof of Corollary 1.2. We assume that the measure µ is as in Theorem 1.1. Let ρn = ρ(Λn) be the +inradius of the lemniscate of pn and let ρ = ρ(Ω−) be the inradius of Ω−. By Theorem 1.1, we +immediately get lim inf ρn ≥ ρ. +Let S be the support of µ. As S ∩ Ω+ = ∅, Theorem 1.3 shows that if m > 0 then Λn ∩ {Uµ ≥ m} is +contained in a union of at most n circles each of radius e−cn. Writing ρn(m) for ρ(Λn ∩ {Uµ < m}) +and ρ(m) for ρ({Uµ < m}), it is then clear that ρn ≤ ρn(m)+2ne−cn ≤ ρ(m)+2ne−cn and therefore, +first letting n → ∞ and then letting m ↓ 0 we see that +lim sup +n→∞ ρn ≤ lim +m↓0 ρ(m). +As Uµ is continuous on C\S, it follows that for any ε > 0 there is m > 0 such that {Uµ < m} ⊆ Ω− +ε , +the ε enlargement of Ω−. Hence, with ρ′(ε) := ρ(Ω− +ε ), we have +lim sup +n→∞ ρn ≤ lim +ε↓0 ρ′(ε). +Under the additional assumption that S ⊆ Ω−, we show that ρ′(ε) ↓ ρ as ε ↓ 0 and that completes +the proof that lim sup ρn ≤ ρ. That ρ′(ε) ↓ ρ requires a proof as inradius is not continuous under +decreasing limits of sets. For example, the inradius of the slit disk D \ [0, 1) is 1/2 but any ε- +enlargement of it has inradius 1. +As Uµ is harmonic on C \ S and S ⊆ Ω− and Uµ(z) ∼ log |z| near ∞, the level set {Uµ = 0} is +a compact set comprised of curves that are real analytic except for a discrete set of points (the +critical points of Uµ are zeros of locally defined analytic functions). It also separates S from ∞. +Thus, {Uµ < 0} can be written as a union of Jordan domains, and there are at most finitely many +components that have inradius more than any given number. +Pick a component V of Ω− that attains the inradius ρ. The boundary of V can have a finite number +of critical points of Uµ. Locally around any such critical point, Uµ is the real part of a holomorphic +function that looks like czp for some p, and hence Uµ = 0 is like a system of equi-angular lines +with angle π/p between successive rays. In particular, there are no cusps. What this shows is that +V satisfies the following “external ball condition”: There is a δ0 > 0 and B < ∞, such that for any +δ < δ0 and each w ∈ ∂V , there is a +(12) +w′ ∈ C \ V such that |w′ − w| = δ and |w′ − z| ≥ δ/B for all z ∈ Ω−. +Now suppose D(z, r) ⊆ Ω− +ε . If ε < δ0/B, we claim that D(z, r−2Bε) ⊆ Ω−, which of course proves +that ρ ≥ ρ′(ε) − Bε, completing the proof. If the claim was not true, then we could find w ∈ ∂V +such that |w − z| ≤ r − 2Bε. Find w′ as in (12) with δ = Bε. Then w′ ̸∈ Ωδ/B = Ωε but +|w′ − z| ≤ |w′ − w| + |w − z| ≤ δ +B + r − 2Bε < r. +This is a contradiction as w′ ∈ D(z, r) ⊆ Ωε. +□ + +INRADIUS OF RANDOM LEMNISCATES +13 +4. PROOF OF THEOREM 1.3 AND COROLLARY 1.4 +A standard net argument can be used to prove the theorem. But we would like to first present a +proof of Corollary 1.4 by a different method, which may be of independent interest. At the end of +the section, we outline the net argument to prove Theorem 1.3. +We will need the following lemma in the proof of Corollary 1.4. +Lemma 4.1. Under the assumptions of Corollary 1.4, there exists c1 > 0 such that +P +� +log |p′ +n(X1)| ≤ m +2 (n − 1) +� +≤ e−c1n. +First we prove the corollary assuming the above Lemma. +Proof of Corollary 1.4. Let Gi be the connected component of Λn containing Xi. Then by Bernstein’s +inequality we have +(13) +|p′ +n(Xi)| ≤ C +n2 +diam(Gi)∥pn∥∂Gi = C +n2 +diam(Gi). +By Lemma 4.1 we have +|p′ +n(Xi)| ≥ exp +�m +2 (n − 1) +� +, +w.o.p. +and we conclude from (13) that +(14) +diam(Gi) ≤ Cn2 exp +� +−m +2 (n − 1) +� +, +w.o.p. +The event Λn ⊂ �n +k=1 Drn(Xk) occurs if diam(Gi) < rn for each i = 1, 2, ..., n. Using (14) and a +union bound, all these events occur with overwhelming probability if we choose rn = exp{−c0n} +for a suitable c0. +□ +It remains to prove Lemma 4.1. +Proof of Lemma 4.1. We have +P +� +log |p′ +n(X1)| ≤ m +2 (n − 1) +� += +� +S +P +� +log |p′ +n(X1)| ≤ m +2 (n − 1) +��X1 = z +� +dµ(z) +(15) += +� +S +P +� n +� +k=2 +log |z − Xk| ≤ m +2 (n − 1) +� +� +�� +� +(∗) +dµ(z). +Let us rewrite the integrand (∗) as +(∗) = P +� +Z ≥ +� +Uµ(z) − m +2 +� +(n − 1) +� +, +Z = (n − 1)Uµ(z) − +n +� +k=2 +log |z − Xk|. +Then we have (with θ to be chosen below) +(∗) = P +� +eθZ ≥ eθ(n−1)(Uµ−m/2)� +(since Uµ ≥ m) +≤ P +� +eθZ ≥ eθ(n−1)(m/2)� +≤ e−θ(n−1)(m/2)EeθZ. + +14 +MANJUNATH KRISHNAPUR, ERIK LUNDBERG, AND KOUSHIK RAMACHANDRAN +Let Zk = − log |z − Xk| + Uµ(z) so that Z = Z2 + . . . + Zn. As Xi are i.i.d., so are Zi and we have +EeθZ = +� +EeθZ2�n−1 +. +We claim that there exist τ < ∞ and θ0 > 0 (not depending on z ∈ S) such that +E[eθZ2] ≤ eτθ2 +for |θ| < θ0. +(16) +Assuming this, the proof can be completed as follows: +(∗) ≤ e−θ(n−1)m/2e(n−1)τθ2 += e− 1 +4 mθ(n−1) +(17) +provided we choose θ < m +4τ . Using this in (15) we obtain +P +� +log |p′ +n(X1)| ≤ m +2 (n − 1) +� +≤ e− 1 +4 mθ(n−1), +which implies the statement in the lemma. +It remains to prove (16). Assumption (D) in definition A yields that for z ∈ S, +P{Z1 > t} = P{|z − X1| ≤ eUµ(z)−t} +≤ Ceε(M−t) +where M = supz∈S Uµ(z). On the other hand, P{Z1 < −t} = 0 for large t, hence by choosing a +smaller ε if necessary, we have the bound +P{|Z1| > t} ≤ 2e−εt. +A random variable satisfying the above tail bound is said to be sub-exponential (see Section 2.7 +in [41]). It is well-known (see the implication (a) +=⇒ +(e) of Proposition 2.7.1 in [41]) that if a +sub-exponential random variable has zero mean, then (16) holds. +□ +Now we outline the argument for the proof of Theorem 1.3 +Proof of Theorem 1.3. The same argument (basically that − log |z − X1| + Uµ(z) has sub-exponential +distribution) that led to (17) shows that there exists θ > 0 +P{log |pn(z)| < 1 +2mn} ≤ e−θn +(18) +for any z ∈ L. Let rn = e− θ +4 n. Then, if z ∈ L \ �n +k=1 B(Xk, rn), we have +|∇ log |pn(z)|| = +�� +n +� +k=1 +1 +z − Xk +�� ≤ +n +rn +. +Therefore, if z ∈ L \ �n +k=1 B(Xk, (1 + m +4 )rn), then combining the bound on the gradient with (18), +we get +P +� +inf +B(z, 1 +4 mrn) +log |pn| ≥ 1 +4mn +� +≥ 1 − e−θn. +Assuming without loss of generality that m ≤ 1, we may choose a net of C/r2 +n points in L such that +every of point of L\�n +k=1 B(Xk, 2rn) is within distance mrn/4 of one of the points of the net. Then, +log |pn| > 1 +4mn everywhere on L\�n +k=1 B(Xk, 2rn), with probability at least 1− C +r2n e−θn ≥ 1−Ce− θ +2 n, +by our choice of rn. +□ + +INRADIUS OF RANDOM LEMNISCATES +15 +5. PROOF OF THEOREM 1.5 +First we claim that Λn ⊆ (1 + ε)D w.o.p. for any ε > 0. Deterministically, Λn ⊆ 2D, since µ is +supported on S1. Further, Uµ(z) = log+ |z|, hence L = {z : 1 + ε ≤ |z| ≤ 2} is a compact subset +of Ω+. By Theorem 1.1 or Theorem 1.3, we see that L ∩ Λn = ∅ w.o.p. proving that Λn ⊆ (1 + ε)D +w.o.p. +Thus, it suffices to consider Λn ∩ D. Consider +gn(z) = +1 +√n +n +� +k=1 +log |z − Xk| +for z ∈ D. As Xk are uniform on S1, it follows that E[log |z − X1|] = 0. Let +K(z, w) = E[(log |z − X1|)(log |w − X1|)] = 1 +2π +� 2π +0 +log |z − eiθ| log |w − eiθ| dθ. +Hence E[gn(z)] = 0 and E[gn(z)gn(w)] = K(z, w). +Let g be the (real-valued) Gaussian process on D with expectation E[g(z)] = 0 and covariance +function E[g(z)g(w)] = K(z, w). Then by the central limit theorem, it follows that +(gn(z1), . . . , gn(zk)) d→ (g(z1), . . . , g(zk)) +for any z1, . . . , zk ∈ D. We observe that gn(0) = 0 and claim that suprD |∇gn| is tight, for any +r < 1. By a well-known criterion for tightness of measures (on the space C(D) endowed with +the topology of uniform convergence on compacts), this proves that gn → g in distribution, as +processes (see Theorem 7.2 in [4]). +To prove the tightness of suprD |∇gn|, fix r < s < 1 and note that ∇gn(z) is essentially the same as +Fn(z) = +1 +√n +�n +k=1 +1 +z−Xk which is holomorphic on D. By Cauchy’s integral formula, for |z| < r, +|Fn(z)|2 = +�� 1 +2π +� 2π +0 +Fn(seiθ) +z − seiθ iseiθdθ +��2 +≤ +� 1 +2π +� 2π +0 +|Fn(seiθ)|2dθ +� � 1 +2π +� 2π +0 +1 +|z − seiθ|2 dθ +� +≤ +1 +(s − r)2 +1 +2π +� 2π +0 +|Fn(seiθ)|2dθ. +The bound does not depend on z, hence taking expectations, +E[(sup +rD +|Fn|)2] ≤ +1 +(s − r)2 +1 +2π +� 2π +0 +E +� +|Fn(seiθ)|2� +dθ +≤ +1 +(s − r)2 +1 +2π +� 2π +0 +E +� +1 +|seiθ − X1|2 +� +dθ +≤ +1 +(s − r)2(1 − s)2 . +The boundedness in L2 implies tightness of the distributions of Fn, as claimed. +In order to formulate a precise statement on almost sure convergence it is necessary to construct +gn and g on a single probability space. One way to accomplish that is by the Skorokhod represen- +tation theorem (see Theorem 6.7 in [4]) from which it follows that gn and g can be constructed on +one probability space so that gn → g uniformly on compacta, a.s. Hence, the proof of Theorem 1.5 +will be complete if we prove the following lemma. + +16 +MANJUNATH KRISHNAPUR, ERIK LUNDBERG, AND KOUSHIK RAMACHANDRAN +Lemma 5.1. Let fn, f; D → R be smooth functions such that {f = 0} ∩ {∇f = 0} = ∅. Suppose fn → f +uniformly on compact sets of D. Then, ρ({fn < 0}) → ρ({f < 0}). +Indeed, applying this to gn, g, we see that ρ(Λn ∩ D) → ρ({g < 0}) almost surely. On the other +hand, for any ε > 0, Theorem 1.3 shows that Λn ∩ ((1 + ε)D)c is contained in a union of n disks of +radius e−cn, w.o.p. Putting these together, ρ(Λn) → ρ({g < 0}) a.s. and hence in distribution. This +completes the proof of the convergence claim in Theorem 1.5. +Proof of Lemma 5.1. For any U ⊆ D, it is clear that ρ(U) − ε ≤ ρ(U ∩ (1 − ε)D) ≤ ρ(U). Applying +this to U = {fn < 0} and U = {f < 0}, we see that to show that ρ({fn < 0}) → ρ({f < 0}), it is +sufficient to show that ρ({fn < 0}∩(1−ε)D) → ρ({f < 0}∩(1−ε)D) for every ε > 0. On (1−ε)D, +the convergence is uniform, hence for any δ > 0, we have {f < −δ} ⊆ {fn < 0} ⊆ {f < δ} for +sufficiently large n. It remains to show that δ �→ ρ({f < δ}) is continuous at δ = 0. +First we show that ρ({f < −δ}) ↑ ρ({f < 0}) as δ ↓ 0. If B(z, r) ⊆ {f < 0}, then for any ε > 0, +the maximum of f on B(z, r − ε) is some −δ < 0. Hence ρ({f ≤ −δ}) ≥ r − ε proving that +ρ({f < −δ}) ↑ ρ({f < 0}). +Next we show that ρ({f ≤ δn}) ↓ ρ({f ≤ 0}) for some δn ↓ 0. Let rn = ρ({f ≤ +1 +n}) and find +zn such that B(zn, rn) ⊆ {f ≤ 1 +n}. Let rn ↓ r0 and zn → z0 without loss of generality. Then if +w ∈ B(z0, r0), then w ∈ B(zn, rn) for large enough n, hence f(w) ≤ 1 +n for large n. Thus f ≤ 0 on +B(z0, r0) showing that ρ({f ≤ 0}) ≥ lim +δ↓0 ρ({f ≤ δ}). +From the assumption that {f = 0} ∩ {∇f = 0} = ∅, we claim that ρ({f ≤ 0}) = ρ({f < 0}). +Indeed, if B(z, r) ⊆ {f ≤ 0}, then in fact B(z, r) ⊆ {f < 0}. Otherwise, we would get w ∈ B(z, r) +with f(w) = 0 which implies that w is a local maximum of f and hence ∇f(w) = 0. +This proves the continuity of δ �→ ρ({f < δ}) at δ = 0, and hence the lemma. +□ +This completes the proof of the first part that ρn = ρ({gn < 0}) converges in distribution to +ρ = ρ({g < 0}). To show that P({ρ < ε}) > 0, it suffices to show that g > 0 on (1−ε)D∩{| Im z| > ε} +with positive probability. To show that P({ρ > +1 +2 − ε}) > 0, it suffices to show that g < 0 in +(1 − ε)D ∩ {| Im z| > ε} with positive probability. We do this in two steps. +(1) There exist u0 : D → R, harmonic with u0(0) = 0 such that u0 < 0 on (1−ε)D∩{| Im z| > ε}. +This is known, see either the proof of Theorem 6.1 of [27] or take log |p| of the polynomial +p constructed in Lemma 5 of Wagner [42]. +(2) For any u : D → R, harmonic with u(0) = 0 and any r < 1 and ε > 0, we claim that +∥g −u∥sup(rD) < ε with positive probability. Applying this to u0 and −u0 from the previous +step show that ρ > 1 +2 − ε with positive probability and ρ < ε with positive probability. +To this end, we observe that the process g can be represented as +g(z) = Re +∞ +� +k=1 +2 +kakzk +where ak are i.i.d. standard complex Gaussian random variables. The covariance of g +defined as above is +E[g(z)g(w)] = +� +k≥1 +1 +k2 (zk ¯wk + wk¯zk) + +INRADIUS OF RANDOM LEMNISCATES +17 +which can be checked to match with the integral expression for K(z, w) given earlier. Given +any harmonic u : D → R with u(0) = 0, write it as +u(z) = Re +� +k≥1 +ckzk +and choose N such that +∥ +� +k>N +ckzk∥sup(rD) < ε. +If both the events +AN = +� +∥ +� +k>N +ak +k zk∥sup(rD) < ε +� +, +BN = +� +|2ak +k +− ck| < ε +N for 1 ≤ k ≤ N +� +occur, then |g − u| < 3ε on rD. As AN and BN are independent and have positive proba- +bility, we also have P(AN ∩ BN) > 0. +6. PROOF OF THEOREM 1.6 +The idea of the proof proceeds along earlier lines: first we fix t > 0 and show that log |pn(z)| is +negative w.o.p. for a fixed z lying on |z| = 1 − t. It then follows from a net argument that the +whole circle (and hence the disk) is contained in Λn w.o.p. +Let t ∈ (0, +1 +100) and fix z with |z| = 1 − t. Taking logarithms, we have as before that +log |pn(z)| = +n +� +k=1 +log |z − Xk|, +except now the roots are no longer i.i.d. Define Ft : C → R by +Ft(w) = +� +� +� +� +� +log 1 +t , +|z − w| ≥ 1 +t +log |z − w|, +t < |z − w| < 1 +t +log t, +|z − w| ≤ t. +Next, we write +log |pn(z)| = +n +� +k=1 +Ft(Xk) + +� +k:|z−Xk|≥ 1 +t +� +log |z − Xk| − log 1 +t +� ++ +� +k:|z−Xk|≤t +(log |z − Xk| − log t) +=: L1 + L2 + L3. +Since the term L3 is negative, we have +(19) +P +� +log |pn(z)| ≥ − t +4n +� +≤ P +� +L1 + L2 ≥ − t +4n +� +We claim that the right hand side of (19) decays exponentially. For that we will need the following +Proposition 6.1. Fix t > 0. There exist constants ct, c2 > 0 such that for all large n, we have +P +� +L1 ≥ − t +2n +� +≤ 5 exp(−ctn), +P(L2 ≥ t +4n) ≤ n exp(−c2n). + +18 +MANJUNATH KRISHNAPUR, ERIK LUNDBERG, AND KOUSHIK RAMACHANDRAN +Assume the Proposition is true for now. Then, it is easy to see that the right hand side of (19) goes +to 0 exponentially with n. Indeed, +P +� +L1 + L2 ≥ − t +4n +� += P +� +L1 + L2 ≥ − t +4n, L2 < t +4n +� ++ P +� +L1 + L2 ≥ − t +4n, L2 ≥ t +4n +� +≤ P +� +L1 ≥ − t +2n +� ++ P +� +L2 ≥ t +4n +� +≤ 5 exp(−ctn) + n exp(−c2n). +which establishes the claim. We now proceed with the proof of Proposition 6.1. +Proof of Proposition 6.1. Step 1: Estimate on L2 +Let Nt = |{k : |z −Xk| ≥ 1 +t }|. If L2 ≥ t +4n, then we must have Nt ≥ 1, which has probability at most +e−cn for some c > 0. To see this, let us recall the following fact about eigenvalues of the Ginibre +ensemble. +Lemma 6.2 (Kostlan [20], [14]). Let λj be the eigenvalues (indexed in order of increasing modulus) of a +Ginbre random matrix (un-normalized). Then, +{|λ1|2, |λ2|2, ..., |λn|2} ∼ {Y1, Y2, ..., Yn}, +where Yj is a sum of j i.i.d. Exp(1) random variables. +Now for the proof of the claim. Since |z| < 1 and t ∈ (0, +1 +100), |z − Xk| ≥ 1 +t implies for instance that +|Xk| > 99. Therefore, by elementary steps and applying Lemma 6.2, we obtain +P(Nt ≥ 1) +≤ P(maxk |Xk| ≥ 99) += P(maxk |Xk|2 ≥ 992) += P(maxk |λk|2 > 992n) += P(maxk Yk > 992n), +where we have used Xj = λj +√n in going from the second to third line above. Then a union bound +and a Cramer-Chernoff estimate gives +P(max +k +Yk > 992n) +≤ nP(Yn > 992n) +≤ n exp(−c2n), +and combining this with the above estimate we obtain +P(Nt ≥ 1) ≤ n exp(−c2n), +as desired. +Step 2: Estimate on L1 +The desired estimate is equivalent to +(26) +P +� +L1 − E(L1) ≥ − t +2n − E(L1) +� +≤ 5 exp(−ctn). +As preparation towards this, observe that 1 +nE(L1) = E +�� +Ftdµn +� +, where µn is the empirical spec- +tral measure defined in (2). By the circular law of random matrices [38], almost surely µn and + +INRADIUS OF RANDOM LEMNISCATES +19 +its expectation both converge to the uniform measure on the unit disk. As a result, taking into +account that Ft is a bounded continuous function, we obtain +(27) +lim +n→∞ +1 +nE(L1) = 1 +π +� +D +Ftdm = |z|2 − 1 +2 ++ t2 +2 , +where the second equality in (27) follows from a computation similar to the one in Example 1.7. +Using |z| = 1 − t, the quantity on the right reduces to −t + t2. Hence, for large n, we have +E(L1) ≤ − 3 +4tn and hence, if the event in (26) holds, then +L1 − E(L1) ≥ t +4n. +Thus, our immediate goal is reduced to showing that the probability of the above event is at +most 5 exp(−ctn) for an appropriate constant ct. We invoke the following result of Pemantle and +Peres [29, Thm. 3.2]. +Theorem 6.3. Given a determinantal point process with n < ∞ points and f a Lipschitz-1 function on +finite counting measures, for any a > 0 we have +P (|f − E(f)| ≥ a) ≤ 5 exp +� +− +a2 +16(a + 2n) +� +. +To say that f is Lipschitz-1 on the space of finite counting measures means that +���f +�k+1 +� +i=1 +δxi +� +− f +� k +� +i=1 +δxi +� ��� ≤ 1 +for any k ≥ 0 and any points x1, . . . , xk. +In our case, as we have recalled, {X1, X2, ..., Xn} is a determinantal point process with exactly n +points. Moreover, L1 is Lipschitz with Lipschitz constant ∥Ft∥sup = log 1 +t . Applying Theorem 6.3 +to L1/ log(1/t), we see that +P +� +L1 − E(L1) ≥ t +4n +� += 5 exp +� +− +t2n2 +256(log(1/t))2( +tn +4 log(1/t) + 2n) +� +≤ 5 exp{−ctn} +where we may take ct = ct2/ log(1/t)2 for a large constant c. This completes the proof of the +proposition. +□ +Now that we have proved the pointwise estimate, the net argument from Lemma 6 can be used +to show that the whole circle |z| = 1 − t lies in the lemniscate w.o.p. The maximum principle then +shows that the corresponding disk lies in the lemniscate w.o.p. This concludes the proof that Λn +contains Dr w.o.p. +We next prove that Λn ⊆ Ds w.o.p. for s > 1. Fix 1 < s′ < s and let δ = s − s′ and ε = 1 +2 log s. We +present the proof in four steps. +(1) |λj| +√n < s′ for all j, w.o.p., i.e., with probability at least 1−e−cn. To see this, invoke Lemma 6.2 +to see that the complementary event has probability less than ne−c(s′)n by the same reason- +ing used in (25), noting that 992 may be replaced by any constant greater than 1. + +20 +MANJUNATH KRISHNAPUR, ERIK LUNDBERG, AND KOUSHIK RAMACHANDRAN +(2) Fix z with |z| = s and let fz,δ(w) = log +� +min{max{|z − w|, δ}, 1 +δ} +� +, a bounded continuous +function. Then by [31] (Theorem 9), +P +� +� +� +�� 1 +n +n +� +j=1 +fz,δ(λj/√n) − +� +D +fz,δ(w)dm(w) +π +�� > ε +� +� +� ≤ e−cε,δn2. +(3) On the event in (1), fz,δ(λj/√n) = log |z − |λj +√n| for all j and all |z| = s. Also, fz,δ(w) = +log |z − w| for all w ∈ D. Hence, w.o.p. +P +��� 1 +n log |pn(z)| − log s +�� > ε +� +≤ e−cε,δn2 + e−cn. +Hence, 1 +n log |pn(z)| > 1 +2ε w.o.p. by the choice of ε = 1 +2 log s. +(4) Let m = 100 +εδ and let z1, . . . , zm be equispaced points on ∂Ds. Then w.o.p. infj≤m 1 +n log |pn(zj)| > +1 +2ε by the previous step. On the event in (1), ∥∇ 1 +n log |pn(z)|∥ ≤ 1 +δ, hence +inf +|z|=s +1 +n log |pn(z)| > 0 +w.o.p. On this event Λn ⊆ Ds. +This concludes the proof of Theorem 1.6. +REFERENCES +[1] R. Ayoub. The lemniscate and Fagnano’s contributions to elliptic integrals. Arch. Hist. Exact Sci., 29(2):131–149, +1984. +[2] I. Bauer and F. Catanese. Generic lemniscates of algebraic functions. Math. Ann., 307(3):417–444, 1997. +[3] S. R. Bell. A Riemann surface attached to domains in the plane and complexity in potential theory. Houston J. Math., +26(2):277–297, 2000. +[4] P. Billingsley. Convergence of probability measures. Wiley Series in Probability and Statistics: Probability and Statistics. +John Wiley & Sons, Inc., New York, second edition, 1999. A Wiley-Interscience Publication. +[5] P. Borwein. The arc length of the lemniscate {|p(z)| = 1}. Proc. Amer. Math. Soc., 123(3):797–799, 1995. +[6] S. Boucheron, G. Lugosi, and P. Massart. Concentration inequalities. Oxford University Press, Oxford, 2013. 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Soc., 144(3):1087–1093, 2016. + diff --git a/2tFQT4oBgHgl3EQf2jaP/content/tmp_files/load_file.txt b/2tFQT4oBgHgl3EQf2jaP/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b93e4289c1ebcd56c50e3ca197d4e6bb6b0f7597 --- /dev/null +++ b/2tFQT4oBgHgl3EQf2jaP/content/tmp_files/load_file.txt @@ -0,0 +1,1194 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf,len=1193 +page_content='INRADIUS OF RANDOM LEMNISCATES MANJUNATH KRISHNAPUR, ERIK LUNDBERG, AND KOUSHIK RAMACHANDRAN ABSTRACT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' A classically studied geometric property associated to a complex polynomial p is the inradius (the radius of the largest inscribed disk) of its (filled) lemniscate Λ := {z ∈ C : |p(z)| < 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' In this paper, we study the lemniscate inradius when the defining polynomial p is random, namely, with the zeros of p sampled independently from a compactly supported probability measure µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' If the negative set of the logarithmic potential Uµ generated by µ is non-empty, then the inradius is bounded from below by a positive constant with overwhelming probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' Moreover, the inradius has a determinstic limit if the negative set of Uµ additionally contains the support of µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' On the other hand, when the zeros are sampled independently and uniformly from the unit circle, then the inradius converges in distribution to a random variable taking values in (0, 1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' We also consider the characteristic polynomial of a Ginibre random matrix whose lemniscate we show is close to the unit disk with overwhelming probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' INTRODUCTION Let p(z) be a polynomial of degree n and Λ be its (filled) lemniscate defined by Λ = {z : |p(z)| < 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' Denote by ρ(Λ) the inradius of Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' By definition, this is the radius of the largest disk that is completely contained in Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' In this paper, we study the inradius of random lemniscates for various models of random polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' The lemniscate {z : |zn − 1| < 1} has an inradius asymptotically proportional to 1/n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' In 1958, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' Erd¨os, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' Herzog, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' Piranian posed a number of problems [10] on geometric properties of polynomial lemniscates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' Concerning the inradius, they asked [10, Problem 3] whether the rate of decay in the example {|zn − 1| = 1} is extremal, that is, whether there exists a positive constant C such that for any monic polynomial of degree n, all of whose roots lie in the closed unit disk, the inradius ρ of its lemniscate Λ satisfies ρ ≥ C n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' This question remains open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' Pommerenke [33] showed in this context that the inradius satisfies the lower bound ρ ≥ 1 2e n2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' Our results, which we state below in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content='4 of the Introduction, show within probabilistic set- tings that the typical lemniscate admits a much better lower bound on its inradius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' Namely, if the zeros of p are sampled independently from a compactly supported measure µ whose logarithmic potential has non-empty negative set, then the inradius of Λ is bounded below by a positive con- stant with overwhelming probability, see Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content='1 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' Let us provide some insight on this result and explain why the logarithmic potential of µ plays an important role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' First, the lemniscate Λ can alternatively be described as the sublevel set { 1 n log |p(z)| < 0} of the discrete logarith- mic potential 1 n log |p(z)| = 1 n � log |z − zk| where zk are the zeros of p(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' For fixed z the sum 1 n � log |z − zk| is a Monte-Carlo approximation for the integral defining the logarithmic potential Uµ(z) of µ, and, in particular, it converges pointwise, by the law of large numbers, to Uµ(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' With the use of large deviation estimates, we can further conclude that each z in the negative set Ω− of Uµ is in Λ with overwhelming probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' The property of holding with overwhelming probabil- ity survives (by way of a union bound) when taking an intersection of polynomially many such events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' This fact, together with a suitable uniform estimate for the derivative p′(z) (for which we can use a Bernstein-type inequality), allows for a standard epsilon-net argument showing that an 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content='13424v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content='PR] 31 Jan 2023 2 MANJUNATH KRISHNAPUR, ERIK LUNDBERG, AND KOUSHIK RAMACHANDRAN arbitrary compact subset of Ω− is contained in Λ with overwhelming probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' Since Ω− is as- sumed nonempty, this leads to the desired lower bound on the inradius, see the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content='1 in Section 3 for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' Under an additional assumption that the negative set Ω− of the logarithmic potential of µ contains the support of µ, the inradius converges to the inradius of Ω− almost surely, see Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content='2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' in particular, the inradius has a deterministic limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' On the other hand, for certain measures µ, the inradius does not have a deterministic limit and rather converges in distribution to a nondegenerate random variable, see Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content='5 addressing the case when µ is uniform measure on the unit circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' We also consider the lemniscate associated to the characteristic polynomial of a random matrix sampled from the Ginibre ensemble, and we show that the inradius is close to unity (in fact the whole lemniscate is close to the unit disk) with overwhelming probability, see Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' See Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content='4 below for precise statements of these results along with some additional results giving further insight on the geometry of Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' Previous results on random lemniscates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' The current paper fits into a series of recent stud- ies investigating the geometry and topology of random lemniscates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' Let us summarize previous results in this direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' We note that the lemniscates studied in the results cited below, in contrast to the filled lemniscates of the current paper, are level sets (as opposed to sublevel sets).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' Partly motivated to provide a probabilistic counterpart to the Erd¨os lemniscate problem on the extremal length of lemniscates [10], [5], [11], [12], the second and third authors in [23] studied the arclength and topology of a random polynomial lemniscate in the plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' When the polynomial has i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' Gaussian coefficients, it is shown in [23] that the average length of its lemniscate approaches a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' They also showed that with high probability the length is bounded by a function with arbitrarily slow rate of growth, which means that the length of a lemniscate typically satisfies a much better estimate than the extremal case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' It is also shown in [23] that the number of connected components of the lemniscate is asymptotically n (the degree of the defining polynomial) with high probability, and there is at least some fixed positive probability of the existence of a “giant component”, that is, a component having at least some fixed positive length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' Of relevance to the focus of the current paper, we note that the proof of the existence of the giant component in [23] shows that for a fixed 0 < r < 1, there is a positive probability that the inradius ρ of the lemniscate satisfies the lower bound ρ > r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' Inspired by Catanese and Paluszny’s topological classification [7] of generic polynomials (in terms of the graph of the modulus of the polynomial with equivalence up to diffeomorphism of the do- main and range), in [9] the second author with M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' Epstein and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' Hanin studied the so-called lemniscate tree associated to a random polynomial of degree n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' The lemniscate tree of a poly- nomial p is a labelled, increasing, binary, nonplane tree that encodes the nesting structure of the singular components of the level sets of the modulus |p(z)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' When the zeros of p are i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' sam- pled uniformly at random according to a probability density that is bounded with respect to Haar measure on the Riemann sphere, it is shown in [9] that the number of branches (nodes with two children) in the induced lemniscate tree is o(n) with high probability, whereas a lemniscate tree sampled uniformly at random from the combinatorial class has asymptotically � 1 − 2 π � n many branches on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' In [21], partly motivated by a known result [11], [43]) stating that the maximal length of a rational lemniscate on the Riemann sphere is 2πn, the second author with A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' Lerario studied the geometry of a random rational lemniscate and showed that the average length on the Riemann sphere is INRADIUS OF RANDOM LEMNISCATES 3 asymptotically π2 2 √n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' Topological properties (the number of components and their nesting struc- ture) were also considered in [21], where the number of connected components was shown to be asymptotically bounded above and below by positive constants times n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' Kabluchko and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' Wigman subsequently established an asymptotic limit law for the number of connected compo- nents in [16] by adapting a method of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' Nazarov and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' Sodin [28] using an integral geometry sandwich and ergodic theory applied to a translation-invariant ensemble of planar meromorphic lemniscates obtained as a scaling limit of the rational lemniscate ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' Motivation for the study of lemniscates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' The study of lemniscates has a long and rich history with a wide variety of applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' The problem of computing the length of Bernoulli’s lemnis- cate played a role in the early study of elliptic integrals [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' Hilbert’s lemniscate theorem and its generalizations [26] show that lemniscates can be used to approximate rather arbitrary domains, and this density property contributes to the importance of lemniscates in many of the applications mentioned below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' In some settings, sequences of approximating lemniscates arise naturally for ex- ample in holomorphic dynamics [25, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' 159], where it is simple to construct a nested sequence of “Mandelbrot lemniscates” that converges to the Madelbrot set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' In the classical inverse problem of logarithmic potential theory—to recover the shape of a two-dimensional object with uniform mass density from the logarithmic potential it generates outside itself—uniqueness has been shown to hold for lemniscate domains [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' This is perhaps surprising in light of Hilbert’s lemniscate the- orem and the fact that the inverse potential problem generally suffers from non-uniqueness [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' Since lemniscates are real algebraic curves with useful connections to complex analysis, they have frequently received special attention in studies of real algebraic curves, for instance in the study of the topology of real algebraic curves [7], [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' Leminscates such as the Arnoldi lemniscate appear in applications in numerical analysis [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' Lemniscates have seen applications in two-dimensional shape compression, where the “fingerprint” of a shape constructed from conformal welding sim- plifies to a particularly convenient form—namely the nth root of a Blaschke product—in the case the two-dimensional shape is assumed to be a degree-n lemniscate [8], [44], [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' Lemniscates have appeared in studies of moving boundary problems of fluid dynamics [18], [24], [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' In the study of planar harmonic mappings, rational lemniscates arise as the critical sets of harmonic polyno- mials [17], [22] as well as critical sets of lensing maps arising in the theory of gravitational lensing [30, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' Lemniscates also have appeared prominently in the theory and application of conformal mapping [3], [15], [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' See also the recent survey [36] which elaborates on some of the more recent of the above mentioned lines of research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' Definitions and Notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' Throughout the paper, µ will denote a Borel probability measure with compact support S ⊂ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' The logarithmic potential of µ is defined by Uµ(z) = � S log |z − w|dµ(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' It is well known that Uµ is a subharmonic function in the plane, and harmonic in C \\ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' For such µ, we denote the associated negative and positive sets of its potential by Ω− = {z ∈ C : Uµ(z) < 0}, Ω+ = {z ∈ C : Uµ(z) > 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' It is easy to see that Ω− is a (possibly empty) bounded open set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' Assumptions on the measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' Let µ be a Borel probability measure with compact support S ⊂ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' We define the following progressively stronger conditions on µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' (A) For each compact K ⊂ C, C(K) = sup z∈K � S (log |z − w|)2 dµ(w) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' 4 MANJUNATH KRISHNAPUR, ERIK LUNDBERG, AND KOUSHIK RAMACHANDRAN (B) There is some C < ∞ and ε > 0 such that for all z ∈ C and all r ≤ 1, we have µ (B(z, r)) ≤ C (log(1/r))2+ε .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' (C) There exists δ > 0 such that sup z∈C � S dµ(w) |z − w|δ < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' (D) There is some C < ∞ and ε > 0 such that for all z ∈ C and all r > 0, we have µ (B(z, r)) ≤ Crε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' Main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' In all theorems (except Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content='6), we have the following setting: Setting: µ is a compactly supported probability measure on C with support S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' The random variables Xi are i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' from the distribution µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' We consider the random polynomial pn(z) := (z − X1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' (z − Xn) and its lemniscate Λn := {z : |pn(z)| < 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' We write ρn = ρ(Λn) for the inradius of Λn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' Throughout the paper, w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' means with overwhelming probability, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=', with probability at least 1 − e−cn for some c > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' The theorems below concern the random lemniscate Λn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' Observe that Λn consists of all z for which log |pn(z)| < 0, or what is the same, 1 n n � k=1 log |z − Xk| < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' By the law of large numbers, the quantity on the left converges to Uµ(z) pointwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' Hence we may expect the asymptotic behaviour of Λn to be described in terms of Uµ and its positive and negative sets Ω+, Ω−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' The first three theorems make this precise under different conditions on the underlying measure µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' Assume that µ satisfies assumption (A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' Suppose that Ω− ̸= ∅ and let ρ = ρ(Ω−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' Fix compact sets K ⊂ Ω−, and L ⊂ Ω+ \\ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' Then for all large n, K ⊂ Λn, w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=', and L ⊂ Λc n w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' In particular, if ρn denotes the inradius of Λn, then ρn ≥ a w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=', ∀a ∈ (0, ρ) Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' In the setting of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content='1, lim inf ρn ≥ ρ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' Further, if S ⊆ Ω−, then ρn → ρ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' Ideally, we would have liked to say that a compact set L ⊆ Ω+ is contained inside Λc n w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' However, this is clearly not true if some of the roots fall inside L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' Making the stronger assumption (D) on the measure and further assuming that Uµ is bounded below by a positive number on L, we show that L is almost entirely contained in Λc n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' Let µ satisfy assumption (D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' Let L be a compact subset of {Uµ ≥ m} for some m > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' Then there exists c0 > 0 such that Λn ∩ L ⊂ n� k=1 B(Xk, e−c0n), w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' INRADIUS OF RANDOM LEMNISCATES 5 In particular, if Uµ ≥ m everywhere, then the whole lemniscate is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' It suffices to assume that Uµ ≥ m on the support of µ, by the minimum principle for potentials (Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content='4 in [34]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' Suppose µ satisfies assumption (D) and Uµ ≥ m on S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' Then there is a c0 > 0 such that Λn ⊂ �n k=1 B(Xk, e−c0n) and ρn ≤ ne−c0n w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' A class of examples illustrating Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content='1 and Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content='3 is given at the end of the section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' What happens when the potential Uµ vanishes on a non-empty open set?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' In this case log |pn| has zero mean, and is (approximately) equally likely to be positive or negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' Because of this, one may expect that the randomness in Λn and ρn persists in the limit and we can at best hope for a convergence in distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' The particular case when µ is uniform on the unit circle is dealt with in the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' Let µ be the uniform probability measure on S1, the unit circle in the complex plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' Then, ρn d→ ρ for some random variable ρ taking values in (0, 1 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' Further, P{ρ < ε} > 0 and P{ρ > 1 2 − ε} > 0 for every ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' As shown in the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content='5, the random function log |pn(z)| converges, after appropri- ate normalization, almost surely to a nondegenerate Gaussian random function on D, and this convergence underlies the limiting random inradius ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' We note that similar methods can be used to study other measures µ for which Uµ vanishes on non-empty open set (such as other instances where µ is the equilibrium measure of a region with unit capacity),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' however the case of the uni- form measure on the circle is rather special,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' as the resulting random function log |pn(z)| as well as its limiting Gaussian random function has a deterministic zero at the origin (which is responsible for the limiting inradius taking values only up to half the radius of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' Another setting where one can rely on convergence of the defining function log |pn(z)| is in the case when the polynomial pn has i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' Gaussian coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' Actually, the convergence in this case is more transparent (and does not require additional tools such as Skorokhod’s Theorem) as pn can already be viewed as the truncation of a power series with i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' This case has a similar outcome as in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content='5, except the value 1/2 is replaced by 1 due to the absence of a deterministic zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' One can ask for results analogous to Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content='1 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content='3 when the zeros are dependent random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' A natural class of examples are determinantal point processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' We consider one special case here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFQT4oBgHgl3EQf2jaP/content/2301.13424v1.pdf'} +page_content=' The Ginibre ensemble is a random set of n points in C with joint density proportional to e− �n k=1 |λk|2 � j −1) volunteers felt that the identity was +captured well by the top two models. +In summary, there is a chronological trend given that the worst +performing model AdaIN-VC is from 2019 and the best StarGANv2-VC +is from 2021. This may indicate that the quality of RT-DF is rapidly +improving. This raises concern, especially since the volunteers were +expecting the attack yet could not accurately tell which clips were +real or fake. Another insight we have is that the presence of artifacts +can help victims identify RT-DFs. However, as quality improves, +we expect that only way to induce significant artifacts will be by +challenging the model. +100 +90 +80 +70 +60 +50 +40 +30 +20 +10 +0 +Ada +Medium +Assem +Fragment +StarGan +Real +0 +10 +20 +30 +40 +50 +60 +70 +80 +90 +100 +Model +1 +2 +3 +4 +5 +Figure 3: RT-DF Quality - The distribution of ratings which +the volunteers gave to each of the RT-DF models and real +voice recordings (1: fake, 5: real). +Ada +Medium +Assem +Fragment +StarGan +0 +10 +20 +30 +40 +Success % +Model +Model +Ada +Medium +Assem +Fragment +StarGan +Per model, on participants who are aware of deepfake possibility +Attack Success Rate +Figure 4: RT-DF Quality - The percent of volunteers fooled +by each RT-DF model, even though they were expecting a +deepfake. +Fragment +Stargan +−2 +0 +2 +0 +20 +40 +60 +80 +0 +20 +40 +60 +80 +Normalized value +Count +Figure 5: RT-DF Identity - A histogram of the normalized +MOS scores for how similar RT-DF audio sounds like the +target identity 𝑡. Positive scores are cases where volunteers +thought a fake audio sounded more like 𝑡 than an authentic +recording of 𝑡. +8 +D-CAPTCHA EVALUATION +In this section, we evaluate the benefit of using a D-CAPTCHA as +opposed to using passive defenses alone. +8.1 +Experiment Setup +8.1.1 +Datasets. To evaluate our system, we recorded 20 English +speaking volunteers to create both speech and challenge-response +datasets, summarized in Table 2: +(D𝑟𝑒𝑎𝑙) 2498 samples of real speech (100-250 random sentences +spoken by each of the 20 volunteers). +(D𝑓 𝑎𝑘𝑒) 1821 samples of RT-DF voice conversion. To create this +dataset we used StarGANv2-VC which was the top performing +model from EXP1a. The model was trained to impersonate + +Conference’17, July 2017, Washington, DC, USA +Yasur et al. +6 of the 20 volunteers from D𝑟𝑒𝑎𝑙, and an additional 14 ran- +dom voice actors from the VCTK dataset. The additional 14 +were added to help the model generalize better, and only the 6 +volunteers’ voices were used to make RT-DFs. +(D𝑟𝑒𝑎𝑙,𝑟) 3317 samples of real responses (attempts at challenges). +A sample of nine tasks were evaluated in total. The following +tasks were performed 30 times per volunteer: sing (S), hum tune +(HT), coughing (Co), vary volume (V), and talk & playback (P), +and the following tasks were performed 5 times per volunteer: +repeat accent (R), clap (Cl), speak with emotion (SE), and vary +speed (VS). +(D𝑓 𝑎𝑘𝑒,𝑟) 16,123 deepfake samples of RT-DF voice conversion ap- +plied to the responses D𝑟𝑒𝑎𝑙,𝑟 using StarGANv2-VC. We did not +convert samples from the same identity (i.e., where 𝑠 = 𝑡) +It took each volunteer over an hour to record their data. The volun- +teers were compensated for their time. For all train-test splits used +in our evaluations, we made sure not to use the same identities in +both the train and test sets. +In addition, we also used public deepfake datasets to train the +realism models R. These datasets were the ASVspoof-DF dataset +[62] with 22,617 real and 15,000 fake samples, and the RITW dataset +[42] with 19,963 real and 11,816 fake samples. +8.1.2 +Models. Our system, when fully automated, consists of 3 +models: R, C and I. The algorithm T does not use a machine +learning model to verify the time constraint. +For the realism model R, we evaluated five different deepfake +detection models: SpecRNet [25] which is a novel neural network +architecture, inspired by RawNet2 [54], which get results compa- +rable to state–of–the-art models despite a significant decrease in +computational requirements. One-Class [63] is a method adapted +from [41] based on a deep residual network ResNet-18 [21]. They +improve and generalize the network performance using One-Class +Softmax activations. GMM-ASVspoof [62] is a Gaussian mixture +model (GMM) which operates on LFCCs features. This model was +a baseline for the in ASVspoof 2021 competition. PC-DARTS [18] +is a convolutional neural network (CNN) that tries to automati- +cally learn the network’s architecture. This work also showed good +results in generalizing to unseen attacks. Finally, we used Local +Outlier Factor (LOF) which is a density-based anomaly detection +model. +We took the union of ASVspoof-DF and RITW and selected 80% +at random for training the models and 10% for validation (early +stopping). The models were tested on the baseline scenario (D𝑟𝑒𝑎𝑙 +and D𝑓 𝑎𝑘𝑒) and our proposed D-CAPTCHA scenario (D𝑟𝑒𝑎𝑙,𝑟 and +D𝑓 𝑎𝑘𝑒,𝑟). +For the task model C, we trained a GMM classifier on the MFCC +features using the baseline model from [62]. One model was trained +per task: to classify between real responses from that task and all +other tasks as well as speech. A 70-30 train-test split was used. +For the identity model I, we used a pre-trained voice recogni- +tion model from the SpeechBrain toolkit [45]. The model uses the +ECAPA-TDNN architecture to classify a speaker. Since we do not +want I to have prior knowledge of 𝑡, we converted the model into +an anomaly detector. Recall that we obtain a voice sample 𝑎𝑡 from +the caller prior to the challenge. This sample is used as a reference +to ensure that the RT-DF is not turned off during the challenge. To +Table 2: The number of samples in each of our datasets +Real: D𝑟𝑒𝑎𝑙 +Fake: D𝑓 𝑎𝑘𝑒 +Speech +2498 +1821 +Real: D𝑟𝑒𝑎𝑙,𝑟 +Fake: D𝑓 𝑎𝑘𝑒,𝑟 +Repeat Accent (R) +98 +570 +Clap (Cl) +99 +551 +Cough (Co) +537 +3,401 +Speak with Emotion (SE) +98 +532 +Hum Tune (HT) +593 +3,325 +Playback Audio (P) +601 +3,420 +Sing (S) +595 +334 +Vary Speed (VS) +98 +570 +Vary Volume (V) +598 +3,420 +Real +Fake +ASVspoof-DF +22,617 +15,000 +RITW +19,963 +11,816 +detect whether the identity of the caller has changed during the +challenge, we compute +I(𝑎𝑡,𝑟𝑐) = ||𝑓 ∗(𝑎𝑡) − 𝑓 ∗(𝑟𝑠)||2 +(2) +where 𝑓 ∗ is the speaker encoding, taken from an inner layer of +the speech recognition model. Smaller scores indicate similarity +between the voice before the challenge and during the challenge. +This technique of comparing speaker encodings has been done in +the past (e.g., [40, 43]). To evaluate I, we create negative pairings +as samples from the same identity (𝑎𝑖,𝑟𝑐,𝑖) and positive pairings as +samples from different identities (𝑎𝑖,𝑟𝑐,𝑗), where +𝑎𝑖,𝑎𝑗 ∈ D𝑟𝑒𝑎𝑙, +𝑟𝑐,𝑖,𝑟𝑐,𝑗 ∈ D𝑟𝑒𝑎𝑙,𝑟 +and 𝑖 ≠ 𝑗. +8.1.3 +Experiments. We performed four experiments: +EXP2a R: A baseline comparison between existing solutions (pas- +sive) and our solution (active) in detecting RT-DFs. +EXP2b C: An evaluation of the task detection model which ensures +that the caller indeed performed the challenge. +EXP2c I: An evaluation of the identity model which ensures that +the caller didn’t just turn off the RT-DF for the challenge. +EXP2d R, C, I: An evaluation of the system end-to-end to evaluate +the performance of the system as a whole. +We do not evaluate T because it is just a restriction that the first +frame of the response 𝑟𝑐 be received within approximately one +second from the start time of the challenge. +To measure the performance of the models, we use the area +under the curve (AUC) and equal error rate (EER) metrics. AUC +measures the general trade-off between the true positive rate (TPR) +and the false positive rate (FPR). An AUC of 1.0 indicates a perfect +classifier while an AUC of 0.5 indicates random guessing. The EER +captures the trade-off between the FPR and the false negate rate +(FNR). A lower EER is better. +8.2 +Experiment Results +8.2.1 +EXP2a (R). The goal of EXP2a was to see if our system can +improve the detection of RT-DFs if the adversary is forced to per- +form a task that is outside of the deepfake model’s capabilities. In +Table 3, we compare the performance of the five deepfake detec- +tors on (1) detecting regular deepfake speech (baseline) and on (2) + +Deepfake CAPTCHA: A Method for Preventing Fake Calls +Conference’17, July 2017, Washington, DC, USA +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +False Positive Rate +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +True Positive Rate +R auc: 0.864 +T&C auc: 0.985 +Co auc: 1.0 +SE auc: 0.938 +HT auc: 0.999 +P auc: 0.998 +S auc: 0.993 +VS auc: 0.963 +V auc: 0.974 +Figure 6: The performance of the task detection model C. +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +False Positive Rate +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +True Positive Rate +Co auc: 0.574 +HT auc: 0.688 +P auc: 0.831 +SE auc: 0.846 +S auc: 0.878 +R auc: 0.89 +T&C auc: 0.904 +VS auc: 0.926 +V auc: 0.942 +Figure 7: The performance of the unsupervised identity de- +tection model I for different tasks. +detecting deepfake challenges. The bold values indicate challenges +which improved the performance of the corresponding model. We +see that with the exception of SpecRNet, all of the detectors benefit +from examining challenges. Overall, the best performing model was +GMM-ASVspoof with the challenges. This means that the challenges +provide a better way to detect RT-DFs. +8.2.2 +EXP2b (C). If an attacker is evasive, he may try to do nothing +instead of the challenge. It’s also possible that the attacker will try +the challenge, but the model will output nothing because it can’t +generate the data. Fig. 6 shows that either way, the task detector C +can tell whether the task was performed or not with high certainty. +8.2.3 +EXP2c (I). Another evasive strategy is where the attacker +turns off the RT-DF while performing the challenge. In this scenario, +we compare the identity of the caller before (𝑎𝑡) and during (𝑟𝑐) the +challenge. In Fig. 7 we present the results of the identity detector I. +Here we can see that the model does quite well, with the exception +of the tasks ‘hum’ and ‘cough’ which do not carry much of the +speaker’s identity. +8.2.4 +EXP2d (R, I, C): D-CAPTCHA. Finally, when executing all +three models, we must consider how the successes and failures of +each model compound together. We set the threshold for each model +(R, I, C) so that the FPR=0.01. We then passed through 3,317 real +responses and 8,758 deepfake responses. Fig. 8 presents the results. +91.9% +93.9% +99.4% +91% +99.6% +93.2% +99.3% 100% +91.7% +90% +92.4% +100% +88.8% +100% +91.3% +99.6% 100% +89.2% +Accuracy +TPR +0.80 +0.85 +0.90 +0.95 +1.00 +Value +100% 100% 99.2% 100% 99.4% 99% 99.3% 100% 98.9% +0% +0% +2% +0% +1.3% +2.2% +1.7% +0% +2.3% +Precision +FPR +Accent +Clap +Cough +Emotion +Hum +Playback +Sing +Speed +Volume +Accent +Clap +Cough +Emotion +Hum +Playback +Sing +Speed +Volume +0.0 +0.3 +0.6 +0.9 +Task +Value +Figure 8: The performance of the ensure D-CAPTCHA sys- +tem (end-to-end). +We found that we were able to achieve a TPR of 0.89-1.00. FPR of +0.0-2.3 and accuracy of 91-100% depending on the selected task. In +contrast, the model which performed the best on deepfake speech +detection (baseline) was SpecRNet with a TPR of 0.66 and accuracy +of 71% when the FPR=0.01. Therefore, D-CAPTCHA significantly +outperforms the baseline and provides a good defense against RT- +DFs audio calls. +9 +FUTURE WORK: VIDEO D-CAPTCHA +As mentioned in the introduction, the same D-CAPTCHA system +outlined in this paper can be applied to video-based RT-DFs as well. +For example, to prevent imposters from joining online meetings +(such as the cases in [48, 59]) we can forward suspicious calls to +a D-CAPTCHA system. There are a wide variety of tasks which +existing models and pipelines cannot handle for similar reasons +to those listed in section 5.1. For example, the caller can be asked +to drop/bounce objects, fold shirt, stroke hair, interact with back- +ground, spill water, pick up objects, perform hand expressions, press +on face, remove glasses, turn around, and so on. These tasks can +easily be turned into challenges to detect video-based RT-DFs. +To demonstrate the potential, we have performed some initial +experiments and will now present some preliminary results. In +our experiment we used a popular zero-shot RT-DF model called +Avatarifye based on the work of [50] to reenact (puppet) a single +photo. We were able to achieve a realistic RT-DF video at 35 fps with +negligible distortions if the face stayed in a frontal position. How- +ever, when we performed some of the mentioned challenges, the +model failed and large distortions appeared. Fig., 9 in the appendix +presents some screenshots of the video during the challenges. +These preliminary results indicate that D-CAPTCHAs can be a +good solution for both RT-DF audio and video calls. +ehttps://github.com/alievk/avatarify-python + +Conference’17, July 2017, Washington, DC, USA +Yasur et al. +Table 3: The AUC and EER of deepfake detectors when used as regular deepfake detectors (baseline) and when used as R with +the challenges. +AUC +Baseline +R +T&C +SE +P +VS +V +S +HT +Co +SpecRNet +0.952 +0.914 +0.538 +0.796 +0.825 +0.922 +0.92 +0.834 +0.701 +0.789 +One-Class +0.939 +0.952 +0.967 +0.941 +0.954 +0.958 +0.957 +0.948 +0.896 +0.832 +GMM-AsvSpoof +0.949 +0.951 +0.978 +0.953 +0.97 +0.957 +0.949 +0.928 +0.949 +0.833 +PC-DARTS +0.551 +0.568 +0.557 +0.611 +0.507 +0.586 +0.579 +0.655 +0.675 +0.635 +LOF +0.678 +0.614 +0.93 +0.635 +0.756 +0.771 +0.824 +0.593 +0.681 +0.982 +EER +Baseline +R +T&C +SE +P +VS +V +S +HT +Co +SpecRNet +0.116 +0.163 +0.475 +0.285 +0.261 +0.155 +0.154 +0.245 +0.354 +0.281 +One-Class +0.128 +0.123 +0.099 +0.133 +0.118 +0.112 +0.104 +0.128 +0.187 +0.259 +GMM-AsvSpoof +0.122 +0.1 +0.071 +0.099 +0.09 +0.092 +0.115 +0.143 +0.131 +0.255 +PC-DARTS +0.449 +0.418 +0.494 +0.386 +0.494 +0.43 +0.437 +0.366 +0.334 +0.415 +LOF +0.326 +0.419 +0.122 +0.412 +0.262 +0.301 +0.26 +0.38 +0.382 +0.051 +Figure 9: Preliminary results showing how the D-CAPTCHA system can help prevent RT-DF video calls. Here a zero-shot +reenactment model called Avatarify breaks the moment the caller performs an action other than basic expressions and talking. +10 +CONCLUSION +Deepfakes are rapidly improving in terms of quality and speed. This +poses a significant threat as attackers are already using real-time +deepfakes to impersonate people over calls. Current defenses use +passive methods to identify deepfakes via their flaws. However, this +approach may have limits as the quality of deepfakes continues +to advance. Instead, in this work we proposed an active defense +strategy: D-CAPTCHA. By challenging the attacker to create con- +tent under four constraints based on practical and technological +limitations, we can force the deepfake model to expose itself. 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ASVspoof 2021: accelerating progress in spoofed and deepfake speech +detection. arXiv preprint arXiv:2109.00537 (2021). +[63] You Zhang, Fei Jiang, and Zhiyao Duan. 2021. One-class learning towards syn- +thetic voice spoofing detection. IEEE Signal Processing Letters 28 (2021), 937–941. +A +ETHICAL DISCLOSURES +The experiments performed in this study have received our institu- +tion’s ethical committee’s approval. All 20 volunteers whose voices +were used to create deepfakes permitted the use of their data for +this purpose. To protect our volunteers, the trained RT-DF voice +models will not be shared. +B +ADDITIONAL FIGURES + +Deepfake CAPTCHA: A Method for Preventing Fake Calls +Conference’17, July 2017, Washington, DC, USA +0.668 +0.811 +0.834 +0.85 +0.854 +0.856 +0.86 +0.863 +0.864 +0.874 +0.88 +0.889 +0.891 +0.891 +0.895 +0.897 +0.904 +0.905 +0.906 +0.906 +0.91 +0.91 +0.911 +0.92 +0.921 +0.921 +0.922 +0.923 +0.924 +0.924 +0.926 +0.93 +0.935 +0.936 +0.938 +0.944 +HT + Co +SE + Co +P + Co +Co + T&C +S + Co +HT + P +R + Co +SE + HT +HT + S +HT + R +SE + P +P + S +HT + T&C +T&C + P +Co + VS +R + P +S + SE +V + Co +P + VS +T&C + SE +SE + R +S + R +T&C + R +T&C + S +R + VS +VS + HT +SE + V +P + V +SE + VS +V + HT +V + R +VS + T&C +T&C + V +VS + S +V + S +VS + V +0.00 +0.25 +0.50 +0.75 +AUC +Pairs of challenges +Figure 11: The performance of I when two challenges are requested, measured in AUC. +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +False Positive Rate +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +True Positive Rate +Receiver Operating Characteristic of gmm +S AUC = 0.9284 +V AUC = 0.9494 +VS AUC = 0.9567 +Co AUC = 0.8328 +P AUC = 0.9704 +HT AUC = 0.9491 +T&C AUC = 0.9784 +SE AUC = 0.9531 +R AUC = 0.9510 +baseline AUC = 0.9489 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +False Positive Rate +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +True Positive Rate +Receiver Operating Characteristic of raw-pc +S AUC = 0.6551 +V AUC = 0.5786 +VS AUC = 0.5859 +Co AUC = 0.6349 +P AUC = 0.5070 +HT AUC = 0.6751 +T&C AUC = 0.5575 +SE AUC = 0.6109 +R AUC = 0.5683 +baseline AUC = 0.5512 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +False Positive Rate +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +True Positive Rate +Receiver Operating Characteristic of SpecRNet +S AUC = 0.8335 +V AUC = 0.9205 +VS AUC = 0.9221 +Co AUC = 0.7889 +P AUC = 0.8255 +HT AUC = 0.7006 +T&C AUC = 0.5378 +SE AUC = 0.7958 +R AUC = 0.9139 +baseline AUC = 0.9518 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +False Positive Rate +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +True Positive Rate +Receiver Operating Characteristic of oneclass +S AUC = 0.9479 +V AUC = 0.9574 +VS AUC = 0.9581 +Co AUC = 0.8324 +P AUC = 0.9542 +HT AUC = 0.8957 +T&C AUC = 0.9668 +SE AUC = 0.9408 +R AUC = 0.9523 +baseline AUC = 0.9393 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +False Positive Rate +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +True Positive Rate +Receiver Operating Characteristic of LOF +S AUC = 0.5934 +V AUC = 0.8241 +VS AUC = 0.7709 +Co AUC = 0.9823 +P AUC = 0.7561 +HT AUC = 0.6810 +T&C AUC = 0.9302 +SE AUC = 0.6355 +R AUC = 0.6139 +baseline AUC = 0.6784 +Figure 12: ROC plots for each deepfake detection model from experiment EXP2a. The bold line shows the baseline (regular +deepfake detection) and the others show the performance on the given task. + diff --git a/3dE1T4oBgHgl3EQfSQOU/content/tmp_files/load_file.txt b/3dE1T4oBgHgl3EQfSQOU/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..91865e8379acb9280442b0c16ae4bdf6242d2a69 --- /dev/null +++ b/3dE1T4oBgHgl3EQfSQOU/content/tmp_files/load_file.txt @@ -0,0 +1,1457 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf,len=1456 +page_content='Deepfake CAPTCHA: A Method for Preventing Fake Calls Lior Yasur,∗ Guy Frankovits,∗ Fred M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Grabovski, Yisroel Mirsky {lioryasu,guyfrank,freddie}@post.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='bgu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='il,yisroel@bgu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='il Ben-Gurion University of the Negev Israel ABSTRACT Deep learning technology has made it possible to generate realistic content of specific individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' These ‘deepfakes’ can now be gen- erated in real-time which enables attackers to impersonate people over audio and video calls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Moreover, some methods only need a few images or seconds of audio to steal an identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Existing de- fenses perform passive analysis to detect fake content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' However, with the rapid progress of deepfake quality, this may be a losing game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' In this paper, we propose D-CAPTCHA: an active defense against real-time deepfakes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' The approach is to force the adversary into the spotlight by challenging the deepfake model to generate con- tent which exceeds its capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' By doing so, passive detection becomes easier since the content will be distorted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' In contrast to existing CAPTCHAs, we challenge the AI’s ability to create content as opposed to its ability to classify content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' In this work we focus on real-time audio deepfakes and present preliminary results on video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' In our evaluation we found that D-CAPTCHA outperforms state- of-the-art audio deepfake detectors with an accuracy of 91-100% depending on the challenge (compared to 71% without challenges).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' We also performed a study on 41 volunteers to understand how threatening current real-time deepfake attacks are.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' We found that the majority of the volunteers could not tell the difference between real and fake audio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' KEYWORDS Deepfake, deep fake, voice cloning, impersonation, CAPTCHA, deep learning, fake calls, social engineering, security ACM Reference Format: Lior Yasur,∗ Guy Frankovits,∗ Fred M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Grabovski, Yisroel Mirsky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Deepfake CAPTCHA: A Method for Preventing Fake Calls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' In Proceedings of ACM Conference (Conference’17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' ACM, New York, NY, USA, 15 pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' https://doi.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='1145/nnnnnnn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='nnnnnnn Attacker Victim Deepfake real fake validate Figure 1: Overview of the proposed defense: the victim re- quests the caller to perform a task which is challenging for a deepfake model to perform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' If the response is distorted or does not contain the task, then the caller is likely a deepfake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' in terms of quality and has been adopted in a variety of applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' For example, deepfake technology is used to enhance productiv- ity [47], education [33] and provide entertainment [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' However, the same technology has been used for unethical and malicious purposes as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' For example, with a deepfake, anyone can imper- sonate a target identity by reenacting the target’s face and/or voice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' This ability has enabled threat actors to perform defamation, black- mail, misinformation, and social engineering attacks on companies and individuals around the world [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' For example, since 2017, the technology has been used to ‘swap’ the identity of individuals into explicit videos for unethical [20] and malicious [35] reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' More recently, in March 2022 during the Russian-Ukraine conflict, a deepfake video was circulated depicting the prime minister of Ukraine telling his troops to give up and stop fighting [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='1 Real-time Deepfakes (RT-DF) Deepfake technology has improved over the last few years in terms of efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' This has enabled attackers to create real-time deep- fakes (RT-DF)a With an RT-DF, an attacker can impersonate people over voice and video calls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' The danger of this emerging threat is that (1) the attack vector is not expected, (2) familiarity can be mistaken as authenticity and (3) the quality of RT-DFs is constantly improving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' aExamples of RT-DF tools: https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='com/iperov/DeepFaceLive https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='com/alievk/avatarify-python https://samsunglabs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='io/MegaPortraits/ https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='respeecher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='com/ arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='03064v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='CR] 8 Jan 2023 Conference’17, July 2017, Washington, DC, USA Yasur et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' To conceptualize this threat, let’s perform the following thought experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Imagine someone receives a call from their mother who is in trouble and urgently needs a money transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' The caller sounds exactly like her, but the situation seems a bit out of place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Under stress and frustration, she hands the phone over to someone who sounds like the victim’s father, who confirms the situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Without hesitation, many would transfer the money even though they’re technically talking to a stranger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Now consider state-actors with considerable amounts of time and resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' They could target workers at power plants and other critical infrastructure by posing as their administrators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Over a phone call, they could convince the worker to change a configuration or reveal confidential informa- tion which would lead to a cyber breach or a catastrophic failure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Attackers could even pose as military officials or politicians leading to a breach of national security.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' These scenarios are plausible because some existing real-time frameworks can impersonate an individual’s face or voice using very little information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' For example, some real-time methods can reenact a face with one sample image [16, 50] and some can clone a voice with just a few seconds of audio [15, 37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Using these tech- nologies, an attacker would only need to call the source voice for a few seconds or scrape the source’s image from the internet to perform the attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='2 The Emerging Threat of RT-DFs Threat actors already understand the utility of RT-DFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' This is evident in recent events where RT-DFs have been used to perform criminal acts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' The first case was discovered in 2019 when a CEO was tricked into transferring $243k due to an RT-DF phone call [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' In 2021, senior European MPs participated in Zoom meetings with someone masquerading as Russian opposition figures [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' In the same year, cyber criminals pulled off a $35 million bank heist involving RT-DF audio calls to a company director, tricking him to perform money transfers [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' In June 2022, the FBI released a warning that cyber criminals are using RT-DFs in job interviews in order to secure remote work positions and gain insider information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Then in August that year, cyber criminals attended Zoom meetings masquerading as the CEO of Binance [59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='3 The Gap in Current Defenses Many methods have been proposed for detecting deepfakes [4, 40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' These methods typically use deep learning models to either (1) detect mistakes or artifacts in generated media, or (2) search for forensic evidence such as a latent noise patterns (examples of these works can be found in section 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' However, there are two fundamental problems with existing defenses: Longevity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Methods which identify semantic errors or artifacts have the assumption that the quality of deepfakes will not significantly improve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' However, it is clearly evident that the quality of deepfakes is improving and at a fast rate [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' There- fore, artifact-based methods have a high potential of becoming obsolete within a short time-frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Evasion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Methods which rely on latent noise patterns can be evaded by applying a post-processor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' For example a deepfake can be passed through a low pass filter, undergo compression or be given additive noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Moreover, these processes are common in audio and video calls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Therefore, the attacker may not need to do anything to remove the forensic evidence in the call.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='4 Real-Time CAPTCHA In this paper, we propose Deepfake-CAPTCHA (D-CAPTCHA): a system for automatically detecting deepfake calls through challenge response analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Instead of passively observing call content, we actively interact with the caller by requesting that he or she to perform a task (the challenge).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' The task is easy for a human to perform but extremely hard for a deepfake model to recreate due to limitations in attack practicality and technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' When a deepfake tries to perform the task, the resulting content (the response) will be severely distorted –making it easier for an anomaly detector, classifier, or even the victim to detect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' In addition, we propose using an identity model and task detection model to mitigate evasion tactics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' The identity model compares the identity of the caller before and during the response to ensure that the caller cannot turn off the RT-DF during the task or splice in content from other identities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Similarly, the task detection model ensures that the caller has indeed performed the task as opposed to doing nothing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Existing CAPTCHA systems, such as reCAPTHCA,b challenge AI to interpret content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' In contrast, we propose a system which chal- lenges AI to create content, with additional constraints on realism, identity, task (complexity), and time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' In this work, we focus on audio-based RT-DF attacks (voice cloning).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' We consider audio RT-DFs a more significant threat over video RT-DFs because it is easier for an attacker to make a phone call than setup a video call with the victim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Also, their occurrences in the wild are increasing [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Therefore, RT-DF audio calls are arguably a bigger threat at this time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' However, we note that the same D-CAPTCHA system proposed in this paper can be applied to video calls as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' In section 9 we present initial results in this domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' In our evaluation, we collected five state-of-the-art audio RT-DF technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' We performed a panel survey to see what the public thinks about their quality and we evaluated the top two models on our defense and on others as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' We found that our method can significantly enhance the performance of state-of-the-art audio- based deepfake detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='5 Contributions In summary, our work has the following contributions: We propose the first active defense against RT-DFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Com- pared to existing artifact-base methods, our approach (1) provides stronger guarantees of detection than using only passive detection and (2) has better longevity because the challenges are extensible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' We define what a D-CAPTCHA is and what constitutes a strong deepfake CAPTCHA: We identify the limitations of existing RT-DF systems and propose four constraints a chal- lenge must present to a caller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' We also present how these constraints can be verified in a response both manually and automatically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' We also provide an initial set of CAPTCHAs and analyze their security and usability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' bhttps://developers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='google.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='com/recaptcha/ Deepfake CAPTCHA: A Method for Preventing Fake Calls Conference’17, July 2017, Washington, DC, USA We evaluated the quality of five state-of-the-art RT-DF voice cloning models with 41 volunteers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Doing so enables us to better understand the current threat which this technology poses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' We provide thorough evaluations on (1) how well the CAPTCHA system performs and (2) how robust it is against an evasive adversary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' 2 BACKGROUND In this work, we focus on mitigating the threat of real-time voice cloning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Furthermore, we focus on methods that perform speech- to-speech voice conversion (VC) [19, 23, 30, 36, 37, 44] as opposed to text-to-speech (TTS) methods such as [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Let 𝑡 be a target identity which we’d like to clone, and 𝑎𝑠 be an audio clip of identity 𝑠 speaking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Content is the part of speech that is independent of a speaker’s vocal anatomy (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=', words, accent, enunciation, and so on).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' The objective of voice cloning is to perform 𝑓𝑡 (𝑎𝑠) = 𝑎𝑔 where 𝑎𝑔 is generated audio containing the content of 𝑎𝑠 in the style of 𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' In an attack, 𝑡 is an individual who is familiar to the victim, and 𝑠 is the attacker (or a voice actor hired by the attacker).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' To convert unbounded audio streams in real-time, audio is pro- cessed as a sequence of short audio frames (approximately 10- 1000ms each).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' In this way, the 𝑖-th input frame 𝑎(𝑖) 𝑡 is converted into 𝑎(𝑖) 𝑔 within one second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' We consider 𝑓𝑡 to be an RT-DF if the pipeline can be executed with no more than a 1 second delay from the microphone to speaker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' In other words, the time it takes for an utterance to be recorded, converted, and played back is no longer than 1 second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Longer delays may raise the victim’s suspicion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Meth- ods which process entire recordings all at once form non-casual systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Therefore, we do not consider them as RT-DF systems (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=', [46]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' There are various levels of flexibility when it comes to prior knowledge of 𝑠 and 𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' For instance, not every model can drive 𝑎𝑔 with content from 𝑠 without prior training on 𝑠.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Many of the audio RT-DF models can be categorized as follows: many-to-many.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Are models which require both the source voice 𝑠 (used in 𝑐) and the target voice 𝑡 to be in 𝑓 ’s training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Since 𝑠 is the attacker, the only challenge is collecting samples of 𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' any-to-many.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Are models which can use any source voice to drive the content in 𝑥𝑔 without retraining the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' any-to-any.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Are models which do not need to see the source 𝑠 or target 𝑡 during training to perform 𝑓𝑡 (𝑐𝑠) = 𝑥𝑔.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' This makes any-to-any models the flexible solution for attackers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' 3 THREAT MODEL There are two ways an adversary can use the RT-DF 𝑓𝑡 maliciously: the adversary can (1) call a victim while impersonating 𝑡 or (2) call a target and threaten to impersonate him.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' The call may take place over the phone through a virtual meeting (such as over Zoom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' We refer to these calls as “fake calls”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='1 Attack Goals There are several attack goals which an adversary can achieve using a fake call: Cyber attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Fake calls can be used in social engineering attacks (SE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' For example, instead of sending spear phishing emails to get employees to install malware, the attacker can call a victims up as their manager and ask them to do it directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' These SE attacks can also be used during an adversary’s reconnaissance on an organization to obtain system information and credentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' For example, the attacker can call a victim posing as a colleague, asking for help to login or claiming that he has "forgotten" some information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Sabotage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' An attacker can impersonate a victim’s supervisor in an attempt to have the victim change some settings or config- urations in a system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' For example, in a chemical processing plant, an adversary can use a manager’s voice to tell a worker to urgently alter the balance of some process –leading to cata- strophic results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Espionage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Fake calls can also be used by state agents as a means for extracting sensitive and confidential information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' For exam- ple, an adversary can gain a political advantage by posing as a politician’s assistant and a military advantage by posing as a military official.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Moreover, sensitive documents and source code can be leaked in a similar manner if the adversary imper- sonates a leading figure who directly asks employees for this material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Finally, by impersonating professionals with LinkedIn profiles, an adversary can obtain remote job interviews which may lead to remote work with a company –ultimately placing an insider within the organization [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Scams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' An attacker can prey upon people and trick them into giv- ing them money.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' For example, the adversary can impersonate a family member of the victim to convince the victim that his family is in danger and needs an urgent money transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Similar schemes can be done on business and banks where the attacker convinces the victim to make a money transfer under false pretexts [12, 51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Blackmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' To coerce a victim to perform an action (pay money, reveal information, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=') an attacker can blackmail the victim using RT-DF technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' For example, the attacker can speak to the victim using the victim’s voice and threaten the victim that calls will be made to reporters, friends, colleagues, or a spouse as the victim if the blackmail terms are not met (similar to a case that happened in Singapore [26]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Defamation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' An adversary can defame the victim by perform- ing embarrassing or unethical acts over calls to the victim’s colleagues or reporters while masquerading as the victim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Misinformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' An attacker can call reporters and do interviews as politicians and other public figures to spread misinformation in the media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='2 Attack Setup The flexibility of the attacker depends on the flexibility of the RT- DF model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' To train the model 𝑓𝑡, the attacker can use one of two common approaches: Batch Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' If the attacker uses conventional learning mod- els such as [23, 30, 36, 44], then the attacker will need to collect a large audio training set of 𝑡 (typically around 20-30 minutes) and train 𝑓 on this data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' This dataset can be obtained from the Internet if 𝑡 is a celebrity (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=', interviews on YouTube).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' If Conference’17, July 2017, Washington, DC, USA Yasur et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' 𝑡 doesn’t have an internet presence, then the dataset may be obtained via long phone calls, wiretaps, and secret recordings (bugs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' These models are usually many-to-many or any-to- many.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Few/Zero-shot Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' When using methods such as [15, 19, 37, 55], the attacker only needs a few seconds of 𝑡’s audio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' In this case, the attacker can make a short phone call to 𝑡 and record his/her voice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' The adversary may also find short video clips on social media or resort to wiretaps and bugs as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' These types of models are usually any-to-any.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' We note that most modern RT-DF technologies do not require labeled data since they are trained in a self-supervised manner [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Regarding quality, batch model training methods are typically preferred over few-shot or zero-shot methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' 4 RELATED WORKS Most audio deepfake detection systems (ADDS) use a common pipeline to detect deepfake audio: given an audio clip 𝑎, the pipeline (1) converts 𝑎 into a stream of one or more audio frames 𝑎(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='𝑎(𝑛), (2) extracts a feature representation from each frame which sum- marizes the frames’ waveforms 𝑥 (1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='𝑥 (𝑛), and then (3) passes the frame(s) through a detector which predicts the likelihood of 𝑎 being real or fake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' The audio features in 𝑥 (𝑖) are either a Short Time Fourier Transform (STFT) [6, 61], spectrogram, Mel Frequency Cepstral Coefficients (MFCC) [27, 52], or the Constant Q Cepstral Coefficients (CQCC) [31, 34] of 𝑎(𝑖).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Some methods simply use the actual waveform of 𝑎(𝑖) [53, 54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' With this representation, an ADDS can either use a classifier [25, 29, 54] or anomaly detector [3, 28] to identify generated audio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' A good summary of modern ADDS can be found in [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' In gen- eral, classifiers are trained on labeled audio data consisting of two classes: real and deepfake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' By providing labeled data, the model can automatically identify the relevant features (semantic or latent) during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' An intuitive example is the case where a deep- fake voice cannot accurately pronounce the letter ‘B’ [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' In this scenario, the model will consider this pattern as a distinguishing feature for that deepfake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' A disadvantage of classifiers is that they follow a closed-world assumption;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' that all examples of the deepfake class are in the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' This assumption requires that detectors be retrained whenever new technologies are released.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' As for the model, some works use classical machine learning models such as SVMs and decision trees [11, 27, 32] while the majority use deep learning architectures such as DNNs [61, 63], CNNs [13, 38], and RNNs [7, 49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' To improve generalization to new deepfakes, some ap- proaches try to train on a diverse set of deepfake datasets (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=', [24]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' However, even with this strategy, ADDS systems still generalize poorly to new audio distributions recorded in new environments and to novel deepfake new technologies [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' In contrast to classifiers, anomaly detectors are trained on real voice data only and flag audio that has abnormal patterns within it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' One approach for anomaly detection is to use the embeddings from a voice recognition model to compare the similarity between real and authentic voices [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Other approaches use one-class machine learning models such as OC-SVMs and statistical models such as Gaussian Mixture Models (GMM) [3, 28, 56, 63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' What’s common with the above defenses is that they are all passive defenses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' This means that they analyze 𝑎 but they do not interact with the caller to reveal the true nature of 𝑎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' In contrast, our proposed method is active in that it can force 𝑓 to try and create content it is not capable of doing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' By ‘pressing’ on the limitations of 𝑓 , we are causing 𝑓 to generate audio with significantly larger artifacts, making it easier for us to detect using classifiers and anomaly detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Our approach also ensures some longevity since the attacker cannot easily overcome the limitations our challenges pose (further discussed in section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Another advantage of our system compared to others is that we know exactly where the anomaly should be in the media stream (due to the challenge response nature of the CAPTCHA protocol).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' This means that our system is more efficient since it only needs to execute its models over specific segments and not entire streams (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=', in contrast to [9]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' The work most similar to ours is rtCAPTCHA [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' In this work the authors perform liveliness detection by (1: challenge) asking the caller to read out a text CAPTCHA, (2: response) verifying that the CAPTCHA was read back correctly, and (3: robustness) verifying that the face and voice match an existing user in a database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' The concept of rtCAPTCHA is that the system assumes that the attacker will not be able to generate a response with the target’s face and voice in real-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' However, with the advent of RT-DFs, this rtCAPTCHA can easily be bypassed since the human attacker can read the text CAPTCHA back through 𝑓𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Moreover, our D- CAPTCHA defense does not require users to register in advance, making the solution widely applicable to many users and scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' 5 DEEPFAKE CAPTCHAS In this section we discuss the limitations of RT-DFs and then use these limitations to define how D-CAPTCHAs work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='1 RT-DF Limitations Current RT-DF models can only generate content within the scope of the task they were trained on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' For example, a model trained to reenact 𝑡’s face in a somewhat frontal position or generate 𝑡’s voice in a calm speaking tone will not be able to generate other content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' This is evident in facial reenactment models such as [50] and DeepFaceLive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' These models have excellent performance in creating faces with frontal poses, but they cannot generate the back of the target’s head.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Similarly, for audio-based RT-DFs, it is hard for the model to identify and then produce certain sounds if the training data, loss functions, and overall pipeline focuses on the perfection of normal speech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' An ideal RT-DF model would be able to create content of 𝑡 performing an arbitrary task, where the content is both realistic and authentic to 𝑡’s identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' However, RT-DF models are not ideal because they are scoped to specific tasks during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' This is because doing so enables the model to perfect the identity and realism in 𝑥𝑔 when driven by 𝑥𝑠.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Therefore, even if out of domain tasks can be anticipated, 𝑓𝑡 cannot be trained recreate them all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' This is due to limitations in technology and practicality: 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='1 Technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' This set of limitations relates to the fact that current technology is not yet capable of creating the ideal RT-DF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Deepfake CAPTCHA: A Method for Preventing Fake Calls Conference’17, July 2017, Washington, DC, USA Inference Speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' The rate at which audio frames can be gener- ated depends on the efficiency of deepfake generation pipeline and the complexity of the model’s architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' However, in or- der to handle a wide variety of different tasks, a model requires significantly more parametersc and possibly more complex fea- ture extractors in its pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' For example, existing RT-DF models would need higher resolution STFTs and MFCCs to capture a wider band of frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Feature Representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' In order to capture certain patterns in the input 𝑎𝑠, a model must extract appropriate feature repre- sentations from the input waveform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Voice tends to use lower frequencies and has a rather consistent spectral envelope com- pared to other sounds such as singing and clapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Existing pipelines use compressed features such as MFCCs or STFTs with lower sample rates (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=', 16-24 KHz [4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' To capture a more dynamic range of frequencies, higher resolution is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' How- ever, increasing input resolution generally makes it harder for a model to converge and increases model complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' To train a model, a loss function must be provided to guide the optimization process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Modern RT-DF systems use at least two loss functions: one for the realism (adversarial loss) and one for preserving the identity of 𝑡 in 𝑎𝑔 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=', perceptual loss) [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' If additional tasks are considered, then the model will likely need additional loss functions to cover each aspect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' How- ever, loss functions compete during optimization and therefore some aspects will suffer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Furthermore, adding loss functions can make it harder for the model to converge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Finally, it’s possible that 𝑎𝑠 may contain a mix of voice and other audio (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=', music or some other voice).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' To work on this audio, the model would have to convert the voice component and not the other audio, and then mix the two components back together in 𝑥𝑔.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' To the best of our knowledge this is an open problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='2 Resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' This set of limitations relates to cases where the desired result is achievable with existing technology, however it may be prohibitively expensive or impractical to obtain it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Data Collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' To make a high quality RT-DF of 𝑡, a significant amount of audio samples of 𝑡 are required (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=', [36] requires 20-30 minutes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' However, it is impractical for an attacker to obtain audio of 𝑡 performing specific tasks other than talking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' If quality can be sacrificed, then zero-shot learning could be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' However, there is still the challenge of (1) gathering an extensive dataset of all possible tasks and (2) training a model that can generalize the samples to new identities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Creating a system that can handle even a subset of arbitrary tasks requires some in-depth knowledge on making generative deep learning models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' This raises the difficulty bar for casual attackers, but not for advanced adversaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Labeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' The process of annotating and labeling large datasets is expensive and time consuming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' This becomes more apparent as the number of classes (tasks) increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' cAs a point of reference, StarGAN [36] is a state-of-the-art audio-based RT-DF models which has about 53 million parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' In contrast, models that produce arbi- trary content (such as DALL-E 2 and Imagen) use 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='5-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='6 billion parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Moreover, methods such as stable-diffusion requires multiple passes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Assets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' The ideal RT-DF model would likely be a complex model to handle the arbitrary tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Executing such a model in real- time would require a powerful GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Depending on the model’s complexity, the GPU may either be prohibitively expensive or simply non-existent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='3 Outlook on RT-DF Limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' We note that the limitations described in this section apply to existing RT-DF systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Although these limitations are hard to overcome, there is no guarantee that future RT-DF technologies will have the same limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' However, we expect that some of the limitations, such as data collection and training, will still apply to novel systems in the near future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Therefore, to gain advantage over the adversary, we suggest that defenses should exploit the limitations of RT-DFs whenever possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='2 D-CAPTCHA According to [2],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' a CAPTCHA is “a cryptographic protocol whose underlying hardness assumption is based on an AI problem.”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' The pro- tocol follows the form of a challenge-response procedure between server 𝐴 (the server/victim) and client 𝐵 (the client/caller),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' where (1) 𝐴 sends challenge 𝑐 to 𝐵,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' (2) 𝐵 sends response 𝑟𝑐 on 𝑐 back to 𝐴,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' and (3) 𝐴 verifies whether 𝑟𝑐 resolves challenge 𝑐: (1) 𝐴 → 𝐵 : 𝑐 (2) 𝐵 → 𝐴 : 𝑟𝑐 (3) 𝐴 : 𝑉 (𝑟𝑐) ∈ {𝑝𝑎𝑠𝑠,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' 𝑓 𝑎𝑖𝑙} For example,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' the popular reCAPTCHA prevents bots from perform- ing automated activities on the web by challenging the client to perform a human skill which is hard for software but easy for hu- mans (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=', decoding distorted letters).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' In contrast,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' a D-CAPTCHA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='challenges a client by requiring the client to create content with ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='the following constraints: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='(1) Realism: The content must be realistic to a human or a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='machine learning model ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='(2) Identity: The content must reflect the identity 𝑡 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='(3) Task: The content must have 𝑡 performing an arbitrary task ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='which is hard to generate ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='(4) Time: The content must be generated in real-time ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='Creating a response to this challenge where 𝑉 (𝑟𝑐) = 𝑝𝑎𝑠𝑠 is hard ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='for existing RT-DF technologies but easy for humans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' In our system the ‘hardness’ of the CAPTCHA directly relates to the limitations of existing RT-DF technology (section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Moreover, just like modern CAPTCHA systems, a D-CAPTCHA system can be easily extended to new limitations of RT-DFs over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' This gives our system flexibility to defend against future threats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='1 Creating a Challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' A challenge demonstrates whether a caller can or cannot create content with realism, identity, task and time constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Realism constraints are necessary to ensure there are no latent or semantic anomalies in the response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Iden- tity constraints are needed to ensure that the attacker isn’t just recording him/herself during the challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Task constraints are required to ensure that the deepfake model tries to operate outside the bounds of its abilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Finally, Time constraints are involved to guarantee that the caller is using an RT-DF model since (1) we don’t want the caller to switch to an offline model and (2) real-time Conference’17, July 2017, Washington, DC, USA Yasur et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Table 1: Examples of audio-based tasks which can be used as challenges in a D-CAPTCHA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Strong challenges are hard for the adversary on all four constraints: realism, identity, complexity and time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' The measures in this list are based on existing RT-DFs methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Playback is where the caller must play some provided audio from his/her phone into the microphone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='Hardness ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='Weakness ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='Effectiveness ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='Task (𝑇) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='Acronym ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='Usability ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='Realism ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='Identity ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='Task ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='Time ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='Evasions ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='Naive Attacker ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='Advanced Attacker ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='Clear Throat ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='CT Hold Musical Note ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='HN Hum Tune ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='HT Laugh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='L Mimic Speaking Style ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='MS Repeat Accent ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='R Sing ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='S Speak with Emotion ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='SE Yawn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='Y Blow Noises ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='BN − bypass − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='Blow on Mic ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='BM − bypass − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='Clap ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='Cl − bypass − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='Click Tongue ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='Clk − bypass − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='Cough ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='Co − bypass − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='Horse Lips ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='HL − bypass − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='Knock ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='K − bypass − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='Playback Audio ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='PA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='− − bypass − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='Raspberry ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='R − bypass − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='Sound Effect ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='SFX − bypass − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='Touch Mic ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='TM − bypass − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='Type ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='T − bypass − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='Whistle ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='W ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='− − bypass − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='Talk & Clap ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='T&C mix − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='Talk & Knock ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='T&K mix − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='Talk & Playback ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='P ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='− mix − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='Talk with Tones ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='TT mix Vary Speed ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='VS mix Vary Volume ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='V mix : high,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' ◦: medium,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' −: low models are more limited since they can only process frames and not entire audio clips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' The core component of a challenge in our system is the task which the caller must perform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Let 𝑇 denote a specific task, such that 𝑇 = ℎ𝑢𝑚 might be “hum a specific song.” We define the set of all possible challenges for task 𝑇 as 𝐶𝑇 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' For example, 𝐶ℎ𝑢𝑚 would be all possible requests for different songs to be hummed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' To select a challenge, (1) random seeds 𝑧0,𝑧1 are generated, (2) 𝑧0 is used to select a random task 𝑇 and (3) 𝑧1 is used to select a random challenge 𝑐 ∈ 𝐶𝑇 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' In Table 1 we present some example tasks which can be used in D-CAPTCHA challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' In the table, we assume that the RT-DF under test has been trained to have the best performance on one task;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' regular talking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Using observations over five state-of-the-art RT-DF models we assess the hardness, weakness, and effectiveness of each task as a challenge (see 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='1 for details on these five models).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Under hardness, we express the difficulty of a modern RT-DF in successfully creating a deepfake of𝑡 given the respective constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' For weakness, we state how an adversary can evade detection if the respective task is chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' For instance, bypass is where the RT-DF is turned off and the attacker speaks directly to our system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' The other case is mix is where the attacker can mix other audio sources into 𝑎𝑔.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' For example, to evade ‘talk & clap’ the attacker creates 𝑎′𝑔 = 𝑎𝑔 + 𝑎𝑐𝑙𝑎𝑝 where 𝑎𝑐𝑙𝑎𝑝 is taken from another microphone so as not to disrupt the RT-DF (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=', execute 𝑓𝑡 (𝑎𝑠 +𝑎𝑚)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Finally, in the table under effectiveness we consider how effective the challenge is given two levels of attackers: naive and advanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' A naive attacker is one which (1) will use existing datasets and only a limited number of samples of 𝑡 to train 𝑓𝑡 and (2) forwards all audio through 𝑓𝑡 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=', if a library is used as-is from GitHub).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' An advanced attacker is one which will collect a practical number of samples on 𝑡 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=', 20 minutes) and is able to mix other audio sources into 𝑎𝑔.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Overall, a strong challenge is a random 𝑐 drawn from a random𝑇 which is hard for the adversary to perform given all four constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='2 Verifying a Challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' To determine whether𝑉 (𝑟𝑐) = 𝑝𝑎𝑠𝑠 or 𝑓 𝑎𝑖𝑙, we must verify whether 𝑟𝑐 adheres to the realism, identity, Task, and time constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' All four constraints can be verified by a human (a moderator or the victim him/herself).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' For exam- ple, if 𝑐 =“say ’I’m hungry’ with anger” but (1) the audio sounds strange/distorted, (2) the voice does not sound like 𝑡, (3) the task is not completed, or (4) it takes too long for the caller to respond, then this would raise suspicion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' However, many users may not trust themselves enough or they may give in to social pretexts and ig- nore the signs –to avoid rejecting a peer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Therefore, we propose an automated way to verify each constraint without prior knowledge of 𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' To verify 𝑟𝑐, we validate each constraint separately: Realism Verification (R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' If an RT-DF attempts to perform𝑐 then 𝑟𝑐 will likely contain distortions and artifacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' This is because (1) the RT-DF is operating outside of its capabilities or (2) be- cause the caller simply is using a poor-quality RT-DF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' These distortions will make it easier for existing anomaly detectors and existing deepfake classifiers to identify the RT-DF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' The output of R is a score on the range [0, ∞) or [0, 1] indicating how unrealistic the content of 𝑟𝑐 is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Identity Verification (I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' To determine if 𝑟𝑐 has the identity 𝑡, we can do as follows: (1) collect a short audio sample 𝑎𝑡 of the Deepfake CAPTCHA: A Method for Preventing Fake Calls Conference’17, July 2017, Washington, DC, USA suspicious?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' no yes challenge 𝑐 + instructions response 𝑟𝑐 drop call evidence (𝑐, 𝑟𝑐) RT-DF Call (3) Response Verification 𝑉 𝑟𝑐 Victim Attacker Deepfake Realism Time Task 𝒯 𝑑 < 𝜙1 Get voice sample 𝑎𝑡 from caller Select challenge seed 𝑧1 (2) Challenge Creation c ∈ 𝐶𝑇 ℛ 𝑟𝑐 < 𝜙2 𝒞 𝑟𝑐, 𝑐 < 𝜙4 true true true (1) Call Forwarding Select task 𝑇 𝑎𝑡 acknowledged?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' true true Ask victim if accept call from 𝑡 given 𝑎𝑡?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Call connected/resumed seed 𝑧0 Identity ℐ 𝑟𝑐, 𝑎𝑡 < 𝜙3 Figure 2: An overview of the proposed D-CAPTCHA system: (1) Calls are forwarded to the system using a blacklist, whitelist, policy or the victim’s intuition, (2) a random D-CAPTCHA 𝑐 with accompanying instructions is generated and send to the caller as a challenge, (3) the response 𝑟𝑐 is verified against the four constraints (time, realism, identity, task) and if all four pass then the call is connected/resumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Otherwise, the call is dropped and evidence is provided to the victim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' caller prior to the challenge and have the victim acknowledge the identity, and (2) use zero-shot voice recognition model to verify that the identity in 𝑎𝑡 and 𝑟𝑐 are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' The reason we have the victim acknowledged 𝑡 in 𝑎𝑡 is to prevent the attacker from switching the identity after the challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Alternatively, interaction with the victim can be avoided if continuous voice verification is used on the caller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' However, doing so would be expensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' The output of I is a similarity score between 𝑎𝑡 and 𝑟𝑐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Task Verification (C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' There are two cases where 𝑟𝑐 would not contain the requested task: (1) the model failed to generate the content and (2) the attacker is trying to evade generating artifacts by performing another task or nothing at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' To en- sure that 𝑟𝑐 contains the task, we can use a machine learning classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' The output of C is the probability that 𝑟𝑐 does not contain the task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Time Verification (T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' The time constraint can be verified by en- suring that the first frame of 𝑟𝑐 is received within roughly 1 second after of the challenge’s start time (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=', after the instruc- tions for 𝑐 are given).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' The output of T is the measured time delay denoted 𝑑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Altogether, we validate𝑟𝑐 if none of the four algorithms (T, R, I, C) exceed their respective thresholds (𝜙1,𝜙2,𝜙3,𝜙4) where each thresh- old has been tuned accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' We invalidate 𝑟𝑐 if any model ex- ceeds its respective threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' The false reject rate can be tuned by weighing the contribution of each constraint, however doing so will compromise the security of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' In summary, validation is performed as follows: 𝑉 (𝑟𝑐) = \uf8f1\uf8f4\uf8f4\uf8f4\uf8f2 \uf8f4\uf8f4\uf8f4\uf8f3 𝑝𝑎𝑠𝑠, T (𝑑) < 𝜙1, R(𝑟𝑐) < 𝜙2, I(𝑟𝑐,𝑎𝑡) < 𝜙3, C(𝑟𝑐,𝑐) < 𝜙4 𝑓 𝑎𝑖𝑙, else (1) We note that a combination of validation methods for each con- straint can be used to increase performance, security and usability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' For example, some verifications can be done with humans, some with algorithms and some with both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' 6 DETECTION FRAMEWORK In this section we present the D-CAPTCHA framework which can be used to protect users (victims) from fake callers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' A summary of the D-CAPTCHA framework can be found in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='1 1: Call Forwarding The very first step is to decide which calls should be forwarded to the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' In high risk settings, a D-CAPTCHA may be used to verify every caller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' However, this is not practical in most settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Instead, calls can be forwarded to the system using blacklists (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=', known offenders) or policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' An example policy is to forward all callers who are not in the victim’s address book, or to screen all calls during working hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Alternatively, call screening can be activated by the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' For example, if a call arrives from an unknown number, the user can choose to forward it to the D-CAPTCHA system if the call is unex- pected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Another option is to let users forward ongoing calls if (1) the caller’s audio sounds strange, (2) the conversation is suspicious, or (3) a sensitive discussion needs to be made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' For example, consider the scenario where a user receives a call from a friend under an odd pretext such as “I’m stuck in Brazil and need money to get out.” Here, the user can increase his/her confidence in the caller’s authenticity after forwarding the call through the D-CAPTCHA system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='2 2: Challenge Creation A random challenge 𝑐 is generated using the approach described in section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' In addition to 𝑐, instructions for the caller are generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Instructions include a list of actions to perform and a start indicator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' For example, an instruction might be “at the tone, 1Conference’17, July 2017, Washington, DC, USA Yasur et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' knock three times while introducing yourself.” The instruction is then converted into an audio message using TTS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' At the start of the challenge, the caller is asked to state his/her name.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' This recording is saved as 𝑎𝑡 and shared with the victim for acknowledgment and with I for identity verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='d Next, the audio instructions are played to the caller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' After playing the instructions, a tone is sounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' The time between the tone and the first audible sounds from the caller is measured and included as part of 𝑟𝑐 for T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' After a set number of seconds, the caller’s recording is saved as 𝑟𝑐 and passed along for verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='3 3: Response Verification The recorded response 𝑟𝑐 and its timing data are sent to T, R, C, and I for constraint verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' If all the algorithms yield scores below their respective thresholds, then 𝑎𝑡 is played to the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' If the user accepts the call with 𝑡 then the D-CAPTCHA is 𝑣𝑎𝑙𝑖𝑑 and the call is connected / resumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' If any of the algorithms produce a score above their threshold, then the call is dropped, and evidence is provided to the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Evi- dence consists of an explanation of why the call was not trusted (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=', information on which constraint(s) failed and to what degree) and playback recordings of 𝑎𝑡, 𝑐, and 𝑟𝑐 accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Although the order which the models are executed does not matter, we can avoid executing redundant models if one model detects the deepfake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Therefore, we suggest the order T → R → C → I to potentially save execution time when detecting a deepfake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' We also note that if higher security is required, then multiple D-CAPTCHAs can be sent out and subsequently verified to reduce the false negative rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='1 Deployment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' In general, the framework can be deployed as an app on the victim’s phone or as a service in the cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' For example, onsite technicians, bankers, and the elderly can have the system screen calls directly on their phones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Call centers and online meeting rooms can use cloud resources to screen callers in waiting rooms (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=', before connecting to a confidential Zoom meeting [48, 59]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='4 Limitations The main limitations of this system are its applicability and usabil- ity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' In terms of deployment, the system must be able to interact with the deepfake so it can only protect against RT-DFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' More- over, since it is an active defense, the CAPTCHA protocol runs the risk of becoming a hindrance to users if not tuned correctly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Regardless, it’s a great solution for screening callers entering high security conversations and meetings in an age where calls cannot be trusted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Finally, the system uses deep learning models in R, I, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Just like other deep learning-based defenses, an attacker can potentially evade these models using adversarial examples [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' However, when trying to evade our system, the attacker must over- come a number of challenges: (1) most calls are made over noisy and compressed channels reducing the impact of the perturbations, (2) performing this attack would require real-time generation of adversarial examples, and (3) R, I, and C would most likely be a dRecall, this is done to prevent attackers from simply turning off the RT-DF during the challenge and using their actual voice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' black box to the attacker, although not impervious, it cannot be easily queried.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' 7 THREAT ANALYSIS In this section, we assess the threat posed by RT-DFs by evaluating the quality of five state-of-the-art RT-DF models in the perspective of 41 volunteers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='1 Experiment Setup 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='1 RT-DF Models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' We surveyed 25 voice cloning papers pub- lished over the last three years which can process audio in real-time as a sequence of frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Of the 25 papers we selected the four recent works which published their source code: AdaIN-VC [15], MediumVC [19], FragmentVC [37] and StarGANv2-VC [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' We also selected ASSEM-VC [30] which is a non-casual model as an additional com- parison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' All works are from 2021 except AdaIN-VC which is from 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' any-to-many.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' StarGANv2-VC is many-to-many model which also works as an any-to-many model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' The audio 𝑎𝑔 is created by passing the spectrogram of 𝑎𝑠 through an encoder-decoder network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' To disentangle content from identity, the decoder also receives an encoding of 𝑎𝑠 taken from a pretrained network which extracts the fundamental frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Finally, the decoder receives reference information on 𝑡 via a style encoder using sample 𝑎𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' ASSEM-VC works in a similar manner except 𝑎𝑠 and a TTS transcript of 𝑎𝑠 are used to generate a speaker independent representation before being passed to the decoder, and the decoder receives reference information on 𝑡 from an identify encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' any-to-any.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' In AdaIN-VC, 𝑎𝑔 is created by disentangling identity from content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' The model (1) passes a sample 𝑎𝑡 through an iden- tity encoder, (2) passes a source frame 𝑎(𝑖) 𝑠 through a content encoder with instance-normalization, and then (3) passes both outputs through a final decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' In MediumVC, 𝑎𝑠 first normal- izes the voice by converting it to a common identity with an any-to-one VC model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' The result is then encoded and passed to a decoder along with an identity encoding (similar to AdaIN-VC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' FragmentVC, extracts the content of 𝑎𝑠 using a Wav2Vec 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='0 model [8] and extracts fragments of 𝑎𝑡 using an encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' A de- coder then uses attention layers to fuse the identity fragments into the content to produce 𝑎𝑔.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' All audio clips in this experiment were generated using the pre- trained models provided by the original authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' To simulate a realistic setting, the clips were passed through a phone filter (a band pass filter on the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='3-3KHz voice range).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='2 Experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' To help quantify the threat of RT-DFs, we per- formed two experiments on a group of 41 volunteers: EXP1a - Quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' The goal of the first experiment was to see how easy it is to identify an RT-DF in the best-case scenario (when the victim is expecting a deepfake).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' EXP1b - Identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' The goal of this experiment was to understand how well RT-DF models are able to clone identities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' In EXP1a, volunteers were asked to rate audio clips on a scale of 1-5 (1: fake, 5: real).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' There were 90 audio clips presented in random Deepfake CAPTCHA: A Method for Preventing Fake Calls Conference’17, July 2017, Washington, DC, USA order: 30 real and 60 fake (12 from each of the five models).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' The clips were about 4-7 seconds long each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' In EXP1b, we selected the top 2 models that performed the best in EXP1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' For each model, we repeated the following trial 8 times: We first let the volunteer listen to two real samples of the target identity as a baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Then we played two real and two fake samples in random order and asked the volunteer to rate how similar their speakers sound compared to the speaker in the baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' If a model has a positive mean opinion score (MOS) in both EXP1 and EXP2 then it is a considerable threat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' This is because it can (1) synthesize high quality speech (2) that sounds like the target (3) all in real-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='2 Experiment Results EXP1a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' To analyze the quality (realism) of the models, we com- pared the MOS scores of the deepfake audio to the MOS of the real audio (both scored blindly).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' 3 we plot the distribution of each model’s MOS compared to real audio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Roughly 20-50% of the volunteers gave the RT-DF audio positive score with StarGANv2-VC having the highest quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' However, opinion scores are subjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Therefore, we need to normalize the MOS to count how many times volunteers were fooled by an RT-DF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' In principle, the range of scores a volunteer 𝑘 has given to real audio captures that volunteer’s ‘trust’ range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Let 𝜇𝑘 𝑟𝑒𝑎𝑙 and 𝜎𝑘 𝑟𝑒𝑎𝑙 be the mean and standard deviation on 𝑘’s scores for real clips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' We estimate that a volunteer would likely be fooled by a clip if he or she scores a clip with a value greater than 𝜇𝑘 𝑟𝑒𝑎𝑙 −𝜎𝑘 𝑟𝑒𝑎𝑙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Using this measure, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' 4 we present the attack success rate for each of the RT-DF models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' We found that StarGANv2-VC has the highest success rate of 46% percent rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' This means that although current RT-DF models are not perfect, they can indeed fool people.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' We note that these results cannot be interpreted as the likelihood of a true RT-DF attack succeeding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' This is because our volunteers were expecting to hear deepfakes and were therefore carefully listening for artifacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' A true victim would likely overlook some artifacts especially when put under pressure by the attacker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' EXP1b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' To analyze the ability of the models to copy identities, we normalized volunteer 𝑘’s scores on fake audio by computing 𝑠𝑐𝑜𝑟𝑒−𝜇𝑘 𝑟𝑒𝑎𝑙 𝜎𝑘 𝑟𝑒𝑎𝑙 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' 5 plots the distribution of the normalized scores on fake audio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' We can see that the volunteers were mostly indecisive, rating some fake clips as more authentic and some as less.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' For the majority of cases (𝑠𝑐𝑜𝑟𝑒 > −1) volunteers felt that the identity was captured well by the top two models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' In summary, there is a chronological trend given that the worst performing model AdaIN-VC is from 2019 and the best StarGANv2-VC is from 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' This may indicate that the quality of RT-DF is rapidly improving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' This raises concern, especially since the volunteers were expecting the attack yet could not accurately tell which clips were real or fake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Another insight we have is that the presence of artifacts can help victims identify RT-DFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' However, as quality improves, we expect that only way to induce significant artifacts will be by challenging the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' 100 90 80 70 60 50 40 30 20 10 0 Ada Medium Assem Fragment StarGan Real 0 10 20 30 40 50 60 70 80 90 100 Model 1 2 3 4 5 Figure 3: RT-DF Quality - The distribution of ratings which the volunteers gave to each of the RT-DF models and real voice recordings (1: fake, 5: real).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Ada Medium Assem Fragment StarGan 0 10 20 30 40 Success % Model Model Ada Medium Assem Fragment StarGan Per model, on participants who are aware of deepfake possibility Attack Success Rate Figure 4: RT-DF Quality - The percent of volunteers fooled by each RT-DF model, even though they were expecting a deepfake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Fragment Stargan −2 0 2 0 20 40 60 80 0 20 40 60 80 Normalized value Count Figure 5: RT-DF Identity - A histogram of the normalized MOS scores for how similar RT-DF audio sounds like the target identity 𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Positive scores are cases where volunteers thought a fake audio sounded more like 𝑡 than an authentic recording of 𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' 8 D-CAPTCHA EVALUATION In this section, we evaluate the benefit of using a D-CAPTCHA as opposed to using passive defenses alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='1 Experiment Setup 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='1 Datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' To evaluate our system, we recorded 20 English speaking volunteers to create both speech and challenge-response datasets, summarized in Table 2: (D𝑟𝑒𝑎𝑙) 2498 samples of real speech (100-250 random sentences spoken by each of the 20 volunteers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' (D𝑓 𝑎𝑘𝑒) 1821 samples of RT-DF voice conversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' To create this dataset we used StarGANv2-VC which was the top performing model from EXP1a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' The model was trained to impersonate Conference’17, July 2017, Washington, DC, USA Yasur et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' 6 of the 20 volunteers from D𝑟𝑒𝑎𝑙, and an additional 14 ran- dom voice actors from the VCTK dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' The additional 14 were added to help the model generalize better, and only the 6 volunteers’ voices were used to make RT-DFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' (D𝑟𝑒𝑎𝑙,𝑟) 3317 samples of real responses (attempts at challenges).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' A sample of nine tasks were evaluated in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' The following tasks were performed 30 times per volunteer: sing (S), hum tune (HT), coughing (Co), vary volume (V), and talk & playback (P), and the following tasks were performed 5 times per volunteer: repeat accent (R), clap (Cl), speak with emotion (SE), and vary speed (VS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' (D𝑓 𝑎𝑘𝑒,𝑟) 16,123 deepfake samples of RT-DF voice conversion ap- plied to the responses D𝑟𝑒𝑎𝑙,𝑟 using StarGANv2-VC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' We did not convert samples from the same identity (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=', where 𝑠 = 𝑡) It took each volunteer over an hour to record their data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' The volun- teers were compensated for their time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' For all train-test splits used in our evaluations, we made sure not to use the same identities in both the train and test sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' In addition, we also used public deepfake datasets to train the realism models R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' These datasets were the ASVspoof-DF dataset [62] with 22,617 real and 15,000 fake samples, and the RITW dataset [42] with 19,963 real and 11,816 fake samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='2 Models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Our system, when fully automated, consists of 3 models: R, C and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' The algorithm T does not use a machine learning model to verify the time constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' For the realism model R, we evaluated five different deepfake detection models: SpecRNet [25] which is a novel neural network architecture, inspired by RawNet2 [54], which get results compa- rable to state–of–the-art models despite a significant decrease in computational requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' One-Class [63] is a method adapted from [41] based on a deep residual network ResNet-18 [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' They improve and generalize the network performance using One-Class Softmax activations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' GMM-ASVspoof [62] is a Gaussian mixture model (GMM) which operates on LFCCs features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' This model was a baseline for the in ASVspoof 2021 competition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' PC-DARTS [18] is a convolutional neural network (CNN) that tries to automati- cally learn the network’s architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' This work also showed good results in generalizing to unseen attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Finally, we used Local Outlier Factor (LOF) which is a density-based anomaly detection model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' We took the union of ASVspoof-DF and RITW and selected 80% at random for training the models and 10% for validation (early stopping).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' The models were tested on the baseline scenario (D𝑟𝑒𝑎𝑙 and D𝑓 𝑎𝑘𝑒) and our proposed D-CAPTCHA scenario (D𝑟𝑒𝑎𝑙,𝑟 and D𝑓 𝑎𝑘𝑒,𝑟).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' For the task model C, we trained a GMM classifier on the MFCC features using the baseline model from [62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' One model was trained per task: to classify between real responses from that task and all other tasks as well as speech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' A 70-30 train-test split was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' For the identity model I, we used a pre-trained voice recogni- tion model from the SpeechBrain toolkit [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' The model uses the ECAPA-TDNN architecture to classify a speaker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Since we do not want I to have prior knowledge of 𝑡, we converted the model into an anomaly detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Recall that we obtain a voice sample 𝑎𝑡 from the caller prior to the challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' This sample is used as a reference to ensure that the RT-DF is not turned off during the challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' To Table 2: The number of samples in each of our datasets Real: D𝑟𝑒𝑎𝑙 Fake: D𝑓 𝑎𝑘𝑒 Speech 2498 1821 Real: D𝑟𝑒𝑎𝑙,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='𝑟 Fake: D𝑓 𝑎𝑘𝑒,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='𝑟 Repeat Accent (R) 98 570 Clap (Cl) 99 551 Cough (Co) 537 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='401 Speak with Emotion (SE) 98 532 Hum Tune (HT) 593 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='325 Playback Audio (P) 601 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='420 Sing (S) 595 334 Vary Speed (VS) 98 570 Vary Volume (V) 598 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='420 Real Fake ASVspoof-DF 22,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='617 15,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='000 RITW 19,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='963 11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='816 detect whether the identity of the caller has changed during the challenge,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' we compute I(𝑎𝑡,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='𝑟𝑐) = ||𝑓 ∗(𝑎𝑡) − 𝑓 ∗(𝑟𝑠)||2 (2) where 𝑓 ∗ is the speaker encoding,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' taken from an inner layer of the speech recognition model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Smaller scores indicate similarity between the voice before the challenge and during the challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' This technique of comparing speaker encodings has been done in the past (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=', [40, 43]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' To evaluate I, we create negative pairings as samples from the same identity (𝑎𝑖,𝑟𝑐,𝑖) and positive pairings as samples from different identities (𝑎𝑖,𝑟𝑐,𝑗), where 𝑎𝑖,𝑎𝑗 ∈ D𝑟𝑒𝑎𝑙, 𝑟𝑐,𝑖,𝑟𝑐,𝑗 ∈ D𝑟𝑒𝑎𝑙,𝑟 and 𝑖 ≠ 𝑗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='3 Experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' We performed four experiments: EXP2a R: A baseline comparison between existing solutions (pas- sive) and our solution (active) in detecting RT-DFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' EXP2b C: An evaluation of the task detection model which ensures that the caller indeed performed the challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' EXP2c I: An evaluation of the identity model which ensures that the caller didn’t just turn off the RT-DF for the challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' EXP2d R, C, I: An evaluation of the system end-to-end to evaluate the performance of the system as a whole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' We do not evaluate T because it is just a restriction that the first frame of the response 𝑟𝑐 be received within approximately one second from the start time of the challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' To measure the performance of the models, we use the area under the curve (AUC) and equal error rate (EER) metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' AUC measures the general trade-off between the true positive rate (TPR) and the false positive rate (FPR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' An AUC of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='0 indicates a perfect classifier while an AUC of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='5 indicates random guessing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' The EER captures the trade-off between the FPR and the false negate rate (FNR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' A lower EER is better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='2 Experiment Results 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='1 EXP2a (R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' The goal of EXP2a was to see if our system can improve the detection of RT-DFs if the adversary is forced to per- form a task that is outside of the deepfake model’s capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' In Table 3, we compare the performance of the five deepfake detec- tors on (1) detecting regular deepfake speech (baseline) and on (2) Deepfake CAPTCHA: A Method for Preventing Fake Calls Conference’17, July 2017, Washington, DC, USA 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='0 False Positive Rate 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='0 True Positive Rate R auc: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='864 T&C auc: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='985 Co auc: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='0 SE auc: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='938 HT auc: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='999 P auc: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='998 S auc: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='993 VS auc: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='963 V auc: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='974 Figure 6: The performance of the task detection model C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='0 False Positive Rate 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='0 True Positive Rate Co auc: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='574 HT auc: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='688 P auc: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='831 SE auc: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='846 S auc: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='878 R auc: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='89 T&C auc: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='904 VS auc: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='926 V auc: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='942 Figure 7: The performance of the unsupervised identity de- tection model I for different tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' detecting deepfake challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' The bold values indicate challenges which improved the performance of the corresponding model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' We see that with the exception of SpecRNet, all of the detectors benefit from examining challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Overall, the best performing model was GMM-ASVspoof with the challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' This means that the challenges provide a better way to detect RT-DFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='2 EXP2b (C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' If an attacker is evasive, he may try to do nothing instead of the challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' It’s also possible that the attacker will try the challenge, but the model will output nothing because it can’t generate the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' 6 shows that either way, the task detector C can tell whether the task was performed or not with high certainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='3 EXP2c (I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Another evasive strategy is where the attacker turns off the RT-DF while performing the challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' In this scenario, we compare the identity of the caller before (𝑎𝑡) and during (𝑟𝑐) the challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' 7 we present the results of the identity detector I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Here we can see that the model does quite well, with the exception of the tasks ‘hum’ and ‘cough’ which do not carry much of the speaker’s identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='4 EXP2d (R, I, C): D-CAPTCHA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Finally, when executing all three models, we must consider how the successes and failures of each model compound together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' We set the threshold for each model (R, I, C) so that the FPR=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' We then passed through 3,317 real responses and 8,758 deepfake responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' 8 presents the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='9% 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='9% 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='4% 91% 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='6% 93.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='2% Accuracy TPR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='95 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='00 Value 100% 100% 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='2% 100% 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='4% 99% 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='3% 100% 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='9% 0% 0% 2% 0% 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='3% 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='2% 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='7% 0% 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='3% Precision FPR Accent Clap Cough Emotion Hum Playback Sing Speed Volume Accent Clap Cough Emotion Hum Playback Sing Speed Volume 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='9 Task Value Figure 8: The performance of the ensure D-CAPTCHA sys- tem (end-to-end).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' We found that we were able to achieve a TPR of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='89-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' FPR of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='0-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='3 and accuracy of 91-100% depending on the selected task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' In contrast, the model which performed the best on deepfake speech detection (baseline) was SpecRNet with a TPR of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='66 and accuracy of 71% when the FPR=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Therefore, D-CAPTCHA significantly outperforms the baseline and provides a good defense against RT- DFs audio calls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' 9 FUTURE WORK: VIDEO D-CAPTCHA As mentioned in the introduction, the same D-CAPTCHA system outlined in this paper can be applied to video-based RT-DFs as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' For example, to prevent imposters from joining online meetings (such as the cases in [48, 59]) we can forward suspicious calls to a D-CAPTCHA system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' There are a wide variety of tasks which existing models and pipelines cannot handle for similar reasons to those listed in section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' For example, the caller can be asked to drop/bounce objects, fold shirt, stroke hair, interact with back- ground, spill water, pick up objects, perform hand expressions, press on face, remove glasses, turn around, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' These tasks can easily be turned into challenges to detect video-based RT-DFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' To demonstrate the potential, we have performed some initial experiments and will now present some preliminary results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' In our experiment we used a popular zero-shot RT-DF model called Avatarifye based on the work of [50] to reenact (puppet) a single photo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' We were able to achieve a realistic RT-DF video at 35 fps with negligible distortions if the face stayed in a frontal position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' How- ever, when we performed some of the mentioned challenges, the model failed and large distortions appeared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=', 9 in the appendix presents some screenshots of the video during the challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' These preliminary results indicate that D-CAPTCHAs can be a good solution for both RT-DF audio and video calls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' ehttps://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Here a zero-shot reenactment model called Avatarify breaks the moment the caller performs an action other than basic expressions and talking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' 10 CONCLUSION Deepfakes are rapidly improving in terms of quality and speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' This poses a significant threat as attackers are already using real-time deepfakes to impersonate people over calls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Current defenses use passive methods to identify deepfakes via their flaws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' However, this approach may have limits as the quality of deepfakes continues to advance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Instead, in this work we proposed an active defense strategy: D-CAPTCHA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' By challenging the attacker to create con- tent under four constraints based on practical and technological limitations, we can force the deepfake model to expose itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' By pro- tecting calls and meetings from deepfake imposters, we believe that this system can significantly improve the security of organizations and individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' ACKNOWLEDGMENTS This work was supported by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='-Israel Energy Center managed by the Israel-U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Binational Industrial Research and Development (BIRD) Foundation and the Zuckerman STEM Leadership Program.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='bbc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='com/news/technology-60780142.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' (Accessed on 11/27/2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' [61] RLMAPC Wijethunga, DMK Matheesha, Abdullah Al Noman, KHVTA De Silva, Muditha Tissera, and Lakmal Rupasinghe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' Deepfake audio detection: a deep learning based solution for group conversations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' In 2020 2nd International Conference on Advancements in Computing (ICAC), Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' IEEE, 192–197.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' [62] Junichi Yamagishi, Xin Wang, Massimiliano Todisco, Md Sahidullah, Jose Patino, Andreas Nautsch, Xuechen Liu, Kong Aik Lee, Tomi Kinnunen, Nicholas Evans, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' ASVspoof 2021: accelerating progress in spoofed and deepfake speech detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' arXiv preprint arXiv:2109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content='00537 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' [63] You Zhang, Fei Jiang, and Zhiyao Duan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' One-class learning towards syn- thetic voice spoofing detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' IEEE Signal Processing Letters 28 (2021), 937–941.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' A ETHICAL DISCLOSURES The experiments performed in this study have received our institu- tion’s ethical committee’s approval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' All 20 volunteers whose voices were used to create deepfakes permitted the use of their data for this purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfSQOU/content/2301.03064v1.pdf'} +page_content=' To protect our volunteers, the trained RT-DF voice models will not be shared.' metadata={'source': 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b/79E1T4oBgHgl3EQfBwKM/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:6fec20320acf111931a9f583465148d858e558d853ff18619140ba3605650c65 +size 4259885 diff --git a/7NE0T4oBgHgl3EQfwAHH/content/tmp_files/2301.02627v1.pdf.txt b/7NE0T4oBgHgl3EQfwAHH/content/tmp_files/2301.02627v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..29a06edf8d11ce399ae195c490e25b658035f600 --- /dev/null +++ b/7NE0T4oBgHgl3EQfwAHH/content/tmp_files/2301.02627v1.pdf.txt @@ -0,0 +1,950 @@ +arXiv:2301.02627v1 [math.RA] 6 Jan 2023 +Pre-Lie algebras, their multiplicative lattice, and +idempotent endomorphisms +Michela Cerqua and Alberto Facchini +Abstract We introduce the notions of pre-morphism and pre-derivation for arbitrary +non-associative algebras over a commutative ring 푘 with identity. These notions +are applied to the study of pre-Lie 푘-algebras and, more generally, Lie-admissible +푘-algebras. Associating with any algebra (퐴, ·) its sub-adjacent anticommutative +algebra (퐴, [−, −]) is a functor from the category of 푘-algebras with pre-morphisms +to the category of anticommutative 푘-algebras. We describe the commutator of two +ideals of a pre-Lie algebra, showing that the condition (Huq=Smith) holds for pre- +Lie algebras. This allows to make use of all the notions concerning multiplicative +lattices in the study of the multiplicative lattice of ideals of a pre-Lie algebra. We +study idempotent endomorphisms of a pre-Lie algebra 퐿, i.e., semidirect-product +decompositions of 퐿 and bimodules over 퐿. +Introduction +The aim of this paper is to present pre-Lie algebras from the point of view of their +multiplicative lattice of ideals, and to study their idempotent endomorphisms. Pre- +Lie algebras were first introduced and studied in [15] by Vinberg. He applied them +to the study of convex homogenous cones. He called “left-symmetric algebras” the +algebras we call pre-Lie algebras in this paper. +We present a notion of pre-morphism and pre-derivation for arbitrary non- +associative algebras over a commutative ring 푘 with identity, and apply it to the +study of pre-Lie 푘-algebras and, more generally, Lie-admissible 푘-algebras. Asso- +ciating with any pre-Lie algebra (퐴, ·) its sub-adjacent Lie algebra (퐴, [−, −]) is a +functor from the category PreL푘,푝 of pre-Lie 푘-algebras with pre-morphisms to the +category of Lie 푘-algebras. We introduce the notion of module 푀 over a pre-Lie +Michela Cerqua e-mail: michela.cerqua@studenti.unipd.it · Alberto Facchini +Dipartimento di Matematica "Tullio Levi Civita", Università di Padova, 35121 Padova, Italy e-mail: +facchini@math.unipd.it +1 + +2 +Michela Cerqua and Alberto Facchini +algebra 퐿 and, like in the case of associative algebras, it is possible to do it in two +equivalent ways, via a suitable scalar multiplication 퐿 × 푀 → 푀 or as a 푘-module +푀 with a pre-morphism 휆: (퐿, ·) → (End(푘푀), ◦). The category of modules over +a pre-Lie 푘-algebra (퐿, ·) is isomorphic to the category of modules over its sub- +adjacent Lie 푘-algebra (퐿, [−, −]). We then consider the commutator of two ideals +in a pre-Lie algebra. In particular we show that the condition (Huq=Smith) holds +for pre-Lie algebras. With the notion of commutator at our disposal, the lattice of +ideals of a pre-Lie algebra becomes a multiplicative lattice [6, 8]. As a consequence +we immediately get the notions of abelian pre-Lie algebra, prime ideal, prime spec- +trum of a pre-Lie algebra, solvable and nilpotent pre-Lie algebras, metabelian and +hyperabelian pre-Lie algebras, centralizer, and center. +We then consider idempotent endomorphisms of a pre-Lie algebra, because they +immediately show what semi-direct products of pre-Lie algebras are, what the action +of a pre-Lie algebra on another pre-Lie algebra is, and lead us to the notion of +bimodule over a pre-Lie algebra. We study the “Dorroh extensions” of pre-Lie +algebras. Like in the associative case, we get a category equivalence between the +category PreL푘 and the category of pre-Lie algebras with identity and with an +augmentation. +1 Preliminary notions on non-associative 풌-algebras +Let 푘 be a commutative ring with identity. In this article, a 푘-algebra is a 푘-module +푘푀 with a further 푘-bilinear operation 푀 × 푀 → 푀, (푥, 푦) ↦→ 푥푦 (equivalently, +a 푘-module morphism 푀 ⊗푘 푀 → 푀). A subalgebra (an ideal, resp.) of 푀 is a +푘-submodule 푁 of 푀 such that 푥푦 ∈ 푁 for every 푥, 푦 ∈ 푁 (푥푛 ∈ 푁 and 푛푥 ∈ 푁 +for every 푥 ∈ 푀 and 푛 ∈ 푁, resp.) As usual, if 푁 is an ideal of 푀, the quotient +푘-module 푀/푁 inherits a 푘-algebra structure. There is a one-to-one correspondence +between the set of all ideals 푁 of 푀 and the set of all congruences on 푀, that is, +all equivalence relations ∼ on 푀 for which 푥 ∼ 푦 and 푧 ∼ 푤 imply 푥 + 푧 ∼ 푦 + 푤, +휆푥 ∼ 휆푦 and 푥푧 ∼ 푦푤 for every 푥, 푦, 푧, 푤 ∈ 푀 and every 휆 ∈ 푘. The opposite 푀op of +an algebra 푀 is defined taking as multiplication in 푀op the mapping (푥, 푦) ↦→ 푦푥. +If 푀 and 푀′ are two 푘-algebras, a 푘-linear mapping 휑: 푀 → 푀′ is a 푘-algebra +homomorphism if 휑(푥푦) = 휑(푥)휑(푦) for every 푥, 푦 ∈ 푀. Clearly, 푘-algebras form a +variety in the sense of Universal Algebra. Moreover, it is a variety of Ω-groups, that +is, a variety which is pointed (i.e., it has exactly one constant) and has amongst its +operations and identities those of the variety of groups. It follows that 푘-algebras form +a semiabelian category. Other examples of Ω-groups are abelian groups, non-unital +rings, commutative algebras, modules and Lie algebras. +If 푀 is any 푘-algebra, its endomorphisms form a monoid, that is, a semigroup +with a two-sided identity, with respect to composition of mappings ◦. A derivation +of a 푘-algebra 푀 is any 푘-linear mapping 퐷 : 푀 → 푀 such that 퐷(푥푦) = (퐷(푥))푦+ +푥(퐷(푦)) for every 푥, 푦 ∈ 푀. For any 푘-algebra 푀, we can construct the 푘-algebra +of derivations Der푘(푀) of the 푘-algebra 푀. Its elements are all derivations of 푀. + +Pre-Lie algebras, their multiplicative lattice, and idempotent endomorphisms +3 +If 푀 is any 푘-algebra and 퐷, 퐷′ are two derivations of 푀, then the composite +mapping 퐷퐷′ is not a derivation of 푀 in general, but 퐷퐷′ − 퐷′퐷 is. Thus, for any +푘-algebra 푀, we can define the Lie 푘-algebra Der푘 (푀) as the subset of End(푘푀) +consisting of all derivations of 푀 with multiplication [퐷, 퐷′] := 퐷퐷′ − 퐷′퐷 for +every 퐷, 퐷′ ∈ Der푘(푀). +It is known that there is not a general notion of representation (or module)over our +(non-associative) 푘-algebras. There is a notion of bimodule over a non-associative +ring due to Eillenberg, and this notion works well for Lie algebras, but is not +convenient in the study of Jordan algebras and alternative algebras. The situation, as +far as modules are concerned, is the following. +1.1 Modules over an associative 풌-algebra. +Given any 푘-algebra 푀, we can consider, for every element 푥 ∈ 푀, the map- +ping 휆푥 : 푀 → 푀, defined by 휆푥(푎) = 푥푎 for every 푎 ∈ 푀. The mapping +휆: 푀 → End(푘푀) is defined by 휆: 푥 ↦→ 휆푥 for every 푥 ∈ +푀. This 휆 is a 푘- +algebra morphism if and only if 푀 is associative. Thus, for any associative 푘-algebra +푀, it is natural to define a left 푀-module as any 푘-module 푘 퐴 with a 푘-algebra +homomorphism 휆: 푀 → End(푘 퐴). Similarly, we can define right 푀-modules as +푘-modules 푘 퐴 with a 푘-algebra antihomomorphism 휌 : 푀 → End(푘 퐴). Here by 푘- +algebra antihomomorphism 휓 : 푀 → 푀′ between two 푘-algebras 푀, 푀′ we mean +any 푘-linear mapping 휓 such that 휓(푥푦) = 휓(푦)휓(푥) for every 푥, 푦 ∈ 푀. Clearly, a +mapping 푀 → 푀′ is a 푘-algebra antihomomorphism if and only if it is a 푘-algebra +homomorphism 푀op → 푀′. It follows that right 푀-modules coincide with left +푀op-modules. More precisely, when we say that right 푀-modules coincide with left +푀op-modules, we mean that there is a canonical category isomorphism between the +category of all right 푀-modules and the category of all left 푀op-modules. Simi- +larly, left 푀-modules coincide with right 푀op-modules. Also, if 푀 is commutative, +then left 푀-modules and right 푀-modules coincide. Finally, left modules 퐴 over +an associative 푘-algebra 푀 can be equivalently defined using, instead of the 푘- +algebra homomorphism 휆: 푀 → End(푘 퐴), a 푘-bilinear mapping 휇: 푀 × 퐴 → 퐴, +휇: (푚, 푎) ↦→ 푚푎, such that (푚푚′)푎 = 푚(푚′푎) for every 푚, 푚′ ∈ 푀 and 푎 ∈ 퐴. +1.2 Modules over a Lie 풌-algebra. +For any 푘-module 퐴 we will denote by 픤픩(퐴) the Lie 푘-algebra End(푘 퐴) of all +푘-endomorphisms of 퐴 with the operation [−, −] defined by [ 푓 , 푔] = 푓 푔 − 푔 푓 . +For any Lie 푘-algebra 푀 and any element 푥 ∈ 푀, the mapping 휆푥 is an element +of the Lie 푘-algebra Der푘 (푀), usually called the adjoint of 푥, or the inner derivation +defined by 푥, and usually denoted by ad푀 푥 instead of 휆푥, and the mapping ad: 푀 → + +4 +Michela Cerqua and Alberto Facchini +Der푘(푀) ⊆ 픤픩(푀), defined by ad: 푥 ↦→ ad푀 푥 for every 푥 ∈ 푀, is a Lie 푘-algebra +homomorphism. +Left modules over a Lie 푘-algebra 푀 are defined as 푘-modules 퐴 with a Lie +푘-algebra homomorphism휆: 푀 → 픤픩(퐴). Similarly, it is possible to define right 푀- +modules as 푘-modules 퐴 with a 푘-algebra antihomomorphism 휌 : 푀 → 픤픩(퐴). But +any Lie 푘-algebra 푀 is isomorphic to its opposite algebra 푀op via the isomorphism +푀 → 푀op, 푥 ↦→ −푥. It follows that the category of right 푀-modules is canonically +isomorphic to the category of left 푀-modules for any Lie 푘-algebra 푀. Therefore +it is useless to introduce both right and left modules, it is sufficient to introduce left +푀-modules and call them simply “푀-modules”. +2 Pre-Lie 풌-algebras +A pre-Lie 푘-algebra is a 푘-algebra (푀, ·) satisfying the identity +(푥 · 푦) · 푧 − 푥 · (푦 · 푧) = (푦 · 푥) · 푧 − 푦 · (푥 · 푧) +(1) +for every 푥, 푦, 푧 ∈ 푀. +For any 푘-algebra (푀, ·), defining the commutator [푥, 푦] = 푥 · 푦 − 푦 · 푥 for every +푥, 푦 ∈ 푀, the algebra (푀, [−, −]) is anticommutative. If (푀, ·) is a pre-Lie algebra, +one gets that (푀, [−, −]) is a Lie algebra, called the Lie algebra sub-adjacent to the +pre-Lie algebra (푀, ·). +Pre-Lie algebras are also called Vinberg algebras or left-symmetric algebras. +This last name refers to the fact that in (1) one exchanges the first two variables on +the left. A right-symmetric algebra is an algebra in which, for every 푥, 푦, 푧 ∈ 푀, +(푥 · 푦) · 푧 − 푥 · (푦 · 푧) = (푥 · 푧) · 푦 − 푥 · (푧 · 푦). It is easily seen that the category of +left-symmetricalgebras and the category of right-symmetricalgebras are isomorphic +(the categorical isomorphism is given by 푀 ↦→ 푀op). +Examples 1 (1) Every associative algebra is clearly a pre-Lie algebra. +(2) Derivations on 푘[푥1, . . . , 푥푛]푛. Let 푘 be a commutative ring with identity, +푛 ≥ 1 be an integer, and 푘[푥1, . . . , 푥푛] be the ring of polynomials in the 푛 indeter- +minates 푥1, . . . , 푥푛 with coefficients in 푘. Let 퐴 be the free 푘[푥1, . . . , 푥푛]-module +푘[푥1, . . . , 푥푛]푛 with free set {푒1, . . . , 푒푛} of generators. As a 푘-module, 퐴 is the +free 푘-module with free set of generators the set { 푥푖1 +1 . . . 푥푖푛 +푛 푒 푗 | 푖1, . . . , 푖푛 ≥ 0, 푗 = +1, . . . , 푛}. Consider the usual derivations of the ring 푘[푥1, . . . , 푥푛]: +휕 +휕푥 푗 +(푥푖1 +1 . . . 푥푖푛 +푛 ) = +� +푥푖1 +1 . . . 푖 푗푥푖푗−1 +푗 +. . . 푥푖푛 +푛 +for 푖 푗 > 0, +0 +for 푖 푗 = 0. +Defineamultiplication on 퐴 setting,forevery 푢 = (푢1, . . . , 푢푛), 푣 = (푣1, . . . , 푣푛) ∈ 퐴, + +Pre-Lie algebras, their multiplicative lattice, and idempotent endomorphisms +5 +푣 · 푢 = ( +푛 +� +푗=1 +푣 푗 +휕푢1 +휕푥 푗 +, . . . , +푛 +� +푗=1 +푣 푗 +휕푢푛 +휕푥 푗 +). +It is then possible to see that 퐴 is a pre-Lie 푘-algebra [2, Section 2.3]. +(3) An example of rank 2. Let 푘 be any commutative ring with identity and +퐿 � 푘 ⊕ 푘 a free 푘-module of rank 2 with free set {푒1, 푒2} of generators. Define +a multiplication on 퐿 setting 푒1푒1 = 2푒1, 푒1푒2 = 푒2, 푒2푒1 = 0, 푒2푒2 = 푒1, and +extending by 푘-bilinearity. Then 퐿 is a pre-Lie 푘-algebra [13]. +(4) Rooted trees. Recall that a tree is an undirected graph in which any two vertices +are connected by exactly one path, or equivalently a connected acyclic undirected +graph. A rooted tree of degree 푛 is a pair (푇, 푟), where 푇 is a tree with 푛 vertices, +and its root 푟 is a vertex of 푇. In the following we will label the vertices of 푇 with +the numbers 1, . . . , 푛, and the root 푟 with 1. +Let 푘 be a commutative ring with identity and T푛 be the free 푘-module with free +set of generators the set of all isomorphism classes of rooted trees of degree 푛. Set +T := +� +푛≥1 +T푛. +Define a multiplication on T setting, for every pair 푇1,푇2 of rooted trees, +푇1 · 푇2 = +� +푣 ∈푉 (푇2) +푇1 ◦푣 푇2, +where 푉(푇2) is the set of vertices of 푇2, and 푇1 ◦푣 푇2 is the rooted tree obtained by +adding to the disjoint union of 푇1 and 푇2 a further new edge joining the root vertex +of 푇1 with the vertex 푣 of 푇2. The root of 푇1 ◦푣 푇2 is defined to be the same as the +root of 푇2. To get a multiplication on T, extend this multiplication by 푘-bilinearity. +Let us give an example. Suppose +푇1 = +1 +2 +3 +and +푇2 = +1 +2 +Then + +6 +Michela Cerqua and Alberto Facchini +푇1 ◦1 푇2 = +1 +2 +3 +4 +5 +and +푇1 ◦2 푇2 = +1 +2 +3 +4 +5 +, +where we have relabelled the vertices of푇1. (If푇1 has 푛 vertices and푇2 has 푚 vertices, +it is convenient to relabel in 푇1 ◦푣 푇2 the vertices 1, . . . , 푛 of 푇1 with the numbers +푚 + 1, . . . , 푚 + 푛, respectively.) Therefore +푇1 · 푇2 = +1 +2 +3 +4 +5 ++ +1 +2 +3 +4 +5 +In this way, one gets a pre-Lie 푘-algebra T [4, 2]. It is a graded 푘-algebra because +T푛 · T푚 ⊆ T푛+푚 for every 푛 and 푚. It can be proved that this is the free pre-Lie +푘-algebra on one generator [4]. (The free generator of T is the rooted tree with one +vertex.) +(5) Upper triangular matrices. This is an interesting example taken from [13], +where all the details can be found. Let 푘 be a commutative ring with identity in which +2 is invertible, and 푛 be a fixed positive integer. Let 푀 be the 푘-algebra of all 푛 × 푛 +matrices, and 푈 be the its subalgebra of upper triangular matrices. Let 휑: 푀 → 푈 +be the the 푘-linear mapping that associates with any matrix 퐴 = (푎푖 푗) ∈ 푀 the +matrix 퐵 = (푏푖 푗) ∈ 푈, where 푏푖 푗 = 푎푖 푗 if 푎푖 푗 is above the main diagonal, 푏푖 푗 = 0 if +푎푖 푗 is below the main diagonal, and 푏푖푖 = 푎푖푖/2 if 푎푖 푗 = 푎푖푖 is on the main diagonal. +Also, for every 퐴 ∈ 푀, let 퐴tr be the transpose of the matrix 퐴. Define an operation +· on 푈 setting, for every 푋,푌 ∈ 푈, 푋 · 푌 := 푋푌 + 휑(푋푌tr + 푌 푋tr). Then (푈, ·) is a +pre-Lie 푘-algebra. + +Pre-Lie algebras, their multiplicative lattice, and idempotent endomorphisms +7 +As we have defined in Section 1, a 푘-algebra homomorphism 휑: 푀 → 푀′ is +a 푘-module morphism such that 휑(푥푦) = 휑(푥)휑(푦) for every 푥, 푦 ∈ 푀. But we +also need another notion. We say that a 푘-module morphism 휑: 푀 → 푀′, where +푀, 푀′ are arbitrary (not-necessarily associative) 푘-algebras, is a pre-morphism if +휑(푥푦) − 휑(푥)휑(푦) = 휑(푦푥) − 휑(푦)휑(푥) for every 푥, 푦 ∈ 푀. +Lemma 1. A mapping 휑: 푀 → 푀′, where (푀, ·), (푀′, ·) are arbitrary 푘-algebras, +is a pre-morphism (푀, ·) → (푀′, ·) if and only if it is a 푘-algebra morphism +(푀, [−, −]) → (푀′, [−, −]). +Proof. If (푀, ·), (푀′, ·) are 푘-algebras and 휑: 푀 → 푀′ is a mapping, then +휑: (푀, ·) → (푀′, ·) +is a pre-morphism if and only if 휑(푎푏) − 휑(푎)휑(푏) = 휑(푏푎) − 휑(푏)휑(푎) for every +푎, 푏 ∈ 푀. This equality can be re-written as 휑(푎푏)−휑(푏푎) = 휑(푎)휑(푏)−휑(푏)휑(푎), +that is, 휑([푎, 푏]) = [휑(푎), 휑(푏)]. +From this lemma and the definition of pre-morphism, we immediately get that: +Lemma 2. (a) Every 푘-algebra morphism is a pre-morphism. +(b) The composite mapping of two pre-morphisms is a pre-morphism. +(c) The inverse mapping of a bijective pre-morphism is a pre-morphism. +In Section 1, we already considered, for any (not-necessarily associative) 푘- +algebra 푀, the mapping 휆: 푀 → End(푘푀), where 휆: 푥 ↦→ 휆푥, 휆푥 : 푀 → 푀, +and 휆푥(푎) = 푥푎. Also, we had already remarked that this mapping 휆 is a 푘-algebra +morphism if and only if 푀 is associative. The mapping 휆 is a pre-morphism if and +only if 푀 is a pre-Lie algebra. +There is a category of 푘-algebras with pre-morphisms, i.e., a category in which +objects are 푘-algebras and the Hom-set of all morphisms 푀 → 푀′ consists of all +pre-morphisms 푀 → 푀′. This category contains as a full subcategory the category +PreL푘,푝 of pre-Lie 푘-algebras (with pre-morphisms). The category PreL푘,푝 contains +as a subcategory the category PreL푘 of pre-Lie algebras with 푘-algebra morphisms, +hence a fortiori the category of associative algebras with their morphisms. +From lemma 1, we get +Theorem 3. Associating with any 푘-algebra (퐴, ·) its sub-adjacent anticommuta- +tive algebra (퐴, [−, −]) is a functor 푈 from the category of 푘-algebras with pre- +morphisms to the category of anticommutative 푘-algebras. +Notice that the functor 푈, viewed as a functor from the category PreL푘,푝 to the +category of Lie 푘-algebras, is fully faithful. Two pre-Lie algebras 퐴, 퐴′ are iso- +morphic in PreL푘,푝 if and only if their sub-adjacent Lie algebras are isomorphic Lie +algebras. Two pre-Lie algebras isomorphic in PreL푘,푝 are not necessarily isomorphic +as pre-Lie algebras. The simplest example is, over the field R of real numbers, the +example of the two R-algebras R × R and C. They are non-isomorphic associative + +8 +Michela Cerqua and Alberto Facchini +commutative 2-dimensional R-algebras, so that their sub-adjacent Lie algebras are +both the 2-dimensional abelian Lie R-algebra. Hence R × R and C are isomorphic +objects in PreLR,푝. All R-linear mappings R × R → C are pre-morphisms. +Remark 4. More generally, a 푘-algebra 퐴 is said to be Lie-admissible if, setting +[푥, 푦] = 푥푦−푦푥, one gets a Lie algebra (퐴, [−, −]). If the associator of a 푘-algebra 퐴 +is defined as (푥, 푦, 푧) = (푥푦)푧 −푥(푦푧) for all 푥, 푦, 푧 in 퐴, then being a pre-Lie algebra +is equivalent to (푥, 푦, 푧) = (푦, 푥, 푧) for all 푥, 푦, 푧 ∈ 퐴. Being a Lie-admissible algebra +is equivalent to +(푥, 푦, 푧) + (푦, 푧, 푥) + (푧, 푥, 푦) = (푦, 푥, 푧) + (푥, 푧, 푦) + (푧, 푦, 푥) +(2) +for every 푥, 푦, 푧 ∈ 퐴. Pre-Lie algebras are Lie-admissible algebras. By lemma 1, +the functor 푈 : (퐴, ·) ↦→ (퐴, [−, −]) is a fully faithful functor from the category of +Lie-admissible 푘-algebras with pre-morphisms to the category of Lie 푘-algebras. +Corresponding to the notion of pre-morphism, there is a notion of pre-derivation. +We say that a 푘-module endomorphism 훿: 푀 → 푀, where 푀 is an arbitrary (not- +necessarily associative) 푘-algebra, is a pre-derivation if +훿(푥푦) − 훿(푥)푦 − 푥훿(푦) = 훿(푦푥) − 훿(푦)푥 − 푦훿(푥) +for every 푥, 푦 ∈ 푀. +Lemma 5. Let 푘 be a commutativering with identity, (퐴, ·) a 푘-algebra, and [−, −] : +퐴 × 퐴 → 퐴 the operation on 퐴 defined by [푥, 푦] := 푥푦 − 푦푥 for every 푥, 푦 ∈ 퐴. Then +a 푘-module endomorphism 훿 of 퐴 is a pre-derivation of (퐴, ·) if and only if it is a +derivation of the 푘-algebra (퐴, [−, −]). +Proof. The 푘-module endomorphism 훿 of 퐴 is a pre-derivation of (퐴, ·) if and only +if 훿(푥푦) − 훿(푥)푦 − 푥훿(푦) = 훿(푦푥) − 훿(푦)푥 − 푦훿(푥), that is, 훿([푥, 푦]) = [훿(푥), 푦] + +[푥, 훿(푦)]. +Proposition 6. (a) Every derivation of a 푘-algebra is a pre-derivation. +(b) If 훿 and 훿′ are two pre-derivationsof a 푘-algebra 퐴, then [훿, 훿′] := 훿◦훿′−훿′◦훿 +is a pre-derivation. +Proof. (a) is trivial, and (b) follows from lemma 5. +Corollary 7. For any 푘-algebra 퐴, the set PreDer푘 (퐴) of all pre-derivations of 퐴 +is a Lie 푘-algebra with the operation [−, −] defined by [훿, 훿′] := 훿 ◦ 훿′ − 훿′ ◦ 훿 for +every 훿, 훿′ ∈ PreDer푘 (퐴). +Proof. The 푘-algebra (PreDer푘(퐴), [−, −]) is the Lie algebra of all derivations of +the 푘-algebra (퐴, [−, −]) (lemma 5). +Proposition 8. Let (퐴, ·) be any 푘-algebra. For every 푥 ∈ 퐴 define a 푘-module +morphism 푑푥 : 퐴 → 퐴 setting 푑푥(푦) := 푥푦 − 푦푥 for every 푦 ∈ 퐴. The following +conditions are equivalent: + +Pre-Lie algebras, their multiplicative lattice, and idempotent endomorphisms +9 +(a) 푑푥 is a pre-derivation for all 푥 ∈ 퐴, that is, the image 푑(퐴) of the mapping +푑 : 퐴 → End(푘 퐴) is contained in PreDer푘(퐴). +(b) The mapping 푑 is a pre-morphism of the 푘-algebra (퐴, ·) into the associative +푘-algebra (End(푘 퐴), ◦). +(c) The 푘-algebra (퐴, ·) is Lie-admissible. +Proof. (a) ⇔ (c) The mapping 푑푥 : (퐴, ·) → (퐴, ·) is a pre-derivation if and only +if the mapping 푑푥 : (퐴, [−, −]) → (퐴, [−, −]) is a derivation by lemma 5, i.e., if +and only if 푑푥([푦, 푧]) = [푑푥(푦), 푧] + [푦, 푑푥(푧)]. Since the mapping 푑푥 is defined by +푑푥(푦) = [푥, 푦], this is equivalent to [푥, [푦, 푧]] = [[푥, 푦], 푧] + [푦, [푥, 푧]], for every +푥, 푦, 푧 ∈ 퐴. This proves that 푑푥 is a pre-derivation for every 푥 ∈ 퐴 if and only if +(퐴, [−, −]) is a Lie algebra, that is, if and only if (퐴, ·) is Lie-admissible. +(b) ⇔ (c) The mapping 푑 is a pre-morphism if and only if 푑푥푦−푑푥◦푑푦 = 푑푦푥−푑푦◦ +푑푥 for every 푥, 푦 ∈ 퐴, that is, if and only if 푑푥푦(푧) −푑푥(푑푦(푧)) = 푑푦푥(푧) −푑푦(푑푥(푧)) +for every 푥, 푦, 푧 ∈ 퐴. This is equivalent to (푥푦)푧 − 푧(푥푦) − 푑푥(푦푧 − 푧푦) = (푦푥)푧 − +푧(푦푥) − 푑푦(푥푧 − 푧푥). An easy calculation shows that this is exactly Condition (2), +i.e., it is equivalent to the fact that 퐴 is Lie-admissible. +If 퐴 is a Lie-admissible 푘-algebra, the mapping 푑푥 is the inner pre-derivation of +퐴 induced by 푥. +3 Pre-Lie algebras are modules over the sub-adjacent Lie algebra +Now we want to give another presentation of pre-Lie algebras, helpful to understand +their structure. +Let 푘 be a commutative ring with identity. Given a pre-Lie 푘-algebra (퐴, ·), we +have already seen in the paragraph after Lemma 2 that the mapping 휆: (퐴, ·) → +End(푘 퐴) is a pre-morphism. Apply to it the functor 푈, getting a Lie 푘-algebra +morphism 퐿 := 푈(휆) : (퐴, [−, −]) → 픤픩(퐴) defined by 퐿 : 푎 ↦→ 휆푎 for every 푎 ∈ 퐴. +This mapping 퐿 is set-theoretically equal to the mapping 휆. In other words, 퐿 defines +a module structure on the 푘-module 푘 퐴, giving it the structure of a module over the +sub-adjacent Lie 푘-algebra (퐴, [−, −]). Moreover, [푥, 푦] = 퐿(푥)(푦) − 퐿(푦)(푥). +This construction can be inverted. Let (퐴, [−, −]) be a Lie 푘-algebra, and suppose +that its sub-adjacent 푘-module 푘 퐴 has a module structure over the Lie algebra +(퐴, [−, −]) via the Lie algebra morphism 퐿 : (퐴, [−, −]) → 픤픩(퐴) and that, for +every 푥, 푦 ∈ 퐴, the condition 퐿(푥)(푦) − 퐿(푦)(푥) = [푥, 푦] holds. Define a new +multiplication · on 퐴 setting 푥 · 푦 = 퐿(푥)(푦) for every 푥, 푦 ∈ 퐴. Then (퐴, ·) turns +out to be a pre-Lie 푘-algebra. These two constructions are one the inverse of the +other. More precisely, fix a Lie 푘-algebra 퐴. Then there is a category isomorphism +between the following two categories S퐴 and M퐴, where: +(1) S퐴 is the category whose objects are all pre-Lie 푘-algebras (퐴, ·) whose +sub-adjacent Lie algebra is the fixed Lie algebra (퐴, [−, −]). The morphisms are all +pre-Lie algebra homomorphisms between such pre-Lie algebras. + +10 +Michela Cerqua and Alberto Facchini +(2) M퐴 is the category whose objects are all pre-Lie 푘-algebra morphisms +퐿 : (퐴, [−, −]) → 픤픩(퐴) such that 퐿(푥)(푦) − 퐿(푦)(푥) = [푥, 푦] for every 푥, 푦 ∈ 퐴. +The morphisms 휑 : 퐿 → 퐿′ between two objects 퐿, 퐿′ of M퐴 are the 푘-module +morphisms 휑: 퐴 → 퐴 for which all diagrams +푀 +푀 +푀 +푀 +휑 +퐿(푎) +퐿′(휑(푎)) +휑 +commute, for every 푎 ∈ 퐴. See [2, Theorem 1.2.7]. +3.1 Modules over a pre-Lie 풌-algebra. +Modules cannot be defined over arbitrary non-associative algebras, but the definition +of pre-Lie algebra immediately suggests us how it is possible to define modules over +a pre-Lie algebra. +A module 푀 over a pre-Lie 푘-algebra 퐴 is any 푘-module 푀 with a 푘-bilinear +mapping ·: 퐴 × 푀 → 푀 such that +(푥 · 푦) · 푚 − 푥 · (푦 · 푚) = (푦 · 푥) · 푚 − 푦 · (푥 · 푚) +(3) +for every 푥, 푦 ∈ 퐴 and 푚 ∈ 푀. +Like in the case of associative algebras, it is possible to equivalently define a +module 푀 over a pre-Lie 푘-algebra (퐴, ·) as any 푘-module 푀 with a pre-morphism +휆: (퐴, ·) → (End(푘푀), ◦). +For instance, if 퐴 is any pre-Lie 푘-algebra and 퐼 is an ideal of 퐼, taking as 푘- +bilinear mapping ·: 퐴 × 퐼 → 퐼 the restriction of the multiplication on 퐴, one sees +immediately that 퐼 is a module over 퐴. +Theorem 9. The category of modules over a pre-Lie 푘-algebra (퐴, ·) and the cate- +gory of modules over its sub-adjacent Lie 푘-algebra (퐴, [−, −]) are isomorphic. +Proof. Modules over the pre-Lie algebra (퐴, ·) are pairs (푘푀, 휆) with 푘 푀 a 푘- +module and 휆: 퐴 → End(푘푀) a pre-morphism, and modules over the Lie algebra +(퐴, [−, −]) are pairs (푘 푀, 휆) with 푘푀 a 푘-module and 휆: (퐴, [−, −]) → 픤픩(푀) a +Lie 푘-algebra morphism. By Lemma 1, they are the same pairs. +Notice that we could have obtained the results in Section 3 in a different way: +every pre-Lie algebra is clearly a module over itself, hence, applying Theorem 9, to +every pre-Lie algebra (퐴, ·) there corresponds a module 퐴푘 over the sub-adjacent +Lie algebra (퐴, [−, −]), that is, a Lie algebra morphism 퐿 : (퐴, [−, −]) → 픤픩(퐴), +and [푥, 푦] = 퐿(푥)(푦) − 퐿(푦)(푥) for every 푥, 푦 ∈ 퐴. + +Pre-Lie algebras, their multiplicative lattice, and idempotent endomorphisms +11 +Also notice that the modules we have defined in this section over a pre-Lie algebra +are left modules. We don’t consider right modules because the definition of pre-Lie +algebra is not right/left symmetric, that is, the opposite of a pre-Lie algebra is not a +pre-Lie algebra. +4 Commutator of two ideals. (Huq=Smith) for pre-Lie algebras +The sum of two ideals of a pre-Lie 푘-algebra 퐴, i.e., their sum as 푘-submodules of +퐴, is an ideal of 퐴, and any intersection of a family of ideals of 퐴 is an ideal of 퐴. +It follows that the set I(퐴) of all ideals of a pre-Lie algebra 퐴 is a complete lattice +with respect to ⊆, and it is a sublattice of the lattice of all 푘-submodules of 퐴푘, hence +I(퐴) is a modular lattice. Moreover, the ideal of 퐴 generated by a subset 푋 of 퐴 is +the intersection of all the ideals of 퐴 that contain 푋. +We now need a notion of commutator of two ideals of a pre-Lie algebra. The +variety V of pre-Lie 푘-algebras is a Barr-exact category, is a variety of Ω-groups, +is protomodular and is semi-abelian [12, Example (2)]. More precisely, pre-Lie +algebras have an underlying group structure with respect to their addition, so that +they have the Mal’tsev term 푝(푥, 푦, 푧) = 푥 − 푦 + 푧. See [5, Proposition 5.3.1]. Notice +that 푝(푝(푥, 푦, 0), 푥, 푦)) = 0 for every 푥, 푦 ∈ 퐴, hence the variety V of pre-Lie +algebras is protomodular by [5, Proposition 3.1.8]. Moreover, 푝 has the property +that 푝(푝(푥, 푦, 푡), 푡, 푧) = 푝(푥, 푦, 푧) for all 푥, 푦, 푧, 푡 ∈ 퐴 (semi-associativity), so V is +semi-abelian by [5, Proposition 5.3.3]. +We want to show that the Huq and the Smith commutators of two ideals of a +pre-Lie 푘-algebra coincide. Recall that in the case of the semi-abelian variety V of +pre-Lie algebras, the Huq commutator of two ideals 퐼 and 퐽 of a pre-Lie algebra 퐴 is +the smallest ideal [퐼, 퐽]퐻 of 퐴 for which there is a well-defined canonical morphism +퐼 × 퐽 → 퐴/[퐼, 퐽]퐻 such that (푖, 0) ↦→ 푖 + [퐼, 퐽]퐻 and (0, 푗) ↦→ 푗 + [퐼, 퐽]퐻 for every +푖 ∈ 퐼 and 푗 ∈ 퐽. That is, [퐼, 퐽]퐻 is the smallest ideal of 퐴 for which the mapping +퐼 × 퐽 → 퐴/[퐼, 퐽]퐻, defined by (푖, 푗) ↦→ 푖 + 푗 + [퐼, 퐽]퐻 for every 푖 ∈ 퐼 and 푗 ∈ 퐽, is +a pre-Lie algebra morphism. +Proposition 10. The Huq commutator [퐼, 퐽]퐻 of two ideals 퐼 and 퐽 of a pre-Lie +algebra 퐴 is the ideal of 퐴 generated by the subset { 푖푗, 푗푖 | 푖 ∈ 퐼, 푗 ∈ 퐽 }. +Proof. The mapping ¯휎 : 퐼 × 퐽 → 퐴/[퐼, 퐽]퐻, defined by (푖, 푗) ↦→ 푖 + 푗 + [퐼, 퐽]퐻, +is a pre-Lie 푘-algebra morphism if and only if it respects multiplication, that is, if +and only if ¯휎((푖, 푗) · (푖′, 푗′)) ≡ ¯휎(푖, 푗) ¯휎(푖′, 푗′) for every (푖, 푗), (푖′, 푗′) ∈ 퐼 × 퐽, that +is, if and only if 푖푖′ + 푗 푗′ ≡ (푖 + 푗)(푖′ + 푗′) modulo [퐼, 퐽]퐻. Hence ¯휎 is a pre-Lie +algebra morphism if and only if 푖푗′ + 푗푖′ ≡ 0 modulo [퐼, 퐽]퐻, i.e., if and only if +푖푗′ + 푗푖′ ∈ [퐼, 퐽]퐻. The conclusion follows immediately. +The Smith commutator in the Mal’tsev variety V (see [11]) can be defined, for +a pre-Lie 푘-algebra 퐴 with Mal’tsev term 푝(푥, 푦, 푧) and two ideals 퐼, 퐽 of 퐴, as the +smallest ideal [퐼, 퐽]푆 of 퐴 for which the function + +12 +Michela Cerqua and Alberto Facchini +푝 : {(푥, 푦, 푧) | 푥 ≡ 푦 +(mod 퐼), 푦 ≡ 푧 +(mod 퐽)} → 퐴/[퐼, 퐽]푆, +defined by 푝(푥, 푦, 푧) = 푥 − 푦 + 푧 + [퐼, 퐽]푆 is a pre-Lie algebra morphism. +Theorem 11. The Smith commutator [퐼, 퐽]푆 of two ideals 퐼 and 퐽 of a pre-Lie +algebra 퐴 is the ideal of 퐴 generated by the subset { 푖푗, 푗푖 | 푖 ∈ 퐼, 푗 ∈ 퐽 }. Hence +Huq=Smith for pre-Lie algebras. +Proof. The mapping 푝 : { (푏 + 푖, 푏, 푏 + 푗) | 푏 ∈ 퐴, 푖 ∈ 퐼, 푗 ∈ 퐽 } → 퐴/[퐼, 퐽]푆 is a +pre-Lie algebra morphism if and only if for every 푏, 푏′ ∈ 퐴, 푖, 푖′ ∈ 퐼, 푗, 푗′ ∈ 퐽, one +has +푝((푏+푖, 푏, 푏+푗)(푏′+푖′, 푏′, 푏′+푗′)) ≡ 푝(푏+푖, 푏, 푏+푗)푝(푏′+푖′, 푏′, 푏′+푗′) +(mod [퐼, 퐽]푆), +that is, 푝((푏 +푖)(푏′ +푖′), 푏푏′, (푏 + 푗)(푏′ + 푗′)) ≡ (푏 +푖 + 푗)(푏′ +푖′ + 푗′) mod[퐼, 퐽]푆. +Equivalently,if and only if 0 ≡ 푖푗′+ 푗푖′ mod[퐼, 퐽]푆. Thereforethe Smith commutator +[퐼, 퐽]푆 of the two ideals 퐼 and 퐽 is the ideal of 퐴 generated by the subset { 푖푗, 푗푖 | +푖 ∈ 퐼, 푗 ∈ 퐽 }. In particolar, [퐼, 퐽]퐻 = [퐼, 퐽]푆. +From now on we will not distinguish between the Huq commutator [퐼, 퐽]퐻 and +the Smith commutator [퐼, 퐽]푆. We will simply call it the commutatorof the two ideals +퐼 and 퐽. Notice that the commutator is commutative, in the sense that [퐼, 퐽] = [퐽, 퐼]. +Let us briefly discuss the structure of this ideal [퐼, 퐽]. It is clear that if 푋 is any +subset of a pre-Lie 푘-algebra 퐴, the ideal ⟨푋⟩ of 퐴 generated by 푋, that is, the +intersection of all the ideals of 퐴 that contain 푋, can be also described as the union +⟨푋⟩ = � +푛≥0 푋푛 of the following ascending chain 푋0 ⊆ 푋1 ⊆ . . . of 푘-submodules of +퐴: 푋0 is the 푘-submodule of 퐴 generated by 푋; given 푋푛, set 푋푛+1 = 푋푛+퐴푋푛+푋푛퐴, +where 퐴푋푛 denotes the set of all finite sums of products 푎푥 with 푎 ∈ 퐴 and 푥 ∈ 푋푛, +and similarly for 푋푛퐴. In the case of the ideal [퐼, 퐽] this specializes as follows: +Proposition 12. Let 퐼 and 퐽 be ideals of a pre-Lie 푘-algebra 퐴. Then +[퐼, 퐽] = 퐼퐽 + +� +푛≥0 +푆푛, +where 푆푛 = ((. . . (((퐽퐼)퐴)퐴) . . . )퐴)퐴 and in 푆푛 there are 푛 factors equal to 퐴 on +the right of the factor J퐼. +Proof. Step 1: 퐴(퐼퐽) ⊆ 퐼퐽. +By Property (1), we have that 퐴(퐼퐽) ⊆ (퐴퐼)퐽 + (퐼퐴)퐽 + 퐼(퐴퐽) ⊆ 퐼퐽. +Step 2: 퐴(퐽퐼) ⊆ 퐽퐼. +From Step 1, by symmetry. +Step 3: 퐴푆푛 ⊆ 푆푛 + 푆푛+1 for every 푛 ≥ 0. +Induction on 푛. Step 2 gives the case 푛 = 0. Suppose that 퐴푆푛 ⊆ 푆푛 + 푆푛+1 +for some 푛 ≥ 0. Then 퐴푆푛+1 = 퐴(푆푛퐴) ⊆ (퐴푆푛)퐴 + (푆푛퐴)퐴 + 푆푛(퐴퐴) ⊆ (푆푛 + +푆푛+1)퐴 + 푆푛+2 + 푆푛+1 = 푆푛+1 + 푆푛+2. + +Pre-Lie algebras, their multiplicative lattice, and idempotent endomorphisms +13 +Step 4: 푆푛퐴 = 푆푛+1. +By definition. +Step 5: (퐼퐽)퐴 ⊆ 퐼퐽 + 푆0 + 푆1. +In fact, (퐼퐽)퐴 ⊆ 퐼(퐽퐴) + (퐽퐼)퐴 + 퐽(퐼퐴) ⊆ 퐼퐽 + 푆1 + 푆0. +Final Step. +Clearly, 퐼퐽 + � +푛≥0 푆푛 is a 푘-submodule of 퐴 that contains 퐼퐽 and 퐽퐼 and is +contained in the ideal generated by 퐼퐽 ∪퐽퐼. Hence it remains to show that it is closed +by left and right multiplication by elements of 퐴. This is proved in Steps 1, 3, 4 and +5. +Now that we have a good notion of commutator of two ideals 퐼 and 퐽 of a +pre-Lie 푘-algebra 퐴, we can introduce the multiplicative lattice of all ideals of +퐴: it is the complete modular lattice I(퐴) of all ideals of 퐴 endowed with the +commutator of ideals. Notice that, trivially, [퐼, 퐽] ⊆ 퐼 ∩ 퐽. As a consequence of +looking at pre-Lie algebras from the point of view of multiplicative lattices, we +immediately get the notions of prime ideal of a pre-Lie 푘-algebra 퐴, (Zariski) prime +spectrum of 퐴, semiprime ideal, abelian pre-Lie algebra, idempotent (=perfect) pre- +Lie algebra, derived series, solvable pre-Lie algebra, lower central series, nilpotent +pre-Lie algebra, 푚-system, 푛-system, hyperabelian pre-Lie algebra, metabelian pre- +Lie algebra, Jacobson radical, centralizer of an ideal, center of a pre-Lie 푘-algebra, +hypercenter. See the next Section 5 and [8, 9, 6, 7]. +Notice that the monotonicity condition holds for our commutator of ideals of a +pre-Lie algebra 퐴, in the sense that if 퐼 ≤ 퐼′ and 퐽 ≤ 퐽′ are ideals of 퐴, then +[퐼, 퐽] ≤ [퐼′, 퐽′]. +Also notice that the description of the commutator in Proposition 12 reduces, in +the case of 퐼 = 퐽 = 퐴, to the equality [퐴, 퐴] = 퐴2 = 퐴퐴. Here 퐴2 is the image of +the 푘-module morphism 휇: 퐴 ⊗푘 퐴 → 퐴 induced by the 푘-bilinear multiplication +of 퐴. +5 The commutator is not associative +In this section we will show that the commutator of ideals in a pre-Lie algebra 퐴 is +not associative in general, that is, if 퐼, 퐽, 퐾 are ideals of 퐴, it is not necessarily true +that [퐼, [퐽, 퐾]] = [[퐼, 퐽], 퐾]. In our example, the algebra 퐴 will be factor algebra +퐴 := T/푃, where T is the pre-Lie algebra of rooted trees of Example 4 in Section 2, +and 푃 is the ideal of T generated by all rooted trees with at least 5 vertices. Such 푃 +is the 푘-submodule of T generated by all rooted trees with at least 5 vertices. The +rooted trees with at most 4 vertices up to isomorphism are + +14 +Michela Cerqua and Alberto Facchini +푣 = +1 +, +푒 = +1 +2 +, +푎 = +1 +2 +3 +, +푏 = +1 +2 +3 +, +푐 = +1 +2 +3 +4 +, +푑 = +1 +2 +3 +4 +, +푓 = +1 +2 +3 +4 +, +푔 = +1 +2 +3 +4 +. +Hence our pre-Lie 푘-algebra 퐴 is eight dimensional, and we will denote by 푣, 푒, 푎, 푏, +푐, 푑, 푓 , 푔 the images in 퐴 of the corresponding rooted trees. That is, we will say that +{푣, 푒, 푎, 푏, 푐, 푑, 푓 , 푔} is a free set of generators for the free 푘-module 퐴. From the +multiplication in T defined in Example 4 of Section 2, we get that the multiplication +table in 퐴 is +푣 +푒 +푎 +푏 +푐 푑 푓 푔 +푣 푒 푎 + 푏 푐 + 2 푓 푓 + 푑 + 푔 0 0 0 0 +푒 푏 푓 + 푔 +0 +0 +0 0 0 0 +푎 푑 +0 +0 +0 +0 0 0 0 +푏 푔 +0 +0 +0 +0 0 0 0 +푐 0 +0 +0 +0 +0 0 0 0 +푑 0 +0 +0 +0 +0 0 0 0 +푓 0 +0 +0 +0 +0 0 0 0 +푔 0 +0 +0 +0 +0 0 0 0 + +Pre-Lie algebras, their multiplicative lattice, and idempotent endomorphisms +15 +From the multiplication table we see that 퐴2 = 퐴퐴 has {푒, 푏, 푑, 푔, 푎, 푓 , 푐} as a set +of generators, and is a seven dimensional free 푘-module. +Now [퐴2, 퐴2] = � +푛≥0(. . . ((퐴2 · 퐴2) · 퐴) · · · · · 퐴) · 퐴, where there are 푛 factors +equal to 퐴 on the right. But, always from the multiplication table, one sees that 퐴2·퐴2 +is generated by 푓 + 푔. Moreover ( 푓 + 푔)퐴 = 0 and 퐴( 푓 + 푔) = 0. Therefore [퐴2, 퐴2] +is one dimension as a free 푘-module, and its free set of generators is { 푓 + 푔}. +Similarly, [퐴2, 퐴] = 퐴 · 퐴2 + � +푛≥1(. . . ((퐴2 · 퐴) · 퐴) · · · · · 퐴) · 퐴, where there +are 푛 + 1 factors equal to 퐴 on the right. From the multiplication table, we see that +퐴 · 퐴2 is generated by 푎 + 푏, 푓 + 푔, 푐 + 2 푓 , 푓 + 푑 + 푔. Also, 퐴2 · 퐴 is generated by +{푏, 푑, 푔, 푓 + 푔}, (퐴2 · 퐴) · 퐴 is generated by 푔, and ((퐴2 · 퐴) · 퐴) · 퐴 = 0. Therefore +[퐴2, 퐴] is the 푘-module generated by 푏, 푑, 푔, 푓 , 푎, 푐 and is six dimensional. It follows +that [퐴2, 퐴]· 퐴 is generated by {푑, 푔}, 퐴·([퐴2, 퐴]) is generated by {푐+2 푓 , 푓 +푑+푔}, +and ([퐴2, 퐴] · 퐴) · 퐴 = 0. From these equalities we get that [[퐴2, 퐴], 퐴] is generated +by {푑, 푔, 푐 + 2 푓 , 푓 + 푑 + 푔}. Equivalently, [[퐴2, 퐴], 퐴] is generated by {푑, 푔, 푓 , 푐} +and is four dimensional. In particular [퐴2, 퐴2] ≠ [[퐴2, 퐴], 퐴]. +Let’s illustrate in detail some of the notions that immediately derive from the +commutative multiplication [−, −] (the commutator) in the multiplicative lattice +I(퐴). +First of all, a pre-Lie 푘-algebra 퐴 is abelian if the commutator of 퐴 and itself is +zero: [퐴, 퐴] = 0. This is equivalent to saying that 푖푗 = 0 for every 푖, 푗 ∈ 퐴. That +is, a pre-Lie algebra (퐴, ·) is abelian if and only if 푥 · 푦 = 0 for every 푥, 푦 ∈ 퐴. +(This is equivalent to requiring that the addition +: 퐴 × 퐴 → 퐴 is a pre-Lie algebra +morphism.) +By definition,an ideal 퐼 of a pre-Lie 푘-algebra 퐴 is prime if it is properly contained +in 퐴 and, for every ideal 퐽, 퐾 of 퐴, [퐽, 퐾] ⊆ 퐼 implies 퐽 ⊆ 퐼 or 퐾 ⊆ 퐼. An ideal 퐼 +of a pre-Lie 푘-algebra 퐴 is semiprime if, for every ideal 퐽 of 퐴, [퐽, 퐽] ⊆ 퐼 implies +that 퐽 ⊆ 퐼. An ideal of 퐴 is semiprime if and only if it is the intersection of a family +of prime ideals (if and only if it is the intersection of all the ideals of 퐴 that contain +it). An ideal 푃 of a pre-Lie 푘-algebra 퐴 is prime if and only if the lattice I(퐴/푃) is +uniform and 퐴/푃 has no non-zero abelian ideal. +Remark 13. Instead of the commutator [퐼, 퐽] of two ideals 퐼 and 퐽, we could have +taken two other “product of ideals” in a pre-Lie 푘-algebra: we could consider the +product 퐼퐽, i.e., the 푘-submodule of 퐴 generated by all products 푖푗, which is a +푘-submodule but not an ideal of 퐴 in general, or the ideal ⟨퐼퐽⟩ generated by the +submodule 퐼퐽. Notice that 퐼퐽 ⊆ ⟨퐼퐽⟩ ⊆ [퐼, 퐽] = ⟨퐼퐽⟩ + ⟨퐽퐼⟩, where the last +equality follows from Proposition 12. Correspondingly, we would have had three +different notions of “prime ideal”. In the next proposition (essentially contained in +[8, Example 3.7]) we prove that these three notions of “prime ideal” coincide: +Proposition 14. The following conditions are equivalent for an ideal 푃 of a pre-Lie +algebra 퐴: +(a) If 퐼, 퐽 are ideals of 퐴 and 퐼퐽 ⊆ 푃, then either 퐼 ⊆ 푃 or 퐽 ⊆ 푃. +(b) If 퐼, 퐽 are ideals of 퐴 and ⟨퐼퐽⟩ ⊆ 푃, then either 퐼 ⊆ 푃 or 퐽 ⊆ 푃. +(c) If 퐼, 퐽 are ideals of 퐴 and [퐼, 퐽] ⊆ 푃, then either 퐼 ⊆ 푃 or 퐽 ⊆ 푃. + +16 +Michela Cerqua and Alberto Facchini +Proof. The implications (a) ⇒ (b) ⇒ (c) follow immediately from the fact that +퐼퐽 ⊆ ⟨퐼퐽⟩ ⊆ [퐼, 퐽]. +(c) ⇒ (a). Let 푃 satisfy condition (c) and fix two ideals 퐼, 퐽 of 퐴 such that 퐼퐽 ⊆ 푃. +Since 푃 is an ideal, it follows that ⟨퐼퐽⟩ ⊆ 푃. Also, [⟨퐽퐼⟩, ⟨퐽퐼⟩] = ⟨⟨퐽퐼⟩⟨퐽퐼⟩⟩ ≤ +⟨퐼퐽⟩ ≤ 푃. From (c), we get that ⟨퐽퐼⟩ ≤ 푃, so that [퐼, 퐽] = ⟨퐼퐽⟩ + ⟨퐽퐼⟩ ≤ 푃. From +(c) again, we get that either 퐼 ⊆ 푃 or 퐽 ⊆ 푃. +Proposition 14 shows that if the pre-Lie algebra 퐴 is an associative algebra, then +this notion of prime ideal coincide with the notion of prime ideal in an associative +algebra. Proposition 12 shows that, for every pair (퐼, 퐽) of ideals of a pre-Lie algebra +퐴, one has [퐼, 퐽] = 퐼퐽 +⟨퐽퐼⟩ = 퐽퐼 +⟨퐼퐽⟩. Also, Step 5 in the proof of that Proposition +shows that one always has that 퐼퐽 + 퐽퐼 + (퐼퐽)퐴 = 퐼퐽 + 퐽퐼 + (퐽퐼)퐴. +A pre-Lie 푘-algebra 퐴 is idempotent (or perfect) if [퐴, 퐴] = 퐴, that is, if 퐴2 = 퐴 +(last paragraph of Section 4). +Given any pre-Lie algebra 퐴, let Spec(퐴) be the set of all its prime ideals. For +every 퐼 ∈ I(퐴), set 푉(퐼) = { 푃 ∈ Spec(퐴) | 푃 ⊇ 퐼 }. Then the family of all subsets +푉(퐼) of Spec(퐴), 퐼 ∈ I(퐴), is the family of all the closed sets for a topology on +Spec(퐴). With this topology, the topological space Spec(퐴) is the (Zariski) prime +spectrum of 퐴, and is a sober space [8]. It is not a spectral space in the sense of +Hochster in general. For instance, if 퐵 is a Boolean ring without identity, then 퐵 is a +pre-Lie algebra, but its prime spectrum is not compact. +If the pre-Lie algebra 퐴 is an associative algebra, then this notion of prime +spectrum coincide with the “standard notion” of prime spectrum of an associative +algebra 퐴, where the points of the spectrum are the prime ideals of 퐴 and the closed +sets are the subsets 푉(퐼) of the spectrum. To tell the truth, there is not a “standard +notion” of prime spectrum of an associative algebra that extends the classical notion +of prime spectrum for commutative associative algebras with identity. There are +several such notions as it is shown in [1] and [14]. For instance, the points of the +spectrum could be the completely prime ideals of 퐴, or the spectrum of 퐴 could be +defined to be the Zariski spectrum of the commutative ring 퐴/[퐴, 퐴], where [퐴, 퐴] +now denotes the ideal of 퐴 generated by all elements 푎푏 − 푏푎. +A pre-Lie 푘-algebra 퐴 is hyperabelian if it has no prime ideal. For instance, +abelian pre-Lie algebras are hyperabelian. +Let 퐴 be a pre-Lie 푘-algebra. The lower central series (or descending central +series) of 퐴 is the descending series +퐴 = 퐴1 ≥ 퐴2 ≥ 퐴3 ≥ . . . , +where 퐴푛+1 := [퐴푛, 퐴] for every 푛 ≥ 1. If 퐴푛 = 0 for some 푛 ≥ 1, then 퐴 is +nilpotent. (Notice that it is not necessary to distinguish between left nilpotency and +rightnilpotency,becausethecommutatoriscommutative,thatis,[퐴푛, 퐴] = [퐴, 퐴푛].) +The derived series of 퐴 [8, Definition 6.1] is the descending series +퐴 := 퐴(0) ≥ 퐴(1) ≥ 퐴(2) ≥ . . . , + +Pre-Lie algebras, their multiplicative lattice, and idempotent endomorphisms +17 +where 퐴(푛+1) := [퐴(푛), 퐴(푛)] for every 푛 ≥ 0. The pre-Lie algebra 퐴 is solvable if +퐴(푛) = 0 for some integer 푛 ≥ 0. It is metabelian if 퐴(2) = 0. +In a multiplicative lattice an element is semisimple if it is the join of a set of +minimal idempotent elements. (An element 푚 of a lattice 퐿 is minimal if, for every +푥 ∈ 퐿, 푥 ≤ 푚 implies 푥 = 푚 or 푥 = 0, that is, if it is minimal in the partially ordered +set 퐿 \ {0}. An element 푒 of a multiplicative lattice 퐿 is idempotent if 푒 · 푒 = 푒). +“Minimal idempotent element” of 퐿 means minimal element of 퐿 \ {0} that is also +an idempotent element. Notice that for a minimal element 푥 ∈ 퐿 either 푥 · 푥 = 푥 or +푥 · 푥 = 0, i.e., minimal elements are either idempotent or abelian. +The Jacobson radical of 퐿 is the meet of the set of all maximal elements 푎 of +퐿 \ {1} with 1 · 1 ̸≤ 푎. The radical is the join of the set of all solvable elements of 퐿. +6 Idempotent endomorphisms, semidirect products of pre-Lie +algebras, and actions +Let 푒 be an idempotent endomorphism of a pre-Lie 푘-algebra 퐴. Then 퐴 = ker(푒) ⊕ +푒(퐴) (direct sum as 푘-modules), where the kernel ker(푒) of 푒 is an ideal of 퐴 and its +image 푒(퐴) is a pre-Lie sub-푘-algebra of 퐴. If there is a direct-sum decomposition +퐴 = 퐼 ⊕ 퐵 as 푘-module of a pre-Lie 푘-algebra 퐴, where 퐼 is an ideal of 퐴 and 퐵 is a +pre-Lie sub-푘-algebra of 퐴, we will say that 퐴is the semidirect product of 퐼 and 퐵. We +are interested in semidirect products because, for any algebraic structure, idempotent +endomorphisms are in one-to-one correspondence with semidirect products and are +related to the notion of action of the structure on another structure, and bimodules. +The proof of the following proposition is elementary. +Proposition 15. Let 퐴 be a pre-Lie 푘-algebra, 퐼 an ideal of 퐴 and 퐵 a pre-Lie +sub-푘-algebra of 퐴. The following conditions are equivalent: +(1) 퐴 = 퐼 ⊕ 퐵 as a 푘-module. +(2) For every 푎 ∈ 퐴, there are a unique 푖 ∈ 퐼 and a unique 푏 ∈ 퐵 such that +푎 = 푖 + 푏. +(3) There exists a pre-Lie 푘-algebra morphism 퐴 → 퐵 whose restriction to 퐵 is +the identity and whose kernel is 퐼. +(4) There is an idempotent pre-Lie 푘-algebra endomorphism of 퐴 whose image +is 퐵 and whose kernel is 퐼. +It is now clear that there is a one-to-one correspondence between the set of all +idempotent endomorphisms of a pre-Lie 푘-algebra 퐴 and the set of all pairs (퐼, 퐵), +where 퐼 is an ideal of 퐴, 퐵 is a pre-Lie sub-푘-algebra of 퐴, and 퐴 is the direct sum +of 퐼 and 퐵 as a 푘-module. +Let us first consider inner semidirect product. Suppose that (퐴, ·) is a pre-Lie +푘-algebra that is a semidirect product of its ideal 퐼 and its pre-Lie sub-푘-algebra 퐵. +Then there is a pre-morphism 휆: (퐵, ·) → (End(퐼푘), ◦), given by multiplying on the + +18 +Michela Cerqua and Alberto Facchini +left by elements of 퐵 (this follows from the fact that every ideal is a module, as we +have already remarked in Section 3.1). Also, there is a 푘-module morphism 휌 : 퐵 → +End(퐼푘), given by multiplying on the right by elements of 퐵, that is, 휌 : 푏 ↦→ 휌푏, +where 휌푏(푖) = 푖 · 푏 for every 푖 ∈ 퐼. Moreover, Identity (1), applied to elements 푥, 푧 +in 퐵 and 푦 ∈ 퐼, can be re-written as 휌푎(휆푏(푖)) − 휆푏(휌푎(푖)) = (휌푎 ◦ 휌푏 − 휌푏·푎)(푖) +for every 푎, 푏 ∈ 퐵 and 푖 ∈ 퐼. Identity (1), applied to elements 푥 in 퐵 and 푦, 푧 ∈ 퐼, +can be re-written as 휆푎(푖) · 푗 − 휆푎(푖 · 푗) = 휌푎(푖) · 푗 − 푖 · 휆푎( 푗) for every 푎 ∈ 퐵 and +푖, 푗 ∈ 퐼. Finally, the same identity (1), applied to elements 푧 in 퐵 and 푥, 푦 ∈ 퐼, can +be re-written as 휌푎(푥 · 푦) − 푥 · 휌푎(푦) = 휌푎(푦 · 푥) − 푦 · 휌푎(푥) for every 푎 ∈ 퐵 and +푖, 푗 ∈ 퐼. +Conversely, for outer semidirect product: +Theorem 16. Let 퐼 and 퐵 be pre-Lie 푘-algebras and (휆, 휌) a pair of 푘-linear map- +pings 퐵 → End(퐼푘) such that: +(a) 휆: (퐵, ·) → (End(퐼푘), ◦) is a pre-morphism. +(b) 휌푎 ◦ 휆푏 − 휆푏 ◦ 휌푎 = 휌푎 ◦ 휌푏 − 휌푏·푎 for every 푎, 푏 ∈ 퐵. +(c) 휆푎(푖) · 푗 − 휆푎(푖 · 푗) = 휌푎(푖) · 푗 − 푖 · 휆푎( 푗) for every 푎 ∈ 퐵 and 푖, 푗 ∈ 퐼. +(d) 휌푎(푖 · 푗) − 푖 · 휌푎( 푗) = 휌푎( 푗 · 푖) − 푗 · 휌푎(푖) for every 푎 ∈ 퐵 and 푖, 푗 ∈ 퐼. +On the 푘-module direct sum 퐼 ⊕ 퐵 define a multiplication ∗ setting +(푖, 푏) ∗ ( 푗, 푐) = (푖 · 푗 + 휆푏( 푗) + 휌푐(푖), 푏 · 푐) +for every (푖, 푏), ( 푗, 푐) ∈ 퐼 ⊕ 퐵. Then (퐼 ⊕ 퐵, ∗) is a pre-Lie 푘-algebra. +Proof. For every 푎, 푏, 푐 ∈ 퐵 and 푥, 푦, 푧 ∈ 퐼 we have that +((푥, 푎) ∗ (푦, 푏)) ∗ (푧, 푐) = (푥 · 푦 + 휆푎(푦) + 휌푏(푥), 푎 · 푏) ∗ (푧, 푐) = += ((푥 · 푦) · 푧 + 휆푎(푦) · 푧 + 휌푏(푥) · 푧 + 휆푎·푏(푧)+ ++휌푐(푥 · 푦 + 휆푎(푦) + 휌푏(푥)), (푎 · 푏) · 푐) +(4) +and +(푥, 푎) ∗ ((푦, 푏) ∗ (푧, 푐)) = (푥, 푎) ∗ (푦 · 푧 + 휆푏(푧) + 휌푐(푦), 푏 · 푐) = += (푥 · (푦 · 푧) + 푥 · 휆푏(푧) + 푥 · 휌푐(푦)+ ++휆푎(푦 · 푧 + 휆푏(푧) + 휌푐(푦)) + 휌푏·푐(푥), 푎 · (푏 · 푐)). +(5) +The difference of (4) and (5) is +((푥 · 푦) · 푧 − 푥 · (푦 · 푧) + 휆푎(푦) · 푧 − 휆푎(푦 · 푧)+ ++휌푏(푥) · 푧 − 푥 · 휆푏(푧) + 휆푎·푏(푧) − (휆푎 ◦ 휆푏)(푧)+ ++휌푐(푥 · 푦) − 푥 · 휌푐(푦) + 휌푐(휆푎(푦)) − 휆푎(휌푐(푦)) + 휌푐(휌푏(푥)) − 휌푏·푐(푥)), +(푎 · 푏) · 푐 − 푎 · (푏 · 푐)). +Similarly, +((푦, 푏) ∗ (푥, 푎)) ∗ (푧, 푐) − (푦, 푏) ∗ ((푥, 푎) ∗ (푧, 푐)) = += ((푦 · 푥) · 푧 − 푦 · (푥 · 푧) + 휆푏(푥) · 푧 − 휆푏(푥 · 푧) + 휌푎(푦) · 푧 − 푦 · 휆푎(푧)+ ++휆푏·푎(푧) − (휆푏 ◦ 휆푎)(푧) + 휌푐(푦 · 푥) − 푦 · 휌푐(푥) + 휌푐(휆푏(푥)) − 휆푏(휌푐(푥))+ ++휌푐(휌푎(푦)) − 휌푎·푐(푦)), (푏 · 푎) · 푐 − 푏 · (푎 · 푐)). + +Pre-Lie algebras, their multiplicative lattice, and idempotent endomorphisms +19 +Hence, for the proof, it suffices to show that +휆푎(푦) · 푧 − 휆푎(푦 · 푧) + 휌푏(푥) · 푧 − 푥 · 휆푏(푧) + 휆푎·푏(푧) − (휆푎 ◦ 휆푏)(푧)+ ++휌푐(푥 · 푦) − 푥 · 휌푐(푦) + 휌푐(휆푎(푦)) − 휆푎(휌푐(푦)) + 휌푐(휌푏(푥)) − 휌푏·푐(푥)) = += 휆푏(푥) · 푧 − 휆푏(푥 · 푧) + 휌푎(푦) · 푧 − 푦 · 휆푎(푧)+ ++휆푏·푎(푧) − (휆푏 ◦ 휆푎)(푧)+ ++휌푐(푦 · 푥) − 푦 · 휌푐(푥) + 휌푐(휆푏(푥)) − 휆푏(휌푐(푥))+ ++휌푐(휌푎(푦)) − 휌푎·푐(푦)). +(6) +Now +휆푎(푦) · 푧 − 휆푎(푦 · 푧) = 휌푎(푦) · 푧 − 푦 · 휆푎(푧) +by hypotheses (c); +휌푏(푥) · 푧 − 푥 · 휆푏(푧) = 휆푏(푥) · 푧 − 휆푏(푥 · 푧) +by hypotheses (c); +휆푎·푏(푧) − (휆푎 ◦ 휆푏)(푧) = 휆푏·푎(푧) − (휆푏 ◦ 휆푎)(푧) by hypotheses (a); +휌푐(푥 · 푦) − 푥 · 휌푐(푦) = 휌푐(푦 · 푥) − 푦 · 휌푐(푥) +by hypotheses (d); +휌푐(휆푎(푦)) − 휆푎(휌푐(푦)) = 휌푐(휌푎(푦)) − 휌푎·푐(푦)) by hypotheses (b); +휌푐(휌푏(푥)) − 휌푏·푐(푥)) = 휌푐(휆푏(푥)) − 휆푏(휌푐(푥)) by hypotheses (b). +Summing up these equalities one gets Equality (6). +Hence the theorem characterises the four properties that an action (휆, 휌), that +is, a pair of 푘-linear mappings 퐵 → End(퐼푘), must have in order to construct the +semidirect product of a pre-Lie 푘-algebra 퐵 acting on a pre-Lie 푘-algebra 퐼. +6.1 Bimodules over a pre-Lie algebra +The most important case of semidirect product is probably when the pre-Lie algebra +퐼 is abelian, i.e., the case where the action, that is, the pair (휆, 휌) of 푘-linear mappings +퐵 → End(퐼푘), is an action of the pre-Lie 푘-algebra 퐵 on a 푘-module 푀. In other +words, when 퐼 is a 퐵-bimodule. Let us be more precise, giving the precise definition +of what a bimodule over a pre-Lie algebra must be: +Definition 17. Let 퐴 be a pre-Lie 푘-algebra. A bimodule over 퐴 is a 푘-module 푀푘 +with a pair (휆, 휌) of 푘-linear mappings 퐴 → End(푀푘) such that: +(a) 휆: (퐴, ·) → (End(푀푘), ◦) is a pre-morphism (that is, 푀 is a module over 퐴). +(b) 휌푎 ◦ 휆푏 − 휆푏 ◦ 휌푎 = 휌푎 ◦ 휌푏 − 휌푏·푎 for every 푎, 푏 ∈ 퐵. +Notice that Conditions (c) and (d) of Theorem 16 are always trivially satisfied +because in this case the 푘-module 푀 is viewed as an abelian pre-Lie algebra, that is, +with null multiplication. This definition already appears, for instance, in [13]. Notice +the nice interpretation of condition (b) given in that paper: In condition (b) the left +hand side 휌푎 ◦ 휆푏 − 휆푏 ◦ 휌푎 describes how far the action is from associativity (for +bimodules over an associative algebra, it is always required to be zero); the right hand +side 휌푎◦휌푏−휌푏·푎 describes how far 휌 is from being a 푘-algebra antihomomorphism. + +20 +Michela Cerqua and Alberto Facchini +6.2 Adjoining the identity to a pre-Lie algebra +The class of pre-Lie algebras contains the class of associative algebras. For asso- +ciative algebras, it is very natural to consider associative algebras with an identity, +and when there is not an identity, to adjoin one. This construction is often called +the “Dorroh extension”. Let’s show that this is possible for pre-Lie algebras as well. +We will see in fact that a more appropriate name for our class of algebras, instead +of “pre-Lie algebras”, would have been “pre-associative algebras”. Adjoining an +identity to a pre-Lie 푘-algebra 퐴 is exactly our semidirect product of the pre-Lie +푘-algebra 푘 acting on the pre-Lie 푘-algebra 퐴. Let’s be more precise. +An identity in a pre-Lie 푘-algebra 퐴 is an element, which we will denote by 1퐴, +such that 푎 · 1퐴 = 1퐴 · 푎 = 푎 for every 푎 ∈ 퐴. If 퐴 has an identity, we will say that 퐴 +is unital. An element 푒 of 퐴 is idempotent if 푒2 := 푒 · 푒 = 푒. The zero of 퐴 is always +an idempotent element of 퐴, and the identity, when it exists, is also an idempotent +element of 퐴. +Let 퐴 be any fixed pre-Lie 푘-algebra. Then the associative commutative ring 푘 is +a pre-Lie 푘-algebra, and there is a one-to-one correspondence between the set of all +the pre-Lie 푘-algebra morphisms 푘 → 퐴 and the set of all idempotent elements of +퐴. For any idempotent element 푒 of 퐴 the corresponding morphism 휑푒 : 푘 → 퐴 is +defined by 휑푒(휆) = 휆푒 for every 휆 ∈ 푘. Conversely, for any morphism 휑: 푘 → 퐴 +the corresponding idempotent element of 퐴 is 휑(1). +For any fixed pre-Lie 푘-algebra 퐴 it is possible to construct the semidirect product +of 푘 acting on 퐴 via the pair (휆, 휌) of 푘-module morphisms 푘 → End(퐴푘) for which +휆훼 = 휌훼 is multiplication by 훼 for all 훼 ∈ 푘. Then the four conditions (a), (b), (c), +(d) of Theorem 16 are all automatically satisfied, and the corresponding semidirect +product is the 푘-module direct sum 퐴 ⊕ 푘 with the multiplication defined by +(푥, 훼)(푦, 훽) = (푥 · 푦 + 훽푥 + 훼푦, 훼훽) +for every (푥, 훼), (푦, 훽) ∈ 퐴 ⊕ 푘. Hence 퐴 ⊕ 푘 becomes a pre-Lie 푘-algebra with +identity (0, 1). The Lie algebra sub-adjacent this pre-Lie algebra 퐴 ⊕ 푘 is the direct +sum of the Lie algebra (퐴, [−, −]) and the abelian Lie algebra 푘. We will denote this +semidirect product by 퐴#푘. +Now let PreL푘,1 be the category of all unital pre-Lie 푘-algebras. Its objects are the +pre-Lie 푘-algebras 퐴 with an identity. Its morphisms 푓 : 퐴 → 퐵 are the 푘-algebra +morphisms 푓 such that 푓 (1퐴) = 1퐵. There is also a further category involved.It is the +category PreL푘,1,푎 of all unital pre-Lie 푘-algebras with an augmentation. Its objects +are all the pairs (퐴, 휀퐴), where 퐴 is a unital pre-Lie 푘-algebra and 휀퐴: 퐴 → 푘 is a +morphism in PreL푘,1 that is a left inverse for 휑1퐴: +푘 +휑1퐴 � 퐴 +휀퐴 �푘. +The morphisms 푓 : (퐴, 휀퐴) → (퐵, 휀퐵) are the morphisms 푓 : 퐴 → 퐵 in PreL푘,1 +such that 휀퐵 푓 = 휀퐴. For instance, the 푘-algebra 퐴#푘 is clearly a unital 푘-algebra with + +Pre-Lie algebras, their multiplicative lattice, and idempotent endomorphisms +21 +augmentation: the augmentation is the canonical projection 휋2 : 퐴#푘 = 퐴 ⊕ 푘 → 푘 +onto the second summand. +It is easy to see that: +Theorem 18. There is a category equivalence 퐹: PreL푘 → PreL푘,1,푎 that associates +with any object 퐴 of PreL푘 the 푘-algebra with augmentation 퐹(퐴) := (퐴#푘, 휋2). +The quasi-inverse of 퐹 is the functor PreL푘,1,푎 → PreL푘, that associates with +each unital pre-Lie 푘-algebra with augmentation (퐴, 휀퐴) the kernel ker(휀퐴) of the +augmentation. +References +1. M. Ben-Zvi, A. Ma and M. Reyes, A Kochen-Specker theorem for integer matrices and +noncommutative spectrum functors, J. Algebra 491 (2017), 28–313. +2. M. Cerqua, “Pre-Lie algebras”, Master Thesis in Math., University of Padua, 2022. +3. Chengming Bai, An Introduction to Pre-Lie Algebras, In “Algebra and Applications 1, Non- +associative algebras and categories”, A. Makhlouf (Ed.), ISTE Ltd, 2020, pp. 245–273. +4. F. Chapoton and N. Livernet, Pre-Lie algebras and the rooted trees operad, International +Mathematics Research Notices 2001 (8) (2001), 395–408. +5. F. Borceux and D. Bourn, “Mal’cev, protomodular, homological and semi-abelian categories”, +Mathematics and Its Applications 566, Kluwer, 2004. +6. A. Facchini, Algebraic structures from the point of view of complete multiplicative lattices, +available at: http://arxiv.org/abs/2201.03295 , accepted for publication in “Rings, Quadratic +Forms, and their Applications in Coding Theory”, Contemporary Math., 2022. +7. A. Facchini and C. A. Finocchiaro, Multiplicative lattices: maximal implies prime, and related +questions, submitted for publication (2022). +8. A. Facchini, C. A. Finocchiaro and G. Janelidze, Abstractly constructed prime spectra, Algebra +Universalis 83 (2022), no. 1, Paper No. 8, 38 pp. +9. A. Facchini, F. de Giovanni and M. Trombetti, Spectra of groups, published online on 5 June +2022 in Algebr. Represent. Theory (2022). +10. A. Facchini and L. Heidari Zadeh, Algebras with a bilinear form, and Idempotent endomor- +phisms, submitted for publication, 2022, available at http://arxiv.org/abs/2210.08230 +11. G. Janelidze, and G. M. Kelly, Central extensions in Mal’tsev varieties, Theory Appl. Categ. +7 (2000), 219–226. +12. G. Janelidze, L. Márki and W. Tholen, Semi-abelian categories, J. Pure Appl. Algebra 168 +(2002), no. 2–3, 367–386. +13. A. Nijenhuis, On a classof common propertiesof some different types of algebras, Nieuw Arch. +Wisk. (3) 17 (1969), 17–46. French translation in Enseign. Math. (2) 14 (1968), 225–277. +14. M. Reyes, Obstructing extensions of the functor Spec to noncommutative rings, Israel +J. Math. 192 (2012), 667–698. +15. E. B. Vinberg, The theory of convex homogeneous cones, Trudy Mosk. Mat. Obshch. 12 +(1963), 303–358. English transl.: Trans. Mosc. Math. Soc. 12 (1963), 340–403. + diff --git a/7NE0T4oBgHgl3EQfwAHH/content/tmp_files/load_file.txt b/7NE0T4oBgHgl3EQfwAHH/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..2bda9ec7ef7ae260c56af17fae68abfc4324fe62 --- /dev/null +++ b/7NE0T4oBgHgl3EQfwAHH/content/tmp_files/load_file.txt @@ -0,0 +1,661 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf,len=660 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content='02627v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content='RA] 6 Jan 2023 Pre-Lie algebras, their multiplicative lattice, and idempotent endomorphisms Michela Cerqua and Alberto Facchini Abstract We introduce the notions of pre-morphism and pre-derivation for arbitrary non-associative algebras over a commutative ring 푘 with identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' These notions are applied to the study of pre-Lie 푘-algebras and, more generally, Lie-admissible 푘-algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Associating with any algebra (퐴, ·) its sub-adjacent anticommutative algebra (퐴, [−, −]) is a functor from the category of 푘-algebras with pre-morphisms to the category of anticommutative 푘-algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' We describe the commutator of two ideals of a pre-Lie algebra, showing that the condition (Huq=Smith) holds for pre- Lie algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' This allows to make use of all the notions concerning multiplicative lattices in the study of the multiplicative lattice of ideals of a pre-Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' We study idempotent endomorphisms of a pre-Lie algebra 퐿, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=', semidirect-product decompositions of 퐿 and bimodules over 퐿.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Introduction The aim of this paper is to present pre-Lie algebras from the point of view of their multiplicative lattice of ideals, and to study their idempotent endomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Pre- Lie algebras were first introduced and studied in [15] by Vinberg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' He applied them to the study of convex homogenous cones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' He called “left-symmetric algebras” the algebras we call pre-Lie algebras in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' We present a notion of pre-morphism and pre-derivation for arbitrary non- associative algebras over a commutative ring 푘 with identity, and apply it to the study of pre-Lie 푘-algebras and, more generally, Lie-admissible 푘-algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Asso- ciating with any pre-Lie algebra (퐴, ·) its sub-adjacent Lie algebra (퐴, [−, −]) is a functor from the category PreL푘,푝 of pre-Lie 푘-algebras with pre-morphisms to the category of Lie 푘-algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' We introduce the notion of module 푀 over a pre-Lie Michela Cerqua e-mail: michela.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content='cerqua@studenti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content='unipd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content='it · Alberto Facchini Dipartimento di Matematica "Tullio Levi Civita", Università di Padova, 35121 Padova, Italy e-mail: facchini@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content='unipd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content='it 1 2 Michela Cerqua and Alberto Facchini algebra 퐿 and, like in the case of associative algebras, it is possible to do it in two equivalent ways, via a suitable scalar multiplication 퐿 × 푀 → 푀 or as a 푘-module 푀 with a pre-morphism 휆: (퐿, ·) → (End(푘푀), ◦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' The category of modules over a pre-Lie 푘-algebra (퐿, ·) is isomorphic to the category of modules over its sub- adjacent Lie 푘-algebra (퐿, [−, −]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' We then consider the commutator of two ideals in a pre-Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' In particular we show that the condition (Huq=Smith) holds for pre-Lie algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' With the notion of commutator at our disposal, the lattice of ideals of a pre-Lie algebra becomes a multiplicative lattice [6, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' As a consequence we immediately get the notions of abelian pre-Lie algebra, prime ideal, prime spec- trum of a pre-Lie algebra, solvable and nilpotent pre-Lie algebras, metabelian and hyperabelian pre-Lie algebras, centralizer, and center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' We then consider idempotent endomorphisms of a pre-Lie algebra, because they immediately show what semi-direct products of pre-Lie algebras are, what the action of a pre-Lie algebra on another pre-Lie algebra is, and lead us to the notion of bimodule over a pre-Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' We study the “Dorroh extensions” of pre-Lie algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Like in the associative case, we get a category equivalence between the category PreL푘 and the category of pre-Lie algebras with identity and with an augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' 1 Preliminary notions on non-associative 풌-algebras Let 푘 be a commutative ring with identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' In this article, a 푘-algebra is a 푘-module 푘푀 with a further 푘-bilinear operation 푀 × 푀 → 푀, (푥, 푦) ↦→ 푥푦 (equivalently, a 푘-module morphism 푀 ⊗푘 푀 → 푀).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' A subalgebra (an ideal, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=') of 푀 is a 푘-submodule 푁 of 푀 such that 푥푦 ∈ 푁 for every 푥, 푦 ∈ 푁 (푥푛 ∈ 푁 and 푛푥 ∈ 푁 for every 푥 ∈ 푀 and 푛 ∈ 푁, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=') As usual, if 푁 is an ideal of 푀, the quotient 푘-module 푀/푁 inherits a 푘-algebra structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' There is a one-to-one correspondence between the set of all ideals 푁 of 푀 and the set of all congruences on 푀, that is, all equivalence relations ∼ on 푀 for which 푥 ∼ 푦 and 푧 ∼ 푤 imply 푥 + 푧 ∼ 푦 + 푤, 휆푥 ∼ 휆푦 and 푥푧 ∼ 푦푤 for every 푥, 푦, 푧, 푤 ∈ 푀 and every 휆 ∈ 푘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' The opposite 푀op of an algebra 푀 is defined taking as multiplication in 푀op the mapping (푥, 푦) ↦→ 푦푥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' If 푀 and 푀′ are two 푘-algebras, a 푘-linear mapping 휑: 푀 → 푀′ is a 푘-algebra homomorphism if 휑(푥푦) = 휑(푥)휑(푦) for every 푥, 푦 ∈ 푀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Clearly, 푘-algebras form a variety in the sense of Universal Algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Moreover, it is a variety of Ω-groups, that is, a variety which is pointed (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=', it has exactly one constant) and has amongst its operations and identities those of the variety of groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' It follows that 푘-algebras form a semiabelian category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Other examples of Ω-groups are abelian groups, non-unital rings, commutative algebras, modules and Lie algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' If 푀 is any 푘-algebra, its endomorphisms form a monoid, that is, a semigroup with a two-sided identity, with respect to composition of mappings ◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' A derivation of a 푘-algebra 푀 is any 푘-linear mapping 퐷 : 푀 → 푀 such that 퐷(푥푦) = (퐷(푥))푦+ 푥(퐷(푦)) for every 푥, 푦 ∈ 푀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' For any 푘-algebra 푀, we can construct the 푘-algebra of derivations Der푘(푀) of the 푘-algebra 푀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Its elements are all derivations of 푀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Pre-Lie algebras, their multiplicative lattice, and idempotent endomorphisms 3 If 푀 is any 푘-algebra and 퐷, 퐷′ are two derivations of 푀, then the composite mapping 퐷퐷′ is not a derivation of 푀 in general, but 퐷퐷′ − 퐷′퐷 is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Thus, for any 푘-algebra 푀, we can define the Lie 푘-algebra Der푘 (푀) as the subset of End(푘푀) consisting of all derivations of 푀 with multiplication [퐷, 퐷′] := 퐷퐷′ − 퐷′퐷 for every 퐷, 퐷′ ∈ Der푘(푀).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' It is known that there is not a general notion of representation (or module)over our (non-associative) 푘-algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' There is a notion of bimodule over a non-associative ring due to Eillenberg, and this notion works well for Lie algebras, but is not convenient in the study of Jordan algebras and alternative algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' The situation, as far as modules are concerned, is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content='1 Modules over an associative 풌-algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Given any 푘-algebra 푀, we can consider, for every element 푥 ∈ 푀, the map- ping 휆푥 : 푀 → 푀, defined by 휆푥(푎) = 푥푎 for every 푎 ∈ 푀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' The mapping 휆: 푀 → End(푘푀) is defined by 휆: 푥 ↦→ 휆푥 for every 푥 ∈ 푀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' This 휆 is a 푘- algebra morphism if and only if 푀 is associative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Thus, for any associative 푘-algebra 푀, it is natural to define a left 푀-module as any 푘-module 푘 퐴 with a 푘-algebra homomorphism 휆: 푀 → End(푘 퐴).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Similarly, we can define right 푀-modules as 푘-modules 푘 퐴 with a 푘-algebra antihomomorphism 휌 : 푀 → End(푘 퐴).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Here by 푘- algebra antihomomorphism 휓 : 푀 → 푀′ between two 푘-algebras 푀, 푀′ we mean any 푘-linear mapping 휓 such that 휓(푥푦) = 휓(푦)휓(푥) for every 푥, 푦 ∈ 푀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Clearly, a mapping 푀 → 푀′ is a 푘-algebra antihomomorphism if and only if it is a 푘-algebra homomorphism 푀op → 푀′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' It follows that right 푀-modules coincide with left 푀op-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' More precisely, when we say that right 푀-modules coincide with left 푀op-modules, we mean that there is a canonical category isomorphism between the category of all right 푀-modules and the category of all left 푀op-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Simi- larly, left 푀-modules coincide with right 푀op-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Also, if 푀 is commutative, then left 푀-modules and right 푀-modules coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Finally, left modules 퐴 over an associative 푘-algebra 푀 can be equivalently defined using, instead of the 푘- algebra homomorphism 휆: 푀 → End(푘 퐴), a 푘-bilinear mapping 휇: 푀 × 퐴 → 퐴, 휇: (푚, 푎) ↦→ 푚푎, such that (푚푚′)푎 = 푚(푚′푎) for every 푚, 푚′ ∈ 푀 and 푎 ∈ 퐴.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content='2 Modules over a Lie 풌-algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' For any 푘-module 퐴 we will denote by 픤픩(퐴) the Lie 푘-algebra End(푘 퐴) of all 푘-endomorphisms of 퐴 with the operation [−, −] defined by [ 푓 , 푔] = 푓 푔 − 푔 푓 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' For any Lie 푘-algebra 푀 and any element 푥 ∈ 푀, the mapping 휆푥 is an element of the Lie 푘-algebra Der푘 (푀), usually called the adjoint of 푥, or the inner derivation defined by 푥, and usually denoted by ad푀 푥 instead of 휆푥, and the mapping ad: 푀 → 4 Michela Cerqua and Alberto Facchini Der푘(푀) ⊆ 픤픩(푀), defined by ad: 푥 ↦→ ad푀 푥 for every 푥 ∈ 푀, is a Lie 푘-algebra homomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Left modules over a Lie 푘-algebra 푀 are defined as 푘-modules 퐴 with a Lie 푘-algebra homomorphism휆: 푀 → 픤픩(퐴).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Similarly, it is possible to define right 푀- modules as 푘-modules 퐴 with a 푘-algebra antihomomorphism 휌 : 푀 → 픤픩(퐴).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' But any Lie 푘-algebra 푀 is isomorphic to its opposite algebra 푀op via the isomorphism 푀 → 푀op, 푥 ↦→ −푥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' It follows that the category of right 푀-modules is canonically isomorphic to the category of left 푀-modules for any Lie 푘-algebra 푀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Therefore it is useless to introduce both right and left modules, it is sufficient to introduce left 푀-modules and call them simply “푀-modules”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' 2 Pre-Lie 풌-algebras A pre-Lie 푘-algebra is a 푘-algebra (푀, ·) satisfying the identity (푥 · 푦) · 푧 − 푥 · (푦 · 푧) = (푦 · 푥) · 푧 − 푦 · (푥 · 푧) (1) for every 푥, 푦, 푧 ∈ 푀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' For any 푘-algebra (푀, ·), defining the commutator [푥, 푦] = 푥 · 푦 − 푦 · 푥 for every 푥, 푦 ∈ 푀, the algebra (푀, [−, −]) is anticommutative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' If (푀, ·) is a pre-Lie algebra, one gets that (푀, [−, −]) is a Lie algebra, called the Lie algebra sub-adjacent to the pre-Lie algebra (푀, ·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Pre-Lie algebras are also called Vinberg algebras or left-symmetric algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' This last name refers to the fact that in (1) one exchanges the first two variables on the left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' A right-symmetric algebra is an algebra in which, for every 푥, 푦, 푧 ∈ 푀, (푥 · 푦) · 푧 − 푥 · (푦 · 푧) = (푥 · 푧) · 푦 − 푥 · (푧 · 푦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' It is easily seen that the category of left-symmetricalgebras and the category of right-symmetricalgebras are isomorphic (the categorical isomorphism is given by 푀 ↦→ 푀op).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Examples 1 (1) Every associative algebra is clearly a pre-Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' (2) Derivations on 푘[푥1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' , 푥푛]푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Let 푘 be a commutative ring with identity, 푛 ≥ 1 be an integer, and 푘[푥1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' , 푥푛] be the ring of polynomials in the 푛 indeter- minates 푥1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' , 푥푛 with coefficients in 푘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Let 퐴 be the free 푘[푥1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' , 푥푛]-module 푘[푥1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' , 푥푛]푛 with free set {푒1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' , 푒푛} of generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' As a 푘-module, 퐴 is the free 푘-module with free set of generators the set { 푥푖1 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' 푥푖푛 푛 푒 푗 | 푖1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' , 푖푛 ≥ 0, 푗 = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' , 푛}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Consider the usual derivations of the ring 푘[푥1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' , 푥푛]: 휕 휕푥 푗 (푥푖1 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' 푥푖푛 푛 ) = � 푥푖1 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' 푖 푗푥푖푗−1 푗 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' 푥푖푛 푛 for 푖 푗 > 0, 0 for 푖 푗 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Defineamultiplication on 퐴 setting,forevery 푢 = (푢1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' , 푢푛), 푣 = (푣1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' , 푣푛) ∈ 퐴, Pre-Lie algebras, their multiplicative lattice, and idempotent endomorphisms 5 푣 · 푢 = ( 푛 � 푗=1 푣 푗 휕푢1 휕푥 푗 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' , 푛 � 푗=1 푣 푗 휕푢푛 휕푥 푗 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' It is then possible to see that 퐴 is a pre-Lie 푘-algebra [2, Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' (3) An example of rank 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Let 푘 be any commutative ring with identity and 퐿 � 푘 ⊕ 푘 a free 푘-module of rank 2 with free set {푒1, 푒2} of generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Define a multiplication on 퐿 setting 푒1푒1 = 2푒1, 푒1푒2 = 푒2, 푒2푒1 = 0, 푒2푒2 = 푒1, and extending by 푘-bilinearity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Then 퐿 is a pre-Lie 푘-algebra [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' (4) Rooted trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Recall that a tree is an undirected graph in which any two vertices are connected by exactly one path, or equivalently a connected acyclic undirected graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' A rooted tree of degree 푛 is a pair (푇, 푟), where 푇 is a tree with 푛 vertices, and its root 푟 is a vertex of 푇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' In the following we will label the vertices of 푇 with the numbers 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' , 푛, and the root 푟 with 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Let 푘 be a commutative ring with identity and T푛 be the free 푘-module with free set of generators the set of all isomorphism classes of rooted trees of degree 푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Set T := � 푛≥1 T푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Define a multiplication on T setting, for every pair 푇1,푇2 of rooted trees, 푇1 · 푇2 = � 푣 ∈푉 (푇2) 푇1 ◦푣 푇2, where 푉(푇2) is the set of vertices of 푇2, and 푇1 ◦푣 푇2 is the rooted tree obtained by adding to the disjoint union of 푇1 and 푇2 a further new edge joining the root vertex of 푇1 with the vertex 푣 of 푇2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' The root of 푇1 ◦푣 푇2 is defined to be the same as the root of 푇2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' To get a multiplication on T, extend this multiplication by 푘-bilinearity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Let us give an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Suppose 푇1 = 1 2 3 and 푇2 = 1 2 Then 6 Michela Cerqua and Alberto Facchini 푇1 ◦1 푇2 = 1 2 3 4 5 and 푇1 ◦2 푇2 = 1 2 3 4 5 , where we have relabelled the vertices of푇1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' (If푇1 has 푛 vertices and푇2 has 푚 vertices, it is convenient to relabel in 푇1 ◦푣 푇2 the vertices 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' , 푛 of 푇1 with the numbers 푚 + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' , 푚 + 푛, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=') Therefore 푇1 · 푇2 = 1 2 3 4 5 + 1 2 3 4 5 In this way, one gets a pre-Lie 푘-algebra T [4, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' It is a graded 푘-algebra because T푛 · T푚 ⊆ T푛+푚 for every 푛 and 푚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' It can be proved that this is the free pre-Lie 푘-algebra on one generator [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' (The free generator of T is the rooted tree with one vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=') (5) Upper triangular matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' This is an interesting example taken from [13], where all the details can be found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Let 푘 be a commutative ring with identity in which 2 is invertible, and 푛 be a fixed positive integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Let 푀 be the 푘-algebra of all 푛 × 푛 matrices, and 푈 be the its subalgebra of upper triangular matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Let 휑: 푀 → 푈 be the the 푘-linear mapping that associates with any matrix 퐴 = (푎푖 푗) ∈ 푀 the matrix 퐵 = (푏푖 푗) ∈ 푈, where 푏푖 푗 = 푎푖 푗 if 푎푖 푗 is above the main diagonal, 푏푖 푗 = 0 if 푎푖 푗 is below the main diagonal, and 푏푖푖 = 푎푖푖/2 if 푎푖 푗 = 푎푖푖 is on the main diagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Also, for every 퐴 ∈ 푀, let 퐴tr be the transpose of the matrix 퐴.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Define an operation on 푈 setting, for every 푋,푌 ∈ 푈, 푋 · 푌 := 푋푌 + 휑(푋푌tr + 푌 푋tr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Then (푈, ·) is a pre-Lie 푘-algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Pre-Lie algebras, their multiplicative lattice, and idempotent endomorphisms 7 As we have defined in Section 1, a 푘-algebra homomorphism 휑: 푀 → 푀′ is a 푘-module morphism such that 휑(푥푦) = 휑(푥)휑(푦) for every 푥, 푦 ∈ 푀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' But we also need another notion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' We say that a 푘-module morphism 휑: 푀 → 푀′, where 푀, 푀′ are arbitrary (not-necessarily associative) 푘-algebras, is a pre-morphism if 휑(푥푦) − 휑(푥)휑(푦) = 휑(푦푥) − 휑(푦)휑(푥) for every 푥, 푦 ∈ 푀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' A mapping 휑: 푀 → 푀′, where (푀, ·), (푀′, ·) are arbitrary 푘-algebras, is a pre-morphism (푀, ·) → (푀′, ·) if and only if it is a 푘-algebra morphism (푀, [−, −]) → (푀′, [−, −]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' If (푀, ·), (푀′, ·) are 푘-algebras and 휑: 푀 → 푀′ is a mapping, then 휑: (푀, ·) → (푀′, ·) is a pre-morphism if and only if 휑(푎푏) − 휑(푎)휑(푏) = 휑(푏푎) − 휑(푏)휑(푎) for every 푎, 푏 ∈ 푀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' This equality can be re-written as 휑(푎푏)−휑(푏푎) = 휑(푎)휑(푏)−휑(푏)휑(푎), that is, 휑([푎, 푏]) = [휑(푎), 휑(푏)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' From this lemma and the definition of pre-morphism, we immediately get that: Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' (a) Every 푘-algebra morphism is a pre-morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' (b) The composite mapping of two pre-morphisms is a pre-morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' (c) The inverse mapping of a bijective pre-morphism is a pre-morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' In Section 1, we already considered, for any (not-necessarily associative) 푘- algebra 푀, the mapping 휆: 푀 → End(푘푀), where 휆: 푥 ↦→ 휆푥, 휆푥 : 푀 → 푀, and 휆푥(푎) = 푥푎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Also, we had already remarked that this mapping 휆 is a 푘-algebra morphism if and only if 푀 is associative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' The mapping 휆 is a pre-morphism if and only if 푀 is a pre-Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' There is a category of 푘-algebras with pre-morphisms, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=', a category in which objects are 푘-algebras and the Hom-set of all morphisms 푀 → 푀′ consists of all pre-morphisms 푀 → 푀′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' This category contains as a full subcategory the category PreL푘,푝 of pre-Lie 푘-algebras (with pre-morphisms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' The category PreL푘,푝 contains as a subcategory the category PreL푘 of pre-Lie algebras with 푘-algebra morphisms, hence a fortiori the category of associative algebras with their morphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' From lemma 1, we get Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Associating with any 푘-algebra (퐴, ·) its sub-adjacent anticommuta- tive algebra (퐴, [−, −]) is a functor 푈 from the category of 푘-algebras with pre- morphisms to the category of anticommutative 푘-algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Notice that the functor 푈, viewed as a functor from the category PreL푘,푝 to the category of Lie 푘-algebras, is fully faithful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Two pre-Lie algebras 퐴, 퐴′ are iso- morphic in PreL푘,푝 if and only if their sub-adjacent Lie algebras are isomorphic Lie algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Two pre-Lie algebras isomorphic in PreL푘,푝 are not necessarily isomorphic as pre-Lie algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' The simplest example is, over the field R of real numbers, the example of the two R-algebras R × R and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' They are non-isomorphic associative 8 Michela Cerqua and Alberto Facchini commutative 2-dimensional R-algebras, so that their sub-adjacent Lie algebras are both the 2-dimensional abelian Lie R-algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Hence R × R and C are isomorphic objects in PreLR,푝.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' All R-linear mappings R × R → C are pre-morphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' More generally, a 푘-algebra 퐴 is said to be Lie-admissible if, setting [푥, 푦] = 푥푦−푦푥, one gets a Lie algebra (퐴, [−, −]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' If the associator of a 푘-algebra 퐴 is defined as (푥, 푦, 푧) = (푥푦)푧 −푥(푦푧) for all 푥, 푦, 푧 in 퐴, then being a pre-Lie algebra is equivalent to (푥, 푦, 푧) = (푦, 푥, 푧) for all 푥, 푦, 푧 ∈ 퐴.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Being a Lie-admissible algebra is equivalent to (푥, 푦, 푧) + (푦, 푧, 푥) + (푧, 푥, 푦) = (푦, 푥, 푧) + (푥, 푧, 푦) + (푧, 푦, 푥) (2) for every 푥, 푦, 푧 ∈ 퐴.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Pre-Lie algebras are Lie-admissible algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' By lemma 1, the functor 푈 : (퐴, ·) ↦→ (퐴, [−, −]) is a fully faithful functor from the category of Lie-admissible 푘-algebras with pre-morphisms to the category of Lie 푘-algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Corresponding to the notion of pre-morphism, there is a notion of pre-derivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' We say that a 푘-module endomorphism 훿: 푀 → 푀, where 푀 is an arbitrary (not- necessarily associative) 푘-algebra, is a pre-derivation if 훿(푥푦) − 훿(푥)푦 − 푥훿(푦) = 훿(푦푥) − 훿(푦)푥 − 푦훿(푥) for every 푥, 푦 ∈ 푀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Let 푘 be a commutativering with identity, (퐴, ·) a 푘-algebra, and [−, −] : 퐴 × 퐴 → 퐴 the operation on 퐴 defined by [푥, 푦] := 푥푦 − 푦푥 for every 푥, 푦 ∈ 퐴.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Then a 푘-module endomorphism 훿 of 퐴 is a pre-derivation of (퐴, ·) if and only if it is a derivation of the 푘-algebra (퐴, [−, −]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' The 푘-module endomorphism 훿 of 퐴 is a pre-derivation of (퐴, ·) if and only if 훿(푥푦) − 훿(푥)푦 − 푥훿(푦) = 훿(푦푥) − 훿(푦)푥 − 푦훿(푥), that is, 훿([푥, 푦]) = [훿(푥), 푦] + [푥, 훿(푦)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' (a) Every derivation of a 푘-algebra is a pre-derivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' (b) If 훿 and 훿′ are two pre-derivationsof a 푘-algebra 퐴, then [훿, 훿′] := 훿◦훿′−훿′◦훿 is a pre-derivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' (a) is trivial, and (b) follows from lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' For any 푘-algebra 퐴, the set PreDer푘 (퐴) of all pre-derivations of 퐴 is a Lie 푘-algebra with the operation [−, −] defined by [훿, 훿′] := 훿 ◦ 훿′ − 훿′ ◦ 훿 for every 훿, 훿′ ∈ PreDer푘 (퐴).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' The 푘-algebra (PreDer푘(퐴), [−, −]) is the Lie algebra of all derivations of the 푘-algebra (퐴, [−, −]) (lemma 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Let (퐴, ·) be any 푘-algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' For every 푥 ∈ 퐴 define a 푘-module morphism 푑푥 : 퐴 → 퐴 setting 푑푥(푦) := 푥푦 − 푦푥 for every 푦 ∈ 퐴.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' The following conditions are equivalent: Pre-Lie algebras, their multiplicative lattice, and idempotent endomorphisms 9 (a) 푑푥 is a pre-derivation for all 푥 ∈ 퐴, that is, the image 푑(퐴) of the mapping 푑 : 퐴 → End(푘 퐴) is contained in PreDer푘(퐴).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' (b) The mapping 푑 is a pre-morphism of the 푘-algebra (퐴, ·) into the associative 푘-algebra (End(푘 퐴), ◦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' (c) The 푘-algebra (퐴, ·) is Lie-admissible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' (a) ⇔ (c) The mapping 푑푥 : (퐴, ·) → (퐴, ·) is a pre-derivation if and only if the mapping 푑푥 : (퐴, [−, −]) → (퐴, [−, −]) is a derivation by lemma 5, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=', if and only if 푑푥([푦, 푧]) = [푑푥(푦), 푧] + [푦, 푑푥(푧)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Since the mapping 푑푥 is defined by 푑푥(푦) = [푥, 푦], this is equivalent to [푥, [푦, 푧]] = [[푥, 푦], 푧] + [푦, [푥, 푧]], for every 푥, 푦, 푧 ∈ 퐴.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' This proves that 푑푥 is a pre-derivation for every 푥 ∈ 퐴 if and only if (퐴, [−, −]) is a Lie algebra, that is, if and only if (퐴, ·) is Lie-admissible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' (b) ⇔ (c) The mapping 푑 is a pre-morphism if and only if 푑푥푦−푑푥◦푑푦 = 푑푦푥−푑푦◦ 푑푥 for every 푥, 푦 ∈ 퐴, that is, if and only if 푑푥푦(푧) −푑푥(푑푦(푧)) = 푑푦푥(푧) −푑푦(푑푥(푧)) for every 푥, 푦, 푧 ∈ 퐴.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' This is equivalent to (푥푦)푧 − 푧(푥푦) − 푑푥(푦푧 − 푧푦) = (푦푥)푧 − 푧(푦푥) − 푑푦(푥푧 − 푧푥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' An easy calculation shows that this is exactly Condition (2), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=', it is equivalent to the fact that 퐴 is Lie-admissible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' If 퐴 is a Lie-admissible 푘-algebra, the mapping 푑푥 is the inner pre-derivation of 퐴 induced by 푥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' 3 Pre-Lie algebras are modules over the sub-adjacent Lie algebra Now we want to give another presentation of pre-Lie algebras, helpful to understand their structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Let 푘 be a commutative ring with identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Given a pre-Lie 푘-algebra (퐴, ·), we have already seen in the paragraph after Lemma 2 that the mapping 휆: (퐴, ·) → End(푘 퐴) is a pre-morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Apply to it the functor 푈, getting a Lie 푘-algebra morphism 퐿 := 푈(휆) : (퐴, [−, −]) → 픤픩(퐴) defined by 퐿 : 푎 ↦→ 휆푎 for every 푎 ∈ 퐴.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' This mapping 퐿 is set-theoretically equal to the mapping 휆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' In other words, 퐿 defines a module structure on the 푘-module 푘 퐴, giving it the structure of a module over the sub-adjacent Lie 푘-algebra (퐴, [−, −]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Moreover, [푥, 푦] = 퐿(푥)(푦) − 퐿(푦)(푥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' This construction can be inverted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Let (퐴, [−, −]) be a Lie 푘-algebra, and suppose that its sub-adjacent 푘-module 푘 퐴 has a module structure over the Lie algebra (퐴, [−, −]) via the Lie algebra morphism 퐿 : (퐴, [−, −]) → 픤픩(퐴) and that, for every 푥, 푦 ∈ 퐴, the condition 퐿(푥)(푦) − 퐿(푦)(푥) = [푥, 푦] holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Define a new multiplication · on 퐴 setting 푥 · 푦 = 퐿(푥)(푦) for every 푥, 푦 ∈ 퐴.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Then (퐴, ·) turns out to be a pre-Lie 푘-algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' These two constructions are one the inverse of the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' More precisely, fix a Lie 푘-algebra 퐴.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Then there is a category isomorphism between the following two categories S퐴 and M퐴, where: (1) S퐴 is the category whose objects are all pre-Lie 푘-algebras (퐴, ·) whose sub-adjacent Lie algebra is the fixed Lie algebra (퐴, [−, −]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' The morphisms are all pre-Lie algebra homomorphisms between such pre-Lie algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' 10 Michela Cerqua and Alberto Facchini (2) M퐴 is the category whose objects are all pre-Lie 푘-algebra morphisms 퐿 : (퐴, [−, −]) → 픤픩(퐴) such that 퐿(푥)(푦) − 퐿(푦)(푥) = [푥, 푦] for every 푥, 푦 ∈ 퐴.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' The morphisms 휑 : 퐿 → 퐿′ between two objects 퐿, 퐿′ of M퐴 are the 푘-module morphisms 휑: 퐴 → 퐴 for which all diagrams 푀 푀 푀 푀 휑 퐿(푎) 퐿′(휑(푎)) 휑 commute, for every 푎 ∈ 퐴.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' See [2, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content='7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content='1 Modules over a pre-Lie 풌-algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Modules cannot be defined over arbitrary non-associative algebras, but the definition of pre-Lie algebra immediately suggests us how it is possible to define modules over a pre-Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' A module 푀 over a pre-Lie 푘-algebra 퐴 is any 푘-module 푀 with a 푘-bilinear mapping ·: 퐴 × 푀 → 푀 such that (푥 · 푦) · 푚 − 푥 · (푦 · 푚) = (푦 · 푥) · 푚 − 푦 · (푥 · 푚) (3) for every 푥, 푦 ∈ 퐴 and 푚 ∈ 푀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Like in the case of associative algebras, it is possible to equivalently define a module 푀 over a pre-Lie 푘-algebra (퐴, ·) as any 푘-module 푀 with a pre-morphism 휆: (퐴, ·) → (End(푘푀), ◦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' For instance, if 퐴 is any pre-Lie 푘-algebra and 퐼 is an ideal of 퐼, taking as 푘- bilinear mapping ·: 퐴 × 퐼 → 퐼 the restriction of the multiplication on 퐴, one sees immediately that 퐼 is a module over 퐴.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' The category of modules over a pre-Lie 푘-algebra (퐴, ·) and the cate- gory of modules over its sub-adjacent Lie 푘-algebra (퐴, [−, −]) are isomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Modules over the pre-Lie algebra (퐴, ·) are pairs (푘푀, 휆) with 푘 푀 a 푘- module and 휆: 퐴 → End(푘푀) a pre-morphism, and modules over the Lie algebra (퐴, [−, −]) are pairs (푘 푀, 휆) with 푘푀 a 푘-module and 휆: (퐴, [−, −]) → 픤픩(푀) a Lie 푘-algebra morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' By Lemma 1, they are the same pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Notice that we could have obtained the results in Section 3 in a different way: every pre-Lie algebra is clearly a module over itself, hence, applying Theorem 9, to every pre-Lie algebra (퐴, ·) there corresponds a module 퐴푘 over the sub-adjacent Lie algebra (퐴, [−, −]), that is, a Lie algebra morphism 퐿 : (퐴, [−, −]) → 픤픩(퐴), and [푥, 푦] = 퐿(푥)(푦) − 퐿(푦)(푥) for every 푥, 푦 ∈ 퐴.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Pre-Lie algebras, their multiplicative lattice, and idempotent endomorphisms 11 Also notice that the modules we have defined in this section over a pre-Lie algebra are left modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' We don’t consider right modules because the definition of pre-Lie algebra is not right/left symmetric, that is, the opposite of a pre-Lie algebra is not a pre-Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' 4 Commutator of two ideals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' (Huq=Smith) for pre-Lie algebras The sum of two ideals of a pre-Lie 푘-algebra 퐴, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=', their sum as 푘-submodules of 퐴, is an ideal of 퐴, and any intersection of a family of ideals of 퐴 is an ideal of 퐴.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' It follows that the set I(퐴) of all ideals of a pre-Lie algebra 퐴 is a complete lattice with respect to ⊆, and it is a sublattice of the lattice of all 푘-submodules of 퐴푘, hence I(퐴) is a modular lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Moreover, the ideal of 퐴 generated by a subset 푋 of 퐴 is the intersection of all the ideals of 퐴 that contain 푋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' We now need a notion of commutator of two ideals of a pre-Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' The variety V of pre-Lie 푘-algebras is a Barr-exact category, is a variety of Ω-groups, is protomodular and is semi-abelian [12, Example (2)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' More precisely, pre-Lie algebras have an underlying group structure with respect to their addition, so that they have the Mal’tsev term 푝(푥, 푦, 푧) = 푥 − 푦 + 푧.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' See [5, Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Notice that 푝(푝(푥, 푦, 0), 푥, 푦)) = 0 for every 푥, 푦 ∈ 퐴, hence the variety V of pre-Lie algebras is protomodular by [5, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content='8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Moreover, 푝 has the property that 푝(푝(푥, 푦, 푡), 푡, 푧) = 푝(푥, 푦, 푧) for all 푥, 푦, 푧, 푡 ∈ 퐴 (semi-associativity), so V is semi-abelian by [5, Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' We want to show that the Huq and the Smith commutators of two ideals of a pre-Lie 푘-algebra coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Recall that in the case of the semi-abelian variety V of pre-Lie algebras, the Huq commutator of two ideals 퐼 and 퐽 of a pre-Lie algebra 퐴 is the smallest ideal [퐼, 퐽]퐻 of 퐴 for which there is a well-defined canonical morphism 퐼 × 퐽 → 퐴/[퐼, 퐽]퐻 such that (푖, 0) ↦→ 푖 + [퐼, 퐽]퐻 and (0, 푗) ↦→ 푗 + [퐼, 퐽]퐻 for every 푖 ∈ 퐼 and 푗 ∈ 퐽.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' That is, [퐼, 퐽]퐻 is the smallest ideal of 퐴 for which the mapping 퐼 × 퐽 → 퐴/[퐼, 퐽]퐻, defined by (푖, 푗) ↦→ 푖 + 푗 + [퐼, 퐽]퐻 for every 푖 ∈ 퐼 and 푗 ∈ 퐽, is a pre-Lie algebra morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' The Huq commutator [퐼, 퐽]퐻 of two ideals 퐼 and 퐽 of a pre-Lie algebra 퐴 is the ideal of 퐴 generated by the subset { 푖푗, 푗푖 | 푖 ∈ 퐼, 푗 ∈ 퐽 }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' The mapping ¯휎 : 퐼 × 퐽 → 퐴/[퐼, 퐽]퐻, defined by (푖, 푗) ↦→ 푖 + 푗 + [퐼, 퐽]퐻, is a pre-Lie 푘-algebra morphism if and only if it respects multiplication, that is, if and only if ¯휎((푖, 푗) · (푖′, 푗′)) ≡ ¯휎(푖, 푗) ¯휎(푖′, 푗′) for every (푖, 푗), (푖′, 푗′) ∈ 퐼 × 퐽, that is, if and only if 푖푖′ + 푗 푗′ ≡ (푖 + 푗)(푖′ + 푗′) modulo [퐼, 퐽]퐻.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Hence ¯휎 is a pre-Lie algebra morphism if and only if 푖푗′ + 푗푖′ ≡ 0 modulo [퐼, 퐽]퐻, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=', if and only if 푖푗′ + 푗푖′ ∈ [퐼, 퐽]퐻.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' The conclusion follows immediately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' The Smith commutator in the Mal’tsev variety V (see [11]) can be defined, for a pre-Lie 푘-algebra 퐴 with Mal’tsev term 푝(푥, 푦, 푧) and two ideals 퐼, 퐽 of 퐴, as the smallest ideal [퐼, 퐽]푆 of 퐴 for which the function 12 Michela Cerqua and Alberto Facchini 푝 : {(푥, 푦, 푧) | 푥 ≡ 푦 (mod 퐼), 푦 ≡ 푧 (mod 퐽)} → 퐴/[퐼, 퐽]푆, defined by 푝(푥, 푦, 푧) = 푥 − 푦 + 푧 + [퐼, 퐽]푆 is a pre-Lie algebra morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Theorem 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' The Smith commutator [퐼, 퐽]푆 of two ideals 퐼 and 퐽 of a pre-Lie algebra 퐴 is the ideal of 퐴 generated by the subset { 푖푗, 푗푖 | 푖 ∈ 퐼, 푗 ∈ 퐽 }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Hence Huq=Smith for pre-Lie algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' The mapping 푝 : { (푏 + 푖, 푏, 푏 + 푗) | 푏 ∈ 퐴, 푖 ∈ 퐼, 푗 ∈ 퐽 } → 퐴/[퐼, 퐽]푆 is a pre-Lie algebra morphism if and only if for every 푏, 푏′ ∈ 퐴, 푖, 푖′ ∈ 퐼, 푗, 푗′ ∈ 퐽, one has 푝((푏+푖, 푏, 푏+푗)(푏′+푖′, 푏′, 푏′+푗′)) ≡ 푝(푏+푖, 푏, 푏+푗)푝(푏′+푖′, 푏′, 푏′+푗′) (mod [퐼, 퐽]푆), that is, 푝((푏 +푖)(푏′ +푖′), 푏푏′, (푏 + 푗)(푏′ + 푗′)) ≡ (푏 +푖 + 푗)(푏′ +푖′ + 푗′) mod[퐼, 퐽]푆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Equivalently,if and only if 0 ≡ 푖푗′+ 푗푖′ mod[퐼, 퐽]푆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Thereforethe Smith commutator [퐼, 퐽]푆 of the two ideals 퐼 and 퐽 is the ideal of 퐴 generated by the subset { 푖푗, 푗푖 | 푖 ∈ 퐼, 푗 ∈ 퐽 }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' In particolar, [퐼, 퐽]퐻 = [퐼, 퐽]푆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' From now on we will not distinguish between the Huq commutator [퐼, 퐽]퐻 and the Smith commutator [퐼, 퐽]푆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' We will simply call it the commutatorof the two ideals 퐼 and 퐽.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Notice that the commutator is commutative, in the sense that [퐼, 퐽] = [퐽, 퐼].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Let us briefly discuss the structure of this ideal [퐼, 퐽].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' It is clear that if 푋 is any subset of a pre-Lie 푘-algebra 퐴, the ideal ⟨푋⟩ of 퐴 generated by 푋, that is, the intersection of all the ideals of 퐴 that contain 푋, can be also described as the union ⟨푋⟩ = � 푛≥0 푋푛 of the following ascending chain 푋0 ⊆ 푋1 ⊆ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' of 푘-submodules of 퐴: 푋0 is the 푘-submodule of 퐴 generated by 푋;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' given 푋푛, set 푋푛+1 = 푋푛+퐴푋푛+푋푛퐴, where 퐴푋푛 denotes the set of all finite sums of products 푎푥 with 푎 ∈ 퐴 and 푥 ∈ 푋푛, and similarly for 푋푛퐴.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' In the case of the ideal [퐼, 퐽] this specializes as follows: Proposition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Let 퐼 and 퐽 be ideals of a pre-Lie 푘-algebra 퐴.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Then [퐼, 퐽] = 퐼퐽 + � 푛≥0 푆푛, where 푆푛 = ((.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' (((퐽퐼)퐴)퐴) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' )퐴)퐴 and in 푆푛 there are 푛 factors equal to 퐴 on the right of the factor J퐼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Step 1: 퐴(퐼퐽) ⊆ 퐼퐽.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' By Property (1), we have that 퐴(퐼퐽) ⊆ (퐴퐼)퐽 + (퐼퐴)퐽 + 퐼(퐴퐽) ⊆ 퐼퐽.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Step 2: 퐴(퐽퐼) ⊆ 퐽퐼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' From Step 1, by symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Step 3: 퐴푆푛 ⊆ 푆푛 + 푆푛+1 for every 푛 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Induction on 푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Step 2 gives the case 푛 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Suppose that 퐴푆푛 ⊆ 푆푛 + 푆푛+1 for some 푛 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Then 퐴푆푛+1 = 퐴(푆푛퐴) ⊆ (퐴푆푛)퐴 + (푆푛퐴)퐴 + 푆푛(퐴퐴) ⊆ (푆푛 + 푆푛+1)퐴 + 푆푛+2 + 푆푛+1 = 푆푛+1 + 푆푛+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Pre-Lie algebras, their multiplicative lattice, and idempotent endomorphisms 13 Step 4: 푆푛퐴 = 푆푛+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' By definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Step 5: (퐼퐽)퐴 ⊆ 퐼퐽 + 푆0 + 푆1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' In fact, (퐼퐽)퐴 ⊆ 퐼(퐽퐴) + (퐽퐼)퐴 + 퐽(퐼퐴) ⊆ 퐼퐽 + 푆1 + 푆0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Final Step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Clearly, 퐼퐽 + � 푛≥0 푆푛 is a 푘-submodule of 퐴 that contains 퐼퐽 and 퐽퐼 and is contained in the ideal generated by 퐼퐽 ∪퐽퐼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Hence it remains to show that it is closed by left and right multiplication by elements of 퐴.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' This is proved in Steps 1, 3, 4 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Now that we have a good notion of commutator of two ideals 퐼 and 퐽 of a pre-Lie 푘-algebra 퐴, we can introduce the multiplicative lattice of all ideals of 퐴: it is the complete modular lattice I(퐴) of all ideals of 퐴 endowed with the commutator of ideals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Notice that, trivially, [퐼, 퐽] ⊆ 퐼 ∩ 퐽.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' As a consequence of looking at pre-Lie algebras from the point of view of multiplicative lattices,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' we immediately get the notions of prime ideal of a pre-Lie 푘-algebra 퐴,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' (Zariski) prime spectrum of 퐴,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' semiprime ideal,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' abelian pre-Lie algebra,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' idempotent (=perfect) pre- Lie algebra,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' derived series,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' solvable pre-Lie algebra,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' lower central series,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' nilpotent pre-Lie algebra,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' 푚-system,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' 푛-system,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' hyperabelian pre-Lie algebra,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' metabelian pre- Lie algebra,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Jacobson radical,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' centralizer of an ideal,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' center of a pre-Lie 푘-algebra,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' hypercenter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' See the next Section 5 and [8, 9, 6, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Notice that the monotonicity condition holds for our commutator of ideals of a pre-Lie algebra 퐴, in the sense that if 퐼 ≤ 퐼′ and 퐽 ≤ 퐽′ are ideals of 퐴, then [퐼, 퐽] ≤ [퐼′, 퐽′].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Also notice that the description of the commutator in Proposition 12 reduces, in the case of 퐼 = 퐽 = 퐴, to the equality [퐴, 퐴] = 퐴2 = 퐴퐴.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Here 퐴2 is the image of the 푘-module morphism 휇: 퐴 ⊗푘 퐴 → 퐴 induced by the 푘-bilinear multiplication of 퐴.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' 5 The commutator is not associative In this section we will show that the commutator of ideals in a pre-Lie algebra 퐴 is not associative in general, that is, if 퐼, 퐽, 퐾 are ideals of 퐴, it is not necessarily true that [퐼, [퐽, 퐾]] = [[퐼, 퐽], 퐾].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' In our example, the algebra 퐴 will be factor algebra 퐴 := T/푃, where T is the pre-Lie algebra of rooted trees of Example 4 in Section 2, and 푃 is the ideal of T generated by all rooted trees with at least 5 vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Such 푃 is the 푘-submodule of T generated by all rooted trees with at least 5 vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' The rooted trees with at most 4 vertices up to isomorphism are 14 Michela Cerqua and Alberto Facchini 푣 = 1 , 푒 = 1 2 , 푎 = 1 2 3 , 푏 = 1 2 3 , 푐 = 1 2 3 4 , 푑 = 1 2 3 4 , 푓 = 1 2 3 4 , 푔 = 1 2 3 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Hence our pre-Lie 푘-algebra 퐴 is eight dimensional, and we will denote by 푣, 푒, 푎, 푏, 푐, 푑, 푓 , 푔 the images in 퐴 of the corresponding rooted trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' That is, we will say that {푣, 푒, 푎, 푏, 푐, 푑, 푓 , 푔} is a free set of generators for the free 푘-module 퐴.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' From the multiplication in T defined in Example 4 of Section 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' we get that the multiplication table in 퐴 is 푣 푒 푎 푏 푐 푑 푓 푔 푣 푒 푎 + 푏 푐 + 2 푓 푓 + 푑 + 푔 0 0 0 0 푒 푏 푓 + 푔 0 0 0 0 0 0 푎 푑 0 0 0 0 0 0 0 푏 푔 0 0 0 0 0 0 0 푐 0 0 0 0 0 0 0 0 푑 0 0 0 0 0 0 0 0 푓 0 0 0 0 0 0 0 0 푔 0 0 0 0 0 0 0 0 Pre-Lie algebras,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' their multiplicative lattice,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' and idempotent endomorphisms 15 From the multiplication table we see that 퐴2 = 퐴퐴 has {푒,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' 푏,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' 푑,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' 푔,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' 푎,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' 푓 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' 푐} as a set of generators,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' and is a seven dimensional free 푘-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Now [퐴2, 퐴2] = � 푛≥0(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' ((퐴2 · 퐴2) · 퐴) · · · · · 퐴) · 퐴, where there are 푛 factors equal to 퐴 on the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' But, always from the multiplication table, one sees that 퐴2·퐴2 is generated by 푓 + 푔.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Moreover ( 푓 + 푔)퐴 = 0 and 퐴( 푓 + 푔) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Therefore [퐴2, 퐴2] is one dimension as a free 푘-module, and its free set of generators is { 푓 + 푔}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Similarly, [퐴2, 퐴] = 퐴 · 퐴2 + � 푛≥1(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' ((퐴2 · 퐴) · 퐴) · · · · · 퐴) · 퐴, where there are 푛 + 1 factors equal to 퐴 on the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' From the multiplication table, we see that 퐴 · 퐴2 is generated by 푎 + 푏, 푓 + 푔, 푐 + 2 푓 , 푓 + 푑 + 푔.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Also, 퐴2 · 퐴 is generated by {푏, 푑, 푔, 푓 + 푔}, (퐴2 · 퐴) · 퐴 is generated by 푔, and ((퐴2 · 퐴) · 퐴) · 퐴 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Therefore [퐴2, 퐴] is the 푘-module generated by 푏, 푑, 푔, 푓 , 푎, 푐 and is six dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' It follows that [퐴2, 퐴]· 퐴 is generated by {푑, 푔}, 퐴·([퐴2, 퐴]) is generated by {푐+2 푓 , 푓 +푑+푔}, and ([퐴2, 퐴] · 퐴) · 퐴 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' From these equalities we get that [[퐴2, 퐴], 퐴] is generated by {푑, 푔, 푐 + 2 푓 , 푓 + 푑 + 푔}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Equivalently, [[퐴2, 퐴], 퐴] is generated by {푑, 푔, 푓 , 푐} and is four dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' In particular [퐴2, 퐴2] ≠ [[퐴2, 퐴], 퐴].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Let’s illustrate in detail some of the notions that immediately derive from the commutative multiplication [−, −] (the commutator) in the multiplicative lattice I(퐴).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' First of all, a pre-Lie 푘-algebra 퐴 is abelian if the commutator of 퐴 and itself is zero: [퐴, 퐴] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' This is equivalent to saying that 푖푗 = 0 for every 푖, 푗 ∈ 퐴.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' That is, a pre-Lie algebra (퐴, ·) is abelian if and only if 푥 · 푦 = 0 for every 푥, 푦 ∈ 퐴.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' (This is equivalent to requiring that the addition +: 퐴 × 퐴 → 퐴 is a pre-Lie algebra morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=') By definition,an ideal 퐼 of a pre-Lie 푘-algebra 퐴 is prime if it is properly contained in 퐴 and, for every ideal 퐽, 퐾 of 퐴, [퐽, 퐾] ⊆ 퐼 implies 퐽 ⊆ 퐼 or 퐾 ⊆ 퐼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' An ideal 퐼 of a pre-Lie 푘-algebra 퐴 is semiprime if, for every ideal 퐽 of 퐴, [퐽, 퐽] ⊆ 퐼 implies that 퐽 ⊆ 퐼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' An ideal of 퐴 is semiprime if and only if it is the intersection of a family of prime ideals (if and only if it is the intersection of all the ideals of 퐴 that contain it).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' An ideal 푃 of a pre-Lie 푘-algebra 퐴 is prime if and only if the lattice I(퐴/푃) is uniform and 퐴/푃 has no non-zero abelian ideal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Remark 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Instead of the commutator [퐼, 퐽] of two ideals 퐼 and 퐽, we could have taken two other “product of ideals” in a pre-Lie 푘-algebra: we could consider the product 퐼퐽, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=', the 푘-submodule of 퐴 generated by all products 푖푗, which is a 푘-submodule but not an ideal of 퐴 in general, or the ideal ⟨퐼퐽⟩ generated by the submodule 퐼퐽.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Notice that 퐼퐽 ⊆ ⟨퐼퐽⟩ ⊆ [퐼, 퐽] = ⟨퐼퐽⟩ + ⟨퐽퐼⟩, where the last equality follows from Proposition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Correspondingly, we would have had three different notions of “prime ideal”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' In the next proposition (essentially contained in [8, Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content='7]) we prove that these three notions of “prime ideal” coincide: Proposition 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' The following conditions are equivalent for an ideal 푃 of a pre-Lie algebra 퐴: (a) If 퐼, 퐽 are ideals of 퐴 and 퐼퐽 ⊆ 푃, then either 퐼 ⊆ 푃 or 퐽 ⊆ 푃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' (b) If 퐼, 퐽 are ideals of 퐴 and ⟨퐼퐽⟩ ⊆ 푃, then either 퐼 ⊆ 푃 or 퐽 ⊆ 푃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' (c) If 퐼, 퐽 are ideals of 퐴 and [퐼, 퐽] ⊆ 푃, then either 퐼 ⊆ 푃 or 퐽 ⊆ 푃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' 16 Michela Cerqua and Alberto Facchini Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' The implications (a) ⇒ (b) ⇒ (c) follow immediately from the fact that 퐼퐽 ⊆ ⟨퐼퐽⟩ ⊆ [퐼, 퐽].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' (c) ⇒ (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Let 푃 satisfy condition (c) and fix two ideals 퐼, 퐽 of 퐴 such that 퐼퐽 ⊆ 푃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Since 푃 is an ideal, it follows that ⟨퐼퐽⟩ ⊆ 푃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Also, [⟨퐽퐼⟩, ⟨퐽퐼⟩] = ⟨⟨퐽퐼⟩⟨퐽퐼⟩⟩ ≤ ⟨퐼퐽⟩ ≤ 푃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' From (c), we get that ⟨퐽퐼⟩ ≤ 푃, so that [퐼, 퐽] = ⟨퐼퐽⟩ + ⟨퐽퐼⟩ ≤ 푃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' From (c) again, we get that either 퐼 ⊆ 푃 or 퐽 ⊆ 푃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Proposition 14 shows that if the pre-Lie algebra 퐴 is an associative algebra, then this notion of prime ideal coincide with the notion of prime ideal in an associative algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Proposition 12 shows that, for every pair (퐼, 퐽) of ideals of a pre-Lie algebra 퐴, one has [퐼, 퐽] = 퐼퐽 +⟨퐽퐼⟩ = 퐽퐼 +⟨퐼퐽⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Also, Step 5 in the proof of that Proposition shows that one always has that 퐼퐽 + 퐽퐼 + (퐼퐽)퐴 = 퐼퐽 + 퐽퐼 + (퐽퐼)퐴.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' A pre-Lie 푘-algebra 퐴 is idempotent (or perfect) if [퐴, 퐴] = 퐴, that is, if 퐴2 = 퐴 (last paragraph of Section 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Given any pre-Lie algebra 퐴, let Spec(퐴) be the set of all its prime ideals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' For every 퐼 ∈ I(퐴), set 푉(퐼) = { 푃 ∈ Spec(퐴) | 푃 ⊇ 퐼 }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Then the family of all subsets 푉(퐼) of Spec(퐴), 퐼 ∈ I(퐴), is the family of all the closed sets for a topology on Spec(퐴).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' With this topology, the topological space Spec(퐴) is the (Zariski) prime spectrum of 퐴, and is a sober space [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' It is not a spectral space in the sense of Hochster in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' For instance, if 퐵 is a Boolean ring without identity, then 퐵 is a pre-Lie algebra, but its prime spectrum is not compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' If the pre-Lie algebra 퐴 is an associative algebra, then this notion of prime spectrum coincide with the “standard notion” of prime spectrum of an associative algebra 퐴, where the points of the spectrum are the prime ideals of 퐴 and the closed sets are the subsets 푉(퐼) of the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' To tell the truth, there is not a “standard notion” of prime spectrum of an associative algebra that extends the classical notion of prime spectrum for commutative associative algebras with identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' There are several such notions as it is shown in [1] and [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' For instance, the points of the spectrum could be the completely prime ideals of 퐴, or the spectrum of 퐴 could be defined to be the Zariski spectrum of the commutative ring 퐴/[퐴, 퐴], where [퐴, 퐴] now denotes the ideal of 퐴 generated by all elements 푎푏 − 푏푎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' A pre-Lie 푘-algebra 퐴 is hyperabelian if it has no prime ideal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' For instance, abelian pre-Lie algebras are hyperabelian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Let 퐴 be a pre-Lie 푘-algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' The lower central series (or descending central series) of 퐴 is the descending series 퐴 = 퐴1 ≥ 퐴2 ≥ 퐴3 ≥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' , where 퐴푛+1 := [퐴푛, 퐴] for every 푛 ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' If 퐴푛 = 0 for some 푛 ≥ 1, then 퐴 is nilpotent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' (Notice that it is not necessary to distinguish between left nilpotency and rightnilpotency,becausethecommutatoriscommutative,thatis,[퐴푛, 퐴] = [퐴, 퐴푛].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=') The derived series of 퐴 [8, Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content='1] is the descending series 퐴 := 퐴(0) ≥ 퐴(1) ≥ 퐴(2) ≥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' , Pre-Lie algebras, their multiplicative lattice, and idempotent endomorphisms 17 where 퐴(푛+1) := [퐴(푛), 퐴(푛)] for every 푛 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' The pre-Lie algebra 퐴 is solvable if 퐴(푛) = 0 for some integer 푛 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' It is metabelian if 퐴(2) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' In a multiplicative lattice an element is semisimple if it is the join of a set of minimal idempotent elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' (An element 푚 of a lattice 퐿 is minimal if, for every 푥 ∈ 퐿, 푥 ≤ 푚 implies 푥 = 푚 or 푥 = 0, that is, if it is minimal in the partially ordered set 퐿 \\ {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' An element 푒 of a multiplicative lattice 퐿 is idempotent if 푒 · 푒 = 푒).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' “Minimal idempotent element” of 퐿 means minimal element of 퐿 \\ {0} that is also an idempotent element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Notice that for a minimal element 푥 ∈ 퐿 either 푥 · 푥 = 푥 or 푥 · 푥 = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=', minimal elements are either idempotent or abelian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' The Jacobson radical of 퐿 is the meet of the set of all maximal elements 푎 of 퐿 \\ {1} with 1 · 1 ̸≤ 푎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' The radical is the join of the set of all solvable elements of 퐿.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' 6 Idempotent endomorphisms, semidirect products of pre-Lie algebras, and actions Let 푒 be an idempotent endomorphism of a pre-Lie 푘-algebra 퐴.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Then 퐴 = ker(푒) ⊕ 푒(퐴) (direct sum as 푘-modules), where the kernel ker(푒) of 푒 is an ideal of 퐴 and its image 푒(퐴) is a pre-Lie sub-푘-algebra of 퐴.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' If there is a direct-sum decomposition 퐴 = 퐼 ⊕ 퐵 as 푘-module of a pre-Lie 푘-algebra 퐴, where 퐼 is an ideal of 퐴 and 퐵 is a pre-Lie sub-푘-algebra of 퐴, we will say that 퐴is the semidirect product of 퐼 and 퐵.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' We are interested in semidirect products because, for any algebraic structure, idempotent endomorphisms are in one-to-one correspondence with semidirect products and are related to the notion of action of the structure on another structure, and bimodules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' The proof of the following proposition is elementary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Proposition 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Let 퐴 be a pre-Lie 푘-algebra, 퐼 an ideal of 퐴 and 퐵 a pre-Lie sub-푘-algebra of 퐴.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' The following conditions are equivalent: (1) 퐴 = 퐼 ⊕ 퐵 as a 푘-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' (2) For every 푎 ∈ 퐴, there are a unique 푖 ∈ 퐼 and a unique 푏 ∈ 퐵 such that 푎 = 푖 + 푏.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' (3) There exists a pre-Lie 푘-algebra morphism 퐴 → 퐵 whose restriction to 퐵 is the identity and whose kernel is 퐼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' (4) There is an idempotent pre-Lie 푘-algebra endomorphism of 퐴 whose image is 퐵 and whose kernel is 퐼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' It is now clear that there is a one-to-one correspondence between the set of all idempotent endomorphisms of a pre-Lie 푘-algebra 퐴 and the set of all pairs (퐼, 퐵), where 퐼 is an ideal of 퐴, 퐵 is a pre-Lie sub-푘-algebra of 퐴, and 퐴 is the direct sum of 퐼 and 퐵 as a 푘-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Let us first consider inner semidirect product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Suppose that (퐴, ·) is a pre-Lie 푘-algebra that is a semidirect product of its ideal 퐼 and its pre-Lie sub-푘-algebra 퐵.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Then there is a pre-morphism 휆: (퐵, ·) → (End(퐼푘), ◦), given by multiplying on the 18 Michela Cerqua and Alberto Facchini left by elements of 퐵 (this follows from the fact that every ideal is a module, as we have already remarked in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Also, there is a 푘-module morphism 휌 : 퐵 → End(퐼푘), given by multiplying on the right by elements of 퐵, that is, 휌 : 푏 ↦→ 휌푏, where 휌푏(푖) = 푖 · 푏 for every 푖 ∈ 퐼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Moreover, Identity (1), applied to elements 푥, 푧 in 퐵 and 푦 ∈ 퐼, can be re-written as 휌푎(휆푏(푖)) − 휆푏(휌푎(푖)) = (휌푎 ◦ 휌푏 − 휌푏·푎)(푖) for every 푎, 푏 ∈ 퐵 and 푖 ∈ 퐼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Identity (1), applied to elements 푥 in 퐵 and 푦, 푧 ∈ 퐼, can be re-written as 휆푎(푖) · 푗 − 휆푎(푖 · 푗) = 휌푎(푖) · 푗 − 푖 · 휆푎( 푗) for every 푎 ∈ 퐵 and 푖, 푗 ∈ 퐼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Finally, the same identity (1), applied to elements 푧 in 퐵 and 푥, 푦 ∈ 퐼, can be re-written as 휌푎(푥 · 푦) − 푥 · 휌푎(푦) = 휌푎(푦 · 푥) − 푦 · 휌푎(푥) for every 푎 ∈ 퐵 and 푖, 푗 ∈ 퐼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Conversely, for outer semidirect product: Theorem 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Let 퐼 and 퐵 be pre-Lie 푘-algebras and (휆, 휌) a pair of 푘-linear map- pings 퐵 → End(퐼푘) such that: (a) 휆: (퐵, ·) → (End(퐼푘), ◦) is a pre-morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' (b) 휌푎 ◦ 휆푏 − 휆푏 ◦ 휌푎 = 휌푎 ◦ 휌푏 − 휌푏·푎 for every 푎, 푏 ∈ 퐵.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' (c) 휆푎(푖) · 푗 − 휆푎(푖 · 푗) = 휌푎(푖) · 푗 − 푖 · 휆푎( 푗) for every 푎 ∈ 퐵 and 푖, 푗 ∈ 퐼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' (d) 휌푎(푖 · 푗) − 푖 · 휌푎( 푗) = 휌푎( 푗 · 푖) − 푗 · 휌푎(푖) for every 푎 ∈ 퐵 and 푖, 푗 ∈ 퐼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' On the 푘-module direct sum 퐼 ⊕ 퐵 define a multiplication ∗ setting (푖, 푏) ∗ ( 푗, 푐) = (푖 · 푗 + 휆푏( 푗) + 휌푐(푖), 푏 · 푐) for every (푖, 푏), ( 푗, 푐) ∈ 퐼 ⊕ 퐵.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Then (퐼 ⊕ 퐵, ∗) is a pre-Lie 푘-algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' For every 푎, 푏, 푐 ∈ 퐵 and 푥, 푦, 푧 ∈ 퐼 we have that ((푥, 푎) ∗ (푦, 푏)) ∗ (푧, 푐) = (푥 · 푦 + 휆푎(푦) + 휌푏(푥), 푎 · 푏) ∗ (푧, 푐) = = ((푥 · 푦) · 푧 + 휆푎(푦) · 푧 + 휌푏(푥) · 푧 + 휆푎·푏(푧)+ +휌푐(푥 · 푦 + 휆푎(푦) + 휌푏(푥)), (푎 · 푏) · 푐) (4) and (푥, 푎) ∗ ((푦, 푏) ∗ (푧, 푐)) = (푥, 푎) ∗ (푦 · 푧 + 휆푏(푧) + 휌푐(푦), 푏 · 푐) = = (푥 · (푦 · 푧) + 푥 · 휆푏(푧) + 푥 · 휌푐(푦)+ +휆푎(푦 · 푧 + 휆푏(푧) + 휌푐(푦)) + 휌푏·푐(푥), 푎 · (푏 · 푐)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' (5) The difference of (4) and (5) is ((푥 · 푦) · 푧 − 푥 · (푦 · 푧) + 휆푎(푦) · 푧 − 휆푎(푦 · 푧)+ +휌푏(푥) · 푧 − 푥 · 휆푏(푧) + 휆푎·푏(푧) − (휆푎 ◦ 휆푏)(푧)+ +휌푐(푥 · 푦) − 푥 · 휌푐(푦) + 휌푐(휆푎(푦)) − 휆푎(휌푐(푦)) + 휌푐(휌푏(푥)) − 휌푏·푐(푥)), (푎 · 푏) · 푐 − 푎 · (푏 · 푐)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Similarly, ((푦, 푏) ∗ (푥, 푎)) ∗ (푧, 푐) − (푦, 푏) ∗ ((푥, 푎) ∗ (푧, 푐)) = = ((푦 · 푥) · 푧 − 푦 · (푥 · 푧) + 휆푏(푥) · 푧 − 휆푏(푥 · 푧) + 휌푎(푦) · 푧 − 푦 · 휆푎(푧)+ +휆푏·푎(푧) − (휆푏 ◦ 휆푎)(푧) + 휌푐(푦 · 푥) − 푦 · 휌푐(푥) + 휌푐(휆푏(푥)) − 휆푏(휌푐(푥))+ +휌푐(휌푎(푦)) − 휌푎·푐(푦)), (푏 · 푎) · 푐 − 푏 · (푎 · 푐)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Pre-Lie algebras, their multiplicative lattice, and idempotent endomorphisms 19 Hence, for the proof, it suffices to show that 휆푎(푦) · 푧 − 휆푎(푦 · 푧) + 휌푏(푥) · 푧 − 푥 · 휆푏(푧) + 휆푎·푏(푧) − (휆푎 ◦ 휆푏)(푧)+ +휌푐(푥 · 푦) − 푥 · 휌푐(푦) + 휌푐(휆푎(푦)) − 휆푎(휌푐(푦)) + 휌푐(휌푏(푥)) − 휌푏·푐(푥)) = = 휆푏(푥) · 푧 − 휆푏(푥 · 푧) + 휌푎(푦) · 푧 − 푦 · 휆푎(푧)+ +휆푏·푎(푧) − (휆푏 ◦ 휆푎)(푧)+ +휌푐(푦 · 푥) − 푦 · 휌푐(푥) + 휌푐(휆푏(푥)) − 휆푏(휌푐(푥))+ +휌푐(휌푎(푦)) − 휌푎·푐(푦)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' (6) Now 휆푎(푦) · 푧 − 휆푎(푦 · 푧) = 휌푎(푦) · 푧 − 푦 · 휆푎(푧) by hypotheses (c);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' 휌푏(푥) · 푧 − 푥 · 휆푏(푧) = 휆푏(푥) · 푧 − 휆푏(푥 · 푧) by hypotheses (c);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' 휆푎·푏(푧) − (휆푎 ◦ 휆푏)(푧) = 휆푏·푎(푧) − (휆푏 ◦ 휆푎)(푧) by hypotheses (a);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' 휌푐(푥 · 푦) − 푥 · 휌푐(푦) = 휌푐(푦 · 푥) − 푦 · 휌푐(푥) by hypotheses (d);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' 휌푐(휆푎(푦)) − 휆푎(휌푐(푦)) = 휌푐(휌푎(푦)) − 휌푎·푐(푦)) by hypotheses (b);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' 휌푐(휌푏(푥)) − 휌푏·푐(푥)) = 휌푐(휆푏(푥)) − 휆푏(휌푐(푥)) by hypotheses (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Summing up these equalities one gets Equality (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Hence the theorem characterises the four properties that an action (휆, 휌), that is, a pair of 푘-linear mappings 퐵 → End(퐼푘), must have in order to construct the semidirect product of a pre-Lie 푘-algebra 퐵 acting on a pre-Lie 푘-algebra 퐼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content='1 Bimodules over a pre-Lie algebra The most important case of semidirect product is probably when the pre-Lie algebra 퐼 is abelian, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=', the case where the action, that is, the pair (휆, 휌) of 푘-linear mappings 퐵 → End(퐼푘), is an action of the pre-Lie 푘-algebra 퐵 on a 푘-module 푀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' In other words, when 퐼 is a 퐵-bimodule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Let us be more precise, giving the precise definition of what a bimodule over a pre-Lie algebra must be: Definition 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Let 퐴 be a pre-Lie 푘-algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' A bimodule over 퐴 is a 푘-module 푀푘 with a pair (휆, 휌) of 푘-linear mappings 퐴 → End(푀푘) such that: (a) 휆: (퐴, ·) → (End(푀푘), ◦) is a pre-morphism (that is, 푀 is a module over 퐴).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' (b) 휌푎 ◦ 휆푏 − 휆푏 ◦ 휌푎 = 휌푎 ◦ 휌푏 − 휌푏·푎 for every 푎, 푏 ∈ 퐵.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Notice that Conditions (c) and (d) of Theorem 16 are always trivially satisfied because in this case the 푘-module 푀 is viewed as an abelian pre-Lie algebra, that is, with null multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' This definition already appears, for instance, in [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Notice the nice interpretation of condition (b) given in that paper: In condition (b) the left hand side 휌푎 ◦ 휆푏 − 휆푏 ◦ 휌푎 describes how far the action is from associativity (for bimodules over an associative algebra, it is always required to be zero);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' the right hand side 휌푎◦휌푏−휌푏·푎 describes how far 휌 is from being a 푘-algebra antihomomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' 20 Michela Cerqua and Alberto Facchini 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content='2 Adjoining the identity to a pre-Lie algebra The class of pre-Lie algebras contains the class of associative algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' For asso- ciative algebras, it is very natural to consider associative algebras with an identity, and when there is not an identity, to adjoin one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' This construction is often called the “Dorroh extension”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Let’s show that this is possible for pre-Lie algebras as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' We will see in fact that a more appropriate name for our class of algebras, instead of “pre-Lie algebras”, would have been “pre-associative algebras”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Adjoining an identity to a pre-Lie 푘-algebra 퐴 is exactly our semidirect product of the pre-Lie 푘-algebra 푘 acting on the pre-Lie 푘-algebra 퐴.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Let’s be more precise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' An identity in a pre-Lie 푘-algebra 퐴 is an element, which we will denote by 1퐴, such that 푎 · 1퐴 = 1퐴 · 푎 = 푎 for every 푎 ∈ 퐴.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' If 퐴 has an identity, we will say that 퐴 is unital.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' An element 푒 of 퐴 is idempotent if 푒2 := 푒 · 푒 = 푒.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' The zero of 퐴 is always an idempotent element of 퐴, and the identity, when it exists, is also an idempotent element of 퐴.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Let 퐴 be any fixed pre-Lie 푘-algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Then the associative commutative ring 푘 is a pre-Lie 푘-algebra, and there is a one-to-one correspondence between the set of all the pre-Lie 푘-algebra morphisms 푘 → 퐴 and the set of all idempotent elements of 퐴.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' For any idempotent element 푒 of 퐴 the corresponding morphism 휑푒 : 푘 → 퐴 is defined by 휑푒(휆) = 휆푒 for every 휆 ∈ 푘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Conversely, for any morphism 휑: 푘 → 퐴 the corresponding idempotent element of 퐴 is 휑(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' For any fixed pre-Lie 푘-algebra 퐴 it is possible to construct the semidirect product of 푘 acting on 퐴 via the pair (휆, 휌) of 푘-module morphisms 푘 → End(퐴푘) for which 휆훼 = 휌훼 is multiplication by 훼 for all 훼 ∈ 푘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Then the four conditions (a), (b), (c), (d) of Theorem 16 are all automatically satisfied, and the corresponding semidirect product is the 푘-module direct sum 퐴 ⊕ 푘 with the multiplication defined by (푥, 훼)(푦, 훽) = (푥 · 푦 + 훽푥 + 훼푦, 훼훽) for every (푥, 훼), (푦, 훽) ∈ 퐴 ⊕ 푘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Hence 퐴 ⊕ 푘 becomes a pre-Lie 푘-algebra with identity (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' The Lie algebra sub-adjacent this pre-Lie algebra 퐴 ⊕ 푘 is the direct sum of the Lie algebra (퐴, [−, −]) and the abelian Lie algebra 푘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' We will denote this semidirect product by 퐴#푘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Now let PreL푘,1 be the category of all unital pre-Lie 푘-algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Its objects are the pre-Lie 푘-algebras 퐴 with an identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Its morphisms 푓 : 퐴 → 퐵 are the 푘-algebra morphisms 푓 such that 푓 (1퐴) = 1퐵.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' There is also a further category involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content='It is the category PreL푘,1,푎 of all unital pre-Lie 푘-algebras with an augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Its objects are all the pairs (퐴, 휀퐴), where 퐴 is a unital pre-Lie 푘-algebra and 휀퐴: 퐴 → 푘 is a morphism in PreL푘,1 that is a left inverse for 휑1퐴: 푘 휑1퐴 � 퐴 휀퐴 �푘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' The morphisms 푓 : (퐴, 휀퐴) → (퐵, 휀퐵) are the morphisms 푓 : 퐴 → 퐵 in PreL푘,1 such that 휀퐵 푓 = 휀퐴.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' For instance, the 푘-algebra 퐴#푘 is clearly a unital 푘-algebra with Pre-Lie algebras, their multiplicative lattice, and idempotent endomorphisms 21 augmentation: the augmentation is the canonical projection 휋2 : 퐴#푘 = 퐴 ⊕ 푘 → 푘 onto the second summand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' It is easy to see that: Theorem 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' There is a category equivalence 퐹: PreL푘 → PreL푘,1,푎 that associates with any object 퐴 of PreL푘 the 푘-algebra with augmentation 퐹(퐴) := (퐴#푘, 휋2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' The quasi-inverse of 퐹 is the functor PreL푘,1,푎 → PreL푘, that associates with each unital pre-Lie 푘-algebra with augmentation (퐴, 휀퐴) the kernel ker(휀퐴) of the augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Ben-Zvi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Ma and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQfwAHH/content/2301.02627v1.pdf'} +page_content=' Reyes, A Kochen-Specker theorem for integer matrices and noncommutative spectrum functors, J.' metadata={'source': 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In this paper we give an explicit presentation of the centre of the restricted rational Cherednik +algebra Hc(Sn ≀Z/ℓZ). More precisely, we describe the centre of the indecomposable blocks of Hc(Sn ≀Z/ℓZ) +in terms of generators and relations. +This presentation is valid for any c such that the Calogero-Moser +space is smooth. In particular, the result holds for generic c. Furthermore, we demonstrate how the explicit +presentation of the centre of Hc(Sn ≀ Z/ℓZ) can be directly derived from the set of ℓ-multipartitions of n. +1. Introduction +Restricted rational Cherednik algebras are objects of interest due to their connections with rational +Cherednik algebras and hence integrable systems, algebraic symplectic geometry and algebraic combina- +torics. Rational Cherednik algebras are defined for any complex reflection group (W, h) and parametrised by +two values t and c. When t ̸= 0 their centre is trivial, but in the case t = 0 the centre is large. To be precise, +the rational Cherednik algebra is a finite dimensional module over its centre. As a consequence, much of +the representation theory of the rational Cherednik algebra can be understood via its centre. The restricted +rational Cherednik algebra is a quotient by a subalgebra contained within the centre and it too has a rich +structure. The main result of this paper (Theorem 9.2) gives an explicit presentation of the centre of the +restricted rational Cherednik algebra for the wreath product groups Sn ≀ Z/ℓZ. +We consider the centre as a direct sum of its indecomposable blocks and find the centre of each of these. +These blocks have the important property that they are isomorphic to a tensor product of two graded rings +A(λ)− and A(λ)+, where λ ∈ Irr W. The ring A(λ)+ is the endomorphism ring of the baby Verma module +for λ and in the symmetric group case is isomorphic to A(λ)− with the opposite grading. In the more general +case of wreath products these rings are still closely related, for precise details see Theorem 3.3. This reduces +the problem of understanding the centre of the restricted rational Cherednik algebra to understanding a +family of endomorphism rings. To achieve our main result we first find the centre of the restricted rational +Cherednik algebra in the symmetric group case Sn. +Theorem 1.1. (9.1) There is an isomorphism of the centre of Hc(Sn) for c ̸= 0 +Z(Hc(Sn)) ∼= +� +λ∈IrrSn +A(λ)− ⊗ A(λ)+. +The algebra A(λ)+ is given by Theorem 7.3 and A(λ)− is isomorphic to A(λ)+ with the opposite grading. +In section 4 we prove Corollary 4.1 that in the symmetric case there is an isomorphism between A(λ)+ and +the ring of functions on the scheme theoretic fibre of a special map denoted π. As we will see this will allow +us to exploit a connection to Schubert cells via the Wronski map. Schubert cells are well understood and +crucially the fibres of the Wronskian map can be explicitly written in terms of generators and relations. This +1 + +is explained in more detail in section 6. Unfortunately this connection only exists for the symmetric group +case and not more general wreath product groups. This is enough to completely describe the endomorphism +rings of the baby Verma modules and hence the entire centre in the symmetric group case. By labelling the +irreducible representations of Sn by partitions of λ ⊢ n we derive Theorem 7.2 which explicitly describes +A(λ)+ in terms of generators and relations. Using elementary results concerning partitions we can show that +A(λ)+ can be directly computed from the Young diagram Dλ of λ. +Theorem 1.2. (7.3) Let λ ⊢ n be a partition. The algebra A(λ)+ is the quotient +A(λ)+ ∼= C[Dλ]/I +by the ideal I that is generated by n homogeneous elements r1, . . . , rn. The rs are ordered so that deg(rs) = s. +The monomials appearing in ri are products of cells which share neither a row or column in Dλ. In other +words if □i,j□k,ℓ is a factor of some monomial in the rs we must have that i ̸= k and j ̸= ℓ. The coefficients +of the generators of I are given by Proposition 7.4. +Since we can describe the rings A(λ)+ explicitly in terms of generators and relations it follows we can +do the same for the indecomposable blocks of Hc(Sn) and hence the entire centre. The main result of this +paper is a generalisation of the first theorem to the wreath product case. +Theorem 1.3. (9.2) There is an isomorphism of the centre of Hc(Sn ≀ Z/ℓZ) for generic c +Z(Hc(Sn ≀ Z/ℓZ)) ∼= +� +λ∈IrrSn≀Z/ℓZ +Ac(λ)− ⊗ Ac(λ)+. +The algebra Ac(λ)+ is given by Theorem 8.2 and Ac(λ)− is isomorphic to Ac(λ∗)+ with the opposite grading. +Similarly to the symmetric case we need only describe the endomorphism rings of the baby Verma modules +to understand the centre. It is important for our purposes to fix a convention for labeling the irreducible +representations of Sn ≀ Z/ℓZ. The irreducible representations of Sn ≀ Z/ℓZ are indexed by ℓ-multipartitions of +n. Further, the ℓ-multipartitions of n are in bijection with the partitions of n with trivial ℓ-core. For precise +definitions of these terms, see section 2. +The wreath product case is more difficult as there is no direct connection to Schubert cells via the Wronski +map. Instead we use an isomorphism due to Bonnafé and Maksimau [9, Theorem 4.21] between the Calogero- +Moser space of Sn≀Z/ℓZ to an irreducible component of the fixed point subspace of the Calogero-Moser space +of Snℓ. The general idea is that we are embedding the wreath product case into a much larger symmetric +case. Section 5 is then dedicated to using this isomorphism to show that the endomorphism rings of the +baby Verma modules for the wreath product Sn ≀ Z/ℓZ are quotients of A(λ)+ for Snℓ. This culminates in +Theorem 5.10 which states that there is an isomorphism A(quoℓ(λ))+ ∼= A(λ)+ +Z/ℓZ. This is a powerful result, +which easily generalises Theorem 7.3 to Theorem 8.2 below. Note that h(i, j) means the hook length of the +cell (i, j) in the Young diagram. +Theorem 1.4. (8.2) Let quoℓ(λ) be the ℓ-quotient of λ ⊢ nℓ. The algebra A(quoℓ(λ))+ is the quotient +A(quoℓ(λ))+ ∼= C[Dℓ +λ]/I +where Dℓ +λ is the subdiagram of Dλ (the younger diagram) excluding the cells (i, j) such that h(i, j) is not +divisible by ℓ. The ideal I is generated by n homogeneous elements rℓ, r2ℓ, . . . , rnℓ. The rsℓ are ordered so +2 + +that deg(rsℓ) = sℓ. The monomials appearing in rsℓ are products of cells which share neither a row or column +in Dℓ +λ. In other words if □i,j□k,m is a factor of some monomial appearing in the rsℓ, we must have that +i ̸= k and j ̸= m. The coefficients of the generators of I are given by Proposition 7.4. +Similarly to the symmetric group case, if we can describe the endomorphism rings of the baby Verma +modules explicitly then we can do the same for the entire centre. In section 9, after proving Theorem 9.2 we +provide an explicit example by calculating the centre of Hc(S2 ≀ Z/2Z). The main results contained within +this paper first appeared in the doctoral thesis of the author [19]. +2. Combinatorics and partitions +This section provides the basic definitions and fundamental results we require for the combinatorics in this +paper. Of particular importance is the notion of ℓ-cores and ℓ-quotients. We will begin with the definition +of a partition before defining ℓ-multipartitions and bead diagrams. +Definition 2.1. Let n be a positive integer. A partition of n is a tuple (λ1, · · · , λn) of non-negative integers +such that λi ≥ λi+1 for all 1 ≤ i ≤ n − 1, and +|λ| := +n +� +i=1 +λi = n. +The length of λ is the positive integer t such that λt ̸= 0 and λt+1 = 0. +Young diagrams are a common way to represent partitions. +The Young diagram for the partition +(λ1, · · · , λn) ⊢ n consists of left aligned rows, with the ith row having λi cells. +We count the columns +from left to right and the rows top to bottom. This means that the cell (2, 3) is the second row down and +the third column along to the right. The hook length is a function that assigns an integer to each cell in the +Young diagram. The hook length is calculated for each cell by summing the cells to the right and the cells +directly below, then adding one, for the cell itself. Below are the Young diagrams of the partitions (4, 0, 0, 0), +(3, 1, 0, 0), (2, 2, 0, 0), (2, 1, 1, 0) and (1, 1, 1, 1), with the hook lengths of each cell written inside. +4 +3 +2 +1 +4 +2 +1 +1 +3 +2 +2 +1 +4 +1 +2 +1 +4 +3 +2 +1 +Let us now give a formula for the hook length of a cell. For a given cell (i, j), let L denote the number of +cells in the jth column, this is called the leg. This is a non-standard use of the term leg, however this version +is better suited for our purposes. The hook length h(i, j) of the cell (i, j) is then +h(i, j) = λi − j + L − i + 1. +(2.1) +Lemma 2.1. Let λ ⊢ n and consider the set P = {d1, · · · , dn}, where di = λi + n − i. Then +|{j | di − j ̸∈ P for 1 ≤ j ≤ di}| = λi. +Proof. Fix i. There are n − i dk’s such that dk ≤ di. Hence there are di − (n + i) = λi + n − i − n + i = λi +many numbers such that di − j ̸∈ P. Therefore |{j | di − j ̸∈ P for 1 ≤ j ≤ di}| = λi. +□ +If we let P and di be the same as in Lemma 2.1 we can prove the following. +3 + +Lemma 2.2. The set of hook lengths of the row i equals the set {j | di − j ̸∈ P}. +Proof. Let us show that the set of hook lengths of row i is contained in the set {j | di − j ̸∈ P}. The result +will then follow from Lemma 2.1. Fix i ∈ {1, 2, · · · , n}. If there exists dm such that di − h(i, j) = dm for +some j then +h(i, j) = di − dm. +Hence +λi − j + L − i + 1 = λi + n − i − λm + n − m, +which simplifies to +L − j + 1 = m − λm. +(2.2) +We now have three cases to consider L > m, L = m and L < m. +First let L > m. Then L = m+k for some positive integer k and so equation (2.2) becomes j = k+λm+1. +Therefore j > λm. Since L > m we have that λm > λL, hence j > λL. But this contradicts the definition of +L. +For the second case, let m = L. Then (2.2) becomes j = λm + 1 = λL + 1 which is clearly a contradiction +as λL ≥ j. +The final case is when L < m. Then L = m−k ≥ i for some positive integer k. Equation (2.2) λm+1 = j+k +now since k is a positive integer we have j ≤ λm. But since m > L we have that j > λm and so we have a +contradiction. Therefore there is no dm such that di − h(i, j) = dm and so h(i, j) ∈ {j | di − j ̸∈ P} +□ +Lemma 2.3. Given a partition λ ⊢ n, we have di − h(i, j) = dk − h(k, j). +Proof. Using formula (2.1) the following calculations gives the result. +di − h(i, j) = λi + n − i − λi + j − L + i − 1 = n + j − L − 1 +and +dk − h(k, j) = λk + n − k − λk + j − L + k − 1 = n + j − L − 1. +□ +Definition 2.2. An ℓ-multipartition of n is an ℓ-tuple (λ1, · · · , λℓ) such that each λi is a partition and +�ℓ +i=1 |λi| = n. +The first column hook lengths of a partition is the set of hook lengths of the cells on the leftmost column +of the Young diagram for a partition. For example the partition (3, 2, 1, 1) has Young diagram +6 +3 +1 +4 +1 +2 +1 +and the first column hook lengths are {6, 4, 2, 1}. The first column hook lengths are important, because the +original partition can always be recovered from the first column hook lengths. To see this, recall the formula +for hook length given above, +h(i, j) = λi − j + L − i + 1. +4 + +The first column hook lengths are given by fixing j = 1. Hence +h(i, 1) = λi − 1 + L − i + 1 = λi + L − i, +and thus λi = h(i, 1) − L + i. +Definition 2.3. We refer to elements of the set Z≤−1×{0, · · · , ℓ−1} as points. A bead diagram is a function +f : Z≤−1 × {0, · · · , ℓ − 1} → {0, 1} which takes the value 1 for only finitely many points. If f(i, j) = 1 then +the point is said to be occupied by a bead. If f(i, j) = 0, then the point (i, j) is empty. +We can construct a bead diagram if given any partition λ and integer ℓ in the following way. Let K denote +the set of first column hook lengths of λ. Then place the beads according to the following rule f(i, j) = 1 if +and only if −(i + 1) · ℓ + j ∈ K. Denote this bead diagram by Bℓ(λ). +There is an easier way to construct the bead diagram associated to a partition, that agrees with the +condition given above. Consider the diagram of ℓ columns of empty beads. Then count left to right and +begin with the top row, the full beads correspond to the first column hook lengths. An example makes this +clear. +Example 2.1. Let ℓ = 3, then we shall write bead diagram for the partition (3, 2, 1, 1). The first column +hook lengths are {6, 4, 2, 1}. Therefore the bead diagram has three columns and the full beads are placed +according to the first column hook lengths counting left to right and top to bottom. We begin counting from +the first empty bead, which denotes the first column hook length 0. Hence B3(3, 2, 1, 1) is +. +Any bead diagram gives rise to a unique partition by following the reverse process, with the caveat that +we begin counting the beads from the first empty bead. For instance consider the bead diagram +. +Since we begin the count from the first empty bead we have first column hook lengths {2, 3, 6} which +corresponds to the partition (4, 2, 2). +Definition 2.4. Let λ ⊢ n and fix a positive integer ℓ ≤ n. Consider the bead diagram Bℓ(λ) of λ with ℓ +columns. If we slide the beads upwards as much as possible we obtain a new bead diagram. The partition +corresponding to this new bead diagram is called the ℓ-core of λ. +5 + +Definition 2.5. Consider the bead diagram Bℓ(λ). The columns can be considered as bead diagrams for +ℓ = 1. Denote the partition defined by the first column by λ1, the second by λ2 and so on. We define the +ℓ-quotient of λ to be the ℓ-multipartition (λ1, · · · , λℓ). +Example 2.2. Let us find the 3-core and 3-quotient of (4, 2, 2). First write B3(4, 2, 2) which is +to find the 3-core we shift all beads up as far as they can go, to obtain the new bead diagram +. +Beginning the count from the first empty bead we get first column hook lengths {1, 2} which corresponds +to the partition (1, 1). To find the 3-quotient we consider the columns of B3(4, 2, 2) as there own bead +diagrams. Then the set of first column hook lengths for λ2 and λ3 are empty. The first bead column has +first column hook lengths {1, 2}. Therefore the 3-quotient of (4, 2, 2) is ((1, 1), (∅), (∅)). +Let us introduce one last piece of notation. Denote by P(n) the set of all partitions of n, and P(n, ℓ) the +set of all ℓ-multipartitions of n. Also, denote by Pλ(n) the set of all partitions of n with ℓ-core λ. A partition +is uniquely determined by its ℓ-core and its ℓ-quotient. The following is [21, Theorem 2.7.30] which allows +us to equate ℓ-multipartitions of n with partitions of nℓ that have trivial ℓ-core. +Theorem 2.1. There is a bijection between the set of partitions of nℓ with trivial ℓ-core and the ℓ-multipartitions +of n +P∅(nℓ) → P(n, ℓ), +given by λ → quo(λ). +3. The blocks of the centre +In this section we will study the essential properties of the indecomposable blocks of the centre of the +restricted rational Cherednik algebra. Namely, we recall that these blocks are isomorphic to a tensor product +of endomorphism rings of baby Verma modules. This fact is of critical importance to later results. Further, +we will show that the blocks are indexed by the irreducible representations of the complex reflection group W +when the centre is a regular algebra. We begin by recalling the definition of the rational Cherednik algebra. +Let W ⊂ GL(h) be a complex reflection group and let V = h ⊕ h∗. There is a natural pairing (−, −) : +h × h∗ → C given by (y, x) := x(y). Then the standard symplectic form ω on V is given by +ω(y1 ⊕ x1, y2 ⊕ x2) = (y1, x2) − (y2, x1). +We denote the symplectic 2-form that is equal to ω on Im(1 − s) and 0 on Ker(1 − s) by ωs. +6 + +Given a complex reflection group (W, h), t ∈ C and a class function c : S → C, where S is the set of +reflections in W. We define the rational Cherednik algebra +Ht,c(W) := T (V ) ⋊ W/⟨x ⊗ y − y ⊗ x − κ(x, y) | ∀x, y ∈ V ⟩ +where +κ(x, y) = t · ω(x, y) · 1 + +� +s∈S +c(s) · ωs(x, y) · s. +The restricted rational Cherednik algebra is the quotient algebra +Hc(W) := H0,c(W)/R+H0,c(W). +Here R+ = C[h]W ++ ⊗ C[h∗]W + C[h]W ⊗ C[h∗]W ++ is an ideal of the central subalgebra C[h∗]W ⊗ C[h]W [17, +Proposition 3.6]. +Let us fix some notation, Zc(W) := Zc(Hc(W)) and Xc(W) := Spec Zc(W). Write the restricted rational +Cherednik algebra as a direct sum of its indecomposable blocks +Hc(W) = +� +j∈I +Bj. +(3.1) +It is shown [17, Proposition 3.6] that there is an inclusion +i : C[h]W ⊗ C[h∗]W ֒→ Zc(W). +The dual map on spectra is then denoted +γ : Xc(W) ։ Spec (C[h]W ⊗ C[h∗]W ) = h/W × h∗/W. +When Xc(W) is smooth the blocks of Hc(W) are in bijection with the points of γ−1(0) [11, Corollary 2.7]. +Therefore, we can replace the indexing set I with γ−1(0) and write (3.1) as +Hc(W) = +� +p∈γ−1(0) +Bp. +(3.2) +The condition that Xc(W) is smooth is therefore fundamentally important. Thankfully, this condition is not +hard to guarantee due to [15, Corollary 1.14] which says the following. +Lemma 3.1. For a suitably generic class function c (one which is a complement to finitely many hyperplanes +on the set of conjugacy classes of reflections in Sn ≀ Z/ℓZ) the variety Xc(Sn ≀ Z/ℓZ) is smooth. +Throughout the rest of this paper it is assumed c is chosen so that Xc(W) is smooth. Also, +we say that c is generic if the algebra Z(Hc(Sn ≀ Z/ℓZ)) is regular. +Let us describe the blocks in (3.1). The algebra Bp is a matrix algebra over the ring of functions at the +points p ∈ γ−1(0) by [17, p. 7], +Bp = Mat|W|(Op). +(3.3) +Here Op = (Zc(W)/R+Zc(W))p is the scheme theoretic fibre of γ at 0 localised at the point p. The next +proposition demonstrates that the blocks can be labeled by the irreducible representations of W. +To prove that the blocks of the centre are labeled by irreducible representations of W we use the baby +Verma modules. Recall, that these are the standard modules for the restricted rational Cherednik algebra. +They are defined for any irreducible representation λ ∈ Irr W as +∆c(λ) = Hc(W) ⊗C[h∗]coW ⋊W λ. +(3.4) +7 + +Here h ⊂ C[h∗]coW acts as 0 on λ. +Proposition 3.1. The blocks of Hc(W) are in bijection with the irreducible representations of W. +Proof. Write +Hc(W) = +� +i∈γ−1(0) +Bi, +a sum of indecomposable submodules. Since ∆(λ) is a Hc(W)-module, +∆(λ) = Hc(W) · ∆(λ) = +� +i∈γ−1(0) +Bi · ∆(λ). +Since ∆(λ) is indecomposable we must have Bi · ∆(λ) = 0 for all i ̸= j for some (unique) j. +Since Op is a local ring it has a unique simple module. Equation (3.3) then implies that Bp also has a +unique simple module. If ∆(λ) = Bp · ∆(λ) and ∆(µ) = Bp · ∆(µ) then the simple module of Bp equals both +L(λ) and L(µ). As the standard modules ∆(λ) have a unique simple quotient L(λ) [17, Proposition 4] this +forces λ = µ. +□ +Let us now show that the centre of Hc(W) is the sum of the centres of the blocks. Using the bijection +from Proposition 3.1, let Bλ correspond to the block Bp. Consider the following maps, the inclusion map +i : Zc(W) → Hc(W), the quotient map q : Hc(W) → Hc(W) defined by q(z) = z + R+Hc(W) and the +projection φp : Hc(W) → Bp. Denote by A(λ) the image of Zc(W) under the composition of these maps +Zc(W) ֒→ Hc(W) ։ Hc(W) ։ Bλ. +(3.5) +We will show that A(λ) = Op. +Lemma 3.2. The image of the centre Zc(W) under the composition of the inclusion and quotient map is +equal to the centre of Hc(W). That is, +q ◦ i(Zc(W)) = Zc(Hc(W)). +Proof. A proof of this can be found in [18, Lemma 2.8] with the condition that the ideal R+Zc(W) is +contained in a maximal ideal corresponding to an Azumaya point. By [14, Theorem 1.7] the Azumaya points +of Hc(W) are precisely the points in the smooth locus of Xc(W), but we have assumed that Xc(W) is +smooth. +□ +Theorem 3.1. The image of Zc(W) under the composition of maps (3.5) is equal to Op. In particular +Op = A(λ). +Proof. By Lemma 3.2 the image of Zc(W) is Zc(Hc(W)). +By [2, Lemma 4.5] the kernel is R+Zc(W) +therefore, Zc(W)/R+Zc(W) = Zc(Hc(W)). Write the block decomposition +Zc(W)/R+Zc(W) = Zc(Hc(W)) = +� +i∈γ−1(0) +Ai ⊂ +� +i∈γ−1(0) +Bi. +Hence Ai = Z(Bi). The image of Zc(W)/R+Zc(W) under the map φp is then Ap. +To localise at the point p we do the following. There is a unique maximal ideal mp ⊂ Ap that corresponds +to the point p and so a maximal ideal A1 ⊕ A2 · · · ⊕ mp ⊕ · · · ⊕ Ar in Zc(W)/R+Zc(W). We make every +element not contained in this ideal invertible. Since we had the block decomposition of Zc(W)/R+Zc(W) +8 + +there is a set of orthogonal idempotents which we shall label e1, · · · , en so that Ai = Aei. Therefore the +maximal ideal A1 ⊕ A2 · · · ⊕ mp ⊕ · · · ⊕ Ar contains every ej ̸= ep. By localising at p we have made ep +invertible and so for any other ei we have ei = eiepe−1 +p += 0. Therefore, +(Zc(W)/R+Zc(W))p = Ap = φp ◦ q ◦ i(Z). +□ +Corollary 3.1. The centre of the block Bλ is A(λ). +Proof. Equation (3.3) implies that Oλ is the centre of Bλ. Therefore, Theorem 3.1 implies that A(λ) is the +center of the block. +□ +A consequence of the above corollary is that if we can describe A(λ) for each λ ∈ Irr W then we have +described the entire centre of Hc(W). The next theorem is [6, Theorem 8.14] and states that the algebras +A(λ) are isomorphic to the tensor product of two endomorphism rings. It is these endomorphism rings that +we will explicitly describe in later sections. We define +Ac(λ)+ := EndHc(W)∆(λ) and Ac(λ)− := EndHc(W)∆∗(λ), +where ∆∗(λ) = Hc(W) ⊗C[h]coW ⋊W λ. +Theorem 3.2. Multiplication induces an isomorphism +Ac(λ)− ⊗C Ac(λ)+ ∼= Ac(λ) +It turns out that in the case of the wreath product we need only describe one of these rings, as we can +then deduce a description of the other. +The final theorem of this section proves the algebra Ac(µ)− := EndHc(W)(∆∗(µ)) is isomorphic to +Ac(λ)+ := EndHc(W)(∆(λ)), but for different generic c and simple modules λ, µ. +It is crucial to the +description of c that we set some notation for Hc(Sn ≀ Z/ℓZ). Fix a generator γ ∈ Z/ℓZ, let γi ∈ Sn ≀ Z/ℓZ +denote the element γ in the ith component. Then γiσij = σijγj. The space h has basis {x1, · · · , xn} with +action γixi = ωxi and γixj = xj for j ̸= i where ω is a primitive ℓth root of unity. The permutations act as +follows σxi = xσ(i). +The defining relations for Hc(Sn ≀ Z/ℓZ) are +[xi, xj] = 0, +[yi, yj] = 0, +[yi, xi] = c(σijγk +i γ−k +j +) +� +i̸=j +ℓ +� +k=0 +σijγk +i γ−k +j ++ +ℓ−1 +� +k=1 +c(γk +i )γk +i +and +[yi, xj] = −c(σijγk +i γ−k +j +) +ℓ +� +k=1 +ωkσijγk +i γ−k +j +. +Then parameter c is defined by +c(σijγk +i γ−k +j +) := c(σijγk +i γ−k +j +) and c(γk +i ) := c(γ−k +i +). +The irreducible modules of Sn ≀ Z/ℓZ are labeled by ℓ-multipartitions of n. If λ = (λ1, · · · , λℓ) then define +λ∗ := (λ1, λℓ, · · · , λ2). +9 + +Theorem 3.3. There is an anti-graded isomorphism +Ac(λ∗)− ∼= Ac(λ)+. +Moreover c is generic if and only if c is generic. +Proof. The map φ : Hc(Sn ≀ Z/ℓZ) → Hc(Sn ≀ Z/ℓZ) given by +φ(xi) = yi, +φ(yi) = −xi, +φ(σ) = σ and φ(γi) = γ−1 +i +for all i and for all σ ∈ Sn is an anti-graded isomorphism of algebras. Recall the baby Verma module (3.4) +∆c(λ) := Hc ⊗C[h]coW ⋊W λ +and define the module +∆∗ +c(λ) := Hc ⊗C[h∗]coW ⋊W λ. +In an abuse of notation let φ denote its restriction to the group algebra CSn≀Z/ℓZ, then λφ = λ∗; this follows +from the construction of irreducible CSn ≀ Z/ℓZ-modules as in [24, Section 5.3]. Since φ(C[h]W ++ ) = C[h∗]W ++ , +this then implies that +∆c(λ)φ = ∆∗ +c(λ∗). +Now M → M φ is a functor Hc-mod→ Hc-mod which is an equivalence. This means that the map +EndHc(∆c(λ)) → EndHc(∆c(λ)φ) = EndHc(∆∗ +c(λ∗)) +is an isomorphism. Implying that +Ac(λ∗)− = EndHc∆∗ +c(λ∗) ∼= EndHc∆c(λ) = Ac(λ)+. +Since φ is an isomorphism, Z(Hc(Sn ≀ Z/ℓZ)) is regular if and only if Z(Hc(Sn ≀ Z/ℓZ)) is regular. Thus c +is generic if and only if c is generic. Therefore, if we know A(λ)+ for generic c and for all λ then we know +A(λ∗)− for generic c and for all λ∗. +□ +Remark 3.1. In the case of the symmetric group ℓ = 1, then c = c and λ∗ = λ. Therefore Ac(λ)+ ∼= Ac(λ)−. +As a consequence of Theorem 3.2 and Theorem 3.3 we can understand Z(Hc(W)) by studying the endo- +morphism rings of baby Verma modules. +4. The symmetric group case +The aim of this section is to show that the endomorphism rings A(λ)+ are isomorphic to the ring of +functions on the preimage of a particular map of spectra denoted π. This is significant, as we shall introduce +the Wronskian map in section 6 and show a connection with π. More precisely, Theorem 6.1 claims that +the ring of functions on the preimage of π at 0 is isomorphic to the ring of functions on the preimage of the +Wronskian map at 0. This will allow us to give the explicit description of A(λ)+ in terms of generators and +relations. +We begin with the following theorem which states two key facts (for any complex reflection group W). +The first is that the baby Verma modules are quotients of the Verma modules. Recall that the Verma module +is defined as ∆c(λ) := Hc(W) ⊗C[h∗]⋊W λ, where h ⊂ C[h∗] acts by 0 on λ. The second, is that for any +irreducible representation λ ∈ IrrW there is a surjection from the centre of the restricted rational Cherednik +algebra onto the endomorphism ring of the Verma module. +10 + +Theorem 4.1. For all λ ∈ IrrW, +(1) ∆(λ) = ∆(λ)/R+∆(λ). +(2) The map defined by multiplication by elements of Zc(W) on ∆(λ) as a Hc(W)-module is a surjection +Zc(W) ։ EndHc(W)(∆(λ)). +Proof. Statement 1 follows from the PBW Theorem [15, Theorem 1.3]. For a proof of 2 see [3, Theorem +1.2]. +□ +Let us consider the implications of the second fact. There is the following composition of maps, the first +being the inclusion map and the second the surjection from Theorem 4.1 (2) +C[h]W ֒→ Zc(W) ։ EndHc(W)(∆(λ)). +This induces a map of spectra +π : Spec EndHc(W)(∆(λ)) → Spec C[h]W = h/W. +(4.1) +From the map (4.1) we see that +C[π−1(0)] ∼= EndHc(W)(∆(λ))/C[h]W ++ EndHc(W)(∆(λ)). +(4.2) +The goal of this section is to prove that the right hand side of (4.2) is in fact isomorphic to A(λ)+. This is +not obvious as the rings A(λ)+ are the endomorphism rings of the baby Verma modules whereas the right +hand side of (4.1) is a quotient of the endomorphism ring of the Verma module. The majority of the rest of +this section will be spent proving the following isomorphism. +EndHc(W)(∆(λ)) ∼= EndHc(W)(∆(λ))/C[h]W ++ EndHc(W)(∆(λ)). +Let +e := +1 +|W| +� +σ∈W +σ +denote the trivial idempotent in CW ⊂ Hc(W). Recall that there is an isomorphism Zc(W) ∼= eHc(W)e +given by the map z → z · e [14, Theorem 3.1], called the Satake isomorphism. +Lemma 4.1. Let A be a finitely generated algebra and e be an idempotent of A. For any A-module M we +have the following isomorphism +eA ⊗A M ∼= eM. +Proof. Define a homomorphism φ : eA ⊗A M → eM as follows φ(ea ⊗ m) = eam. We shall prove this is an +isomorphism. It is clearly surjective and a morphism so we need only prove that it is injective. We prove that +the kernel of φ is 0. If φ(ea⊗m) = 0 then eam = 0, but note that ea⊗m = e2a⊗m = e⊗eam = e⊗0 = 0. +□ +The following is a standard fact [5, Corollary 1.6.3]. +Theorem 4.2. The spherical Cherednik algebra eHc(W)e is Morita equivalent to the rational Cherednik +algebra Hc(W) if and only if e · M = 0 implies that M = 0 for all M ∈ Hc(W)-mod. +In the proof of [5, Corollary 1.6.3] the equivalence is given explicitly, which we summarise in the following +Theorem. +11 + +Theorem 4.3. The functor +e : Hc(W)−mod → eHc(W)e−mod +is an equivalence of categories if and only if e · M = 0 implies that M = 0 for all M ∈ Hc(W)−mod. +Lemma 4.2. The Zc(W)-module e∆(λ) is cyclic. +Proof. In [3, Theorem 4.1] it is shown that e∆(λ) is a cyclic EndHc(W)(∆(λ))-module. We also know from +Theorem 4.1 that Zc(W) surjects onto EndHc(W)(∆(λ)), hence e∆(λ) is a cyclic Zc(W)-module. +□ +We can now prove that the endomorphism ring of the baby Verma module is a quotient of the endomor- +phism ring of the corresponding Verma module. In particular they are exactly the quotients we desire. +Theorem 4.4. There is an isomorphism +EndHc(W)(∆(λ)) ∼= EndHc(W)(∆(λ))/C[h]W ++ EndHc(W)(∆(λ)). +Proof. For brevity, write H = Hc(W). We make use of Theorem 4.2, that the spherical Cherednik algebra +eHe is Morita equivalent to H, hence eHe−mod ∼= H−mod. Given an endomorphism f ∈ EndH(∆(λ)) we +have an endomorphism +f ∈ EndH(∆(λ)/C[h]W ++ ∆(λ)) +where f(m + C[h]W ++ ) = f(m) + C[h]W ++ ∆(λ). In this way we have a map +φ : EndH(∆(λ)) → EndH(∆(λ)/C[h]W ++ ∆(λ)). +We wish to show that Kerφ = R+EndH(∆(λ)). Since eHe is Morita equivalent to H we have the following +commutative diagram +EndH(∆(λ)) +EndH(∆(λ)/C[h]W ++ ∆(λ)) +EndeHe(e∆(λ)) +EndeHe(e∆(λ)/eC[h]W ++ e∆(λ)) +φ +∼ += +∼ += +By Lemma 4.2, e∆(λ) is a cyclic eHe-module. Hence e∆(λ) ∼= eHe/I, where I is the annihilator of the +generator. Therefore, +EndeHe(e∆(λ)) ∼= EndeHe(eHe/I) ∼= eHe/I. +Similarly, +EndeHe(e∆(λ)/eC[h]W ++ e∆(λ)) ∼= EndeHe((eHe/I)/(eC[h]W ++ eHe/I)), +and +EndeHe((eHe/I)/(eC[h]W ++ eHe/I)) ∼= eHe/C[h]W ++ eHe + I. +12 + +Hence we have a new commutative diagram +EndH(∆(λ)) +EndH(∆(λ)/C[h]W ++ ∆(λ)) +EndeHe(e∆(λ)) +EndeHe(e∆(λ)/eC[h]W ++ e∆(λ)) +eHe/I +eHe/C[h]W ++ eHe + I +∼ += +∼ += +∼ += +∼ += +. +It is easy to see from the diagram that the kernel of the bottom map is C[h]W ++ eHe + I. Then, via a simple +diagram chasing argument, we see that Kerφ = C[h]W ++ EndH(∆(λ)). Hence +EndH(∆(λ)/C[h]W ++ ∆(λ)) ∼= EndH(∆(λ))/C[h]W ++ EndH∆(λ)), +and by Theorem 4.1, EndH(∆(λ)) ∼= EndH(∆(λ)/C[h]W ++ ∆(λ)). Therefore +EndH(∆(λ)) ∼= EndH(∆(λ))/C[h]W ++ EndH∆(λ)). +□ +Applying Theorem 4.4 proves the following important corollary. +Corollary 4.1. There is an isomorphism of algebras C[π−1(0)] ∼= A(λ)+. +Proof. From the map (4.1) we see that +C[π−1(0)] = EndHc(W)(∆(λ))/C[h]W ++ EndHc(W)(∆(λ)). +Hence, by Theorem 4.4, C[π−1(0)] ∼= EndHc(W)(∆(λ)) = A(λ)+. +□ +Corollary 4.1 reduces the problem of understanding the blocks of Hc(W) to understanding the scheme +theoretic fiber of the map π. +5. Wreath product case +Describing the centre of the restricted rational Cherednik algebra of Sn ≀ Z/ℓZ requires more finesse than +the symmetric group case. As a result of Theorem 3.3 and Theorem 3.2, to describe the centre we need only +understand the endomorphism rings of the baby Verma modules. We cannot simply use Corollary 4.1, as we +shall see in Section 6 the connection between π and the Wronskian only holds in the case of the symmetric +group. Instead we will need to construct two maps πnℓ and πn,ℓ for the symmetric group and the wreath +product group respectively. Then we will prove a generalisation of Corollary 4.1. Key to all of this is the +following isomorphism [9, Theorem 4.21] which provides a connection between Zc(Sn) and Zc(Sn ≀ Z/ℓZ) via +their respective Calogero-Moser spaces. +Theorem 5.1. Let σ ∈ Z/ℓZ ⊂ C× be a root of unity and assume Xc(W) is smooth. Then Xc(W)σ, the +subscheme of Xc(W) fixed by the action of σ, is smooth. For each irreducible component X0 ⊂ Xc(W)σ +13 + +there exists a reflection subquotient W ′ ⊂ W and conjugacy function c such that there is a C×-equivariant +isomorphism of varieties +X0 ∼= Xc(W ′). +(5.1) +Since the parameters c and c will always be generic and our results independent of the previous result, +we write c for c in the remainder of this section. +It is important at this stage to fix notation for the irreducible representations of Sn ≀ Z/ℓZ. There is a +bijection between the ℓ-multipartitions of n and the irreducible representations of Sn ≀ Z/ℓZ [24, p. 221]. +Then, by Theorem 2.1, it makes sense to denote an irreducible representation of Sn ≀ Z/ℓZ as quoℓ(λ), for +a partition λ ⊢ nℓ with trivial ℓ-core. The reason why we choose to label the irreducible representations by +the ℓ-quotients will become clearer later in the section, particularly in light of Lemma 5.5. Note that when +ℓ = 1 we are back in the symmetric case and quoℓ(λ) = λ. +In an identical manner to (4.1) we can construct two maps +πn,ℓ : Spec EndHc(Sn≀Z/ℓZ)∆(quoℓ(λ)) → Cn/(Sn ≀ Z/ℓZ), +and +πnℓ : (Spec EndHc(Snℓ)(∆(λ)))Z/ℓZ → (Cnℓ/Snℓ)Z/ℓZ +The strategy is to embed Spec End∆(quoℓ(λ)) into Xc(Sn ≀ Z/ℓZ) then, using (5.1), realise it as a subvariety +of Xc(Snℓ)Z/ℓZ. From this point on we will write EndHc(Sn≀Z/ℓZ)∆(quoℓ(λ)) as End∆(quoℓ(λ)) for brevity. +We embed Spec End∆(quoℓ(λ)) into Xc(Sn ≀ Z/ℓZ) by proving it is equal to a subvariety called the +attracting set. +Definition 5.1. Let X be an affine scheme over C with a C×-action and assume that XC× is finite. An +attracting set for the C×-action is defined to be Ωp := {x ∈ X | limt→∞ t·x = xp} where xp is a fixed point. +To prove that Spec End∆(quoℓ(λ)) can be identified with an attracting set in Xc(Sn ≀Z/ℓZ) we show that +two equalities hold. The first is +Spec End∆(quoℓ(λ)) = SuppZc(Sn≀Z/ℓZ)(∆(quoℓ(λ)), +(5.2) +which is relatively straight forward. The second is +SuppZc(Sn≀Z/ℓZ)(∆(quoℓ(λ)) = Ωquoℓ(λ), +(5.3) +which requires significantly more work than equation (5.2). From now on we shorten Zc(Sn ≀ Z/ℓZ) to Zc. +Lemma 5.1. There is an equality of supports +SuppZc∆(quoℓ(λ)) = SuppZce∆(quoℓ(λ)). +Proof. Since e∆(quoℓ(λ)) ⊂ ∆(quoℓ(λ)) we have annZc∆(quoℓ(λ)) ⊂ annZce∆(quoℓ(λ)). Hence +SuppZce∆(quoℓ(λ)) ⊂ SuppZc∆(quoℓ(λ)). +It remains to show the reverse inclusion. Consider p ∈ SuppZc∆(quoℓ(λ)). Then ∆(quoℓ(λ)) ⊗Zc (Zc)p ̸= 0. +Therefore, to prove the reverse inclusion all we need show is that e∆(quoℓ(λ))⊗Zc(Zc)p ̸= 0. By Theorem 4.3, +the functor +e · − : Hc(Sn ≀ Z/ℓZ))−mod → eHc(Sn ≀ Z/ℓZ))e−mod +14 + +is an equivalence and hence maps the non-zero objects in Hc(Sn ≀ Z/ℓZ))−mod to non-zero objects in +eHc(Sn ≀ Z/ℓZ))e−mod. Therefore ∆(quoℓ(λ)) ⊗Zc (Zc)p ̸= 0 if and only if e∆(quoℓ(λ)) ⊗Zc (Zc)p ̸= 0. +□ +We can now prove that the equality (5.2) holds. +Theorem 5.2. There is an equality of varieties +Spec End∆(quoℓ(λ)) = SuppZc ∆(quoℓ(λ)). +Proof. Note that e∆(quoℓ(λ)) is a cyclic module Zc-module by Lemma 4.2. Therefore Zc/I ∼= e∆(quoℓ(λ)) +as left Zc-modules for some ideal I. +Since Hc(Sn ≀ Z/ℓZ) is Morita equivalent to eHc(Sn ≀ Z/ℓZ)e and +eHc(Sn ≀ Z/ℓZ)e ∼= Zc we have +EndHc(Sn≀Z/ℓZ)(∆(quoℓ(λ))) ∼= EndeHc(Sn≀Z/ℓZ)ee∆(quoℓ(λ)) ∼= EndZcZc/I ∼= Zc/I. +Therefore +Spec EndHc(Sn≀Z/ℓZ)(∆(quoℓ(λ))) ∼= Spec Zc/I, +and since Zc is commutative we see that Spec Zc/I ∼= SuppZc(e∆(quoℓ(λ))). +Then Lemma 5.1 implies +SuppZc(e∆(quoℓ(λ))) ∼= SuppZc(∆(quoℓ(λ))). Hence +Spec EndHc(Sn≀Z/ℓZ)(∆(quoℓ(λ))) = SuppZc(∆(quoℓ(λ))). +□ +Several technical results are required to prove the equality (5.3). We start with the following theorem [22, +Theorem 13.4]. +Theorem 5.3. Let R be a Noetherian local ring with maximal ideal m. Let S be the associated graded of R +with respect to the m-adic filtration. Then R is regular if and only if S is a polynomial ring. +Let X be an affine algebraic variety over C that admits a C×-action. This induces a grading on C[X]. +Also note that Z/ℓZ ⊂ C× by identifying the cyclic group of order ℓ with the ℓth roots of unity. Recall that +the fixed point locus is defined as +XZ/ℓZ = {x ∈ X | g · x = x ∀g ∈ Z/ℓZ} +which can be equivalently defined as +Spec +� +C[X] +⟨f − g · f | g ∈ Z/ℓZ and f ∈ C[X]⟩ +� +. +Proposition 5.1. Fix a Z/ℓZ-stable subspace V ⊂ C[X] such that C[X] is generated by V . The subspace V +decomposes into V Z/ℓZ ⊕ VZ/ℓZ, where V Z/ℓZ is the invariant subspace under the action of Z/ℓZ and VZ/ℓZ +is its Z/ℓZ-stable complement. Then +XZ/ℓZ = Spec +� C[X] +⟨VZ/ℓZ⟩ +� +. +Proof. We wish to show that ⟨VZ/ℓZ⟩ = ⟨{f − g · f | g ∈ Z/ℓZ, f ∈ C[X]}⟩. We first show that ⟨VZ/ℓZ⟩ ⊂ +⟨{f − g · f | g ∈ Z/ℓZ, f ∈ C[X]}⟩. Fix a basis of VZ/ℓZ, {x1, x2, · · · , xk} and generator s ∈ Z/ℓZ such that +s · xi = waixi where w is a primitive ℓth root of unity and ai is an integer. Since xi is not a fixed point there +15 + +is a g ∈ Z/ℓZ such that g · xi = µxi for some scalar µ ̸= 1. Then consider the function (1 − µ)−1(xi − g · xi). +We have +(1 − µ)−1(xi − g · xi) = (1 − µ)−1(xi − µxi) = (1 − µ)−1(1 − µ)xi = xi. +Hence xi ∈ ⟨{f − g · f | g ∈ Z/ℓZ, f ∈ C[X]}⟩. +Now we show {f − g · f | g ∈ Z/ℓZ, f ∈ C[X]} ⊂ ⟨VZ/ℓZ⟩. If f − g · f ̸= 0 then without loss of generality f +is a monomial and there exists 1 ≤ i ≤ k such that xi divides f. Then xi must also divide g · f as g simply +scales xi therefore f − g · f ∈ ⟨VZ/ℓZ⟩. +□ +Define +C[X](0) := +C[X] +⟨C[X]̸=0⟩. +The next lemma refines the statement of Proposition 5.1, claiming that the fixed point subscheme can be +written as follows +XZ/ℓZ = Spec C[X](0). +Lemma 5.2. Assume that X is smooth. Then XZ/ℓZ = Spec C[X](0) is smooth. In particular, C[X](0) is +reduced. +Proof. We first show equality of sets +XZ/ℓZ = Spec C[X](0). +Let f ∈ C[X] be homogeneous of degree d, g ∈ Z/ℓZ and p ∈ XZ/ℓZ then we have +f(p) = f(g−1 · p) = (g · f)(p) = gdf(p) +hence f(p) = 0 if d ̸= 0 mod ℓ. +Therefore ⟨C[X]̸=0⟩ is contained in the maximal ideal defining p and +XZ/ℓZ ⊂ Spec C[X](0). +Conversely Z/ℓZ acts trivially on C[X](0) and so every point in Spec C[X](0) is fixed by Z/ℓZ. Hence +Spec C[X](0) ⊂ XZ/ℓZ. +It remains to show that Spec C[X](0) is reduced. We must show that the localisation of C[X](0) at each +point is regular. By Theorem 5.3 it is enough to show that the tangent cone of Spec C[X](0) at a fixed point +p ∈ XZ/ℓZ is a polynomial ring. Since X is regular at p the tangent cone at p of X is equal to V := Tp(X) +as a Z/ℓZ-module. By [16, Theorem 5.2] the tangent cone of XZ/ℓZ at a point p is equal to V Z/ℓZ. It is then +clear that +V Z/ℓZ = Spec C[V ](0) +is affine space and C[V ](0) is a polynomial ring. +□ +The next stage in proving (5.3) is showing that the irreducible components of Xc(SnZ/ℓZ) are the at- +tracting sets. To do so we need two powerful theorems, Theorem 5.4 and Theorem 5.5 these are [8, Theorem +2.3] and [8, Theorem 2.5] respectively. These use the concept of definite actions, for the readers benefit we +include the definition here. +Definition 5.2. Let G be an algebraic torus, then G ∼= C× × · · · × C× = (C×)n for some positive integer n. +Any G-module can be written as a direct sum of one dimensional G-modules. If V is a G-module, then we +can find a basis {vi} of V such that +(g1 · · · gn) · vi = gsi1 +1 +· · · gsin +n vi for (g1 · · · gn) ∈ G. +16 + +The module V is positive (respectively negative) if +(1) sij ≥ 0 (respectively sij ≤ 0) for all i, j. +(2) For every i ∈ I there exists j such that sij ̸= 0. +The module is non-negative (respectively non-positive) if (1) is satisfied. The module is fully definite (re- +spectively definite) if there exists an isomorphism G ∼= C× × · · · × C× such that the module is positive +(respectively non-negative). +Definition 5.3. Let η : G × X → X be an action of a torus on X and let a ∈ XG be a closed point. The +action of η on a is fully definite (respectively definite) if the G-module Ta(X) is fully definite (respectively +definite). +In the above Ta(X) denotes the tangent space at a. +Theorem 5.4. Let X be irreducible and reduced. Let G be an algebraic torus. If the action of G on X is +definite at a ∈ XG then XG is irreducible +Theorem 5.5. Let G be an algebraic torus. Let the action of G on X be definite at a. If X is irreducible +then there exists an open G-invariant neighbourhood U of a which is G-isomorphic to (U ∩ XG) × V , where +V is a finite-dimensional (fully definite) G-module and the action of G on (U ∩ XG) × V is induced by the +trivial action of G on U ∩XG and the linear action on V (determined by the given structure of a G-module). +More specifically, in the proof of Theorem 5.5 the vector space V is defined to be the G-module complement +of Ta(XG) in Ta(X). In our case XG is a finite set and so Ta(XG) is zero and V = Ta(X). +The following lemma tells us that the irreducible components are precisely the attracting sets and also +that the attracting sets are equal to their own tangent space at the fixed point. Following the notation of +the previous two theorems set G = C×. +Lemma 5.3. Assume XC× is finite and non-empty and that limt→∞ t · x exists for all x ∈ X. If the action +of C× is definite at each fixed point then +(1) +X = +� +p∈XC× +Ωp, +where Ωp is the attracting set of p. The sets Ωp are the irreducible components of the space X. +(2) Ωp ∼= Tp(Ωp) as varieties. +Proof. 1.) Since limt→∞ t·x exists for each x ∈ X and limits are unique it follows that X is a disjoint union +of the sets Ωp. To see that the sets Ωp are the irreducible components note that for an arbitrary irreducible +component L we must have that LC× contains a single point. This is because of Theorem 5.4, which states +that if L is irreducible then so is LC×, but this is clearly not the case if it consists of more than one fixed +point. Furthermore it must contain at least one point, as if L is an irreducible component it equals its closure +and the closure of any non-empty C×-stable subset of X contains some fixed point. +If LC× = {p} then the fact that L is closed implies that limt→∞ t · x = p for all x ∈ L. Hence L ⊂ Ωp. +Conversely if x ∈ Ωp then C× · x is an irreducible subvariety containing x. +Since X is smooth, L is a +connected component of X. So p ∈ C× · x ∩ L ̸= ∅ implies C× · x ⊂ L and hence Ωp ⊂ L. +17 + +2.) Since p is a unique fixed point of Ωp we apply Theorem 5.5 to Ωp to conclude that there exists an open +neighbourhood U ⊂ Ωp containing p such that U ∼= (U ∩ XC×) × V . By the hypothesis we have ΩC× +p += p +and the discussion above states that V = Tp(Ωp), hence U ∼= {p} × Tp(Ωp) ∼= Tp(Ωp). Now we show that +Ωp ⊂ U. Let x ∈ Ωp. Then limt→∞ t · x = p. Since U is an open neighbourhood of p we must have that +t · x ∈ U for some t. Recall that U is C×-invariant and so if t · x ∈ U then we must have x ∈ U. Hence +U = Ωp and Ωp ∼= Tp(Ωp). +□ +Proposition 5.2. Assume that Y is a smooth, affine scheme over C with Y C× finite. Let C[Y ] = A and +Y + = +� +y ∈ Y +�� lim +t→∞ t · y exists +� +. +Then: +(1) Y + is a closed subset of Y , defined by the vanishing of the (reduced) ideal ⟨A<0⟩. +(2) Y + = � +p∈Y C× Ωp, where Ωp ∼= (TpY )>0 as C×-varieties. +(3) If IA0(p) = {a ∈ A0 | a(p) = 0} then Ωp is the closed subset of Y defined by the reduced ideal +⟨A<0, IA0(p)⟩. +Proof. (1) Similarly to the proof of Lemma 5.2 we have that ⟨A<0⟩ vanishes on Y +. Indeed if f ∈ A<0 is +homogeneous of degree r < 0, y ∈ Y + and t ∈ C× then +f(t · y) = t−rf(y) +and +lim +t→∞ f(t · y) = lim +t→∞ t−rf(y). +If f(y) ̸= 0 then the limit of f(t·y) does not exist. This is a contradiction. Therefore ⟨A<0⟩ vanishes on Y +. +Clearly Spec (A/⟨A<0⟩) ⊂ Y hence Spec (A/⟨A<0⟩)C× ⊂ Y C× is finite. The ring A/⟨A<0⟩ is non-negatively +graded, so Spec (A/⟨A<0⟩)+ = Spec (A/⟨A<0⟩). By Lemma 5.3 we see that Spec (A/⟨A<0⟩) is a disjoint +union of attracting sets, in particular all of its limits exist and so it is the the vanishing ideal defining Y +. +Now we must check that the ideal ⟨A<0⟩ is reduced. Since it is homogeneous, the radical of A/⟨A<0⟩ is +homogeneous. Hence, if it is not zero there exists a fixed point that is not reduced. Therefore, it suffices to +show that for every p ∈ (Spec A/⟨A<0⟩)C×, the local ring (A/⟨A<0⟩)p is reduced. +Let m denote the maximal ideal corresponding to p, so m is stable under C×. We show that Am/A<0Am +is a regular local ring. Let (T ∗ +p Y )<0 ⊂ m/m2 be the subspace spanned by all negative weight vectors and +choose N ⊂ m a homogeneous vector space lift of (T ∗ +p Y )<0. We fix another homogeneous vector space +lift V ⊂ m of m/m2 that contains N. A basis of V is a regular system of parameters for Am. If n is the +augmentation ideal of SymV then the map SymV → A induces graded isomorphisms φq : SymV/nq → A/mq +and A/mq = Am/(mAm)q for all q ≥ 1 as A is regular at m. Since C× acts semisimply on A, the quotient +A ։ A/mq induces surjections Ai ։ (A/mq)i for all i. Hence (A/mq)<0 = (A<0 + mq)/mq. Therefore φq +restricts to +SymV<0 + nq +nq += (SymV/nq)<0 ∼= (A/mq)<0 = A<0 + mq +mq +. +Since N is a subspace of V defined by being the lift of the space of negative weightvectors we have NSymV ⊂ +(SymV )<0SymV . Since the action of C× on V is linear, NSymV = (SymV<0)SymV as any negativley +graded vector in SymV can be broken into a sum of monomials, which in particular are negativley graded +18 + +weightvectors. Now we argue that (A<0A+mq)/mq = (NA+mq)/mq. Clearly NA+mq/mq ⊂ A<0A+mq/mq +so we show the opposite inclusion. First note +A<0 + mq +mq += φq +�SymV<0 + nq +nq +� +⊂ φq +�NSymV + nq +nq +� += NA + mq +mq +. +Now note NA+mq +mq +is an ideal hence A<0A+mq +mq +⊂ NA+mq +mq +. Since A<0A+mq +mq += A<0Am+mq +mq +, [20, Lemma 2.1] +implies that A<0Am is generated by the regular sequence N. Since Am is regular this implies that Am/A<0Am +is a regular local ring. +(2) This is simply an application of Lemma 5.3 as Y + is smooth and the action of C× is definite at each +fixed point. +(3) Let a ∈ A0 and y ∈ Ωp, then by Lemma 5.2 this is reduced. +a(y) = t0a(y) = (t · a)(y) = a(t−1y) +hence a(y) is a constant. Therefore IA0(p) vanishes on Ωp and the zero set of ⟨IA0(p)⟩ is equal to Ωp as +sets. +□ +As explained in [3, p. 5] the fixed points set Xc(Sn ≀ Z/ℓZ)C× is precisely γ−1(0). Therefore, by Proposi- +tion 3.1, we can identify Irr Sn ≀ Z/ℓZ +∼ +−→ Xc(Sn ≀ Z/ℓZ)C× by the map quoℓ(λ) �→ xquoℓ(λ). It is now possible +to prove the equality (5.3). +Lemma 5.4. Assume that Xc(Sn ≀ Z/ℓZ) is smooth. Then there is an equality of varieties +SuppZc∆(quoℓ(λ)) = Ωquoℓ(λ). +Proof. Throughout this proof denote Xc(Sn ≀ Z/ℓZ) by X. Note that the Verma module is positively graded +with the degree zero part equal to 1 ⊗ quoℓ(λ). Let z ∈ Zc(Sn ≀ Z/ℓZ) be a negatively graded element. Then +z · x ⊗ quoℓ(λ) = zx ⊗ quoℓ(λ) = x(z ⊗ quoℓ(λ)) = 0, +as z ⊗ quoℓ(λ) has negative degree and ∆(quoℓ(λ)) is positively graded. +Therefore the annihilator of +∆(quoℓ(λ)) contains all the negatively graded elements of Zc(Sn ≀Z/ℓZ). If we denote the ideal generated by +the negatively graded elements by I− then I− ⊂ annZc∆(quoℓ(λ)). Hence SuppZc∆(quoℓ(λ)) ⊂ V (I−). By +Proposition 5.2 (1) and (2) we see that SuppZc∆(quoℓ(λ)) is contained in one of the connected components +of X+. Since xquoℓ(λ) ∈ SuppZc(∆(quoℓ(λ))) we have SuppZc∆(quoℓ(λ)) ⊂ Ωquoℓ(λ). We argue that this +containment is actually an equality by proving that dim SuppZc∆(quoℓ(λ)) = dim Ωquoℓ(λ). This suffices +since Ωquoℓ(λ) is an irreducible variety and SuppZc∆(quoℓ(λ)) a closed subset of Ωquoℓ(λ). +By Theorem 5.2 we have the equality SuppZc∆(quoℓ(λ)) = Spec End∆(quoℓ(λ)) and therefore the dimen- +sions are equal, dim SuppZc∆(quoℓ(λ)) = dim Spec End∆(quoℓ(λ)). But dim Spec End∆(quoℓ(λ)) is equal +to the Krull dimension of End∆(quoℓ(λ)). Since End∆(quoℓ(λ)) is a finite free module over C[h]Sn≀Z/ℓZ +it has Krull dimension equal to dim h by [7, Corollary 1.4.5]. From Lemma 5.3, we see that Ωquoℓ(λ) ∼= +Tquoℓ(λ)(Ωquoℓ(λ)), so we need show that dim Tquoℓ(λ)(Ωquoℓ(λ)) ≤ dim h. +Since xquoℓ(λ) is a fixed point we have the following inclusions of C×-submodules Tquoℓ(λ)(xquoℓ(λ)) ⊂ +Tquoℓ(λ)(Ωquoℓ(λ)) ⊂ Tquoℓ(λ)(X). +We can decompose Tquoℓ(λ)(X) = T− ⊕ T0 ⊕ T+ into the negatively +graded part, the degree zero part and the positively graded part. From [16, Theorem 5.2] we have that +Tquoℓ(λ)(xquoℓ(λ)) = Tquoℓ(λ)(XC×) = T0. Now Lemma 5.2 says that XC× is smooth hence Tquoℓ(λ)(xquoℓ(λ)) = +{0} and so T0 = {0}. The fixed point xquoℓ(λ) is in the smooth locus and [10, Theorem 7.8] implies that +19 + +Tquoℓ(λ)(X) is a symplectic vector space. The symplectic form on Tquoℓ(λ)(X) is C×-invariant hence its +non-degeneracy forces dim T− = dim T+. +Since X is smooth we have dim X = dim Tquoℓ(λ)(X). +Since +Zc(Sn ≀ Z/ℓZ) = C[X] is a finite free module over C[h]Sn≀Z/ℓZ ⊗ C[h∗]Sn≀Z/ℓZ we have +dim Zc(Sn ≀ Z/ℓZ) = dim C[h]Sn≀Z/ℓZ ⊗ C[h∗]Sn≀Z/ℓZ = 2 dim h. +This means that dim X = 2 dim h. Therefore dim T+ = dim h. Since Tquoℓ(λ)(Ωquoℓ(λ)) is positively graded +we have Tquoℓ(λ)(Ωquoℓ(λ)) ⊂ T+ hence dim Tquoℓ(λ)(Ωquoℓ(λ)) ≤ dim h. +□ +Thus the spectrum of the endomorphism ring of any given Verma module is equal to a corresponding +attracting set. Therefore, Spec End∆(quoℓ(λ)) can be realised as a subvariety of Xc(Sn ≀ Z/ℓZ). +Theorem 5.6. For any quoℓ(λ) ∈ Irr Sn ≀ Z/ℓZ we have an isomorphism of varieties +Spec End∆(quoℓ(λ)) = Ωquoℓ(λ). +Proof. Theorem 5.2 states Spec End∆(quoℓ(λ)) = SuppZc∆(quoℓ(λ)) for any quoℓ(λ) ∈ Irr Sn ≀ Z/ℓZ and, +by Lemma 5.4, SuppZc∆(quoℓ(λ)) = Ωquoℓ(λ). The claim follows. +□ +Recall that at the beginning of this section we claimed that it would be necessary to use the isomorphism +from Theorem 5.1 to generalise the results of section 4. A more refined version of the statement is given +below, in terms of the symmetric group and wreath product group. First, let us introduce some required +notation. Let Cℓ[n] denote the set of ℓ-cores such that for each γ ∈ Cℓ[n] +|γ| ≤ n and |γ| = n mod ℓ. +The following theorem is [9, Theorem 4.21], applied to the particular case of W = Snℓ. Recall that Xc(Snℓ) +admits a C×-action. Therefore, the group Z/ℓZ can be considered acting on Xc(Snℓ) by identifying Z/ℓZ +with the ℓth roots of unity. +Theorem 5.7. Assume that Xc(Snℓ) is smooth. Then Xc(Snℓ)Z/ℓZ is smooth and: +(1) There is a bijection γ → I(γ) between Cℓ[nℓ] and the irreducible components of Xc(Snℓ)Z/ℓZ such +that xλ ∈ I(γ) if and only if coreℓ(λ) = γ for λ ∈ P[nℓ]. +(2) Let γ ∈ Cℓ[nℓ] and r = (n − |γ|)/ℓ. There is an isomorphism of varieties +iγ : Xc(Sr ≀ Z/ℓZ) → I(γ). +This satisfies iγ(xl +µ) = x(quo♭ +ℓ)−1(µ) for all µ ∈ Pℓ[r] with ℓ-core γ. +We apply the above theorem to the case when γ = ∅. It is immediate that the fixed points xλ with λ +having trivial ℓ-core all lie in the irreducible component I(∅). Furthermore, there is an isomorphism +Xc(Sn ≀ Z/ℓZ) ∼= I(∅). +Theorem 5.7 also describes where the fixed points are mapped under the isomorphism. Since, in our case, +the quotient map quoℓ : P[nℓ] → Pℓ[n] is a bijection by [9, Lemma 4.7], we have i∅(xquoℓ(λ)) = xλ. This fact +will be of upmost importance to us and so we record it as a lemma. +Lemma 5.5. There is a C×-equivariant isomorphism i∅ : Xc(Sn ≀Z/ℓZ) → I(∅) such that under the labeling +of the fixed points we have i∅(xquoℓ(λ)) = xλ for λ ∈ P[nℓ]. +20 + +The above lemma lets us prove the following. +Proposition 5.3. The map i∅ restricts to a C×-equivariant isomorphism of attracting sets +i∅ : Ωquoℓ(λ) ∼= ΩZ/ℓZ +λ += Ωλ ∩ Xc(Snℓ)Z/ℓZ. +Proof. Since the map i∅ is C×-equivariant it maps attracting sets to attracting sets hence +i∅(Ωquoℓ(λ)) ⊂ ΩZ/ℓZ +λ +and +i−1 +∅ (ΩZ/ℓZ +λ +) ⊂ Ωquoℓ(λ) +and so i∅ : Ωquoℓ(λ) → ΩZ/ℓZ +λ +is a bijective morphism. Since a bijective morphism between smooth varieties +is an isomorphism, the result follows. +□ +To summarise we have an equality of varieties +Spec End∆(quoℓ(λ)) = Ωquoℓ(λ), +and now an isomorphism +Ωquoℓ(λ) ∼= ΩZ/ℓZ +λ +. +This gives us a way to relate the endomorphism rings of the Verma modules for the symmetric and wreath +product groups. Unfortunately, this is not enough to arrive at an explicit presentation of the endomorphism +rings of the baby Verma modules. To do that we must understand a particular isomorphism explicitly. In +[14, Theorem 11.16] Etingof and Ginzburg construct an isomorphism between Zc(Sn ≀ Z/ℓZ) and a suitable +Calogero-Moser space. We now focus on describing this map and showing it has the properties we desire. +Recall the Calogero-Moser space can be defined as the quotient variety +MZ/ℓZ,n,c = MZ/ℓZ,n,c/PGLZ/ℓZ,n,c +where +MZ/ℓZ,n,c = {∇1, ∇2 ∈ End(Cn ⊗ CZ/ℓZ)| [∇1, ∇2] = kℓ · o ⊗ eZ/ℓZ + IdCn ⊗ c′ for some o ∈ O}. +We require an explicit understanding of the isomorphism i∅ by Bonnafé and Maksimau. In their paper this +map is given by the inclusion map between the Calogero-Moser spaces associated to Xc(Sn ≀ Z/ℓZ) and +Xc(Snℓ). +By [14, Theorem 1.7] there is an identification of Spec Zc(Sn ≀ Z/ℓZ) with IrrHc(Sn ≀ Z/ℓZ) given by the +assignment +p → Hc(Sn ≀ Z/ℓZ)e ⊗Z p, +∀p ∈ MaxSpec Zc(Sn ≀ Z/ℓZ) +(5.4) +where p is viewed as a homomorphism Zc(Sn ≀ Z/ℓZ) → C. Consider an irreducible Hc(Sn ≀ Z/ℓZ)-module +E. Let Γn−1 denote the subgroup of Γ = Sn ≀ Z/ℓZ that stabilises the first basis vector x1 in h. Let EΓn−1 +denote the subspace of E fixed by Γn−1. Clearly x1 and y1 commute with the action of Γn−1. Therefore we +can define operators X, Y ∈ EndC(EΓn−1) via the action of x1 and y1 on E respectively. The isomorphism +φ : IrrHc(Sn ≀ Z/ℓZ) → MZ/ℓZ,n,c [14, Theorem 11.16] is given by φ(E) = (X, Y ). +Consider the open set U in IrrHc(Sn ≀ Z/ℓZ) where the action of the elements xi − ωkxj are invertible; U +is an open set. Let (λ, µ) ∈ C2n with λℓ +i ̸= λℓ +j for all i ̸= j. Let O(λ,µ) denote the orbit of (λ, µ) under the +21 + +group Sn ≀ Z/ℓZ. This is a free orbit, so |O(λ,µ)| = n!ℓn. Up to isomorphism, each representation E in U is +of the form E(λ,µ) = C[Oλ,µ]. A basis of E(λ,µ) is given by the characteristic equations +χs(a, b) = + + + +1 if s · (a, b) = (λ, µ) +0 else +for s ∈ Sn ≀ Z/ℓZ. +The subspace EΓn−1 is then ℓn-dimensional with basis χs1,iγr +1 for 1 ≤ i ≤ a and +0 ≤ r ≤ ℓ − 1. The action of Hc(Sn ≀ Z/ℓZ) on E(λ,µ) is given by +xi · F(a, b) = aiF(a, b), yi · F(a, b) = bi · F(a, b) + c0 +� +j̸=i +ℓ−1 +� +k=0 +si,jγk +i γ−k +j +F(a, b) +ω−kaj − ai ++ +ℓ−1 +� +k=1 +ckγk +i F(a, b) +aiωk − ai +. +and +(w · F)(a, b) = F(w−1 · a, w−1 · b), +for w ∈ Sn ≀Z/ℓZ and ω a primitive ℓth root of unity. We must check that these equations satisfy the defining +relations of the rational Cherednik algebra at t = 0. These relations can be written [12, p. 22] as +[xi, xj] = 0, +[yi, yj] = 0 +[yi, xi] = c0 +� +i̸=j +ℓ−1 +� +k=0 +sijsk +i s−k +j ++ +ℓ−1 +� +k=1 +cks−k +j +[yi, xj] = −c0 +ℓ−1 +� +k=0 +sijωksk +i s−k +j . +The check is just a straight forward computation which we include for completeness +[yi, xt] · F(a, b) = yi · xt · F(a, b) − xt · yi · F(a, b) += xt + +bi · F(a, b) + c0 +� +j̸=i +ℓ−1 +� +k=0 +si,jγk +i γ−k +j +F(a, b) +ω−kaj − ai ++ +ℓ−1 +� +k=1 +ckγk +i F(a, b) +aiωk − ai + + − yi · atF(a, b) += atbiF(a, b) + c0 +� +j̸=i,j̸=t +ℓ−1 +� +k=0 +si,jγk +i γ−k +j +atF(a, b) +ω−kaj − ai ++ c0 +ℓ−1 +� +k=0 +si,tγk +i γ−k +t +ωkaiF(a, b) +ω−kat − ai ++ +ℓ−1 +� +k=1 +atckγk +i atF(a, b) +aiωk − ai +− + +atbi · F(a, b) + c0 +� +j̸=i +ℓ−1 +� +k=0 +atsi,jγk +i γ−k +j +F(a, b) +ω−kaj − ai ++ +ℓ−1 +� +k=1 +atckγk +i F(a, b) +aiωk − ai + + += c0 +ℓ−1 +� +k=0 +(ωkai − at)si,tγk +i γ−k +t +ωkaiF(a, b) +ω−kat − ai += c0 +ℓ−1 +� +k=0 +ωksi,tγk +i γ−k +t +F(a, b). +Now we check the relation +[yi, xi] = c0 +� +i̸=j +ℓ−1 +� +k=0 +sijsk +i s−k +j ++ +ℓ−1 +� +k=1 +cks−k +j . +So +[yi, xi] · F(a, b) = yi · xi · F(a, b) − xi · yi · F(a, b) +22 + += xi · + +bi · F(a, b) + c0 +� +j̸=i +ℓ−1 +� +k=0 +si,jγk +i γ−k +j +F(a, b) +ω−kaj − ai ++ +ℓ−1 +� +k=1 +ckγk +i F(a, b) +aiωk − ai + + − yi · aiF(a, b) += aibiF(a, b) + c0 +� +j̸=i +ℓ−1 +� +k=0 +si,jγk +i γ−k +j +ω−kajF(a, b) +ω−kaj − ai ++ +ℓ−1 +� +k=1 +ckγk +i ωkaiF(a, b) +aiωk − ai +) +−ai + +bi · F(a, b) + c0 +� +j̸=i +ℓ−1 +� +k=0 +si,jγk +i γ−k +j +F(a, b) +ω−kaj − ai ++ +ℓ−1 +� +k=1 +ckγk +i F(a, b) +aiωk − ai + + += c0 +� +j̸=i +ℓ−1 +� +k=0 +(ω−kaj − ai)si,jγk +i γ−k +j +F(a, b) +ω−kaj − ai ++ +ℓ−1 +� +k=1 +(ωkai − ai)ckγk +i F(a, b) +aiωk − ai += c0 +� +j̸=i +ℓ−1 +� +k=0 +si,jγk +i γ−k +j +F(a, b) + +ℓ−1 +� +k=1 +ckγk +i F(a, b) +as required. +Theorem 5.8. Let φ : Spec Zc(Sn≀Z/ℓZ) → MZ/ℓZ,n,c be the map defined above. Then we have the following +equality +φ∗(tr(X)k) = + + + +ℓ(xk +1 + ... + xk +n) if ℓ|k +0 else +Proof. We begin by noting that Zc(Sn ≀ Z/ℓZ) is reduced since [10, Proposition 7.2] states that eHc(Sn ≀ +Z/ℓZ)e ∼= Zc(Sn ≀ Z/ℓZ) is a domain. Hence for f, g ∈ Zc(Sn ≀ Z/ℓZ), f = g if and only if f(p) = g(p) for +all p ∈ maxSpec Zc(Sn ≀ Z/ℓZ). Furthermore Spec Zc(Sn ≀ Z/ℓZ) is irreducible, therefore f = g if and only +if f(p) = g(p) for all p ∈ U ⊂ Spec Zc(Sn ≀ Z/ℓZ), where U is the open set defined above. We also use the +identification of Spec Zc(Sn ≀ Z/ℓZ) with IrrHc(Sn ≀ Z/ℓZ) as in equation (5.4) above. Fix an irreducible +module E(λ,µ) ∈ U. +We shall first calculate φ∗(tr(X))k(E(λ,µ)). +As described in Section 2.9, EΓn−1 +(λ,µ) is +isomorphic as a C(Z/ℓZ)-modules to n copies of C(Z/ℓZ). Therefore, EΓn−1 +(λ,µ) can be viewed a sum of vector +spaces V0 ⊕ · · · ⊕ Vℓ−1 where Vi is n copies of the irreducible representation of Z/ℓZ where the generator s +acts by ωi. If we denote the action of x1 on Vi by Xi then note that Xi : Vi → Vi+1 as +s · Xi(v) = s · x1v = ωx1s · v = ωi+1x1v = ωi+1Xi(v) if v ∈ Vi. +Recall that φ(E) = (X, Y ), where X = x1 acting on EΓn−1. Then we see that as a matrix, +X = + + +0 +0 +. . . +Xℓ−1 +X0 +0 +. . . +0 +0 +... +0 +0 +0 +. . . +Xℓ−2 +0 + + +. +(5.5) +Hence φ∗(tr(X)k)(E(λ,µ)) = tr(Xk) and if ℓ ̸ | k then tr(Xk) = 0 as Xk has every entry on the main diagonal +equal to 0. However if ℓ|k then write k = mℓ and tr(Xk) = ℓtr((X0 · · · Xℓ−1)m). For each Xi, we have +Xi = diag(λ1, · · · , λn), hence +tr((X0 · · · Xℓ−1)m) = λmℓ +1 ++ · · · + λmℓ +n += λk +1 + · · · + λk +n. +Substituting back in we find +tr(Xk) = ℓ(λk +1 + · · · + λk +n). +23 + +Now we must check that ℓ(xk +1 + · · · + xk +n)(u) = ℓ(λk +1 + · · · + λk +n)(u) for all u ∈ E. For each F ∈ E(λ,µ), +(xk +1 + · · · + xk +n · F)(a, b) = (λk +1 + · · · + λk +n)F(a, b) hence xk +1 + · · · + xk +n acts by scalar multiplication on E(λ,µ) +by λk +1 + · · · + λk +n. +□ +Another map we must explicitly understand is the following. +Lemma 5.6. There is an isomorphism +α : Cn/(Sn ≀ Z/ℓZ) +∼ +−→ (Cnℓ/Snℓ)Z/ℓZ +given by +α(a1, a2, · · · , an) = (a1, ωa1, ω2a1, · · · , ωℓ−1an) +Recall the map i∅ introduced in Lemma 5.5, we must break this into a composition of three maps. Consider +the inclusion map on quiver varieties +i∅ : MZ/ℓZ,n,c → MZ/Z,nℓ,c +given by sending (X0, . . . Xℓ−1) to the matrix X of equation (5.5) and (Y0, . . . , Yℓ−1) to + + +0 +0 +. . . +Yℓ−1 +Y0 +0 +. . . +0 +0 +... +0 +0 +0 +. . . +Yℓ−2 +0 + + +. +Then the map i∅ of Lemma 5.5 is given by φ−1 +Snℓ ◦ i∅ ◦ φSn≀Z/ℓZ. With all the appropriate maps introduced +we can perform the following diagram chase. +Theorem 5.9. There is an isomorphism Xc(Sn ≀Z/ℓZ) → Y to a connected component of Xc(Snℓ)Z/ℓZ such +that the following diagram commutes +Xc(Sn ≀ Z/ℓZ) +Xc(Snℓ)Z/ℓZ +Cn/(Sn ≀ Z/ℓZ) +(Cnℓ/Snℓ)Z/ℓZ +φ−1 +Snℓ ◦ i∅ ◦ φSn≀Z/ℓZ +πn,ℓ +πnℓ +α +Proof. The first step is to unpack the diagram by introducing the Calogero-Moser spaces +Xc(Sn ≀ Z/ℓZ) +Xc(Snℓ) +Cn/(Sn ≀ Z/ℓZ) +(Cnℓ/Snℓ)Z/ℓZ +MZ/ℓZ,n,c +MZ/Z,nℓ,c +φSn≀Z/ℓZ +i∅ +φ−1 +Snℓ +πn,ℓ +πnℓ +α +. +24 + +It is easier understand the duals of the maps in the diagram above and since a diagram commutes if and +only if its dual does we shall prove this instead. We must therefore prove the commutivity of the following +diagram +Zc(Sn ≀ Z/ℓZ) +Zc(Snℓ) +C[Cn/(Sn ≀ Z/ℓZ)] +C[(Cnℓ/Snℓ)Z/ℓZ] +C[MZ/ℓZ,n,c] +C[MZ/Z,nℓ,c] +φ∗ +Sn≀Z/ℓZ +i +∗ +∅ +(φ∗)−1 +Snℓ +in,ℓ +inℓ +α∗ +. +First note that C[(Cnℓ/Snℓ)Z/ℓZ] are the symmetric polynomials fixed under the action of the ℓth roots of +unity. Hence it is generated by elements of the form +xkℓ +1 + · · · + xkℓ +nℓ where k ∈ Z≥0. +Now α∗(f)(p) = f(α(p)), where p ∈ Cn/Sn ≀ Z/ℓZ. Consider an arbitrary generator xkℓ +1 + · · · + xkℓ +nℓ, then +α∗(xkℓ +1 + · · · + xkℓ +nℓ)(a1, a2, · · · , an) = (xkℓ +1 + · · · + xkℓ +nℓ)(a1, ωa1, · · · , ωℓ−1an) += akℓ +1 + (ωa1)kℓ + (ω2a1)kℓ + · · · + (ωℓ−1an)kℓ += (1 + ωkℓ + (ω2)kℓ · · · + (ωℓ−1)kℓ)(akℓ +1 + akℓ +2 + · · · + xkℓ +n ) = ℓ(akℓ +1 + · · · + akℓ +n ). +Hence +α∗(xkℓ +1 + · · · + xkℓ +nℓ) = ℓ(xkℓ +1 + · · · + xkℓ +n ). +Recall the map π is the dual of the inclusion map so +in,ℓ(ℓ(xkℓ +1 + · · · + xkℓ +n )) = ℓ(xkℓ +1 + · · · + xkℓ +n ). +Now we must chase the diagram the other way. The first map inℓ is also the inclusion map hence +inℓ(xkℓ +1 + · · · + xkℓ +nℓ) = xkℓ +1 + · · · + xkℓ +nℓ. +By Theorem 5.8, we have that φ∗ +Snℓ(tr(X)k) = xk +1 + · · · + xk +nℓ, therefore (φ∗ +Snℓ)−1(xkℓ +1 + · · · + xkℓ +nℓ) = tr(X)kℓ. +Then we have by definition +(i +∗ +∅)−1(tr(X)kℓ) = tr(i∅(X)kℓ). +Therefore we complete the proof by showing φ∗ +Sn≀Z/ℓZ(tr(i∅(X)kℓ) = ℓ(xkℓ +1 + · · · + xkℓ +n ). This is precisely the +statement of Theorem 5.8. +□ +Before presenting the main theorem of this section we fix some notation. Recall by Corollary 4.1 that +A(λ)+ ∼= C[π−1(0)] and, we denote the functions on the fixed point locus as +A(λ)+ +Z/ℓZ := C[π−1 +nℓ (0)]/⟨f − s · f | s ∈ Z/ℓZ, f ∈ C[π−1 +nℓ (0)]⟩. +Theorem 5.10. There is an isomorphism of algebras +A(quoℓ(λ))+ ∼= A(λ)+ +Z/ℓZ. +25 + +Proof. By definition, A(quoℓ(λ))+ := End∆(quoℓ(λ)) = C[Spec End∆(quoℓ(λ))]. By Proposition 5.3 there +is an isomorphism +i∅ : Ωquoℓ(λ) ∼= Ωλ ∩ Xc(Snℓ)Z/ℓZ. +Therefore Theorem 5.9 implies that there is a commutative diagram +Ωquoℓ(λ) +ΩZ/ℓZ +λ +Cn/(Sn ≀ Z/ℓZ) +(Cnℓ/Snℓ)Z/ℓZ +φ−1 +Snℓ ◦ i∅ ◦ φSn≀Z/ℓZ +πn,ℓ +πnℓ +α +. +By Theorem 5.6, Spec End∆(quoℓ(λ)) = Ωquoℓ(λ) and Spec End∆(λ) = Ωλ. Hence the diagram becomes +Spec End∆(quoℓ(λ)) = Ωquoℓ(λ) +ΩZ/ℓZ +λ += (Spec End∆(λ))Z/ℓZ +Cn/(Sn ≀ Z/ℓZ) +(Cnℓ/Snℓ)Z/ℓZ +i∅ +πn,ℓ +πnℓ +α +. +Since α and i∅ are both isomorphisms we have π−1 +n,ℓ(0) ∼= (π−1 +nℓ (0))Z/ℓZ. +Therefore there is an algebra +isomorphism C[π−1 +n,ℓ(0)] ∼= C[π−1 +nℓ (0)Z/ℓZ]. Finally, Corollary 4.1 implies that +A(quoℓ(λ))+ ∼=C[π−1 +n,ℓ(0)] +∼=C[π−1 +nℓ (0)Z/ℓZ] +∼=C[π−1 +nℓ (0)]/⟨f − s · f | s ∈ Z/ℓZ, f ∈ C[π−1 +nℓ (0)]⟩ +hence A(quoℓ(λ))+ = A(λ)+ +Z/ℓZ. +□ +Theorem 5.10 allows us to understand the endomorphism rings of the baby Verma modules for the wreath +product group in terms of the symmetric case. As we will see in the next section this will allow us to easily +generalise the explicit presentation of Zc(Sn) to Zc(Sn ≀ Z/ℓZ). +6. The Wronskian, Schubert cells and an Explicit presentation of A(λ)+ +Here we will introduce the Wronskian and explain why it allows us to give an explicit presentation of +A(λ)+ and therefore, an explicit presentation of Zc(Sn). The Wronskian of a set of polynomials {f1, · · · , fn} +26 + +is the determinant +Wr(f1, f2, · · · , fn) := det + + +f1 +f2 +f3 +. . . +fn +f (1) +1 +f (1) +2 +f (1) +3 +. . . +f (1) +n +... +... +... +... +... +f (n−1) +1 +f (n−1) +2 +f (n−1) +3 +. . . +f (n−1) +n + + +There is a closely related map called the Wronski map, it is actually this which we will be most interested in. +As we will explain next there is a connection to Schubert cells, a much studied object. Importantly, Schubert +cells are objects which can be described explicitly in terms of generators and relations. In their paper [25] +Mukhin, Tarasov and Varachenko describe how to write the ring of functions on a Schubert cell labelled by +a partitions λ ⊢ n explicitly. Let us describe their construction. Following the notation of Lemma 2.1 the +Schubert cell Ωqe +λ consists of subspaces X with basis +fi = udi + +di +� +j=1, di−j̸∈P +fi,judi−j. +(6.1) +The algebra C[Ωqe +λ ] is a free polynomial algebra with generators fij, i.e. +C[Ωqe +λ ] = C[fij, i = 1, . . . , n, j = 1, . . . di, di − j ̸∈ P]. +Note that this is a graded algebra, we set deg(fij) = j. The Wronskian of a basis of X is a polynomial of +degree n. We write +Wr(f1, · · · , fn) = un + r1un−1 + · · · + rn. +The Wronski map is defined on elements by +Wrλ(X) = (a1, · · · , an) if Wr(X) = un + +n +� +i=1 +(−1)iaiun−i. +Therefore the scheme theoretic fibre of the Wronski map is +C[Wr−1 +λ (a)] ∼= C[Ωqe +λ ]/Iλ,a, +(6.2) +where Iλ,a is the ideal generated by the rs − (−1)sas. +Thanks to the isomorphism (6.2) we see that the scheme theoretic fibre of the Wronski map can be written +explicitly as we know how to do so for the ring of functions on the Schubert cell. It only remains for us to +connect the notion of the Wronski map with the map π from sections 4 and 5. This is done below in the +following theorem, but it is extremely important to note this holds for the symmetric group case only. +Theorem 6.1. If W = Sn, there is an isomorphism of varieties +π−1(0) ∼= Wr−1 +λ (0). +where π is the map on spectra π : Spec End∆(λ) → h/Sn induced by the inclusion map C[h]Sn ֒→ End∆(λ). +Proof. See [4, Proposition 6.4]. +□ +Thanks to the results from section 4 we easily deduce the following. +Theorem 6.2. There is an isomorphism of algebras +A(λ)+ ∼= C[Wr−1 +λ (0)]. +27 + +Proof. Theorem 6.1 states π−1(0) ∼= Wr−1 +λ (0), hence C[π−1(0)] = C[Wr−1 +λ (0)]. By Corollary 4.1 we have +that A(λ)+ ∼= C[π−1(0)] and therefore A(λ)+ ∼= C[Wr−1 +λ (0)]. +□ +Consider the case a = 0 in (6.2), Theorem 6.2 implies that A(λ)+ is the quotient of C[Ωqe +λ ] by the ideal +generated by the coefficients rs of the polynomial Wr(f1, · · · , fn). Let us now describe how to find the algebra +A(λ)+ explicitly in terms of generators and relations in the symmetric case. Given λ ⊢ n define positive +integers di = λi + n − i and denote the set of these by P = {d1, · · · , dn}. Then calculate the Wronskian +Wr(f1, f2, · · · , fn) = det + + +f1 +f2 +f3 +. . . +fn +f (1) +1 +f (1) +2 +f (1) +3 +. . . +f (1) +n +... +... +... +... +... +f (n−1) +1 +f (n−1) +2 +f (n−1) +3 +. . . +f (n−1) +n + + += un + r1un−1 + · · · + rn +of the polynomials +fi = udi + +di +� +j=1, di−j̸∈P +fijudi−j. +The algebra A(λ)+ is given by taking the polynomial algebra generated by the fij and quotienting by the +coefficients rs of the Wronskian. +Example 6.1. For the partition λ = (3, 2) we have d1 = 7, d2 = 5, d3 = 2, d4 = 1 and d5 = 0. Therefore +f1(u) = u7 + f11u6 + f13u4 + f14u3, f2(u) = u5 + f21u4 + f22u3, f3(u) = u2, f4(u) = u and f5(u) = 1. Let +us calculate the Wronskian, which is +det + + +u7 + f11u6 + f13u4 + f14u3 +u5 + f21u4 + f22u3 +u2 +u +1 +7u6 + 6f11u5 + 4f13u3 + 3f14u2 +5u4 + 4f21u3 + 3f22u2 +2u +1 +0 +42u5 + 30f11u4 + 12f13u2 + 6f14u +20u3 + 12f21u2 + 6f22u +2 +0 +0 +210u4 + 120f11u3 + 24f13u + 6f14 +60u2 + 24f21u + 6f22 +0 +0 +0 +840u3 + 360f11u2 + 24f13 +120u + 24f21 +0 +0 +0 + + +. +Hence +Wr(f1(u), f2(u), f3(u), f4(u), f5(u)) = 25200u5 + (14400f11 + 30240f21)u4 ++(11520f11f21 + 10080f22)u3 + (−2880f13 + 4320f11f22)u2 − 1440f14u + (−288f14f21 + 288f13f22). +Then A(3, 2)+ is the quotient by the ideal generated by the coefficients, this can be equivalently written as +A(3, 2)+ = C[f11, f13, f14, f21, f22]/(10f11 − 21f21, 8f11f21 + 7f22, 2f13 − 3f11f22, f14, f14f21 − f13f22). +Hence +A(3, 2)+ ∼= C[f11]/(f 5 +11). +To be able to construct the algebras A(λ)+ directly from the partition λ ⊢ n will require a greater +understanding of both the Wronski map and the dimension of A(λ)+. It is the latter of these that we now +focus on, we shall address the former in the next section. +The following results are inspired by the formula +sλ(1, q, · · · , qn) = +�n +i=1(1 − qi) +� +(i,j)∈Dλ(1 − qh(i,j)). +28 + +This can be found in [26, p. 364]. The term sλ denotes the Schur function associated to the partition λ ⊢ n, +Dλ denotes the Young diagram of λ and h(i, j) is the hook length of the box (i, j). A similar formula allows +us to calculate the graded dimension of A(λ)+. +We now record two general lemmata about the Wronskian and certain homogeneous polynomials. +Lemma 6.1. If f1, · · · , fn, is a family of homogeneous polynomials and Wr(f1, · · · , fn) ̸= 0 then the Wron- +skian is homogeneous and deg(Wr(f1, · · · , fn)) = � +i(deg(fi)) − (n−1)(n) +2 +. +Proof. We proceed by induction on n, the case n = 1 being trivial. Suppose the lemma holds for all positive +integers less than n. Then +det + + +f1 +f2 +f3 +. . . +fn +f (1) +1 +f (1) +2 +f (1) +3 +. . . +f (1) +n +... +... +... +... +... +f (n−1) +1 +f (n−1) +2 +f (n−1) +3 +. . . +f (n−1) +n + + += +n +� +i=1 +(−1)i+1fi + + +f (1) +1 +ˆf (1) +i +. . . +f (1) +n +f (2) +1 +ˆf (2) +i +. . . +f (2) +n +... +... +... +... +f (n−1) +1 +ˆf (n−1) +i +. . . +f (n−1) +n + + +where theˆsymbol denotes an omitted column. By the inductive hypothesis, each of the components of the +sum is a homogeneous polynomial of degree +deg(fi) + +n +� +j,j̸=i +deg(f (1) +j +) − (n − 2)(n − 1) +2 += deg(fi) + +n +� +j,j̸=i +deg(fj) − (n − 1) − (n − 2)(n − 1) +2 +. +This then simplifies to +n +� +i=1 +deg(fi) − (n − 1) − (n − 2)(n − 1) +2 += +n +� +i=1 +deg(fi) − (n − 1)n +2 +. +□ +In the following lemma it is important to recall that when considering the polynomials defined in 6.1 the +generators fij have a degree. For instance the polynomial +u3 + f12u +is homogeneous as both u3 and f12u have degree 3. +Lemma 6.2. Let λ ⊢ n be a partition of n, and {f1, · · · , fn} the family of polynomials as defined in (6.1). +The Wronskian Wr(f1, · · · , fn) is a homogeneous polynomial of degree n. +Proof. By Lemma 6.1 if the Wronskian is non-zero then it is a homogeneous polynomial with degree +� +i(deg(fi)) − (n−1)(n) +2 +. We have deg(fi) = di = λi + n − i. Hence +n +� +i +(deg(fi)) − (n − 1)(n) +2 += +n +� +i +λi + n − i − (n − 1)(n) +2 += +n +� +i +λi + +n +� +i +n + +n +� +i +−i − (n − 1)(n) +2 +, +and +n +� +i +λi + +n +� +i +n + +n +� +i +−i − (n − 1)(n) +2 += n + n2 − n(n + 1) +2 +− (n − 1)(n) +2 += n. +□ +It will be important to us that A(λ)+ is a complete intersection. to prove this we need to use the following +non-trivial fact which is can be found in [25, Lemma 3.11]. +29 + +Lemma 6.3. The algebra A(λ)+ is finite dimensional. +Lemma 6.4. For any λ ⊢ n, the algebra A(λ)+ is a complete intersection. +Proof. The algebra A(λ)+ has n generators and n relations. Therefore, it is a complete intersection if and +only if its Krull dimension is zero. Since A(λ)+ is finite dimensional as a vector space it is Artinian and +therefore every prime ideal is maximal [1, Proposition 8.1]. Hence it has Krull dimension zero. +□ +Assume that C[x1, . . . , xm]/(g1, · · · , gn) is graded with deg(xi) = ai > 0, so that each gj is homogeneous, +of degree bj say. The Hilbert-Poincaré polynomial of A is defined to be +P(A, q) := +� +i≥0 +(dim Ai)qi. +Lemma 6.5. If A is a graded complete intersection then +P(A, q) = +�t +i=1(1 − qbi) +�n +j=1(1 − qaj). +Proof. Since A is a complete intersection we can write +A = C[x1, . . . , xm]/(g1, . . . , gt). +Let S = C[x1, . . . , xm]. +The Koszul resolution [13, Chapter 17] can be used to resolve A as a graded +S-module. Let V = SpanC{g1, . . . , gt}, a graded vector space. Then +0 → ∧tV ⊗C S → ∧t−1V ⊗C S → · · · → ∧0V ⊗ S → A → 0 +is an exact sequence of graded S-modules. The maps in the Kozul resolution are dk : ∧tV ⊗CS → ∧t−1V ⊗CS +and defined on elements +dk(v1 ∧ · · · ∧ vt ⊗ f) = +t +� +i=1 +(−1)i+1v1 ∧ · · · ∧ ˆvi ∧ · · · ∧ vt ⊗ vif, +here ˆvi means the term vi is omitted. This implies (by the “Euler-Poincaré principle”) that +P(A, q) = +t +� +j=0 +(−1)j+1P(∧jV ⊗C S, q) += + + +t +� +j=0 +(−1)j+1P(∧jV, q) + + P(S, q). +We have +P(S, q) = + + +n +� +j=1 +(1 − qaj) + + +−1 +and +t +� +j=0 +(−1)j+1P(∧jV, q) = +t� +i=1 +(1 − qbi). +□ +We can now present the formula for calculating the graded dimension of A(λ)+. Recall that Dλ denotes +the Young diagram for a partition λ and h(i, j) is the hook length of the cell (i, j). +Theorem 6.3. For any A(λ)+, we have +� +i≥0 +(dim A(λ)+ +i )qi = +�n +i=1(1 − qi) +� +(i,j)∈Dλ(1 − qh(i,j)). +30 + +Proof. Since A(λ)+ is a complete intersection by Lemma 6.4, Lemma 6.5 implies that +� +i≥0 +(dim A(λ)+ +i )qi = P(A(λ)+, q) = +�t +i=1(1 − qbi) +�n +j=1(1 − qaj). +Lemma 2.2 says that �n +j=1(1 − qaj) = � +(i,j)∈Dλ(1 − qh(i,j)). By definition of A(λ)+, +t� +i=1 +(1 − qbi) = +n +� +i=1 +(1 − qi). +Hence, +� +i≥0 +(dim A(λ)+ +i )qi = +�n +i=1(1 − qi) +� +(i,j)∈Dλ(1 − qh(i,j)). +□ +The following example highlights how easy this formula is to use. +Example 6.2. The Young diagram for the partition λ = (3, 1, 0, 0) is +4 +2 +1 +1 +and therefore +� +i≥0 +(dim A(λ)+ +i )qi = (1 − q)(1 − q2)(1 − q3)(1 − q4) +(1 − q4)(1 − q2)(1 − q)(1 − q) = 1 − q3 +1 − q = 1 + q + q2. +Hence A(λ)+ consists of a one dimensional space in degrees 0, 1 and 2. +The formula for the graded dimensions allows us to calculate the dimension of the entire algebra A(λ)+. +Theorem 6.4. The dimension of A(λ)+ is given by the hook length formula +dim A(λ)+ = +n! +� +(i,j)∈Dλ h(i, j). +Proof. We have the formula +� +i≥0 +(dim Ai)qi = +�n +i=1(1 − qi) +� +(i,j)∈Dλ(1 − qh(i,j)). +We can use L’Hopitals rule to evaluate the formula when q = 1. Clearly the left hand side gives the dimension +of A. Repeated applications of L’Hoptials rule to the right hand side gives +n! +� +(i,j)∈Dλ h(i, j). +□ +31 + +7. Calculating A(λ)+ directly from the partition +As demonstrated in Section 6 we have an algorithm for calculating the algebras A(λ)+ using the Wron- +skian. Here we will refine this result and show that calculating the Wronskian is unnecessary. In fact, the +algebra A(λ)+ can be derived directly from the partition λ. Key to providing an explicit presentation of +A(λ)+ directly from a partition λ ⊢ n is understanding how the coefficients of the terms appear in the +Wronskian. We shall split the problem of understanding the coefficients into two distinct cases. We say that +terms of the form fi,j are linear and terms of the form fi1,j1fi2,j2 · · · fim,jm for m > 1 are non-linear. We +begin with the simpler task of understanding the linear terms. +The first question we want to answer is how often do linear terms appear in the coefficients of the +Wronskian for a given partition of n? It is fairly straight forward to see that each linear term can appear in +only one coefficient. Recall that algebra A(λ)+ can be written as +C[fij, i = 1, . . . , n, j = 1, . . . , di, di − j ̸∈ P] +(r1, . . . , rn) +where the rs are homogeneous elements and deg(rs) = s. Abusing terminology we say that “a monomial m +in the fi,j appears in rs” if the coefficient of m in rs is non-zero. +Lemma 7.1. If the linear term fi,j appears as a monomial in one of the elements rs, then j = s. +Proof. By Lemma 6.2 the Wronskian is a homogeneous polynomial of degree n. Also A(λ)+ is a complete +intersection and therefore the coefficient of ui is non-zero for all 0 ≤ i ≤ n. In other words, ri ̸= 0 for all i. +Therefore, a linear coefficient of ui has degree n − i. Since the linear term fi,j has degree j we conclude that +it can only appear in rj. +□ +The following is a partial converse to Lemma 7.1. +Lemma 7.2. Consider a finite dimensional commutative ring +A = C[x1, . . . , xn] +(r1, . . . , rn) , +where the relations ri are homogeneous and do not contain any constant terms. For each 1 ≤ j ≤ n, there +exist k ≥ 1 and i such that ri contains the monomial xk +j . +Proof. Argue by contradiction. Assume that xk +1 does not appear in any ri for k ≥ 1. Since the algebra is +finite dimensional and positively graded, we must have xm +1 = 0 for some m and hence +xm +1 = +� +l +clrl +for cl ∈ C. Since every monomial with non-zero coefficient in rj is divisible by some xi with i ̸= 0, there is +a well defined evaluation morphism evc : A → C that sends x1 to some constant c ̸= 0 and xi to 0 for i > 1. +Then +cm = evc(xm +1 ) = evc +�� +l +clrl +� += +� +l +clevc(rl) = 0, +which is a contradiction. +□ +Lemma 7.3. Let f1, · · · , fn be the set of pairwise distinct polynomials as in (6.1). All terms in the Wronskian +Wr(f1, · · · , fn) of the form f k1 +i,j · · · f kz +u,v have ks ≤ 1 for all 1 ≤ s ≤ z. +32 + +Proof. By writing the Wronskian +Wr(f1, · · · , fn) = det + + +f1 +f2 +f3 +. . . +fn +f (1) +1 +f (1) +2 +f (1) +3 +. . . +f (1) +n +... +... +... +... +... +f (n−1) +1 +f (n−1) +2 +f (n−1) +3 +. . . +f (n−1) +n + + +the statement becomes clearer. Each term in the determinant is some product of terms which do not share +a row or column. Fix an fi,j, then we see that it only appears in the ith column. Since the product of the +terms in the Wronskian cannot share a column we see that if f k +i,j appears then k = 1. +□ +Lemma 7.3 states that in the case of the Wronskian our coefficients can not contain terms of a higher power +than 1, for instance we cannot have f 2 +11 appearing in the relations. It is also clear that the coefficients of the +Wronskian contain no constant terms, since they are homogeneous of positive degree. These observations +give us the following lemma. +Lemma 7.4. Given a partition λ ⊢ n, each fi,j appears as a linear term of a coefficient in the Wronskian. +Proof. Follows from Lemma 7.3 and Lemma 7.2. +□ +Now that we have proven that each linear term must appear we can strengthen Lemma 7.1. Using the +same notation as before we have the following. +Proposition 7.1. Each linear term fi,j appears in rj and with non-zero coefficient. +Proof. Follows from Lemma 7.1 and Lemma 7.4. +□ +Proposition 7.1 completely solves the problem of understanding the position of the linear terms. The next +step is to prove an analogous statement for the non-linear terms. We will find that almost all non-linear +terms appear in the coefficients, except for a specific few. We will need the following results first. +Lemma 7.5. Let {f1, · · · , fn} be the set of polynomials defined as in (6.1). Assume there are polynomials +fi and fj such that fi contains a term of the form fi,suk and fj contains a term fj,tuk. Then the Wronskian +Wr(f1, · · · fn) contains no monomial divisible by fi,sfj,t. +Proof. The determinant is a sum of multiples of elements of different columns and different rows in the +matrix. The terms fi,s and fj,t only appear in the columns i and j respectively. Hence, all the terms with +form fi,sfj,t appearing in the determinant come from an expression of the form F(f (a) +i +f (b) +j +− f (b) +i +f (a) +j +). Here +F is some multiple of entries from different rows and columns excluding columns i and j. An easy calculation +gives fi,su(a)fj,tu(b) − fi,su(b)fj,tu(a) = 0. +□ +There is a useful recursive formula for the Wronskian that can be found in [23, Proposition 1]. +Proposition 7.2. The recursive formula for the Wronskian is given by +Wr(f1, · · · , fn) = f n +1 Wr +��f2 +f1 +�′ +, · · · , +�fn +f1 +�′� +where deg(fi) < deg(fj) for i < j. +33 + +We will use this formula to prove two lemmata that will be crucial in showing that most non-linear terms +are non-zero. +Lemma 7.6. Let {f1, · · · , fn} be a set of monomials in one variable such that deg(fi) > deg(fj) for i < j, +and fi ̸= 0 for all i. The Wronskian Wr(f1, · · · , fn) is non-zero. +Proof. Proceed by induction on n. The case where n = 1 is obvious. Assume the statement is true for all +m < n, then use the recursive formula given in Proposition 7.2 +Wr(f1, · · · , fn) = f n +1 Wr +��f2 +f1 +�′ +, · · · , +�fn +f1 +�′� +. +Clearly Wr(( f2 +f1 )′, · · · , ( fn +f1 )′) satisfies the assumptions of the lemma. Therefore Wr(( f2 +f1 )′, · · · , ( fn +f1 )′) is non- +zero. Since f n +1 ̸= 0, +Wr(f1, · · · , fn) = f n +1 Wr +��f2 +f1 +�′ +, · · · , +�fn +f1 +�′� +̸= 0. +□ +Lemma 7.7. Let {f1, · · · , fn} be a set of monomials in one variable, such that deg fi ̸= deg fj for all i ̸= j +and fi ̸= 0 for all i. The Wronskian Wr(f1, · · · , fn) is non-zero. +Proof. We see from Lemma 7.6 above that if deg(fi) > deg(fj) for i < j then the Wronskian is non-zero. +Since the monomials all have pairwise different degrees there is a matrix A that permutes the columns of the +matrix such that the monomials are in order of ascending degree in the first row. Then det Wr(f1, . . . , fn) = +det(A) det(Wr(f ′ +1, . . . , f ′ +n)) where f ′ +i > f ′ +j for i < j. +Since A is an invertible matrix its determinant is +non-zero and so det Wr(f1, . . . , fn) ̸= 0. +□ +This is the last result we require to state precisely where and which non-linear terms appear in the +coefficients of the Wronskian. There is just one convention we must establish first. Recall Lemma 2.2, which +stated that the set +{j | di − j ̸∈ P for 1 ≤ j ≤ di} +is equal to the set of hook lengths in the ith row of length λi. Also recall that +fi = udi + +di +� +j=1, di−j̸∈P +fi,judi−j. +Lemma 2.2 implies there is a bijection between the polynomials fi,j for a fixed i and the cells in the ith row +of λ. This bijection sends the cell (i, j) to fi,h(i,j). To demonstrate this bijection let us consider an example. +Example 7.1. Take the partition (3, 2). The Young diagram is +4 +3 +1 +2 +1 +. +The cells of the first row, read left to right, are mapped to f1,4, f1,3 and f1,1 respectively. The cells of the +second row are similarly mapped to the generators f2,2 and f2,1. +We say that two generators fi,j and fs,t share a row or column if they share a row or column in the Young +diagram under this mapping. +34 + +Theorem 7.1. Fix a partition λ ⊢ n and let f1, · · · , fn be as in (6.1). +The non-linear coefficients of +Wr(f1, · · · , fn) are all non-zero except for monomials divisible by fi,jfs,t where i = j or h(i, j) = h(s, t). In +other words, they have no factors that share a row or column in the partition diagram. +Proof. The first thing to note is that if two generators fi,j and fs,t share the same row in the partition +then i = s and they must appear in the same column in the Wronskian. Hence, they cannot appear in the +determinant. Assume now that fi,j and fs,t appear in the same column of Dλ. Lemma 2.3 says that +da − h(a, b) = dc − h(c, b) +holds for all a, b and c. In particular if fi,j and fs,t share the same column in the Young diagram then +di − j = ds − t. From the definition of the fi in (6.1), we see that in fi the monomial fi,j is the coefficient of +udi−j. Likewise fs,t is the coefficient of uds−t in fs. Lemma 7.5 then implies that there is no monomial in +the Wronskian that is divisible by fi,jfs,t. We need only prove that the other nonlinear terms are non-zero. +We prove this for products of two monomials, the general case follows from a similar argument using the +coefficients in Proposition 7.3. Assume that fi,j and fs,t share neither a row nor a column in the Young +diagram. In the determinant fi,jfs,t will be the coefficient of the un−j−t term. This observation lets us +see that when deciding if this is nonzero we need only consider entries in the Wronskian that are scalars +or powers of u. +That is we exclude all fy,z where fy,z ̸= fi,j or fs,t. +Hence we need only check that +Wr(ud1, · · · , fi,judi−j, · · · , fs,tuds−t, · · · , udn) is non-zero. This simplifies +Wr(ud1, · · · , fi,judi−j, · · · , fs,tuds−t, · · · , udn) = fi,jfs,tWr(ud1, · · · , udi−j, · · · , uds−t, · · · , udn). +Since fi,j and fs,t do not share a column in Dλ, Lemma 2.3 implies that di − j ̸= ds − t. Therefore, the +degrees of all the monomials are pairwise different and non-zero. In this case Lemma 7.7 implies that the +determinant is non-zero. +□ +These results can be improved upon by giving a formula for calculating the scalar coefficients of the linear +and non-linear terms in the Wronskian. Let us explain some necessary notation. Recall that the recursive +formula in Proposition 7.2 is given for a particular order of entries, namely that they are increasing in degree. +This is often not the case, and so we must permute the columns of the Wronskian first. Let A be the matrix +that permutes in the desired order. Note that det(A) = σ(A), where σ is the sign function. +Proposition 7.3. The scalar coefficient of a given term fi1j1 . . . fimjm is +σ(A) +� +idj and 1≤k≤m +(dik − jk − dl) +� +di−ik>dj−jl +(di − ik − dj − jl). +Proof. The proof is by induction, on the size of Wronskian, the case m = 1 is clear. Assume the statement +holds for all m up to n − 1. Consider the case m = n. We have +Wr(ud1, . . . , fi1j1udi1−j1, . . . , ud +n) = det(A)Wr(ud +n, . . . , fi1j1udi1−j1, . . . , ud1) += udnWr((udn−1/udn)′, . . . , fi1j1(udi1 −j1/udn)′, . . . , (ud +1/udn)′) += udnWr((dn−1 − dn)udn−1−dn−1, . . . , fi1j1(di1 − j1 − dn)udi1−j1−dn−1, . . . , (d1 − dn)ud1−dn−1) += +� +1≤idn +(dik − jk − dn)fi1j1 . . . fimjmWr(udn−1−dn−1, . . . , udi1−j1−dn−1, . . . , ud1−dn−1). +35 + +and therefore += σ(A) +� +idj and 1≤k≤m +(dik − jk − dl) +� +di−ik>dj−jl +(di − ik − dj − jl). +□ +By defining a new term e we can greatly simplify this expression. +Proposition 7.4. Consider the nonlinear term fi1j1 . . . fimjm ordered so that ia < ia+1. Define +eik = +� +dik − jk +if 1 ≤ k ≤ m +dik +else. +Then the scalar coefficient of the term of fi1j1 . . . fimjm in the Wronskian is +� +1≤idj and 1≤k≤m +(dik − jk − dl) +� +di−ik>dj−jl +(di − ik − dj − jl) +(7.1) +is the coefficient of fi1j1 . . . fimjm. By definition of the ei, the term +� +1≤i 8 Gyr are a factor of ∼ 4 +lower than in more recent infallers while controlling for total stellar mass. Nine of the 16 galaxies have spatially extended cold disks, +and most of them show positive or zero age gradients; stars in the inner disk are ∼ 2 − 5 Gyr younger than that in the outer disk, in +contrast to the expectation of inside-out growth. Our results indicate that the assembly of cold disks in galaxies is strongly affected +by their infall into clusters, by either removal of gas in outer regions or even tidally stripping or heating part of the pre-existing disks. +Star formation in outer disks can stop quickly after the galaxy falls into the cluster, while star formation in the inner disks can last for +a few Gyrs more, building the positive age gradient measured in cold disks. +Key words. galaxies: kinematics and dynamics – galaxies: elliptical and lenticular, cD – galaxies: stellar population – galaxies: +formation – galaxies: structure – galaxies: evolution +1. Introduction +Galaxies in cluster environments have different star formation +histories and display different morphology types with respect to +galaxies in the field. Generally speaking, galaxies in clusters are +redder, more likely to be quenched, and exhibit an elliptical mor- +phology (e.g., Dressler 1980; Butcher & Oemler 1984; Dressler +et al. 1997; Peng et al. 2010; Lewis et al. 2002; Blanton et al. +⋆ Email:ycding@shao.ac.cn +⋆⋆ Corr author: lzhu@shao.ac.cn +2005; Alpaslan et al. 2015). Galaxy evolution is mainly affected +by three physical processes related to the cluster environment. +First, in a process known as “harassment”, tidal forces induced +by the central halo (Gunn & Gott 1972; Toomre & Toomre 1972; +Barnes & Hernquist 1992; Bournaud et al. 2004) or neighbour- +ing flyby galaxies can strip away part of the stellar and gas par- +ticles of the satellite galaxies (Gunn & Gott 1972). Second, the +interaction with the intracluster medium (ICM) known as “ram +pressure” can strip away the hot gas and sometimes even the cold +gas in the disk of the satellite galaxies (Gunn & Gott 1972; Abadi +Article number, page 1 of 31 +arXiv:2301.05532v1 [astro-ph.GA] 13 Jan 2023 + +A&A proofs: manuscript no. main +et al. 1999; Yun et al. 2019). Third, the accretion rate is much +lower in the cluster environment and it could cause the cut-off of +gas supply to a galaxy leading to “strangulation” (Larson et al. +1980; Balogh et al. 2000; Kawata & Mulchaey 2008). All three +processes could ultimately lead to the cessation of star formation +and affect the morphology evolution of the satellite galaxies in +the cluster. +The timescale of the star-formation quenching in cluster en- +vironments and the relative contribution of the aforementioned +physical processes are still under debate. The current gas con- +tent in cluster galaxies provides some direct clues. Maier et al. +(2019b) calculated the fraction of star-forming galaxies of seven +nearby clusters through Hα observation with Local Cluster Sub- +structure Survey (LoCuSS). By comparing with the Millennium +simulations, they suggested that star formation in cluster galax- +ies can last for 1-2 Gyr after their infall into the cluster, but +then they quickly get quenched. This “slow-then-rapid quench- +ing” is consistent with the strangulation scenario. On the other +hand, Reynolds et al. (2022) found evidence that HI gas in the +outer regions of the galaxies in the Hydra I cluster is partially re- +moved. However, these galaxies still lie on the star-forming main +sequence and gas removal is not yet affecting the inner star form- +ing disks. In the same cluster, Wang et al. (2021) modeled the +ram-pressure stripping strength through the HI mass from WAL- +LABY observation. They found that the ram pressure stripping +can significant change the total HI gas mass of satellite galaxies +within 600 Myr after falling into the cluster. +Cosmological hydrodynamical simulations allow us to inves- +tigate the details of the infalling of galaxies into the cluster as +well as of the gas removal and structure formation of galaxies. +Joshi et al. (2020) found that the pre-exisiting stellar disks in +cluster galaxies are largely disrupted by impulsive tidal shock- +ing and stripping at pericentres within 0.5-4 Gyr after falling +into the massive clusters with M200 ∼ 1014−14.3 M⊙ in the Il- +lustris TNG100 simulation. Meanwhile, all the cluster galax- +ies quenched their star formation after the disk disruption. The +quenching timescale could be 1-1.5 Gyr for gas-poor massive +galaxies, while the star formation could last for ∼4.5 Gyr for gas- +rich galaxies and a dynamically cold disk could regrow. Similar +results apply also to Fornax-mass analogues in the TNG50 sim- +ulations (Galán-de Anta et al. 2022) +The detailed study of S0 galaxies in TNG100 simulation +shows two major formation scenarios (Deeley et al. 2021): S0 +galaxies in clusters formed via starvation or stripping, whereas +S0s in field formed via mergers. This finding is also supported by +observations (Coccato et al. 2020; Coccato et al. 2022). Star for- +mation quenching is different in these two formation scenarios +(Deeley et al. 2021): when a blue and gas-rich galaxy falls into +a cluster, its gas in the outer regions is quickly stripped, while +the star formation in the inner regions could last for a long time, +whereas in case of galaxy-galaxy merger, galaxies start quench- +ing from the inner regions, and star formation continues in the +outer ring-like regions. +These studies suggest that cluster environment may signifi- +cantly affect the internal dynamical structure of galaxies, espe- +cially the persistent growth of dynamically cold disks. The frac- +tion of stars in cold disks and their age distributions may im- +print the long-term star-formation quenching process happening +in cluster galaxies. We expect that the disk stars are older in the +inner regions and younger in the outer ones for a normal inside- +out growth (e.g., Bird et al. 2013; Stinson et al. 2013). Cluster +quenching may turn over the age gradient in the stellar disks, the +age difference between inner and outer regions can thus tell us +the star formation duration in galaxies after falling into the clus- +ter. Although some other studies show that complicated physical +processes in field galaxies might cause similar age gradient (La- +gos et al. 2022). +In the past decades, integral-field unit (IFU) surveys, such as +SAURON (de Zeeuw et al. 2002), ATLAS3D (Cappellari et al. +2011), CALIFA (Sánchez et al. 2012), SAMI (Bryant et al. +2015), and MaNGA (Bundy et al. 2015), have provided us the +stellar kinematics, age, and metallicity maps for thousands of +nearby galaxies. Early-type galaxies (ETGs) show a large vari- +ety of kinematical structures (Cappellari et al. 2007; Emsellem +et al. 2007, 2011), some lenticular galaxies have a rapidly rotat- +ing disk similar to that of spiral galaxies. This suggests that gas +removal by cluster environment may play a major role in trans- +forming spirals into lenticulars (Cappellari 2016). Overall, ETGs +are found to be old and with shallow age gradients (McDermid +et al. 2015; González Delgado et al. 2015; Zibetti 2019), while +late-type galaxies (LTGs) generally have negative age gradients, +which is consistent with the inside-out scenario (González Del- +gado et al. 2015). The environmental dependence of the galaxy +age gradient is still controversial. A positive age gradient was +found in SAURON and MaNGA samples (e.g., Goddard et al. +2017; Li et al. 2018; Kuntschner et al. 2010), with a mixture of +galaxies in cluster and field environments. On the other hand, +there is no significant difference found for age gradient of ETGs +in different environments (Zheng et al. 2017; Ferreras et al. 2019; +Santucci et al. 2020), with the above surveys covering the inner +1Re ∼ 2Re for most galaxies. +The Fornax3D survey (Sarzi et al. 2018) measured high- +quality age and metallicity maps covering at least the inner 2Re +of 23 ETGs in the Fornax cluster. These galaxies show signifi- +cant positive age gradients between Re and 2Re (Spavone et al +2022), while they have negative metallicity gradients, which are +shallower than the control sample galaxies in the field. On the +other hand, positive age gradients together with negative metal- +licity gradients are widely found in the dwarf galaxies of the +Local Group (e.g., Koleva et al. 2011; Kirby et al. 2011; Zhuang +et al. 2021). These positive age gradients could be the direct con- +sequence of ram pressure stripping (Genina et al. 2019; Deeley +et al. 2021), although others argue theses are the result of gas- +rich mergers for dwarf galaxies (Yozin & Bekki 2012). +The observation of external galaxies provides integrated +properties along the line-of-sight. The stellar kinematics and +populations of a galaxy are a combination of different dynam- +ical components, displaying a range of possible physical origins. +As learned from the simulations aforementioned, the luminosity +fraction and age distribution of stellar disk might be a good probe +of environmental affects. However, there are relatively small disk +fractions in ETGs (Zhu et al. 2018). The age and metallicity +maps of the whole galaxy may not reveal the properties of the +disk component. +Dynamical models are powerful tools to uncover the galax- +ies’ underlying mass profiles, as well as the internal 3D struc- +tures which could lead to physical-motivated decomposition of +different components. There are a few methods widely used +for modelling the IFU kinematic data of external galaxies, in- +cluding the Jeans-Anisotripic-MGE (JAM) models (e.g., Cappel- +lari 2008; Li et al. 2017) , the particle-based Made-to-Measure +(M2M) models (e.g., Syer & Tremaine 1996; de Lorenzi et al. +2007; Long & Mao 2010, 2012; Zhu et al. 2014), and the +orbit-superposition Schwarzschild method used in this work +(Schwarzschild 1979). The Schwarzschild’s orbit superposition +method can be used in different geometries such as the spheri- +cal systems (Richstone & Tremaine 1985; Breddels et al. 2013; +Kowalczyk et al. 2017), axisymmetric systems (Cretton & van +Article number, page 2 of 31 + +Y. Ding et al.: F3D: the environmental effects on the assembly of disks +den Bosch 1999; Gebhardt et al. 2000; Valluri et al. 2004; Cap- +pellari et al. 2006a; Thomas et al. 2007; Saglia et al. 2016), +and triaxial systems (van den Bosch et al. 2008; Neureiter et al. +2021). The orbit-superposition model reconstructs the backbone +of galaxies without ad-hoc assumptions of the underlying dis- +tribution functions, it has been widely used to uncover galax- +ies’ underlying dark matter distributions (e.g., Cappellari et al. +2006b; Yang et al. 2020) , central black hole mass (e.g., van der +Marel et al. 1998; Cretton & van den Bosch 1999; Verolme et al. +2002; Gebhardt et al. 2003; Valluri et al. 2004; Krajnovi´c et al. +2009; Ahn et al. 2018; Thater et al. 2022b), and internal stel- +lar orbit distributions (e.g., Zhu et al. 2018b,a; Jin et al. 2020). +Recently it has been modified to include the barred structures ex- +plicitly, thus also for uncovering the bar pattern speed (Vasiliev +& Valluri 2020; Tahmasebzadeh et al. 2022). +Based on the Schwarzschild method, a population-orbit su- +perposition method (Poci et al. 2019; Zhu et al. 2020) was re- +cently developed by tagging age and metallicity to the orbits, +we can thus simultaneously fit the stellar kinematic, age, and +metallicity maps from IFU survey, and obtain the internal stel- +lar orbit distribution of galaxies as well as the age and metallic- +ity distributions. This allows for a physically motivated chemo- +dynamical decomposition of galaxies. In this work, we apply +the population-orbit superposition method to galaxies in For- +nax cluster with the IFU data obtained with MUSE/VLT in the +context of the Fornax3D survey (Iodice et al. 2019). Using this +modeling technique, age-dispersion profiles of dynamically de- +composed cold disks in a few edge-on galaxies have been ob- +tained (Poci et al. 2019; Poci et al. 2021) and a dynamically de- +composed “hot inner stellar halo” has been used to weigh and +time the ancient massive mergers the galaxies experienced in +NGC 1380 and NGC 1427 (Zhu et al. 2022). +In this paper, we set our focus on dynamically cold disks. By +studying the luminosity fraction, age, and metallicity of these +stars, we aim to quantify how the cluster environment has af- +fected the formation of cold disks. This results will also lead to +a direct and in-depth comparison with galaxies in cosmological +simulations and, thus, to a better understanding of the physical +processes driving galaxy evolution in cluster environments. This +work is undertaken in the context of ΛCDM cosmology, with +density parameters of Ωm = 0.3089, ΩΛ = 0.6911, Ωb = 0.0486, +normalization σ8 = 0.8159 and spectral index ns = 0.9667 +(Planck Collaboration et al. 2014). +The paper is organized as follows. We introduce the data set +in § 2. We describe the population-orbit superposition model and +its relevance to the orbital decomposition in § 3. We show the +orbital decomposition for all galaxies and present the surface- +brightness, age, and metallicity radial profiles of each compo- +nent in § 4. We present an explanation of how the cold disk age +can be used as a novel proxy for galaxy infall time into a cluster +in § 5. We further show the dependence of cold disk properties +on galaxy stellar mass and cluster environment in § 6. We discuss +the results in § 7 and present our summary in § 8. +2. Sample and Data +We set our focus on galaxies in the Fornax cluster, which has a +virial radius of Rvir ∼ 0.7 Mpc , virial mass of Mvir ∼ 7 × 1013 +M⊙, and a distance of D ∼ 20 Mpc (Diaferio 1999; Drinkwater +et al. 2001). +We used the deep photometric data of the Fornax Deep Sur- +vey (FDS, Venhola et al. (2018)). FDS observed all the galaxies +in the area of 9 square degrees around the core of the cluster, with +the VLT Survey Telescope down to a surface-brightness level of +27 mag arcsec−2 in the r band (Iodice et al. 2016). +We used the spectroscopic data of the Fornax3D survey +(Sarzi et al. 2018). Using the MUSE instrument on the VLT, For- +nax3D observed all the 23 ETGs and 10 LTGs within the virial +radius of the cluster down to 25 mag arcsec−2 in B band. MUSE +has a field-of-view of 1×1 arcmin2 and spatial scale of 0.2 arcsec +pixel−1. The wavelength covers the range of 4650 Å to 9300 Å, +with a resolution of FWHM7000Å = 2.5 Å. Twenty-eight galax- +ies were observed by more than one MUSE pointing. The data +extend to at least 2Re for 20 galaxies. +2.1. Stellar kinematics maps +To obtain reliable stellar kinematics, we first bin the spectra in +nearby pixels to reach a target signal-to-noise ratio (S/N) taken +a Voronoi binning scheme (Cappellari & Copin 2003). The stel- +lar kinematics of all the galaxies in the Fornax3D survey was +published in Iodice et al. (2019) with target S/N=40. The only +difference in the kinematic maps we use here is that different +S/N for different galaxies were chosen as indicated in Table 1. +We took a target S/N = 100 or 200 for the bright galaxies +with more than one MUSE pointing and S/N =60 or 40 for the +remaining galaxies. This results in a number of bins from ∼ 100 +to ∼ 1000 for each galaxy with a spatial resolution from ∼ 100 +pc in the inner region to ∼ 1 kpc in the outer faint regions. This +choice is convenient for dynamical modeling without too many +bins, while the spatial resolution is still good enough to meet our +science goals for resolving sub-kpc scale structures. +The stellar kinematics was then extracted by applying the +pPXF full-spectral fitting (Cappellari & Emsellem 2004) to +the wavelength range 4750-5500 Å. This yielded high-quality +maps of the stellar mean velocity, V, velocity dispersion, σ, +and higher order velocity moments parameterized through the +Gauss-Hermite (GH) coefficients h3 and h4 (Gerhard 1993; Rix +et al. 1997). +Such stellar kinematic maps were derived in a consistent way +for the 15 ETGs and 5 LTGs included in this paper. We did +not consider the 8 ETGs – FCC 213 (i.e., the BCG), FCC 090, +FCC 184, FCC 190, FCC 193, FCC 219, FCC 277, and FCC 310 +– due to existence of bright foreground stars, limited data cov- +erage or the presence of strong bar, and the 5 LTGs FCC 113, +FCC 176, FCC 267, FCC 285, and FCC 306 due to the pres- +ence of a strong bar or irregular morphology. We excluded the +strongly barred galaxies because our current dynamical models +cannot fit their kinematic maps well with bars that are not in- +cluded explicitly. However, we did keep the three weakly barred +galaxies (FCC 179, FCC 182, and FCC 263), since their stel- +lar kinematic maps are not strongly affected by the bar (see fig- +ures in Appendix C) and our current kinematic models can still +provide reasonably good fits. Some basic properties of the final +sample of galaxies are listed in Table 1. +2.2. Age and metallicity maps +We used the age and metallicity maps derived in Martín-Navarro +et al. (2021). Here, we briefly describe how they were derived. +To obtain high-quality maps of the age and metallicity, the +spectra were spatially rebinned to reach a minimum of S/N = +100 for each galaxy. We note that the spaxels with S/N< 5 were +not included in this process in order to retain the highest qual- +ity of data. Thus, for most of the galaxies, only the inner region +map was available. The single stellar population synthesis mod- +Article number, page 3 of 31 + +A&A proofs: manuscript no. main +Table 1. Properties of the sample galaxies. +Object +Type +Distance +Re +M⋆ +i +PA +S/N +[Mpc] +[kpc] +[1010M⊙] +[◦] +[◦] +(1) +(2) +(3) +(4) +(5) +(6) +(7) +(8) +FCC 083 +E5 +19.2 +3.46 +2.27 +68.2±13.3 +139.7 +100 +FCC 119 +S0 +20.9 +1.17 +0.05 +64.9±5.4 +45.4 +40 +FCC 143 +E3 +19.3 +1.07 +0.28 +64.6±7.1 +124.6 +60 +FCC 147 +E0 +19.6 +2.40 +2.40 +33.3±6.0 +120.9 +100 +FCC 148 +S0 +19.9 +2.74 +0.58 +75.8±3.1 +89.2 +60 +FCC 153 +S0 +20.8 +1.92 +0.76 +87.5±0.1 +82.6 +100 +FCC 161 +E0 +19.9 +2.77 +2.63 +33.1±3.3 +0.0 +100 +FCC 167 +S0/a +21.2 +5.47 +9.85 +78.0±2.1 +5.0 +100 +FCC 170 +S0 +21.9 +1.54 +2.25 +84.9±1.5 +-42.0 +100 +FCC 177 +S0 +20.0 +3.48 +0.85 +85.8±1.3 +177.8 +100 +FCC 179 +Sa +20.9 +2.91 +1.58 +69.9±0.7 +64.34 +100 +FCC 182 +SB0 +19.6 +0.96 +0.15 +63.2±4.4 +169.3 +60 +FCC 249 +E0 +20.9 +0.93 +0.98 +37.8±4.1 +150.6 +60 +FCC 255 +S0 +20.9 +1.34 +0.31 +78.1±5.8 +172.9 +60 +FCC 263 +SBcdIII +20.9 +2.64 +0.04 +66.4±1.9 +87.9 +60 +FCC 276 +E4 +19.6 +4.33 +1.81 +66.6±9.0 +75.4 +100 +FCC 290 +ScII +20.9 +4.70 +0.64 +52.8±1.1 +122.5 +60 +FCC 301 +E4 +19.7 +1.13 +0.20 +82.0±5.7 +155.8 +60 +FCC 308 +Sd +20.9 +3.60 +0.04 +73.5±1.4 +88.5 +60 +FCC 312 +Scd +20.9 +10.62 +1.48 +81.0±0.9 +136.2 +100 +Notes. Name (1); Hubble type (2) taken from (Ferguson 1989); distance (3); Effective radius (4); and stellar mass (5) is from Iodice et al. (2019) +and Martín-Navarro et al. (2021). The inclination angle (6) obtained from our orbit-superposition model. The positional angle (7) determined by +photometric isophotes and adopted for the multi-Gaussian expansion fitting and target S/N used in binning the spectra to obtain kinematic maps +(8). +els (SSP) by Vazdekis et al. (2016) based on the MILES stellar +library by Sánchez-Blázquez et al. (2006) at a constant spectral +resolution of 2.51 Å (FWHM) (Falcón-Barroso et al. 2011) were +used for the analysis. +The luminosity-weighted age and metallicity maps were de- +rived in a two-step spectral fitting of the wavelength range be- +tween 4800 Å and 6400 Å, considering the initial mass func- +tion (IMF) variation. First pPXF was run with an un-regularized +combination of MILES SSPs in order to measure V and σ. A +second pPXF was run again with stellar kinematics fixed to that +measured in the first step, and this time regularizing the age- +metallicity-IMF slope plane. The mean luminosity-weighted age +and metallicity of the spectra were measured using this regu- +larized second pPXF run. The metallicity, Z, is added up lin- +early rather than in the commonly used logarithmic way. Tem- +plates with a variable IMF were intentionally included to ac- +count for the possible effect of the IMF in the age determina- +tion. The abundance ratio [α/Fe] was not fitted, but we used the +so-called base MILES models which inherit the [α/Fe]-[M/H] +relation of the solar neighborhood (Vazdekis et al. 2015). This +choice avoids non-local equilibrium uncertainties introduced by +the theoretical response functions needed to compute stellar pop- +ulation models with variable abundance ratios, uncertainties that +can be particularly problematic for Balmer lines and thus for the +age determination. We note that in Martín-Navarro et al. (2021), +the metallicity maps were also derived in a third step with in- +dex fitting when determining the IMF. However, we used the +luminosity-weighted metallicity maps derived from the pPXF +fitting, in a consistent way as the luminosity-weighted age maps. +Alternatively, we have another version of the age and +metallicity maps for FCC 153, FCC 167, FCC 170, and +FCC 177, which was obtained by pPXF fitting regularizing +age-metallicity-[α/Fe] assuming a Kroupa IMF (Pinna et al. +2019a,b). +We compared the two versions of age and metallicity maps +for these four galaxies, they generally show the same age and +metallicity gradients but with a systematic offset of 1-2 Gyr in +age. This offset will not affect our results. The version of the +age and metallicity maps by Martín-Navarro et al. (2021) is used +consistently for all galaxies in this paper. +Finally, we have the age and metallicity maps for 16 galaxies, +including all the 15 ETGs and the LTG FCC 179 only. The re- +Article number, page 4 of 31 + +Y. Ding et al.: F3D: the environmental effects on the assembly of disks +Table 2. Properties of the sample galaxies extracted by the models. +Object +Mtot,e +fDM,e +fcold,e +fcold,2e +⟨tcold,e⟩ +⟨thot,e⟩ +⟨Zcold,e⟩ +⟨Zhot,e⟩ +∇tcold,e +∇thot,e +∇Zcold,e +∇Zhot,e +tinfall +[1010M⊙] +[Gyr] +[Gyr] +[Z⊙] +[Z⊙] +[Gyr/Re] +[Gyr/Re] +Z⊙/Re +Z⊙/Re +[Gyr ago] +(1) +(2) +(3) +(4) +(5) +(6) +(7) +(8) +(9) +(10) +(11) +(12) +(13) +(14) +FCC 083 2.67±0.19 +0.40±0.05 +0.18±0.02 +0.19±0.02 +11.8±0.6 +11.4±0.5 +0.72±0.24 0.38±0.20 -2.52±1.73 +-2.92±0.48 +-0.99±0.67 +-0.91±0.15 +8.7±3.1 +FCC 119 0.04±0.02 +0.66±0.05 +0.17±0.06 +0.12±0.03 +5.9±1.1 +4.2±0.5 +0.36±0.05 0.43±0.05 18.24±5.75 +10.02±1.14 +-1.22±0.28 +-0.7±0.07 +2.4±2.3 +FCC 143 0.15±0.03 +0.24±0.06 +0.06±0.01 +0.05±0.01 +11.9±1.3 +13.0±0.5 +0.98±0.71 0.36±0.19 “‘ +-2.05±0.44 +“‘ +-0.49±0.25 +11.4±1.2 +FCC 147 2.88±0.30 +0.19±0.03 +0.11±0.02 +0.12±0.02 +12.7±0.8 +13.5±0.3 +0.91±0.33 0.46±0.13 -2.07±1.87 +-1.01±0.26 +-1.19±0.84 +-1.32±0.04 +9.6±3.0 +FCC 148 0.56±0.03 +0.47±0.03 +0.27±0.02 +0.33±0.02 +4.9±0.6 +5.7±0.6 +0.90±0.11 0.95±0.18 0.97±2.05 +4.13±0.87 +-1.37±0.6 +-1.47±0.1 +3.8±2.4 +FCC 153 1.03±0.14 +0.43±0.01 +0.32±0.01 +0.49±0.01 +9.4±0.2 +9.9±0.6 +1.52±0.09 0.51±0.19 3.69±2.11 +2.85±0.6 +-0.81±0.50 +-1.24±0.34 +8.9±1.7 +FCC 161 1.33±0.07 +0.25±0.04 +0.10±0.02 +0.09±0.02 +11.6±1.1 +12.9±0.4 +0.49±0.14 0.47±0.04 “‘ +-1.41±0.23 +“‘ +-0.37±0.03 +9.0±3.0 +FCC 167 12.16±0.87 0.25±0.03 +0.12±0.01 +0.20±0.01 +11.6±0.1 +13.6±0.1 +1.95±0.01 1.61±0.01 -4.08±1.88 +-0.32±0.28 +-0.3±0.83 +-0.73±0.13 +8.8±3.1 +FCC 170 1.31±0.3 +0.14±0.02 +0.03±0.01 +0.15±0.01 +13.2±0.6 +13.2±0.1 +1.05±0.44 0.47±0.03 -0.1±0.20 +-0.5±0.08 +-0.02±0.30 +-0.48±0.08 +9.8±3.0 +FCC 177 0.74±0.05 +0.62±0.03 +0.32±0.01 +0.30±0.01 +6.7±0.4 +8.1±0.8 +1.25±0.20 0.81±0.23 -0.12±2.80 +5.71±0.40 +0.65±0.85 +-1.30±0.13 +6.3±2.2 +FCC 179 2.2±0.09 +0.19±0.02 +0.31±0.02 +0.47±0.03 +5.0±0.5 +7.2±0.7 +1.12±0.15 0.77±0.05 1.59±1.59 +-2.45±1.03 +-1.11±0.67 +-0.22±0.10 +3.7±2.4 +FCC 182 0.05±0.01 +0.34±0.08 +0.05±0.01 +0.03±0.01 +11.6±1.3 +10.3±0.8 +0.81±0.46 0.44±0.09 “‘ +-0.84±0.24 +“‘ +-0.3±0.05 +10.4±0.9 +FCC 249 0.60±0.02 +0.17±0.03 +0.07±0.03 +0.05±0.02 +13.3±0.7 +12.0±0.5 +1.26±1.18 0.37±0.13 “‘ +-1.97±0.25 +“‘ +-0.41±0.21 +12.3±1.7 +FCC 255 0.11±0.01 +0.44±0.03 +0.12±0.02 +0.16±0.02 +3.7±2.5 +8.2±1.5 +0.98±0.27 0.37±0.05 6.42±3.51 +3.5±0.93 +-0.64±0.32 +-0.45±0.06 +1.0±1.0 +FCC 263 0.21±0.02 +0.88±0.01 +0.08±0.01 +0.10±0.02 +“‘ +“‘ +“‘ +“‘ +“‘ +“‘ +“‘ +“‘ +“‘ +FCC 276 4.23±0.46 +0.33±0.04 +0.04±0.01 +0.03±0.01 +13.1±0.7 +12.7±0.3 +0.79±0.52 0.35±0.05 “‘ +-2.01±0.37 +“‘ +0.74±0.06 +9.4±3.0 +FCC 290 2.08±0.13 +0.68±0.05 +0.25±0.03 +0.37±0.04 +“‘ +“‘ +“‘ +“‘ +“‘ +“‘ +“‘ +“‘ +“‘ +FCC 301 0.15±0.01 +0.28±0.05 +0.07±0.01 +0.11±0.01 +6.4±0.9 +7.8±0.8 +0.87±0.01 0.38±0.05 1.66±3.80 +0.3±0.65 +-1.16±0.36 +0.35±0.04 +6.5±2.1 +FCC 308 0.33±0.03 +0.51±0.03 +0.09±0.01 +0.09±0.01 +“‘ +“‘ +“‘ +“‘ +“‘ +“‘ +“‘ +“‘ +“‘ +FCC 312 6.16±0.72 +0.63±0.06 +0.32±0.04 +0.32±0.05 +“‘ +“‘ +“‘ +“‘ +“‘ +“‘ +“‘ +“‘ +“‘ +Notes. Name (1); total dynamical mass (2); dark matter fraction (3); 1Re luminosity fraction of the cold disk (4); 2Re luminosity fraction of the cold disk (5); mean age of the cold disk (6); mean +age of the non-disk component (7); mean metallicity of the cold disk (8); mean metallicity of the non-disk component (9); age gradient of the cold disk (10); age gradient of the non-disk component +(11); metallicity gradient of the cold disk (12); metallicity gradient of non-disk component (13). All values in columns (2)-(4) and (6)-(13) are calculated within the effective radius Re, except for +(5) which is extracted within 2Re . Columns (10) and (12) of FCC 170 and FCC 255 are extracted within the whole data coverage as their disks have very low surface-brightness at r < Re. For +six ETGs FCC 143, FCC 161, FCC 182, FCC 249, FCC 276, and FCC 308, we only show the age and metallicity gradient of non-disk component because of the low surface-brightness at r < Re. +Column (13) gives the infall time into the cluster based on the cold-disk age using a correlation calibrated with TNG50 simulations. For four LTGs FCC 263, FCC 290, FCC 308, and FCC 312, we +only have the stellar kinematics and thus only orbital models without coloring. +Article number, page 5 of 31 + +A&A proofs: manuscript no. main +maining 4 LTGs of our sample appear to be dominated by young +stars at all radii, for which the uncertainties of age and metallic- +ity derived from the spectra fitting are large (Ge et al. 2018), we +will build their dynamical models without associating age and +metallicity to the stellar orbits. +3. Population-orbital superposition method +We follow Zhu et al. (2020)1 in constructing population-orbit +superposition models that simultaneously fit the luminosity den- +sity, kinematic, age, and metallicity maps of each galaxy. From +the Best-fit models, we obtain the internal stellar orbit distribu- +tion associated with the age and metallicity distributions. Based +on the model, we then decomposed each galaxy into a dynam- +ically cold disk and a dynamically hot non-disk component, +based on circularity distribution of the orbits, and extracted the +face-on surface-brightness, age, and metallicity radial profiles +for each component separately. We used FCC 177 as an example +to illustrate the whole process as shown in Figs. 1 and 2. +The construction of a population-orbit superposition model +consists of four steps. First, constructing the model of gravita- +tional potential. Second, calculating the orbital library under the +gravitational potential. Third, fitting the luminosity density and +kinematic maps by weighting the orbits. Fourth, fitting the age +and metallicity maps by “colorin” the orbits. The method has +been presented in great detail and carefully validated in Zhu et al. +(2020). Here, we just briefly describe some of the key steps. +3.1. Constructing the gravitational potential +The gravitational potential combines the contribution of the +stars, dark matter halo, and a central black hole. +To construct the stellar mass distribution, we use the multi- +Gaussian expansion (MGE, Cappellari (2002)) to fit the r-band +image from FDS. The galaxies in our sample have a variety of +morphological and kinematic properties, which can be classified +in three categories: galaxies which are generally flat and with +strong rotation indicating an extended disk, galaxies which are +generally round but flatter, and with stronger rotation in the inner +regions indicating an inner disk, galaxies with no clear evidence +of a disk. For the first two categories, we consider an axisymmet- +ric solution by adopting a constant positional angle (PA) for the +Gaussians to match that of the photometric and kinematic signa- +ture of the inner or outer disk component. Photometric PA from +the disk region is taken, which is generally consistent with the +kinematic PA in that region. For the third category, we still con- +sider axisymmetric solution, but we did not put any extra con- +strain on the PA; thus, the photometric PA directly derived from +the whole galaxy image was taken. The best-fit Gaussian param- +eters are listed in the Appendix for all the galaxies. +Then we de-project the 2D Gaussian components to 3D by +assuming a set of viewing angles (θ, ψ, φ), where θ and φ define +the orientation of the line of sight with respect to the principal +axes of the galaxy and ψ is chosen to specify the rotation of the +galaxy around the line-of-sight in the projected sky-plane. By +combining the 3D Gaussian components, we obtained the 3D +luminosity density distribution of the galaxy and we multiply +a stellar mass-to-light ratio M∗/L to get the 3D mass density +distribution. The gravitational potential could then be calculated +1 We have fixed the bug in the orbit mirroring as reported in Quen- +neville et al. (2022) although our results are unaffected as confirmed in +Thater et al. (2022a). +by the classical Chandrasekhar formula (van den Bosch et al. +2008). +In practice, we do not directly use the three viewing angles +as free parameters. In van den Bosch et al. (2008), three intrinsic +shape parameters (p, q, u) are used instead of the viewing angles +(θ, ψ, φ), where q = Z/X, p = Y/X, and u = X′/X, where X, +Y, Z are the intrinsic long, intermediate, and short axis of the +galaxy and X′ is the projected major axis. The conversion of the +two sets of parameters follows Eq. 10 in van den Bosch et al. +(2008). Exploring the space of intrinsic shape parameters is more +efficient than that of the three viewing angles. For instance, the +deprojection of an axisymmetric system would have no constrain +on the parameter φ but have a finite axis-ratio between Y and X. +In this work, we follow the approach by van den Bosch et al. +(2008) and allow for some degree of triaxiality of the galaxy by +setting a non-unity u. +We use a spherical Navarro-Frenk-White (NFW, Navarro +et al. (1996)) dark matter halo, with one free parameter dark +matter virial mass M200, while concentration C is fixed by the +correlation between M200 and C (Dutton & Macciò 2014). Al- +though there is a large scatter in the M200 − C plane for real +galaxies, the choice of a fixed C will not significantly affect our +results because the two parameters are degenerated and will not +be constrained separately with our kinematic data covering out +to 2 − 3Re. The potential also includes a central black hole char- +acterized by a Plummer potential (van den Bosch et al. 2008), +with the back hole mass, M•, fixed according the M•-σ relation +(Kormendy & Ho 2013). This choice does not affect our results, +because the black hole sphere of influence is mostly not resolved +by our kinematic data and we cannot directly constrain the black +hole mass. +In summary, we have five free "hyper-parameters," for the +gravitational potential: the mass-to-light ratio, M∗/L, three pa- +rameters on the intrinsic shape of stellar distribution, p, q, and u, +and the dark matter virial mass, M200. +3.2. Computing the orbit library +For each model, with a set of hyper-parameters, we calculated an +orbit library with tens of thousands orbits. The orbits sampling +follows the way described in van den Bosch et al. (2008). +We first sampled regular orbits according to a separable tri- +axial potential. The orbits are sampled from the three integrals +of motion: energy, E, second integral of motion, I2, and third in- +tegral of motion, I3 (Binney & Tremaine 2008). We sample two +sets of 55 × 11 × 11 combinations of (E, I2, I3) as initial condi- +tions, which include co-rotating and counter-rotating orbits. +Box orbits are crucial for supporting the triaxial structures, +we sampled another set of box orbits by constructing the ini- +tial conditions on equipotential surfaces with the energy, E, two +spherical angles, θ and φ, which gives another set of 55×11×11 +orbits. +The number of orbits, especially across E, we sample here +is larger than that used in previous works (e.g., van den Bosch +et al. 2008; Zhu et al. 2018a; Jin et al. 2019). For instance, 21 × +7 × 7 orbits were used in fitting the CALIFA data (Zhu et al. +2018a). Because the kinematic data used in this work have larger +spatial coverage and higher spatial resolution than those from the +previous IFU surveys, more freedom is required by the model to +fit the data. To reduce the Poisson noise of the model, we dither +every orbit by slightly perturbing the initial conditions to give 53 +orbits. +Article number, page 6 of 31 + +Y. Ding et al.: F3D: the environmental effects on the assembly of disks +Fig. 1. Best-fit population-orbit superposition model of FCC 177. From top to bottom: Maps of the data, model, and residuals (model−data). From +left to right: Maps of the surface-brightness, SB, mean velocity, V, velocity dispersion, σ, Gauss-Hermite coefficients h3 and h4, light-weighted +age and metallicity of the stars. Similar plots for the other sample galaxies are shown in Figs. B.1-B.19. +Fig. 2. Orbital decomposition of FCC 177. Probability density distribution p(r, λz) in the left panel, age distribution p(r, t) in the central panel, +and metallicity distribution p(r, Z) in the right panel of the stellar orbits in the phase space of time-averaged radius, r, versus circularity, λz. The +probability densities are normalized to unity within the data coverage. All the distributions are averages of multiple Best-fit models that fall within +the 1σ confidence level of the model hyper-parameters. The dashed line marks our orbit-based division into two components: a dynamically cold +disk component (λz ≥ 0.8) and a dynamically hot non-disk component (λz < 0.8). The shadow regions are beyond the data coverage of age and +metallicity maps. Similar plots for the other galaxies are shown in Figs. B.1-B.19. +3.3. Fitting stellar luminosity density and kinematics +We fit the luminosity density and stellar kinematics of the galaxy +by weighting the orbits. The 2D surface-brightness, 3D luminos- +ity density deprojected from the MGE, and kinematic maps are +used as model constrains. The kinematic maps include the line- +of-sight mean velocity, V, velocity dispersion, σ, Gauss-Hermite +coefficients, h3 and h4. It is worth noting that we did not fit V +and σ maps directly, but the Gauss-Hermite coefficients h1, h2, +h3, and h4. We extracted similar luminosity and kinematic maps +from the model superposed by orbits, then obtained a solution +of the orbit weights by minimizing the χ2 between the data and +model using a non-negative least squares (NNLS) method (Law- +son & Hanson 1974; Lawson & Hanson 1995). +We used an optimized grid searching process (Zhu et al. +2018b) to adjust the free hyper-parameters of the gravitational +potential. We started with a model with an initial guess of the +hyper-parameters, then we performed an iterative searching pro- +cess with intervals of 0.05, 0.02, 0.01, 0.01, and 0.05 for the +hyper-parameters, M∗/L, p, q, u, and log M200. We note that the +concentration parameter, C, in the NFW dark matter halo is fixed +by the M200−C relation. After the previous sampled models were +completed, we selected the best-fit models with χ2 − χ2 +min < 200 +and sampled the new models around the selected ones. We con- +tinued the iterative process until an area of χ2 minimum was +found and all models within 3 − σ confidence level around the +minimum χ2 were calculated. At the end, we calculated a few +hundred models for each galaxy, the resulting parameter grid for +FCC 177 is shown in Fig. C.1 in the appendix. The models with +least-χ2 are selected as the Best-fit model. In Fig. 1, we show +the Best-fit model of FCC 177, where the model matches the +observed kinematic data in great detail. Considering the model +uncertainties, we defined the 1σ confidence level in a similar +way as Zhu et al. (2018): +δχ2 = χ2 − χ2 +min < f +� +2 × nGH × Nobs , +(1) +where nGH = 4 is the number of stellar kinematic moments and +Nobs is the number of bins in the kinematic data. We adopt f = 1 +for the galaxies with Nobs < 200, similar to the previous models +for the CALIFA data; for galaxies with a much larger value for +Nobs, we found that the χ2 fluctuation caused by numerical noise +Article number, page 7 of 31 + +SB [log 10 Lofpc?] +V [km/s] +α [kmfs] +h3 +h4 +t [Gyr] +Z [ZZ] +50 +50 +20 +4D +a.D +0.1 +0.2 +a.D +0.2 +8 +O.D +LD +1.5 +[] +Data +20 +[B] 1 +21 +Model +0 +20 +P/[P- pW] +op/[Po eowo] +[h3mode +H - h3dabaJfdh3 +[h4. +(tmcdel +-ZdabaJfdZ +0.05D-0.025 0.DO0 0.D25 0.D51 5.02.50.D +25 +5D +5.02.5 +0.D25 +2.5 +5.D +5.02.5 +2.5 +O.D +2.5 +5.D +5.0 +-2.5 +5.D +[] +EE +EEEEE +FEEEEF +FEEEE +FF +25 +0 +25 +50 +25 +50 +25 +25 +0 +25 +4 +51 +25 +0 +F1 +X [arcsec] +X [arcsec] +X [arcsec] +X [arcsec] +X [arcsec] +X [arcsec] +X [arcsec]Probabilitydensity +t [Gyr] +Z[ZZo] +0-0'0 +0.01 +0.02 +4 +8 +12 +0 +1 +2 +1.0 +0.5 +0.0 +0.5 +-1.0 + +20 +40 +60 +器心 +20 +40 +60 +器心 +20 +40 +60 +80 +rarcsec +rarcsec +r[arcsec]A&A proofs: manuscript no. main +is much higher and thus we adopted f = 4, following the results +of a bootstrapping analysis. 2 +Once we fit the stellar kinematics, we obtained the intrin- +sic stellar orbit distribution of the galaxy. We use the circularity, +λz, and time-averaged radius, r, to characterize the orbits, where +λz is defined as the orbital angular momentum around the z di- +rection, normalized by the maximum that is allowed by a circu- +lar orbit with the same binding energy. Here, λz ∼ 1 represents +highly rotating short-axis tube orbits and the λz ∼ 0 represents +mostly long-axis tube or box orbits. The stellar orbit distribution +from the Best-fit model of FCC 177 is shown in the left panel of +Fig. 2. +3.4. Tagging orbits with age and metallicity +Next, we fit the age and metallicity maps by tagging age and +metallicities to the orbits. We took the models within the 1σ con- +fidence level selected by the kinematic fitting as given by Eq. (1). +For each model, we have the stellar orbit distribution. We applied +a Voronoi binning scheme to the orbits in the phase-space of λz +vs. r and decompose them into ∼ 100 orbital bundles. Orbits +with similar r and λz are included in the same orbital bundle and +each bundle has a minimum orbital weight of 0.005. +After the Voronoi binning, we assume that each orbital bun- +dle k has a single value of age, tk, and metallicity, Zk. Then the +age, ti +obs, and metallicity, Zi +obs, of each observational aperture, i, +can be expressed as: +ti +obs = ΣNbundle +k=1 +tk f i +k/Σk f i +k, +(2) +Zi +obs = ΣNbundle +k=1 +Zk f i +k/Σk f i +k, +k = 1, ..., Nbundle, +(3) +where Nbundle is the total number of orbital bundles and f i +k is the +luminosity contribution of orbital bundle, k, at an aperture, i. +We applied a Bayesian statistical analysis (Python package +pymc3)(Salvatier et al. 2016) to first solve tk by fitting the ob- +served age map and then to solve Zk by fitting the observed +metallicity map. To use the Bayesian theorem to compute the +posterior probability of a model, it is necessary to include the +prior probability and data likelihood. For the prior probability of +tk, we adopted a bounded normal distribution, where the mean +value µ(tk) is randomly sampled around the average of the ob- +served ti +obs. We adopted a student’s t-distribution for the data +likelihood (Salvatier et al. 2016), which allows for some outliers +in the data and results in a robust fitting. For each tk, we ran a +chain with 2000 steps and take the last 500 steps to calculate the +mean and 1σ values of tk. +For a few galaxies, we did not get a good match of the age +maps with the above fitting process. Therefore, we tried a sec- +ond fitting round with µ(tk) in the bounded normal distribution +chosen as the tk value obtained from the first fitting round (Zhu +et al. 2020). We obtained a good match of age maps for all the +galaxies after the second round of fitting. +2 For galaxies with Nobs ∼ 1000, the δχ2 is obtained by a bootstrap- +ping process in the following: in a single model with fixed potential and +orbit library, we perturb the kinematic data with its errors and fit the +model to the perturbed data for many times. The standard deviation of +χ2 obtained from these fittings are taken as the χ2 fluctuation caused by +numerical noise of the model. Unlike the classic statistical analysis for +analytic models, numerical noise is dominating the χ2 in our models, +and it could be different for data with different spatial resolution. The +confidence level we adopt is not motivated by robust statistical consid- +eration (more discussion on it could see Lipka & Thomas (2021)), but +practically it works well in covering the true values in our model test. +Fig. 3. Surface-brightness, age, and metallicity radial profiles of the +whole galaxy, the dynamically cold disk, and the dynamically hot non- +disk component for FCC 177. We show the profiles of the whole galaxy +(left panels) with the blue solid curve, and the profiles of dynamically +cold disk and dynamically hot non-disk component (right panels) with +the blue solid curve and black dashed curve. The shadowed areas in- +dicate the scatter of the profiles of models that fall within the 1σ con- +fidence level. The age and metallicity profiles of the dynamically cold +disks are considered reliable and shown in the regions where the dynam- +ically cold disk contributes at least 10% of the total surface brightness. +Then we fit the metallicity map. For the prior probability of +Zk, we adopt a log-normal distribution, where the mean value +µ(log(Zk)) is set by a build-in age-metallicity relation using tk +obtained from the fitting of age map and following Zhu et al. +(2020). The student’s t-distribution is adopted again for the data +likelihood, and we run a chain with 2000 steps and take the last +500 steps to calculate the mean and 1σ values of Zk. +The Best-fit model of FCC 177 is shown in Fig.1. At the +end, we obtained models that simultaneously fit well the surface- +brightness, stellar kinematics, age, and metallicity maps of all +the galaxies. We note that the age and metallicity maps we used +have different binning schemes and have less data coverage than +the kinematic data. For this reason, the age and metallicity of the +orbits beyond the data coverage are not well constrained. +3.5. Orbital decomposition +In Fig. 2, we show the probability density, age, and metallicity +distributions of the orbits of the best-fit model of FCC 177 in +the phase-space of λz versus r. We consider only the age and +metallicity distributions within the data coverage (r ≲ 45 arcsec) +to be reliable. The models are usually more noisy (than smooth) +in the phase-space of λz vs. r. As we tested with mock data in +Zhu et al. (2020), not all the substructures in the model are real, +but we can trust the model considering the distributions of orbits +and of these age and metallicity in a statistical way. +Article number, page 8 of 31 + +Whole +Cold +[zd/ 6ol +Hot +0 +-2 +12 +Age [Gyr] +10 +8 +6 +2.0 +Metallicity [ZIZ o ] +1.5 +1.0 +0.5 +0.0 +0.00 +0.25 +0.50 +0.75 +1.00 +1.25 +0.00 +0.25 +0.50 +0.75 +1.00 +1.25 +R/Re +R/ReY. Ding et al.: F3D: the environmental effects on the assembly of disks +Based on the stellar orbit distribution, we coarsely decom- +pose the galaxy into two components: the orbits with λz ≥ 0.8 +are taken as a dynamically-cold component which is usually a +disk. All the remaining orbits with λz < 0.8 are considered as +a dynamically-hot non-disk component. We note that here the +dynamically cold component is defined as in Zhu et al. (2018a), +whereas the dynamically hot component actually includes their +dynamically warm, hot, and counter-rotating orbits. In Fig. 3, +we show the radial profiles of surface-brightness, age and metal- +licity of the whole galaxy for FCC 177, as well as those pro- +files for the dynamically cold and hot components, separately. +At r ∼ 0.3Re, the cold disk is about 2 Gyr younger and sig- +nificantly more metal-rich than the hot component and the age +difference has risen to ∼ 4 Gyr at r ∼ 0.7Re. However, there is +a young and metal-rich nuclear star cluster in the galactic cen- +ter (r ≲ 0.2Re) which contributes to the hot component. The +luminosity-weighted mean age and metallicity of the hot compo- +nent are dominated by this young and metal-rich central regions, +as a result, the difference of mean age (metallicity) of the cold +disk and the hot component becomes smaller than that revealed +in the profiles at r ≳ 0.3Re. Within Re, the stars on the cold disk +orbits have an luminosity-weighted average age of 6.7 Gyr and +an luminosity-weighted metallicity of 1.25 Z⊙) while the stars on +dynamically warmer orbits have an average age of 8.1 Gyr and +an average metallicity of 0.81 Z⊙. +4. Results of the orbital decomposition for the +sample galaxies +We have 20 galaxies with orbit-superposition models. For 16 +of those (15 ETGs and one LTG, FCC 179), we also have age +and metallicity maps and subsequently build a population-orbit +model as well. In this section, we show the surface-brightness, +age, and metallicity radial profiles of the dynamically cold disk +component and dynamically hot non-disk component for all 16 +galaxies with their orbits tagged with age and metallicity. For the +four LTGs without orbits tagged with age and metallicity, we are +limited to the surface-brightness profiles of each component. +4.1. Surface-brightness, age, and metallicity radial profiles +As illustrated by the best-fit model of FCC 177, we decomposed +each model into a dynamically cold disk (with λz ≥ 0.8) and a +dynamically hot non-disk component (including all the remain- +ing orbits with λz < 0.8). Then, we reconstruct the 3D luminos- +ity, age and metallicity distributions of each orbital component. +We project each orbital component to be face-on and then ex- +tract the radial profiles of surface-brightness, age, and metallicity +from the projected maps. +We note that the model already takes into the account the +information for the 2D maps of the observational data. When +we extract the radial profiles from the face-on projection, the +radial profile along different directions are the same for the disk +which is axisymmetric, while very similar for the hot non-disk +component which could be moderately triaxial. For each galaxy, +we have tens to hundreds of models within the 1σ confidence +level. The aforementioned radial profiles are extracted for each +model. We take the average of these profiles from models within +the 1σ confidence level as the mean profile, and their scatter as +the 1σ uncertainty. +There are significant variations of internal structure from +galaxy to galaxy. We define the luminosity fraction of cold disk +as: +fcold = +λz≥0.8 +� +k +fk, +(4) +where fk is the luminosity fraction of orbital bundle k. The +cold-disk fraction of a galaxy varies with radius, we calculated +fcold(r < Re) and fcold(r < 2Re) with orbits within Re and 2Re, +respectively. +We classify the galaxies into two categories: 11 of them have +a relatively large cold-disk fraction and their cold disks have an +extended surface-brightness, whereas the other 9 galaxies have a +relatively small cold-disk fraction and their cold disks are con- +centrated in the very inner regions. Although the two categories +are classified by eye, this is fairly consistent with a separation +based on the cold-disk fraction fcold(r < 2Re): the galaxies in the +first category generally have fcold(r < 2Re) > 0.1, most galax- +ies in the second category have fcold(r < 2Re) < 0.1, but a few +> 0.1. We take this visual classification for practical reasons that +we can obtain reliable age and metallicity gradients for the ex- +tended cold disks in the following analysis, but not for the con- +centrated ones. +The age and metallicity observed at any radius is a combi- +nation of different components. Our model tests in Zhu et al. +(2020) show that we cannot recover the age and metallicity of +a component at radii where it contributes ≤ 10% to the total +luminosity. For the galaxies with extended disks, the surface- +brightness of the cold disk component contributes at least 10% +of the total surface-brightness over a radial region extending to +at least one Re. Reliable age and metallicity profiles of the disk +are thus obtained over extended radial regions. On the contrary, +for the galaxies with small cold-disk fractions and concentrated +cold components, the age and metallicity radial profiles of the +cold component are only derived in inner radial regions. +In Fig. 4, we show the radial profiles of surface-brightness +of the whole galaxy and of the dynamically cold and hot com- +ponents, for all the 20 galaxies, as well as the age and metal- +licity radial profiles for the 16 galaxies with their orbits tagged +with age and metallicity. The galaxies with extended cold disks +are shown in the left panels, while those with concentrated cold +components are shown in the right panels. The dynamically cold +and hot components are well separated with different surface- +brightness, age, and metallicity radial profiles, which will be dis- +cussed in the following sections. +We note that the surface-brightness profiles of the cold +disks do not follow the traditional exponential law: the surface- +brightness in the central regions is much lower than the inward +extrapolation of the exponential profile. This finding is consis- +tent with the results of orbital decomposition of the CALIFA +galaxies (Zhu et al. 2018) and with those from direct spectrum +fitting, where bulge and disk are separated through their stellar +populations (Breda et al. 2020). +4.2. Quantitative description of the radial profiles +We quantify the luminosity fraction fcold of the cold disk compo- +nent within Re (or 2Re), with respect to the cumulative luminos- +ity of the galaxy within that radius. The 1σ uncertainty of fcold is +calculated by the scatter of the models within the 1σ confidence +level. We calculate fcold for all the 20 sample galaxies. +For the 16 galaxies with their orbits tagged with age and +metallicity, we obtain the mean value and gradient of the radial +profiles of age and metallicity of each component. We calculated +Article number, page 9 of 31 + +A&A proofs: manuscript no. main +Fig. 4. Surface-brightness, age, and metallicity radial profiles of the galaxy, dynamically cold disk, and dynamically hot non-disk component for +the sample galaxies. We divided the galaxies into two groups: 11 galaxies with a dynamically cold disk extended out to large radii (left panels) and +9 galaxies with a dynamically cold disk concentrated in the inner regions (right panels). The radial profiles are color-coded by the galaxy stellar +mass. The shadowed areas indicates the scatter of the profiles of models that fall within the 1σ confidence level of the model hyper-parameters. +Dashed lines refer to the four LTGs without orbits tagged with age and metallicity. The age and metallicity radial profiles of the dynamically cold +disks are considered reliable and shown in the regions where the dynamically cold disk contributes at least 10% of galaxy surface-brightness. +Therefore, these profiles are available only for a limited radial range for the galaxies with a concentrated cold disk. +the mean age and mean metallicity of the cold disk by +⟨tcold⟩ = 1 +fcold +λz≥0.8 +� +k +tk fk, +(5) +⟨Zcold⟩ = 1 +fcold +λz≥0.8 +� +k +Zk fk, +(6) +where tk is the age of the orbital bundle, k. The lumonisity- +weighted mean age ⟨thot⟩ and metallicity ⟨Zhot⟩ of the hot compo- +nent with the orbits λz < 0.8 are calculated in a similar manner. +Note that the mean age is luminosity-weighted. Due to the dif- +ferent surface-brightness profiles of the cold and hot component, +the difference in mean age of these two components sometimes +appears differently from that seen in their age profiles, similar to +the case of FCC 177 (as we explained in Section 3.5). +The gradients of age and metallicity are calculated as fol- +lows. We first uniformly interpolated 100 data points along the +radial profile within the data coverage to smooth the profile, then +we calculated the linear slope (gradient) between adjacent data +points. In the end, we took the average of the gradients from the +data points. We obtain the uncertainty of the gradient by boot- +strapping within the shadowed regions of the profiles as shown +in Fig. 4. All these parameters, together with other basic param- +eters obtained from the population-orbit superposition models, +are included in Table 2. +5. Cold-disk age as a proxy of galaxy infall time into +a cluster +5.1. Calibration from simulations +It is hard to accurately estimate the infall time into the cluster +for each galaxy from observations, although we can statistically +infer the likelihood of being ancient or recent infallers from their +projected position and line-of-sight velocity in the cluster (Iodice +et al. 2019). By analyzing the cluster galaxies in the cosmologi- +cal simulation Illustris TNG50, we find that the cold-disk age is +tightly correlated with the infall time of the galaxy into the clus- +ter, as a result of star formation quenching in disks associated +with galaxy infall into the cluster (Ding et al., in prep.). +To make direct comparison between the Fornax and TNG50 +galaxies. We defined cold disks of TNG50 galaxies in exactly +the same way as here for the Fornax cluster galaxies, with the +probability density distribution of stars in the phase-space of cir- +cularity λz versus r calculated from the 6D position-velocity in- +formation of particles in the simulation. We note that we use +λz and r of the particles’ orbits, not directly the instantaneously +values of each particle, following previous work in comparing +the stellar orbit distribution of observed and simulated galaxies +(Zhu et al. 2018a). The cold-disk fraction and cold-disk age are +thus defined exactly the same way as described in Section 4. All +galaxies in the 14 clusters with virial mass M200 > 1013.3 M⊙ +in the TNG50 simulations are chosen. We define the galaxies’ +infall time into the cluster as the time when it first reaches the +virial radius r200 of the cluster at that time. For the pre-processed +galaxies, we define their infall time as their first time of falling +into the pre-cluster. +Article number, page 10 of 31 + +FCC255 +FCC119 +FCC177 +FCC182 +FCC148 +FCC263 +FCC153 +FCC301 +FCC290 +FCC143 +FCC170 +FCC308 +FCC083 +FCC249 +FCC312 +FCC161 +FCC179 +FCC276 +FCC147 +FCC167Y. Ding et al.: F3D: the environmental effects on the assembly of disks +Fig. 5. Infall time into the Fornax cluster of the sample galaxies for which we obtained the age of the dynamically cold disk. Left panel: Correlation +between the galaxy infall time tinfall and age of the dynamically cold disk tcold for four different mass bins: 8 < log M∗/M⊙ < 8.6 (light cyan), +8.6 < log M∗/M⊙ < 9.2 (cyan), 9.2 < log M∗/M⊙ < 10 (blue), and 10 < log M∗/M⊙ < 12 (dark blue). Dashed lines mark the 1σ confidence limits +of each correlation. Circles and crosses correspond to galaxies with extended and concentrated dynamically cold disks, respectively. Each galaxy +is plotted by the age of its dynamically cold disk and infall time given by the median of the correlation corresponding to its stellar mass. The error +bars corresponds to the 1σ uncertainty of the infall time inferred from the correlation. Right panel: Distribution of the left-panel galaxies in the +phase-space of the projected line-of-sight velocity of the galaxy normalized by line-of-sight velocity dispersion of the cluster VLOS/σLOS versus +the projected clustercentric radius of the galaxy normalized by the cluster virial radius Rproj/Rvir. Each galaxy is color-coded according to its infall +time except for the 4 LTGs without age and metallicity information shown in gray. The symbols are the same as in the left panel and the LTGs are +marked by gray symbols. The boundaries of the regions A, B, C, D, and E are defined as in Rhee et al. (2017). +We found that the correlation between the cold-disk age +and the infall time of the galaxy into the cluster does not +strongly depend on the cluster mass. We divided all the galax- +ies in the 14 clusters into four mass bins with log10 M∗/M⊙ ∈ +(8, 8.6), (8.6, 9.2), (9.2, 10), and (10, 11), respectively. In the left +panel of Fig. 5, we show the running median and ±1σ scatter of +the correlation in the four mass bins color-coded by the stellar +mass. The 1σ scatter is 0.63, 0.63, 0.94, and 1.34 Gyr from the +low to high mass bins. The correlation is tighter for the least- +massive galaxies. The scatter becomes significantly larger in the +most massive galaxies because a significant fraction of them are +already quenched or had little gas left before they fell into the +cluster. +For galaxies with the cold-disk age obtained from the +population-orbit superposition model, we can thus use this corre- +lation to infer their infall time into the Fornax cluster. As shown +in Fig. 5, we obtained the infall time of each galaxy by means +of the correlation for the mass bin including its stellar mass. The +values inferred from the median and 1σ scatter of the correlation +are taken as the median and 1σ uncertainty of the galaxy infall +time tinfall. They are listed in the last column of Table 2. +5.2. Comparison with galaxy locations in the projected +phase-space +We cross-checked the infall time inferred from the cold-disk +age with the galaxy location in the phase-space VLOS/σLOS vs. +Rproj/Rvir, as shown in the right panel of Fig. 5. Here, VLOS/σLOS +is the projected line-of-sight velocity of the galaxy normalized +by line-of-sight velocity dispersion of the cluster and Rproj/Rvir +is the projected clustercentric radius of the galaxy normalized +by the cluster virial radius. The galaxies are color-coded by the +Table 3. Fractions of ancient, intermediate, and recent infallers in the +regions of E, D, and B+C we find in our sample and a comparison with +that predicated from Rhee et al. (2017) in brackets. +Region +Ancient +Intermediate +Recent +E +85.7% (50%) +14.3% (20%) +0% (25%) +D +50% (20%) +0% (30%) +50% (50%) +B+C +40% (30%) +20% (20%) +40% (50%) +infall time inferred from the cold-disk age in the left panel of +Fig. 5. +Following Iodice et al. (2019), we define ancient infallers +as galaxies that fell into the cluster at 8-12 Gyr ago, intermedi- +ate infallers at 4-8 Gyr ago, and recent infallers at 0-4 Gyr ago. +We divided the phase-space into five regions like in Iodice et al. +(2019), with the fraction of ancient infallers decreaseing from +regions E and D to B+C (based on Rhee et al. (2017)). About +50% of the galaxies in region E are assumed to be ancient in- +fallers in our sample, while six out of seven galaxies in region E +are ancient infallers and one is intermediate infaller. Galaxies in +region D are supposed to be dominated by recent and intermedi- +ate infallers, while we have two ancient infallers and two recent +infallers. Galaxies in region B+C are suggested to be dominated +by recent infallers, while we have two ancient, one intermedi- +ate, and two recent infallers. The fractions of ancient, intermedi- +ate and ancient infallers in different regions we obtained and in +comparison to that predicted from Rhee et al. (2017) are shown +in Table 3. +Although we have a low amount of statistics at hand, we re- +main consistent with Rhee et al. (2017) in that the fraction of +ancient infallers is highest in region E and then lower in D and +B+C. However, the fraction of ancient infallers in our sample is +Article number, page 11 of 31 + +8< log10M+/M。 < 8.6 +Extended +B +Concentrated +8.6 9 Gyr) have low cold-disk fractions +of fcold(r < Re) ≲ 0.1 at all mass regions. +In the right panel, we show cold-disk fraction as a function of +galaxy infall time by dividing the galaxies into three mass bins. +For galaxies with similar stellar mass, the cold-disk fraction de- +creases from recent to ancient infallers. Although there is some +scatter, especially in the case of FCC 153, with tinfall ∼ 9 Gyr but +a high cold-disk fraction of fcold(< 1Re) > 0.3. +Similar figures but for cold-disk fraction within 2 Re are +shown in Fig. 7. The cold-disk fraction within 2 Re is about +0.1 higher than that within 1 Re for the galaxies with extended +disks. The cold-disk fractions of fcold(< 1Re) ∼ 0.3 and fcold(< +2Re) ∼ 0.4 at M∗ ∼ 1010 M⊙ for the recent infallers are a factor +of ∼ 3× or ∼ 4× higher than the ancient infallers mostly with +both fcold(< 1Re) < 0.1 and fcold(< 2Re) < 0.1. +6.2. Cold-disk age and age gradient +We further studied the age and age gradient of cold disks in these +galaxies. We calculated the age gradients of cold disks ∇tcold for +the nine galaxies with extended disks and colored models. As we +can see from Fig. 4, the age profiles of disks are diverse in the +inner regions and they become flat at the outer regions for most +galaxies. We calculated the age gradients with profiles at r < Re +consistently for all galaxies, except for FCC 170 and FCC 255, +the age gradient is calculated within the whole data coverage as +their disks have very low surface-brightness at r < Re. +In the left panel of Fig. 8, we show the mean age of the cold +disk versus stellar mass for the nine ETGs with extended disks +and with colored model. The disks are generally older in the +more massive galaxies. In the right panel of Fig. 8, we show the +age gradient of the cold disk versus stellar mass colored by the +galaxy infall time into the cluster. We get a positive- or zero age +gradient of the cold disk for most galaxies. The least-massive +galaxies show significantly positive ∇tcold for r < Re and the +age profiles become shallower at the outer regions. Three of the +most massive galaxies FCC 083, FCC 147, and FCC 167 still +have negative ∇tcold at r < Re, while their age profiles become +flat or even turn to be positive (for FCC 167) at the outer regions. +6.3. Cold-disk metallicity and metallicity gradient +In the left panel of Fig. 9, we show the mean metallicity of the +cold disks versus stellar mass M∗ colored by the galaxy infall +time into the cluster. We only show the nine galaxies with ex- +tended cold disks and colored models. The cold disks appear to +be similarly metal rich from low mass to high mass galaxies. The +average metallicity of the cold disks is ⟨Zcold⟩ = 0.91 ± 0.6 Z⊙, +with no significant correlation with galaxy stellar mass or infall +time into the cluster. Comparing to the general metallicity-mass +relation of galaxies (Gallazzi et al. 2005), the cold disks in galax- +ies with M∗ ≲ 1010 M⊙ in our sample are systematically more +metal-rich than the general population of galaxies, while in more +massive galaxies, they are are more similar in composition. +In the right panel of Fig. 9, we show the metallicity gradi- +ent of the cold disks versus stellar mass. Metallicity gradients +are calculated in a way similar to age gradients and only for the +nine galaxies with extended disks and colored models. Almost +all these galaxies show negative metallicity gradients for the cold +disks, which are more metal-rich in the inner regions than in the +outer one. There is no significant correlation seen between the +metallicity gradient of cold disk and stellar mass – other than the +change of metallicity gradients of the whole galaxy from pos- +itive to negative for low to high mass galaxies (Goddard et al. +2017; Zhuang et al. 2019), however, with low statistics of only +nine galaxies included. +6.4. Cold versus hot galaxy components +We compare the age and metallicity of the dynamically cold disk +and dynamically hot non-disk component in Fig. 10. The stars +in the cold disks are as old as those of the hot components for +the ancient infallers, whereas they are significantly younger than +those of the hot components in recent infallers. The cold disks +are in general more metal-rich that the hot components, which +could explain the difference between our results on cold disk +and the general mass-metallicity relation (Gallazzi et al. 2005), +as shown in Fig. 9. +We compared the age and metallicity gradients of the two +components in Fig. 11. For the galaxies with negative age gra- +dients in cold disks, they also have negative age gradients in the +hot component, while for the galaxies with positive age gradi- +ent in cold disks, the age gradients in their hot component could +be very different; whereas both the cold disk and hot component +have negative metallicity gradients in most galaxies and are gen- +erally consistent with each other. +The age and metallicity gradients of the cold disks are similar +to that of the whole galaxy, as shown in Fig. 12. The signs of +the gradients in cold disks are the same as what is seen for the +whole galaxy in most cases. In this sense, the gradients directly +measured for the whole galaxy could statistically reflect that in +the cold disks, for galaxies with extended cold disks. +Article number, page 12 of 31 + +Y. Ding et al.: F3D: the environmental effects on the assembly of disks +Fig. 6. Dependence of cold-disk fraction (within 1Re) on galaxy stellar mass and infall time. Left panel: Cold-disk fraction of the sample galaxies +as a function of stellar mass, color-coded by infall time into the Fornax Cluster to highlight the ancient (red), intermediate (white), and recent (blue) +infallers. Grey symbols correspond to the four LTGs without colored model which are considered as recent infallers. The grey curve represents +the cold-disk fraction with 1Re of CALIFA galaxies (Zhu et al. 2018b). Circles and crosses correspond to galaxies with extended and concentrated +dynamically cold disks, respectively. Right panel: Cold-disk fraction within 1Re as a function of infall time color-coded by stellar mass, divided +into three mass bins: log M∗/M⊙ < 9.2 (light blue), 9.2 < log M∗/M⊙ < 10 (blue), and log M∗/M⊙ > 10 (dark blue). The four LTGs without +colored model are plotted with an infall time upper limit of 4 Gyr ago. +Fig. 7. Dependence of cold-disk fraction (within 2Re) on galaxy stellar mass and infall time. Details are similar to Fig. 6, but for cold-disk fractions +within 2 Re, which are about 0.1 higher than that within 1 Re for galaxies with extended disks. +7. Discussion +We find that the most ancient infallers in the Fornax cluster have +low cold-disk fractions of fcold ≲ 0.1, which is lower by about +a factor of 4 than those of the recent and intermediate infallers +in the Fornax cluster and of the field galaxies (Zhu et al. 2018b), +with similar stellar mass. Although there is some scatter, espe- +cially with regard to FCC 153, which is potentially an ancient +infaller, but with a high cold-disk fraction. +Our results indicate that a galaxy’s infall into a cluster has a +strong effect on the assembly of the cold disk, which might be +caused by the cut-off of the gas accretion from the cluster en- +vironment or removal of the gas by ram pressure and tidal strip- +ping. Both processes explain the above results: galaxies that have +earlier fallen into the cluster have their gas earlier removed or cut +off, thus their disks are older and contribute a smaller fraction to +total luminosity; galaxies that have recently fallen into the clus- +ter have more time to grow their disk and the disk stars could be +younger. However, it should be noted that we have the luminos- +ity and not the mass fraction of the disk. The higher luminosity +fraction of the cold disks in recent infallers could be partially +caused by the lower stellar mass-to-light ratio in younger stellar +disks. +It has been argued that ancient infallers could have a dis- +tinct evolutionary history before entering the cluster environ- +ment, which determined their morphology, kinematics, and gas +Article number, page 13 of 31 + +Extended +O× log10M+/M。< 9.2 + Concentrated +O× 9.2 10.0O× log10M+/M。< 9.2 +Extended +× 9.2 10.0A&A proofs: manuscript no. main +Fig. 8. Mean age (left panel) and age gradient (right panel) of the extended cold disks in the sample galaxies as a function of stellar mass and +infall time into the Fornax Cluster. Dashed line in the right panel marks the zero age gradient. +Fig. 9. Mean metallicity (left panel) and metallicity gradient (right panel) of the extended cold disks in the sample galaxies as a function of stellar +mass and infall time into the Fornax Cluster. The black curve and grey shadow represent the metallicity-stellar mass relation and the 1σ uncertainty +for the galaxies in Gallazzi et al. (2005). Symbols are same as Fig. 8. +content “at infall“ (e.g., Han et al. 2018; Su et al. 2021). This +could be part of the reason for the difference we see between +the ancient and recent infallers at present time. But this prepro- +cessing cannot explain all the differences. Numerical simulations +show that dynamically cold disks are often partially removed or +heated by tidal disruption in cluster (Joshi et al. 2020). With the +control of galaxy properties at the time of infall, the galaxies that +fall into a cluster result in significantly smaller disk fraction at +redshift zero than those not. In this sense, the ancient infallers +now in the cluster center would suffer most from the tidal heat- +ing or disruption, thus their cold-disk fraction could be further +reduced. +We find positive or flat age gradients in the extended disks, +which is opposite to what expected from a normal inside-out disk +growth. The turnover of disk-age gradients could be the result of +the continuous star formation in the galaxy inner regions, while +the star formation in the outer regions is quickly reduced after +falling into the cluster. This is generally consistent with the sce- +nario found in the EAGLE simulations (Pfeffer et al. 2022). In +five of our nine galaxies with extended disks, the stars in the in- +ner disk region are about 2-5 Gyr younger than those in the outer +disk region, which indicates that the star formation in galaxy in- +ner regions could last for a few Gyrs more after the galaxy falls +into the cluster. None of the 15 ETGs of our sample exhibit any +detectable ongoing star formation: stars in the inner disks are +younger, but their star formation stopped a few Gyrs ago. On +the other hand, the inner regions are usually blended with an +old bulge component, the age gradient of the whole galaxy is +thus not so obviously negative as the disk only. Lasting star for- +mation in galaxy inner regions is usually difficult to be directly +observed in galaxies at present time, while hopefully might be +directly observable at high redshift (Maier et al. 2019a). +Article number, page 14 of 31 + +Extended +young inside vs. old outside +old inside vs. young outsideExtended +poor inside vs. rich outside +Gallazzi+2005 +rich inside vs. poor outsideY. Ding et al.: F3D: the environmental effects on the assembly of disks +Fig. 10. Comparison of the mean ages (left panel) and mean metallicities (right panel) of the dynamically cold disks and dynamically hot non-disk +components. Circles and crosses correspond to galaxies with extended and concentrated cold disks, respectively. The mean ages and metallicities +are calculated for r < Re (filled circles). Black dashed lines represent y = x. +Fig. 11. Comparison of the age gradient (left panel) and metallicity gradient (right panel) of the dynamically cold disks and dynamically hot +non-disk components, for the galaxies with extended disks. Black dashed lines represent y = x. Blue dashed lines represent zero gradients. +Strong positive age gradients have been reported for the +SAURON ETGs in the mass range of 1010−12 M⊙ using the +method based on line indices (Kuntschner et al. 2010). How- +ever, in larger sample of MaNGA galaxies, the age gradients of +ETGs are found to be generally shallow (Goddard et al. 2017). +Positive age gradients are mostly found in low mass ETGs with +M∗ ≲ 1010.5 (Li et al. 2018) by using the technique of full spec- +tra fitting to derive the stellar populations. As shown in Li et al. +(2018), the strong positive age gradients reported in Kuntschner +et al. (2010) could disappear when uniformly using the same +technique. As we are also using the full spectra fitting, a com- +parison with these recent results from MaNGA galaxies might +offer a more fair outcome. We note that the age gradients found +in MaNGA galaxies are different in Goddard et al. (2017) and +Li et al. (2018), while the samples these authors studied are also +different. +We found positive age gradients in the cold disks of least- +massive galaxies, namely, M∗ ≲ 1010.5, whereas the disks that +have negative age gradients in the inner regions and tend to be +flat in the outer regions of most massive galaxies. This is consis- +tent with the positive age gradient in low-mass MaNGA ETGs +(Li et al. 2018), considering age gradients of the cold disks are +statistically consistent with that of the whole galaxies. The differ- +ence of age gradients in low mass and high mass ETGs could be +explained by the difference of gas content in low and high mass +galaxies at infall. The low mass galaxies could have large gas +fractions when they fall into the cluster. The gas in the outer re- +gions is stripped, but the gas kept in inner regions form a signif- +icant amount of new stars that finally turn-over the age gradient. +On the other hand, massive galaxies could have already formed +most of their stars before entering the cluster. They have small +gas fractions when falling into the cluster, thus the age gradient +Article number, page 15 of 31 + +Extended +× ConcentratedA&A proofs: manuscript no. main +Fig. 12. Comparison of the age gradient (left panel) and metallicity gradient (right panel) of the dynamically cold disks and the whole galaxy, for +the galaxies with extended disks. Symbols are the same as in Fig. 11. +is less likely to be turned over by the formation of new stars after +entering the cluster. Although with limited statistics, there is no +galaxy with significant negative age gradient in our sample with +M∗ ≲ 1010.5. Approximately flat gradients for such galaxies are +found in the MaNGA sample studied by Goddard et al. (2017), +while the MaNGA sample studied in Li et al. (2018) includes +galaxies in different environment, and shows large scatter of age +gradients of ETGs in the same mass range, which may indicate +the effects of different environments. Although other physical +processes, such as mergers, could also cause new star formation +in the center of these relatively low mass ETGs in the field (Yozin +& Bekki 2012; Lagos et al. 2022). +Similar ages of the disk and hot components in ancient in- +fallers indicate that stars also formed on dynamically warmer or- +bits after falling. Star formation onto dynamically warmer orbits +is possible if the gas is compacted into the inner regions of the +galaxy, resulting in a high gas density in the galaxy center after +falling into the cluster. As shown in Fig. 4, we also find younger +stars in the very center of the hot components for a few galaxies, +which is consistent with the aforementioned scenario. +8. Conclusions +We built orbit-superposition models for 20 galaxies (15 ETGs ++ 5 LTGs) in the Fornax cluster observed with MUSE/VLT +in the context of the Fornax3D survey. We built a population- +orbit superposition model for 16 sample galaxies by coloring +the orbits with age and metallicity. By simultaneously fitting the +surface-brightness, stellar kinematic, age, and metallicity maps, +we obtain the internal stellar orbit distribution, as well as age +and metallicity distribution of each galaxy. Based on the Best- +fit population-orbit superposition model, we decompose each +galaxy into a dynamically cold disk (λz ≥ 0.8) and a dynami- +cally hot non-disk component (λz < 0.8). Then, we obtain the +surface-brightness, age, and metallicity radial profiles for each +component by projecting them face-on. We analyze the depen- +dence of the cold disk properties on galaxy stellar mass and infall +time into the Fornax cluster. Our main results are as follows: +1. We estimate the galaxy infall time into the cluster based +on a tight correlation with cold-disk age calibrated with the cos- +mological simulation TNG 50. The infall time inferred in this +way is statistically consistent with the galaxy’s location in the +cluster’s observational phase-space diagram. +2. The ancient infallers have significantly lower luminosity +fractions of the cold disk component, regardless of stellar mass. +The recent and intermediate infallers have cold-disk fraction in- +creasing as a function of stellar mass. They reach fcold(< Re) ∼ +0.3 and fcold(< 2Re) ∼ 0.4 at M∗ ∼ 1010 M⊙, consistent with +the CALIFA galaxies in the field. Most ancient infallers have +fcold ≲ 0.1, which is a factor of ∼ 4 lower than that of recent +infallers of the same stellar mass. +3. Nine of the 16 galaxies with population-orbit superposi- +tion have extended cold disks. The five least-massive galaxies +have positive or flat age gradients in their cold disks. Indeed, the +stars in the inner disk regions are about 2-5 Gyr younger than +those in the outer disk regions. In contrast, the four more mas- +sive galaxies, which might have a lower gas content at infall, +have negative age gradients in the inner disk regions and flat age +profiles in the outer disk regions. +Our results indicate that the star formation rate in disks can +be significantly reduced after the galaxies fall into a cluster in +general. While the star formation in the outer regions of a galaxy +could stop shortly after it falls into the cluster, star formation in +the inner regions should last for longer time causing the stars in +the inner disk regions to be 2-5 Gyr younger than those in the +outer disk regions. This is generally consistent with the scenario +found for the Virgo cluster (Crowl & Kenney 2008), while we +have been able for the first time to quantify the age difference +between the inner and outer regions of cold disks. +By taking advantage of the high-quality MUSE datacubes +and structure decomposition based on the population-orbit su- +perposition models, we have been able to isolate and accurately +study the dynamically cold disk component. This allows for an +in-depth and direct comparison of the galactic structures with +galaxies formed in cosmological simulations, which will help us +to further understand the physical processes driving galaxy evo- +lution. +Acknowledgements. The models and corresponding results presented in this pa- +per would not have been possible without the VLT-MUSE data granted through +the award of DDT telescope time (ESO programme 296.B-5054(A)). LZ ac- +knowledges the support from the National Key R&D Program of China under +Article number, page 16 of 31 + +Y. Ding et al.: F3D: the environmental effects on the assembly of disks +grant No. 2018YFA0404501, National Natural Science Foundation of China +under grant No. Y945271001, and CAS Project for Young Scientists in Basic +Research under grant No. YSBR-062. GvdV acknowledges funding from the +European Research Council (ERC) under the European Union’s Horizon 2020 +research and innovation programme under grant agreement No 724857 (Consol- +idator Grant ArcheoDyn). L.C. acknowledges financial support from Comunidad +de Madrid under Atracción de Talento grant 2018-T2/TIC-11612 and the Span- +ish Ministerio de Ciencia, Innovación y Universidades through grant PGC2018- +093499-B-I00. J. F-B, IMN and FP acknowledge support through the RAVET +project by the grant PID2019-107427GB-C32 from the Spanish Ministry of +Science, Innovation and Universities (MCIU), and through the IAC project +TRACES which is partially supported through the state budget and the regional +budget of the Consejería de Economía, Industria, Comercio y Conocimiento of +the Canary Islands Autonomous Community. The F3D data is based on obser- +vations collected at the European Southern Observatory under ESO programme +296.B-5054(A), and available in the ESO Science Archive Facility. EMC is sup- +ported by MIUR grant PRIN 2017 20173ML3WW-001 and Padua University +grants DOR2019-2021. +References +Abadi, M. G., Moore, B., & Bower, R. G. 1999, MNRAS, 308, 947 +Ahn, C. P., Seth, A. C., Cappellari, M., et al. 2018, ApJ, 858, 102 +Alpaslan, M., Driver, S., Robotham, A. S. G., et al. 2015, MNRAS, 451, 3249 +Balogh, M. L., Navarro, J. F., & Morris, S. L. 2000, ApJ, 540, 113 +Barnes, J. 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C. 2021, +ApJ, 920, 63 +Zibetti, S. 2019, in The Art of Measuring Galaxy Physical Properties, 34 +Article number, page 18 of 31 + +Y. Ding et al.: F3D: the environmental effects on the assembly of disks +Appendix A: Mass density MGE of all galaxies +Σ +σ +q +[M⊙ /pc2] +[arcsec] +26077.2 +0.3800 +0.670 +6911.64 +1.330 +0.673 +2183.15 +3.290 +0.667 +643.410 +8.000 +0.576 +199.660 +15.24 +0.666 +60.8800 +30.28 +0.612 +24.9400 +57.44 +0.554 +6.48000 +91.07 +0.867 +Table A.1. MGE parametrization of the stellar mass distribution of +FCC 083. The central mass surface density (1), rms (2), and axial ra- +tio (3) of all the model Gaussians are given. +Σ +σ +q +[M⊙ /pc2] +[arcsec] +505.05 +0.570 +0.894 +129.23 +4.600 +0.846 +39.090 +10.40 +0.941 +7.2100 +10.48 +0.503 +9.4600 +25.18 +0.783 +Table A.2. MGE parametrization of the stellar mass distribution of +FCC 119. Details are same as Table A.1. +Σ +σ +q +[M⊙ /pc2] +[arcsec] +17884.1 +0.2900 +0.787 +3400.83 +1.020 +0.738 +1153.89 +2.030 +0.837 +426.680 +4.330 +0.697 +104.830 +7.060 +0.990 +59.5300 +9.470 +0.462 +50.9900 +13.76 +0.977 +7.08000 +30.06 +0.943 +Table A.3. MGE parametrization of the stellar mass distribution of +FCC 143. Details are same as Table A.1. +Σ +σ +q +[M⊙ /pc2] +[arcsec] +54287.4 +0.2800 +0.871 +11087.4 +1.080 +0.845 +3380.12 +2.730 +0.896 +944.320 +7.150 +0.889 +288.790 +14.53 +0.903 +90.3000 +24.10 +0.891 +43.3400 +42.76 +0.947 +6.38000 +98.90 +0.910 +Table A.4. MGE parametrization of the stellar mass distribution of +FCC 147. Details are same as Table A.1. +Σ +σ +q +[M⊙ /pc2] +[arcsec] +35949.7 +0.2500 +0.673 +3819.35 +0.8500 +0.717 +1382.12 +2.010 +0.629 +704.410 +4.320 +0.552 +385.580 +15.15 +0.385 +59.4800 +31.93 +0.436 +4.55000 +67.21 +0.664 +Table A.5. MGE parametrization of the stellar mass distribution of +FCC 148. Details are same as Table A.1. +Σ +σ +q +[M⊙ /pc2] +[arcsec] +22889.8 +0.2800 +0.534 +1889.26 +1.300 +0.582 +732.830 +4.810 +0.487 +165.960 +8.680 +0.672 +1454.73 +17.72 +0.0660 +269.200 +23.37 +0.152 +77.8500 +34.30 +0.264 +10.3700 +48.75 +0.452 +Table A.6. MGE parametrization of the stellar mass distribution of +FCC 153. Details are same as Table A.1. +Σ +σ +q +[M⊙ /pc2] +[arcsec] +41054.9 +0.2700 +0.990 +6726.13 +0.8900 +0.990 +1976.01 +2.240 +0.990 +878.060 +5.350 +0.990 +456.600 +10.56 +0.990 +154.630 +19.45 +0.990 +80.1700 +35.79 +0.990 +7.38000 +84.77 +0.862 +Table A.7. MGE parametrization of the stellar mass distribution of +FCC 161. Details are same as Table A.1. +Σ +σ +q +[M⊙ /pc2] +[arcsec] +16508.58 +0.584 +0.864 +2537.55 +1.917 +0.999 +2872.85 +5.143 +0.727 +463.50 +13.820 +0.807 +485.79 +14.448 +0.541 +120.08 +43.797 +0.269 +150.33 +47.212 +0.524 +27.77 +58.192 +0.972 +10.88 +120.764 +0.542 +Table A.8. MGE parametrization of the stellar mass distribution of +FCC 167. Details are same as Table A.1. +Article number, page 19 of 31 + +A&A proofs: manuscript no. main +Σ +σ +q +[M⊙ /pc2] +[arcsec] +8967.82 +0.4600 +0.877 +1007.78 +2.500 +0.645 +158.910 +11.75 +0.493 +316.810 +23.26 +0.145 +78.0600 +30.28 +0.315 +15.6900 +47.55 +0.510 +Table A.10. MGE parametrization of the stellar mass distribution of +FCC 177. Details are same as Table A.1. +Σ +σ +q +[M⊙ /pc2] +[arcsec] +14717.035 +1.318 +0.582 +2904.596 +3.360 +0.851 +512.611 +10.647 +0.658 +281.527 +37.423 +0.336 +28.881 +47.429 +0.493 +Table A.11. MGE parametrization of the stellar mass distribution of +FCC 179. Details are same as Table A.1. +Σ +σ +q +[M⊙ /pc2] +[arcsec] +2579.35 +0.2400 +0.600 +641.810 +0.8400 +0.684 +549.780 +2.370 +0.516 +400.120 +3.290 +0.862 +68.7500 +6.680 +0.990 +42.9800 +12.07 +0.996 +5.95000 +24.20 +0.990 +Table A.12. MGE parametrization of the stellar mass distribution of +FCC 182. Details are same as Table A.1. +Σ +σ +q +[M⊙ /pc2] +[arcsec] +36302.10 +0.15000 +0.72549 +11913.97 +0.37207 +0.35867 +12273.34 +0.37328 +0.92307 +8826.98 +0.68704 +0.32285 +9179.47 +0.97482 +0.79328 +4462.54 +1.90931 +0.74662 +3495.96 +2.83135 +0.74571 +988.46 +5.90353 +0.59377 +590.84 +6.86747 +0.75000 +360.95 +7.31068 +0.97943 +26.06 +16.27466 +0.99990 +434.80 +16.30921 +0.18070 +162.51 +26.67680 +0.18070 +162.51 +26.67696 +0.18070 +162.51 +26.67736 +0.18070 +4.03 +32.84468 +0.99990 +21.70 +36.36404 +0.25361 +59.66 +36.80145 +0.25867 +13.12 +46.28263 +0.39076 +Table A.9. MGE parametrization of the stellar mass distribution of +FCC 170. Details are same as Table A.1. +Σ +σ +q +[M⊙ /pc2] +[arcsec] +25539.2 +0.2600 +0.841 +5431.22 +1.010 +0.775 +1957.88 +1.830 +0.907 +876.430 +3.080 +0.990 +307.980 +5.340 +0.990 +186.070 +8.180 +0.990 +51.7400 +15.45 +0.990 +9.10000 +45.49 +0.990 +Table A.13. MGE parametrization of the stellar mass distribution of +FCC 249. Details are same as Table A.1. +Σ +σ +q +[M⊙ /pc2] +[arcsec] +3301.56 +0.3100 +0.510 +309.560 +1.910 +0.699 +472.760 +6.970 +0.322 +180.810 +12.80 +0.405 +48.2500 +21.67 +0.591 +7.15000 +43.10 +0.855 +Table A.14. MGE parametrization of the stellar mass distribution of +FCC 255. Details are same as Table A.1. +Σ +σ +q +[M⊙ /pc2] +[arcsec] +199.583 +2.438 +0.999 +244.042 +6.932 +0.471 +127.639 +12.327 +0.501 +42.153 +24.796 +0.522 +1.213 +71.410 +0.571 +Table A.15. MGE parametrization of the stellar mass distribution of +FCC 263. Details are same as Table A.1. +Σ +σ +q +[M⊙ /pc2] +[arcsec] +66551.4 +0.26000 +0.690 +15406.9 +1.0000 +0.660 +4284.05 +2.2900 +0.720 +1621.34 +4.8800 +0.728 +576.060 +11.500 +0.695 +222.270 +24.030 +0.696 +49.4000 +50.390 +0.688 +12.6200 +107.64 +0.769 +Table A.16. MGE parametrization of the stellar mass distribution of +FCC 276. Details are same as Table A.1. +Article number, page 20 of 31 + +Y. Ding et al.: F3D: the environmental effects on the assembly of disks +Σ +σ +q +[M⊙ /pc2] +[arcsec] +2494.579 +0.355 +0.794 +558.645 +2.237 +0.836 +225.838 +6.090 +0.752 +102.083 +32.165 +0.614 +60.018 +46.855 +0.711 +Table A.17. MGE parametrization of the stellar mass distribution of +FCC 290. Details are same as Table A.1. +Σ +σ +q +[M⊙ /pc2] +[arcsec] +8660.98 +0.2500 +0.482 +1943.09 +2.390 +0.414 +954.810 +3.660 +0.551 +249.790 +7.690 +0.561 +68.4400 +13.88 +0.850 +10.9000 +30.71 +0.870 +Table A.18. MGE parametrization of the stellar mass distribution of +FCC 301. Details are same as Table A.1. +Σ +σ +q +[M⊙ /pc2] +[arcsec] +91.394 +0.920 +0.749 +184.543 +11.217 +0.382 +55.719 +32.469 +0.345 +6.282 +91.447 +0.333 +Table A.19. MGE parametrization of the stellar mass distribution of +FCC 308. Details are same as Table A.1. +Σ +σ +q +[M⊙ /pc2] +[arcsec] +374.086 +0.917 +0.665 +284.956 +7.942 +0.342 +243.135 +31.433 +0.184 +67.028 +62.940 +0.222 +9.274 +151.033 +0.260 +Table A.20. MGE parametrization of the stellar mass distribution of +FCC 312. Details are same as Table A.1. +Appendix B: Best-fit models of all galaxies +Appendix C: Grid parameters of FCC 177 +Article number, page 21 of 31 + +A&A proofs: manuscript no. main +Fig. B.1. Best-fit population-orbit superposition model (top panels) and orbital decomposition (bottom panels) of FCC 083. The content of the top +and bottom panels is the same as in Figs. 1 and 2, respectively. +Fig. B.2. Best-fit population-orbit superposition model (top panels) and orbital decomposition (bottom panels) of FCC 119. The content of the top +and bottom panels is the same as in Figs. 1 and 2, respectively. +Article number, page 22 of 31 + +SB [log 10 La/pc?] +V [km/s] +α [km/s] +h3 +h4 +t [Gyr] +z [ZZ] +3 +4 +-101 +0 +1i0 +T +125 +0.1 +O.D +0.1 +0.1 +a.D +n.1 +12 +0.5 +L'D +15 +[] +20 +Data +0. +-20 +40 +[] +20 +Model +0 +-20 +AP[ - PWA] +[omadel Odetn]fdo +Eyp/[qPey - pawey] +ryp/[qappy - pawpy] +[tmadel - tcabn]/dt +[Zmodel -Zdaba]/dZ +0.05D-0.025 0.D00.D250.D55.0 +-2.5 +2.5 +5.D +5.0 +-2.5 +2.5 +5D +5.0 +-2.5 +O.D +2.5 +5.D +2.5 +5.D +O.D +2.5 +5.D +5.0 +2.5 +O.D +5.D +Residual +[eare] +0. +20 +FEE +25 +25 +25 +-25 +25 +-25 +25 +25 +50 +] +X [arcsec] +(4] +X [arcsec] +X [arcsec] +X [arcsec] +X [arcsec] +X [arcsec] +X [arcsec]Probability density +t [Gyr] +z[ZZo] +0.000 0.005 0.010 0.015 0.020 +8 +10 +12 +14 +0.0 +0.5 +1.0 +1.0 +FCC083 +0.5 +0.0 +0.5 +-1.0 + +20 +40 +60 +80 +0 +20 +40 +60 +80 +0 +20 +40 +60 +80 +r[arcsec] +r[arcsec] +r[arcsec]SB [log 10 La/p22] +V [km/s] +α [kmfs] +h3 +h4 +t [Gyr] +z [ZZo] +L50 +175 +200 +225 +-10 +0 +1D +20 +1B +18 +20 +0.2 +O.D +0.2 +-0.1 +0.1 +8 +O.D +0.2 +0.4 +O.B +0.B +白 +[] +Data +-10 +[] +Model +-10 +AP/[PA- PWA] +op[Po pwo] +cyp/[qapey - 1ipoey] +4p/[qappy - i9pup4] +[tmadel - taaba]fdt +(Zmod -Zdala]/dZ +0.05-0.025 0.000 0.025 0.050 - +5.0 +-2.5 +D +2.5 +5D +5.0 +-2.5 +0.D25 +5.0 +2.50.D +25 +5D +5.0 -2.50.0 +25 +5D +5.0 -2.50.025 +5D +5.0 +-2.50025 +5.D +Residual +[] +10 +FEEEE +20 +-10 +1D +20 +-20 +0 +in +-20 +-10 +0 +i +20 +-20 +-i0 +0 +1D +20 +-10 +20 +-20 +-10 +0 +in +-20 +-10 +0 +20 +X [arcsec] +X [arcsec] +X [arcsec] +X [arcsec] +X [arcsec] +X [arcsec] +X [arcsec]Probability density +t[Gyr] +Z[ZZo] +0-0'0 +0.01 +0.02 +0.03 +0.04 +2.5 +5.0 +10.0 12.5 +0.0 +0.2 +0.4 +0.6 +1.0 +- +FCC119 +0.5 +0.0 +0.5 +-1.0 + +5 +10 +15 +20 0 +5 +10 +15 +20 0 +10 +15 +20 +rarcsec +rarcsec +rarcsecY. Ding et al.: F3D: the environmental effects on the assembly of disks +Fig. B.3. Best-fit population-orbit superposition model (top panels) and orbital decomposition (bottom panels) of FCC 143. The content of the top +and bottom panels is the same as in Figs. 1 and 2, respectively. +Fig. B.4. Best-fit population-orbit superposition model (top panels) and orbital decomposition (bottom panels) of FCC 147. The content of the top +and bottom panels is the same as in Figs. 1 and 2, respectively. +Article number, page 23 of 31 + +Probability density +t [Gyr] +Z[ZZo] +0.00 +0.02 +0.04 +0.06 +8 +10 +12 +14 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +FCC143 +0.5 +0.0 +0.5 +-1.0 + +20 +40 +0 09 +20 +40 +60 Q +20 +40 +60 +r[arcsec] +r[arcsec] +r[arcsec]SB [log 10 Lafp2?] +V [km/s] +α [kmfs] +h3 +h4 +t [Gyr] +Z [ZZ] +3 +50 +0 +50 +150 +2010 +0.1 +O.D +a.1 +0.1 +0.1 +12 +13 +0.5 +LD +1.5 +[ae] 1 +21 +-20 +[] +Mod +白 +叫 +[SBmadel - SBdeta]/SBdalta +AP/(PA-Ppa"A) +(omedel - odeta ]fdo +E4P/[q8Pe4 - 9p04E4] +14p/[qePpy - 9pp4] +(tmadel - taaba]/dt +[Zmodd Zdeta]/dZ +0.05D-0.025 0.000 0.025 0.D50 5.0 +2.5 +O.D +25 +5.D +-2.50.D25 +5.D +2.5 +O.D +5D +5.02.50.D25 +5D +5.0-2.50.D25 +5.D +5.0 +2.50.D +25 +5.D +4D +Residual +[] +20 +20 +40 +-20 +40 +-20 +40 +20 +P:1 +40 +20 +40 +-20 +40 +-20 +40 +20 +X [arcsec] +X [arcsec] +X [arcsec] +X [arcsec] +X [arcsec] +X [arcsec] +X [arcsec]Probability density +t [Gyr] +Z[ZZo] +0.00 +0.01 +0.02 +0.03 +0.04 +10 +11 +12 +13 +14 +0.0 +0.5 +1.0 +1.5 +1.0 +FCC147 +0.5 +0.0 +0.5 +-1.0 + +20 +40 +60 +8 心 +20 +40 +60 +8 心 +20 +40 +60 +80 +r[arcsec] +rarcsec +r[arcsec]SB [log 10 Lsfp2?] +V[km/s] +α [kmfs] +h3 +h4 +t [Gyr] + [ZZ] +2 +3 +-20 +0 +-0.1 +a.D +0.1 +-0.1 +a.D +a.1 +11 +12 +13 +0.5 +LD +[ae] +Data +30 +Model +-30 +AP[明A - PWA] +op/[qapo - powo] +E4p/[qPey - p0wey] +[tmadel taaba]fdt +[Zmoded -Zdala ]/dZ +0.05D0.025 0.D00 0.D25 0.D50 +5.0 +2.5 +O.D +2.5 +5.D +5.0 +-2.5 +O.D +2.5 +5.D +5.0 +2.5 +O.D +2.5 +5.D +5.0 +-2.50.D +2.5 +5.D +2.5 +5.D +5.0 +-2.5 +2.5 +5.D +20 +Residual +[aBae] +30 +二 +TT +OZ- +0 +20 +OZ- +0 +-20 +(4] +Oz- +] +-20 +-20 +0 +20 +0 +X[arcsec] +X [arcsec] +X [arcsec] +X [arcsec] +X [arcsec] +X[arcsec] +X [arcsec]A&A proofs: manuscript no. main +Fig. B.5. Best-fit population-orbit superposition model (top panels) and orbital decomposition (bottom panels) of FCC 148. The content of the top +and bottom panels is the same as in Figs. 1 and 2, respectively. +Fig. B.6. Best-fit population-orbit superposition model (top panels) and orbital decomposition (bottom panels) of FCC 153. The content of the top +and bottom panels is the same as in Figs. 1 and 2, respectively. +Article number, page 24 of 31 + +SB [log 10 Lafp2?] +V [km/s] +α [km/s] +h3 +h4 +t [Gyr] +Z [ZZ] +3 +50 +50 +40 +N'T +a.D +0.2 +-0.1 +a.D +α.1 +4 +LD +1.5 +[aea] +20 +Data +0 +[] +20 +Model +0 +20 +AP[A- PWA] +[omadel Odeta ]/do +E4p/[q8pey - 18p04ey] +- tab/dt +[Zmodd -Zdaba]/dZ +0.05D-0.025 0.000 0.025 0.050 +5.0 +2.5 +O.D +2.5 +5.D +5.0 +2.5 +2.5 +5.D +5.0 +2.5 +O.D +2.5 +5.D +5.0 +2.5 +O.D +5.D +O.D +5.D +O.D +5.D +Residual +[Ba] +-20 +25 +25 +F1 +5 +25 +0 +25 +F1 +25 +0 +50 +75 +-25 +0 +25 +25 +25 +25 +75 +25 +... +T5 +X [arcsec] +X [arcsec] +X [arcsec] +X [arcsec] +X [arcsec] +X [arcsec] +X [arcsec]Probability density +t [Gyr] +Z[ZZo] +0.000 0.005 0.010 0.015 0.020 +2.5 +7.5 +10.0 +0 +1 +2 +1.0 +FCC148 +Circularity Az +0.5 +0.0 +0.5 +-1.0+ +10 +20 +30 +40 0 +10 +20 +30 +40 0 +10 +20 +30 +40 +r[arcsec] +r[arcsec] +r[arcsec]SB [log 10 La/pc?] +V [kn/s] +α [kmfs] +h3 +h4 +t [Gyr] +z [ZZs] +3 +COT- +0 +25 +50 +0.2 +a.D +0.2 +-0.1 +a.D +a.1 +i +12 +a.D +LD +L5 +0 +20 +F: +20 +[Vmodel - Vdata ]/dV +[omadel - Odeba ]fdo +[h4model h4data ]/dh4 +(tmadel taaba]/dt +(Zmodd -Zdala]/dZ +0.050-0.025 0.000 0.D25 0.050 5.02.50.0 +25 +5.D +5.0-2.5 +O.D +5.D +5.02.5 +O.D +5.D5.02.5 +5.D +5.02.5 +2.5 +BITE +FEEEE +++++? +EEE +FEE +-75 +75 +-75 +75 +-75 +x[arcsec] +Xarcsec] +xarcsec] +X [arcsec] +xarcsec] +X [arcsec] +X [arcsec]Probability density +t [Gyr] +Z[ZZo] +0-0'0 +0.01 +0.02 +0.03 +6 +8 +10 +12 +14 +0 +i +2 +3 +4 +1.0 +FCC153 +0.5 +0.0 +0.5 +-1.0 + +20 +40 +60 +器心 +20 +40 +60 +器心 +20 +40 +60 +80 +r[arcsec] +rarcsec +rarcsecY. Ding et al.: F3D: the environmental effects on the assembly of disks +Fig. B.7. Best-fit population-orbit superposition model (top panels) and orbital decomposition (bottom panels) of FCC 161. The content of the top +and bottom panels is the same as in Figs. 1 and 2, respectively. +Fig. B.8. Best-fit population-orbit superposition model (top panels) and orbital decomposition (bottom panels) of FCC 167. The content of the top +and bottom panels is the same as in Figs. 1 and 2, respectively. +Article number, page 25 of 31 + +SB [log 10 Lofpc?] +[sf] A +α [kmfs] +h3 +h4 +t [Gyr] +z [ZZa] +2 +3 +-25 +10 +125 +a.i +a.D +a.1 +12 +13 +0.2 +0.4 +0.B +a.B +[] +20 +Data +-20 +40 +[B] 1 +Model +0 +-20 +40 +[SBmadel - SBdeta]/SBdeta +[Vmodel - Vdala ]/dV +op(Po powo] +Eyp/("Pey pouey) +typ/[qappy - apoupy] +[tmadel - taaba]fdt +[Zmodel -Zdala ]/dZ +0.050-0.025 0.000 0.025 0.050 -5.0 +-2.5 +2.5 +5D +5.0 +2.5 +O.D +2.5 +5.D +5.0 +2.5 +O.D +2.5 +5.D +5.0 +2.5 +5D +5.0 +2.5 +O.D +25 +5.D +5.0 +-2.5 +O.D +2.5 +5.D +Residual +[B] +20 +-20 +LLLLL +FLLL +40 +0 +40 +20 +0 +40 +0 +AD +40 +20 +0 +4D +40 +-20 +40 +40 +-20 +0 +20 +40 +40 +-20 +0 +X [arcsec] +X [arcsec] +X [arcsec] +[oesa] x +X [arcsec] +X [arcsec] +X [arcsec]Probability density +t [Gyr] +Z[ZZo] +0.00 +0.02 +0.04 +10 +12 +14 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +FCC161 +Circularity Az +0.5 +0.0 +0.5 +-1.0 + +20 +40 +60 +器心 +20 +40 +60 +器心 +20 +40 +60 +80 +r[arcsec] +r [arcsec] +r[arcsec]SB [log 10 Lsfp2?] +V[km/s] +α [kmfs] +h3 +h4 +t [Gyr] +Z [ZZ] +200 +0 +50 +150 +2010 +0.2 +O.D +0.2 +-0.1 +O.D +0.1 +12 +LD +1.5 +2D +BL] 1 +25 +[] +SAdebaJ/SBdab +AP/[PA - P] +op/[Po - ppo4o] +E4p/(8ey - pa"e4] +(tmadel - taaba]/dt +[Zmpddl - Zdala]/dZ +-2.5 +O.D +2.5 +5.D +5.0-2.5 +O.D +25 +5.D +5.0 +-2.5 +O.D +25 +5D +5.0 +2.5 +O.D +2.5 +5.D +5.02.5 +2.5 +5.D +5.02.5 +O.D +5.D +TLLLLLL +101 +50 +-100 +101 +101 +50 +101 +50 +100 +50 +-100 +50 +X [arcsec] +X [arcsec] +X [arcsec] +X [arcsec] +X [arcsec] +X [arcsec] +X [arcsec]Probability density +t [Gyr] +Z[ZZo] +0.00 +0.01 +0.02 +8 +10 +12 +14 +1 +2 +3 +1.0 +FCC167 +0.5 +0.0 +0.5 +-1.0 + +50 +100 +150 +50 +100 +150 +50 +100 +150 +r[arcsec] +r[arcsec] +r[arcsec]A&A proofs: manuscript no. main +Fig. B.9. Best-fit population-orbit superposition model (top panels) and orbital decomposition (bottom panels) of FCC 170. The content of the top +and bottom panels is the same as in Figs. 1 and 2, respectively. +Fig. B.10. Best-fit population-orbit superposition model (top panels) and orbital decomposition (bottom panels) of FCC 179. The content of the +top and bottom panels is the same as in Figs. 1 and 2, respectively. We find a possible weak bar structure at |X| ≲ 20 arcsec and |Y| ≲ 10 arcsec. +Article number, page 26 of 31 + +SB [log 10 La/pc?] +V [km/s] +α [kmfs] +h3 +h4 +t [Gyr] +z [ZZa] +110 +2 +4 +-101 +1rio +150 +0.2 +a.D +0.2 +-0.1 +a.D +1i +12 +0 +a.D +50 +LD +[] +Data +25 +[] +Model +0 +25 +AP/[A - PWA] +[h3 + h3dabaJ/dh3 +14p/[qappy - 9popy] +tabJ/dt +5.0 +5.D +5.0 +5.D +5.0 +O.D +5.D +5.0 +Residua +[4] +L +05 +50 +0 +50 +50 +0 +50 +50 +50 +50 +50 +50 +0 +50 +X [arcsec] +X [arcsec] +X [arcsec] +X [arcsec] +X [arcsec] +X [arcsec] +X [arcsec]Probabilitydensity +t [Gyr] +Z[ZZo] +0.00 +0.01 +0.02 +12 +13 +14 +0.0 +0.5 +1.0 +1.5 +2.0 +1.0 +FCC170 +0.5 +0.0 +0.5 +-1.0 + +20 +40 +60 +器心 +20 +40 +60 +8 心 +20 +40 +60 +80 +r[arcsec] +rarcsec +r[arcsec]SB [log 10 La/p2?] +V[km/s] +α [kmfs] +h3 +h4 +t [Gyr] +z [ZZ] +101 +10 +50 +0 +-0.2 +a.D +0.2 +.D +a.1 +0.5 +LD +1.5 +2:1 +B +20 +BL +0 +20 +40 +[SE, +1- SBdebaJ/SBdabe +AP/(BPA-Ppa"A) +Odeta Jfdo +[h3. + h3daba J/dh3 +[h4. + h4dabaJ/dh4 +caba]/dt +aJfdz +0.05D0.025 0.D00 0.D25 0.D50 5.0 +2.5 +O.D +2.5 +5D +5.0 +O.D +2.5 +5D +2.5 +O.D +2.5 +5.D +5.0 +O.D +2.5 +5.D +5.0 +2F +5.D +5.0 +20 +50 +50 +50 +50 +50 +50 +X [arcsec] +X [arcsec] +X [arcsec] +X [arcsec] +X [arcsec] +X [arcsec] +X [arcsec]Probability density +t [Gyr] +Z[ZZo] +000'0 +0.0050.0100.015 +2.5 +5.0 +7.5 +10.0 +12.5 +0.5 +1.0 +1.5 +1.0 +FCC179 +0.5 +0.0 +0.5 +-1.0 + +20 +40 +60 +80 +1000 +20 +40 +60 +80 +1000 +20 +40 +60 +80 +100 +r [arcsec] +r[arcsec] +r[arcsec]Y. Ding et al.: F3D: the environmental effects on the assembly of disks +Fig. B.11. Best-fit population-orbit superposition model (top panels) and orbital decomposition (bottom panels) of FCC 182. The content of the +top and bottom panels is the same as in Figs. 1 and 2, respectively. +Fig. B.12. Best-fit population-orbit superposition model (top panels) and orbital decomposition (bottom panels) of FCC 249. The content of the +top and bottom panels is the same as in Figs. 1 and 2, respectively. +Article number, page 27 of 31 + +SB [log 10 Ls/pc?] +[sf] A +α [kmfs] +h3 +h4 +t [Gyr] +z [ZZ] +-10 +1D +40 +0.05 +0.DO +0.D5 +0.1 +O.D +a.1 +12 +0.2 +0.4 +O.B +0.B +1 +[B]1 +Data +10 +[3a] 1 +Model +[SBmadel - SBdeha]/SBdala +AP/[A - PA] +[h3model h3data ]/dh3 +t4p/[qappy - 3poy] +[tmadel taaba]fdt +[Zmodd - Zdala]/dZ +0.05D-0.025 0.D000.D250.D5 +2.5 +5.D +5.0 +-2.5 +5.D +5.0 +5.D +2.5 +O.D +5.D +2.5 +O.D +5.0 +21 + Residual +20 +-20 +-20 +20 +0 +-20 +-20 +-20 +20 +X [arcsec] +X [arcsec] +X [arcsec] +X [arcsec] +X [arcsec] +X [arcsoc] +X [arcsec]Probability density +t[Gyr] +Z[ZZo] +0.00 +0.01 +0.02 +0.03 +6 +8 +10 +12 +14 +0.000.250.500.751.00 +1.0 +FCC182 +0.5 +0.0 +0.5 +-1.0+ +5 +10 +15 +20 +25 0 +5 +10 +15 +20 +25 0 +5 +10 +15 +20 +25 +r [arcsec] +r [arcsec] +r[arcsec]SB [log 10 La/p2?] +V [km/s] +α [kmfs] +h3 +h4 +t [Gyr] +z [ZZ] +50 +50 +-0.1 +O.D +a.1 +0.2 +a.D +ii +12 +13 +aD +0.2 +0.4 +O.B +[aBae] X +Data +[] +Model +[SBmedel SBdeta]/SBdati +AP[A- PWA] +op[Po w] +E4p/[q8pey - poe4] +(tmodel taba]fdt +[Zmodel -Zdata]/dZ +0.050-0.025 0.000 0.025 0.050 5.0 +2.5 +5.D +5.0 +-2.5 +2.5 +5.D +5.0 +5.0 +50 +5.0 +2.5 +O.D +2.5 +5D +5.0 +2.5 +O.D +25 +5.0 +Residual +[] +20 +20 +0 +20 +0 +20 +24 +-20 +X [arcsec] +X [arcsec] +X [arcsec] +X [arcsec] +X [arcsec] +X [arcsec] +X [arcsec]Probability density +t [Gyr] +Z[ZZo] +0-0'0 +0.01 +0.02 +0.03 +10 +12 +14 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +FCC249 +Circularity Az +0.5 +0.0 +0.5 +-1.0 +10 +20 +30 0 +10 +20 +30 0 +10 +20 +30 +rarcsec +rarcsec +r[arcsec]A&A proofs: manuscript no. main +Fig. B.13. Best-fit population-orbit superposition model (top panels) and orbital decomposition (bottom panels) of FCC 255. The content of the +top and bottom panels is the same as in Figs. 1 and 2, respectively. +Fig. B.14. Best-fit orbit superposition model (left panels) and orbital decomposition (right panels) of FCC 263. Details are similar to Figs. 1 and 2 +as we show for FCC 177, but we do not show the age and metallicity. +Article number, page 28 of 31 + +SB [log 10 La/pc?] +V [km/s] +α [kmfs] +h3 +h4 +t [Gyr] +z [ZZ] +25 +25 +40 +0.1 +a.D +α.1 +0.1 +aD +0.1 +0.D0 +0.25 +0.50 +0.75 +[] 1 +Data +-20 +[] +Model +0 +-10 +-30 +[omedel - odeha ]fdo +[h3, +1 - h3daba J/dh3 +tha. +npcel=h4dh1fdh4 +-thdi +5.0 +5D +5D +5D +5.0 +2.5 +O.D +2.5 +5D +Residual +[B] 1 +-20 +-30 +-20 +20 +P:1 +40 +20 +40 +20 +40 +-20 +20 +21 +X [arcsec] +X [arcsec] +X [arcsec] +X [arcsec] +X [arcsec] +X [arcsec] +X [arcsec]Probabilitydensity +t[Gyr] +Z[ZZo] +0.00 +0.01 +0.02 +5 +10 +0.00 0.25 0.50 0.75 1.00 +1.0 +FCC255 +0.5 +0.0 +0.5 +- +- += +1.0 + +10 +20 +30 +40 0 +10 +20 +30 +40 0 +10 +20 +30 +40 +r[arcsec] +rarcsec +r[arcsec]Y. Ding et al.: F3D: the environmental effects on the assembly of disks +Fig. B.15. Best-fit population-orbit superposition model (top panels) and orbital decomposition (bottom panels) of FCC 276. The content of the +top and bottom panels is the same as in Figs. 1 and 2, respectively. +Fig. B.16. Best-fit orbit superposition model (left panels) and orbital decomposition (right panels) of FCC 290. Details are similar to Figs. 1 and 2 +as we show for FCC 177, but we do not show the age and metallicity. +Article number, page 29 of 31 + +SB [log 10 Lo/p22] +V [km/s] +α [kmfs] +h3 +h4 +t [Gyr] +Z [ZZs] +50 +50 +10 +150 +0.2 +O.D +0.2 +0.2 +a.D +0.2 +12 +o.D +0.5 +LD +[]1 +Data +0 +[B] 1 +Model +40 +[SBmedel SBdrtn]/SBdala +-VdabaJ/dv +op[ - +[h3model + h3daba J/dh3 +- h4dab J/dh4 +(tmadel - tab]fdt +[Zmoded -Zdaba ]/dZ +0.05D-0.025 0.D00 0.D25 0.D51 - +5.0 +2.5 +5.D +5.D +5.0 +2.5 +5.0 +O.D +5.D +5.0 +2.5 +O.D +5.D +[] +0 +-20 +75 +05 +25 +0 +75 +05 +75 +05- +-25 +] +-75 +50 +-25 +0 +75 +25 +75 +-25 +75 +05 +X [arcsec] +X [arcsec] +X [arcsec] +X [arcsec] +X [arcsec] +X [arcsec] +X [arcsec]Probability density +t[Gyr] +Z[ZZo] +0.00 +0.02 +0.04 +0.06 +10 +12 +14 +0.000.250.500.751.00 +1.0 +FCC276 +0.5 +0.0 +0.5 +-1.0 + +20 +40 +60 +80 +0 +20 +40 +60 +80 +0 +20 +40 +60 +80 +r[arcsec] +rarcsec +r[arcsec]A&A proofs: manuscript no. main +Fig. B.17. Best-fit population-orbit superposition model (top panels) and orbital decomposition (bottom panels) of FCC 301. The content of the +top and bottom panels is the same as in Figs. 1 and 2, respectively. +Fig. B.18. Best-fit orbit superposition model (left panels) and orbital decomposition (right panels) of FCC 308. Details are similar to Figs. 1 and 2 +as we show for FCC 177, but we do not show the age and metallicity. +Fig. B.19. Best-fit orbit superposition model (left panels) and orbital decomposition (right panels) of FCC 312. Details are similar to Figs. 1 and 2 +as we show for FCC 177, but we do not show the age and metallicity. +Article number, page 30 of 31 + +SB [log 10 Lafpc?] +V [km/s] +α [kmfs] +h3 +h4 +t [Gyr] +z [ZZ] +2 +0 +40 +0.1 +a.D +a.1 +-0.1 +a.D +4 +8 +a.Do +0.25 +0.50 +0.75 +[] +Data +10 +20 +[] +Model +[Vmodd-Vdaba]/dV +[e- +(h3. +odel - h3daba Jfdh3 +pyp[qaPpy - gppy] +[tmadel - taaba]fdt +[Zmpddl - Zdala]fdZ +0.05D-0.025 0.DO0 0.D25 0.D50 5.0 +2.5 +O.D +2.5 +5.D +5.0 +O.D +5.D +5.0 +O.D +5.0 +P2F +5D +5.0 +2.5 +O.D +2.5 +5D +5.0 +2.5 +O.D +2.5 +5D +Residual +[B] +20 + OZ- +-20 +-20 +20 +-20 +20 +-20 +2 +-20 +20 +20 +X [arcsec] +X [arcsec] +X [arcsec] +X [arcsec] +X [arcsec] +X [arcsec] +X [arcsec]Probability density +t [Gyr] +Z[ZZo] +0.0000.0050.010 0.015 +5.0 +7.5 +10.012.5 +0.00.2 +0.4 +0.6 +1.0 +1.0 +FCC301 +0.5 +0.0 +0.5 +-1.0 + +10 +20 +30 +0 +10 +20 +30 +10 +20 +30 +r[arcsec] +r[arcsec] +r[arcsec]Y. Ding et al.: F3D: the environmental effects on the assembly of disks +0.2 +0.4 +0.6 +0.8 +1.0 +Yr +0.960 +0.965 +0.970 +0.975 +0.980 +0.985 +0.990 +0.995 +1.000 +umin +0.2 +0.4 +0.6 +0.8 +1.0 +Yr +0.92 +0.93 +0.94 +0.95 +0.96 +0.97 +0.98 +0.99 +1.00 +pmin +0.2 +0.4 +0.6 +0.8 +1.0 +Yr +0.100 +0.105 +0.110 +0.115 +0.120 +0.125 +0.130 +qmin +0.2 +0.4 +0.6 +0.8 +1.0 +Yr +2.2 +2.4 +2.6 +2.8 +3.0 +3.2 +3.4 +log(M200/Mstar) +2.25 +2.50 +2.75 +3.00 +3.25 +log(M200/Mstar) +0.960 +0.965 +0.970 +0.975 +0.980 +0.985 +0.990 +0.995 +1.000 +2.25 +2.50 +2.75 +3.00 +3.25 +log(M200/Mstar) +0.92 +0.93 +0.94 +0.95 +0.96 +0.97 +0.98 +0.99 +1.00 +2.25 +2.50 +2.75 +3.00 +3.25 +log(M200/Mstar) +0.100 +0.105 +0.110 +0.115 +0.120 +0.125 +0.130 +0.10 +0.11 +0.12 +0.13 +qmin +0.960 +0.965 +0.970 +0.975 +0.980 +0.985 +0.990 +0.995 +1.000 +0.10 +0.11 +0.12 +0.13 +qmin +0.92 +0.93 +0.94 +0.95 +0.96 +0.97 +0.98 +0.99 +1.00 +0.92 +0.94 +0.96 +0.98 +1.00 +pmin +0.960 +0.965 +0.970 +0.975 +0.980 +0.985 +0.990 +0.995 +1.000 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +( +2 +r +min +2 +r )/(2Nobs)1/2 +Fig. C.1. Optimization over the 5D parameter-space for FCC 177. Each points indicates an exploration of the parameter-space with the smaller +orbit sampling. Color represents the χ2 of each parameter set. The best fitting parameter is indicated by the yellow most point with a cross symbol. +Article number, page 31 of 31 + diff --git a/A9E5T4oBgHgl3EQfSw_w/content/tmp_files/load_file.txt b/A9E5T4oBgHgl3EQfSw_w/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..58d16e25348db2654a63c3d11895bb1d741cdb89 --- /dev/null +++ b/A9E5T4oBgHgl3EQfSw_w/content/tmp_files/load_file.txt @@ -0,0 +1,3725 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf,len=3724 +page_content='Astronomy & Astrophysics manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' main ©ESO 2023 January 16, 2023 The Fornax3D project: Environmental effects on the assembly of dynamically cold disks in Fornax cluster galaxies Y.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Astrofísica,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Universidad de La Laguna,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Calle Astrofísico Francisco Sánchez s/n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' 38206 La Laguna,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Tenerife,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Spain 12 Research Centre for Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Astrophysics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' and Astrophotonics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Department of Physics and Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Macquarie University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' NSW 2109,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Australia 13 Max-Planck-Institut für Astronomie,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Königstuhl 17,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' 69117 Heidelberg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Germany 14 Armagh Observatory and Planetarium,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' College Hill,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Armagh BT61 9DG,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' UK 15 Centre for Astrophysics Research,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' University of Hertfordshire,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' College Lane,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Hatfield AL10 9AB,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' UK Received XXXXX;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' accepted XXXXX ABSTRACT We apply a population-orbit superposition method to 16 galaxies in the Fornax cluster observed with MUSE/VLT in the context of the Fornax3D project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' By fitting the luminosity distribution, stellar kinematics, and age and metallicity maps simultaneously, we obtained the internal stellar orbit distribution, as well as the age and metallicity distribution of stars on different orbits for each galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Based on the model, we decompose each galaxy into a dynamically cold disk (orbital circularity λz ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='8) and a dynamically hot non-disk component (orbital circularity λz < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='8), and obtain the surface-brightness, age, and metallicity radial profiles of each component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' The galaxy infall time into the cluster is strongly correlated with galaxy cold-disk age with older cold disks in ancient infallers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' We quantify the infall time tinfall of each galaxy with its cold-disk age using a correlation calibrated with TNG50 cosmological simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' For galaxies in the Fornax cluster, we found that the luminosity fraction of cold disk in galaxies with tinfall > 8 Gyr are a factor of ∼ 4 lower than in more recent infallers while controlling for total stellar mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Nine of the 16 galaxies have spatially extended cold disks, and most of them show positive or zero age gradients;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' stars in the inner disk are ∼ 2 − 5 Gyr younger than that in the outer disk, in contrast to the expectation of inside-out growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Our results indicate that the assembly of cold disks in galaxies is strongly affected by their infall into clusters, by either removal of gas in outer regions or even tidally stripping or heating part of the pre-existing disks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Star formation in outer disks can stop quickly after the galaxy falls into the cluster, while star formation in the inner disks can last for a few Gyrs more, building the positive age gradient measured in cold disks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Key words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' galaxies: kinematics and dynamics – galaxies: elliptical and lenticular, cD – galaxies: stellar population – galaxies: formation – galaxies: structure – galaxies: evolution 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Introduction Galaxies in cluster environments have different star formation histories and display different morphology types with respect to galaxies in the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Generally speaking, galaxies in clusters are redder, more likely to be quenched, and exhibit an elliptical mor- phology (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=', Dressler 1980;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Butcher & Oemler 1984;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Dressler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' 1997;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Peng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Lewis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Blanton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' ⋆ Email:ycding@shao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='cn ⋆⋆ Corr author: lzhu@shao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='cn 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Alpaslan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Galaxy evolution is mainly affected by three physical processes related to the cluster environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' First, in a process known as “harassment”, tidal forces induced by the central halo (Gunn & Gott 1972;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Toomre & Toomre 1972;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Barnes & Hernquist 1992;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Bournaud et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' 2004) or neighbour- ing flyby galaxies can strip away part of the stellar and gas par- ticles of the satellite galaxies (Gunn & Gott 1972).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Second, the interaction with the intracluster medium (ICM) known as “ram pressure” can strip away the hot gas and sometimes even the cold gas in the disk of the satellite galaxies (Gunn & Gott 1972;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Abadi Article number, page 1 of 31 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='05532v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='GA] 13 Jan 2023 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' main et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Yun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Third, the accretion rate is much lower in the cluster environment and it could cause the cut-off of gas supply to a galaxy leading to “strangulation” (Larson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' 1980;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Balogh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Kawata & Mulchaey 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' All three processes could ultimately lead to the cessation of star formation and affect the morphology evolution of the satellite galaxies in the cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' The timescale of the star-formation quenching in cluster en- vironments and the relative contribution of the aforementioned physical processes are still under debate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' The current gas con- tent in cluster galaxies provides some direct clues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Maier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' (2019b) calculated the fraction of star-forming galaxies of seven nearby clusters through Hα observation with Local Cluster Sub- structure Survey (LoCuSS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' By comparing with the Millennium simulations, they suggested that star formation in cluster galax- ies can last for 1-2 Gyr after their infall into the cluster, but then they quickly get quenched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' This “slow-then-rapid quench- ing” is consistent with the strangulation scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' On the other hand, Reynolds et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' (2022) found evidence that HI gas in the outer regions of the galaxies in the Hydra I cluster is partially re- moved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' However, these galaxies still lie on the star-forming main sequence and gas removal is not yet affecting the inner star form- ing disks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' In the same cluster, Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' (2021) modeled the ram-pressure stripping strength through the HI mass from WAL- LABY observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' They found that the ram pressure stripping can significant change the total HI gas mass of satellite galaxies within 600 Myr after falling into the cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Cosmological hydrodynamical simulations allow us to inves- tigate the details of the infalling of galaxies into the cluster as well as of the gas removal and structure formation of galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Joshi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' (2020) found that the pre-exisiting stellar disks in cluster galaxies are largely disrupted by impulsive tidal shock- ing and stripping at pericentres within 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='5-4 Gyr after falling into the massive clusters with M200 ∼ 1014−14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='3 M⊙ in the Il- lustris TNG100 simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Meanwhile, all the cluster galax- ies quenched their star formation after the disk disruption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' The quenching timescale could be 1-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='5 Gyr for gas-poor massive galaxies, while the star formation could last for ∼4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='5 Gyr for gas- rich galaxies and a dynamically cold disk could regrow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Similar results apply also to Fornax-mass analogues in the TNG50 sim- ulations (Galán-de Anta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' 2022) The detailed study of S0 galaxies in TNG100 simulation shows two major formation scenarios (Deeley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' 2021): S0 galaxies in clusters formed via starvation or stripping, whereas S0s in field formed via mergers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' This finding is also supported by observations (Coccato et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Coccato et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Star for- mation quenching is different in these two formation scenarios (Deeley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' 2021): when a blue and gas-rich galaxy falls into a cluster, its gas in the outer regions is quickly stripped, while the star formation in the inner regions could last for a long time, whereas in case of galaxy-galaxy merger, galaxies start quench- ing from the inner regions, and star formation continues in the outer ring-like regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' These studies suggest that cluster environment may signifi- cantly affect the internal dynamical structure of galaxies, espe- cially the persistent growth of dynamically cold disks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' The frac- tion of stars in cold disks and their age distributions may im- print the long-term star-formation quenching process happening in cluster galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' We expect that the disk stars are older in the inner regions and younger in the outer ones for a normal inside- out growth (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=', Bird et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Stinson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Cluster quenching may turn over the age gradient in the stellar disks, the age difference between inner and outer regions can thus tell us the star formation duration in galaxies after falling into the clus- ter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Although some other studies show that complicated physical processes in field galaxies might cause similar age gradient (La- gos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' In the past decades, integral-field unit (IFU) surveys, such as SAURON (de Zeeuw et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' 2002), ATLAS3D (Cappellari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' 2011), CALIFA (Sánchez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' 2012), SAMI (Bryant et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' 2015), and MaNGA (Bundy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' 2015), have provided us the stellar kinematics, age, and metallicity maps for thousands of nearby galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Early-type galaxies (ETGs) show a large vari- ety of kinematical structures (Cappellari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Emsellem et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' 2007, 2011), some lenticular galaxies have a rapidly rotat- ing disk similar to that of spiral galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' This suggests that gas removal by cluster environment may play a major role in trans- forming spirals into lenticulars (Cappellari 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Overall, ETGs are found to be old and with shallow age gradients (McDermid et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' González Delgado et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Zibetti 2019), while late-type galaxies (LTGs) generally have negative age gradients, which is consistent with the inside-out scenario (González Del- gado et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' The environmental dependence of the galaxy age gradient is still controversial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' A positive age gradient was found in SAURON and MaNGA samples (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=', Goddard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Kuntschner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' 2010), with a mixture of galaxies in cluster and field environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' On the other hand, there is no significant difference found for age gradient of ETGs in different environments (Zheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Ferreras et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Santucci et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' 2020), with the above surveys covering the inner 1Re ∼ 2Re for most galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' The Fornax3D survey (Sarzi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' 2018) measured high- quality age and metallicity maps covering at least the inner 2Re of 23 ETGs in the Fornax cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' These galaxies show signifi- cant positive age gradients between Re and 2Re (Spavone et al 2022), while they have negative metallicity gradients, which are shallower than the control sample galaxies in the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' On the other hand, positive age gradients together with negative metal- licity gradients are widely found in the dwarf galaxies of the Local Group (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=', Koleva et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Kirby et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Zhuang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' These positive age gradients could be the direct con- sequence of ram pressure stripping (Genina et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Deeley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' 2021), although others argue theses are the result of gas- rich mergers for dwarf galaxies (Yozin & Bekki 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' The observation of external galaxies provides integrated properties along the line-of-sight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' The stellar kinematics and populations of a galaxy are a combination of different dynam- ical components, displaying a range of possible physical origins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' As learned from the simulations aforementioned, the luminosity fraction and age distribution of stellar disk might be a good probe of environmental affects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' However, there are relatively small disk fractions in ETGs (Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' The age and metallicity maps of the whole galaxy may not reveal the properties of the disk component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Dynamical models are powerful tools to uncover the galax- ies’ underlying mass profiles, as well as the internal 3D struc- tures which could lead to physical-motivated decomposition of different components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' There are a few methods widely used for modelling the IFU kinematic data of external galaxies, in- cluding the Jeans-Anisotripic-MGE (JAM) models (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=', Cappel- lari 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' 2017) , the particle-based Made-to-Measure (M2M) models (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=', Syer & Tremaine 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' de Lorenzi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Long & Mao 2010, 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' 2014), and the orbit-superposition Schwarzschild method used in this work (Schwarzschild 1979).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' The Schwarzschild’s orbit superposition method can be used in different geometries such as the spheri- cal systems (Richstone & Tremaine 1985;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Breddels et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Kowalczyk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' 2017), axisymmetric systems (Cretton & van Article number, page 2 of 31 Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Ding et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' : F3D: the environmental effects on the assembly of disks den Bosch 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Gebhardt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Valluri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Cap- pellari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' 2006a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Thomas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Saglia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' 2016), and triaxial systems (van den Bosch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Neureiter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' The orbit-superposition model reconstructs the backbone of galaxies without ad-hoc assumptions of the underlying dis- tribution functions, it has been widely used to uncover galax- ies’ underlying dark matter distributions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=', Cappellari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' 2006b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' 2020) , central black hole mass (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=', van der Marel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Cretton & van den Bosch 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Verolme et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Gebhardt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Valluri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Krajnovi´c et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Ahn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Thater et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' 2022b), and internal stel- lar orbit distributions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=', Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' 2018b,a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Jin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Recently it has been modified to include the barred structures ex- plicitly, thus also for uncovering the bar pattern speed (Vasiliev & Valluri 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Tahmasebzadeh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Based on the Schwarzschild method, a population-orbit su- perposition method (Poci et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' 2020) was re- cently developed by tagging age and metallicity to the orbits, we can thus simultaneously fit the stellar kinematic, age, and metallicity maps from IFU survey, and obtain the internal stel- lar orbit distribution of galaxies as well as the age and metallic- ity distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' This allows for a physically motivated chemo- dynamical decomposition of galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' In this work, we apply the population-orbit superposition method to galaxies in For- nax cluster with the IFU data obtained with MUSE/VLT in the context of the Fornax3D survey (Iodice et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Using this modeling technique, age-dispersion profiles of dynamically de- composed cold disks in a few edge-on galaxies have been ob- tained (Poci et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Poci et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' 2021) and a dynamically de- composed “hot inner stellar halo” has been used to weigh and time the ancient massive mergers the galaxies experienced in NGC 1380 and NGC 1427 (Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' In this paper, we set our focus on dynamically cold disks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' By studying the luminosity fraction, age, and metallicity of these stars, we aim to quantify how the cluster environment has af- fected the formation of cold disks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' This results will also lead to a direct and in-depth comparison with galaxies in cosmological simulations and, thus, to a better understanding of the physical processes driving galaxy evolution in cluster environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' This work is undertaken in the context of ΛCDM cosmology, with density parameters of Ωm = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='3089, ΩΛ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='6911, Ωb = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='0486, normalization σ8 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='8159 and spectral index ns = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='9667 (Planck Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' We introduce the data set in § 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' We describe the population-orbit superposition model and its relevance to the orbital decomposition in § 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' We show the orbital decomposition for all galaxies and present the surface- brightness, age, and metallicity radial profiles of each compo- nent in § 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' We present an explanation of how the cold disk age can be used as a novel proxy for galaxy infall time into a cluster in § 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' We further show the dependence of cold disk properties on galaxy stellar mass and cluster environment in § 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' We discuss the results in § 7 and present our summary in § 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Sample and Data We set our focus on galaxies in the Fornax cluster, which has a virial radius of Rvir ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='7 Mpc , virial mass of Mvir ∼ 7 × 1013 M⊙, and a distance of D ∼ 20 Mpc (Diaferio 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Drinkwater et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' We used the deep photometric data of the Fornax Deep Sur- vey (FDS, Venhola et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' (2018)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' FDS observed all the galaxies in the area of 9 square degrees around the core of the cluster, with the VLT Survey Telescope down to a surface-brightness level of 27 mag arcsec−2 in the r band (Iodice et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' We used the spectroscopic data of the Fornax3D survey (Sarzi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Using the MUSE instrument on the VLT, For- nax3D observed all the 23 ETGs and 10 LTGs within the virial radius of the cluster down to 25 mag arcsec−2 in B band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' MUSE has a field-of-view of 1×1 arcmin2 and spatial scale of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='2 arcsec pixel−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' The wavelength covers the range of 4650 Å to 9300 Å, with a resolution of FWHM7000Å = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='5 Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Twenty-eight galax- ies were observed by more than one MUSE pointing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' The data extend to at least 2Re for 20 galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Stellar kinematics maps To obtain reliable stellar kinematics, we first bin the spectra in nearby pixels to reach a target signal-to-noise ratio (S/N) taken a Voronoi binning scheme (Cappellari & Copin 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' The stel- lar kinematics of all the galaxies in the Fornax3D survey was published in Iodice et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' (2019) with target S/N=40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' The only difference in the kinematic maps we use here is that different S/N for different galaxies were chosen as indicated in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' We took a target S/N = 100 or 200 for the bright galaxies with more than one MUSE pointing and S/N =60 or 40 for the remaining galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' This results in a number of bins from ∼ 100 to ∼ 1000 for each galaxy with a spatial resolution from ∼ 100 pc in the inner region to ∼ 1 kpc in the outer faint regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' This choice is convenient for dynamical modeling without too many bins, while the spatial resolution is still good enough to meet our science goals for resolving sub-kpc scale structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' The stellar kinematics was then extracted by applying the pPXF full-spectral fitting (Cappellari & Emsellem 2004) to the wavelength range 4750-5500 Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' This yielded high-quality maps of the stellar mean velocity, V, velocity dispersion, σ, and higher order velocity moments parameterized through the Gauss-Hermite (GH) coefficients h3 and h4 (Gerhard 1993;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Rix et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' 1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Such stellar kinematic maps were derived in a consistent way for the 15 ETGs and 5 LTGs included in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' We did not consider the 8 ETGs – FCC 213 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=', the BCG), FCC 090, FCC 184, FCC 190, FCC 193, FCC 219, FCC 277, and FCC 310 – due to existence of bright foreground stars, limited data cov- erage or the presence of strong bar, and the 5 LTGs FCC 113, FCC 176, FCC 267, FCC 285, and FCC 306 due to the pres- ence of a strong bar or irregular morphology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' We excluded the strongly barred galaxies because our current dynamical models cannot fit their kinematic maps well with bars that are not in- cluded explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' However, we did keep the three weakly barred galaxies (FCC 179, FCC 182, and FCC 263), since their stel- lar kinematic maps are not strongly affected by the bar (see fig- ures in Appendix C) and our current kinematic models can still provide reasonably good fits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Some basic properties of the final sample of galaxies are listed in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Age and metallicity maps We used the age and metallicity maps derived in Martín-Navarro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Here, we briefly describe how they were derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' To obtain high-quality maps of the age and metallicity, the spectra were spatially rebinned to reach a minimum of S/N = 100 for each galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' We note that the spaxels with S/N< 5 were not included in this process in order to retain the highest qual- ity of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Thus, for most of the galaxies, only the inner region map was available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' The single stellar population synthesis mod- Article number, page 3 of 31 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' main Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Properties of the sample galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Object Type Distance Re M⋆ i PA S/N [Mpc] [kpc] [1010M⊙] [◦] [◦] (1) (2) (3) (4) (5) (6) (7) (8) FCC 083 E5 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='46 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='27 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='2±13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='3 139.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='7 100 FCC 119 S0 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='05 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='9±5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='4 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='4 40 FCC 143 E3 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='28 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='6±7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='1 124.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='6 60 FCC 147 E0 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='40 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='40 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='3±6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='0 120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='9 100 FCC 148 S0 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='9 2.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='91 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='58 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='9±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='7 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='34 100 FCC 182 SB0 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='15 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='2±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='4 169.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='3 60 FCC 249 E0 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='93 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='98 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='8±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='1 150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='6 60 FCC 255 S0 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='31 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='1±5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='8 172.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='9 60 FCC 263 SBcdIII 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='04 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='4±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='9 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='9 60 FCC 276 E4 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='33 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='81 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='6±9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='0 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='4 100 FCC 290 ScII 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='9 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='64 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='8±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='1 122.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='5 60 FCC 301 E4 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='20 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='0±5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='7 155.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='8 60 FCC 308 Sd 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='04 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='5±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='4 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='5 60 FCC 312 Scd 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='9 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='62 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='48 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='0±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='9 136.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='2 100 Notes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Name (1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Hubble type (2) taken from (Ferguson 1989);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' distance (3);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Effective radius (4);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' and stellar mass (5) is from Iodice et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' (2019) and Martín-Navarro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' The inclination angle (6) obtained from our orbit-superposition model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' The positional angle (7) determined by photometric isophotes and adopted for the multi-Gaussian expansion fitting and target S/N used in binning the spectra to obtain kinematic maps (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' els (SSP) by Vazdekis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' (2016) based on the MILES stellar library by Sánchez-Blázquez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' (2006) at a constant spectral resolution of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='51 Å (FWHM) (Falcón-Barroso et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' 2011) were used for the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' The luminosity-weighted age and metallicity maps were de- rived in a two-step spectral fitting of the wavelength range be- tween 4800 Å and 6400 Å, considering the initial mass func- tion (IMF) variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' First pPXF was run with an un-regularized combination of MILES SSPs in order to measure V and σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' A second pPXF was run again with stellar kinematics fixed to that measured in the first step, and this time regularizing the age- metallicity-IMF slope plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' The mean luminosity-weighted age and metallicity of the spectra were measured using this regu- larized second pPXF run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' The metallicity, Z, is added up lin- early rather than in the commonly used logarithmic way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Tem- plates with a variable IMF were intentionally included to ac- count for the possible effect of the IMF in the age determina- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' The abundance ratio [α/Fe] was not fitted, but we used the so-called base MILES models which inherit the [α/Fe]-[M/H] relation of the solar neighborhood (Vazdekis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' This choice avoids non-local equilibrium uncertainties introduced by the theoretical response functions needed to compute stellar pop- ulation models with variable abundance ratios, uncertainties that can be particularly problematic for Balmer lines and thus for the age determination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' We note that in Martín-Navarro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' (2021), the metallicity maps were also derived in a third step with in- dex fitting when determining the IMF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' However, we used the luminosity-weighted metallicity maps derived from the pPXF fitting, in a consistent way as the luminosity-weighted age maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Alternatively, we have another version of the age and metallicity maps for FCC 153, FCC 167, FCC 170, and FCC 177, which was obtained by pPXF fitting regularizing age-metallicity-[α/Fe] assuming a Kroupa IMF (Pinna et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' 2019a,b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' We compared the two versions of age and metallicity maps for these four galaxies, they generally show the same age and metallicity gradients but with a systematic offset of 1-2 Gyr in age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' This offset will not affect our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' The version of the age and metallicity maps by Martín-Navarro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' (2021) is used consistently for all galaxies in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Finally, we have the age and metallicity maps for 16 galaxies, including all the 15 ETGs and the LTG FCC 179 only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' The re- Article number, page 4 of 31 Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Ding et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' : F3D: the environmental effects on the assembly of disks Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Properties of the sample galaxies extracted by the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Object Mtot,e fDM,e fcold,e fcold,2e ⟨tcold,e⟩ ⟨thot,e⟩ ⟨Zcold,e⟩ ⟨Zhot,e⟩ ∇tcold,e ∇thot,e ∇Zcold,e ∇Zhot,e tinfall [1010M⊙] [Gyr] [Gyr] [Z⊙] [Z⊙] [Gyr/Re] [Gyr/Re] Z⊙/Re Z⊙/Re [Gyr ago] (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) FCC 083 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='67±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='40±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='18±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='19±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='02 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='8±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='6 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='4±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='72±0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='0 FCC 290 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='08±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='68±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='25±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='37±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='04 “‘ “‘ “‘ “‘ “‘ “‘ “‘ “‘ “‘ FCC 301 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='15±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='28±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='07±0.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='87±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='38±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='66±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='3±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='65 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='16±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='35±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='04 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='5±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='1 FCC 308 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='33±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='51±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='09±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='09±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='01 “‘ “‘ “‘ “‘ “‘ “‘ “‘ “‘ “‘ FCC 312 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='16±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='72 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='63±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='32±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='32±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='05 “‘ “‘ “‘ “‘ “‘ “‘ “‘ “‘ “‘ Notes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Name (1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' total dynamical mass (2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' dark matter fraction (3);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' 1Re luminosity fraction of the cold disk (4);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' 2Re luminosity fraction of the cold disk (5);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' mean age of the cold disk (6);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' mean age of the non-disk component (7);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' mean metallicity of the cold disk (8);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' mean metallicity of the non-disk component (9);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' age gradient of the cold disk (10);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' age gradient of the non-disk component (11);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' metallicity gradient of the cold disk (12);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' metallicity gradient of non-disk component (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' All values in columns (2)-(4) and (6)-(13) are calculated within the effective radius Re, except for (5) which is extracted within 2Re .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Columns (10) and (12) of FCC 170 and FCC 255 are extracted within the whole data coverage as their disks have very low surface-brightness at r < Re.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' For six ETGs FCC 143, FCC 161, FCC 182, FCC 249, FCC 276, and FCC 308, we only show the age and metallicity gradient of non-disk component because of the low surface-brightness at r < Re.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Column (13) gives the infall time into the cluster based on the cold-disk age using a correlation calibrated with TNG50 simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' For four LTGs FCC 263, FCC 290, FCC 308, and FCC 312, we only have the stellar kinematics and thus only orbital models without coloring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Article number, page 5 of 31 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' main maining 4 LTGs of our sample appear to be dominated by young stars at all radii, for which the uncertainties of age and metallic- ity derived from the spectra fitting are large (Ge et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' 2018), we will build their dynamical models without associating age and metallicity to the stellar orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Population-orbital superposition method We follow Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' (2020)1 in constructing population-orbit superposition models that simultaneously fit the luminosity den- sity, kinematic, age, and metallicity maps of each galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' From the Best-fit models, we obtain the internal stellar orbit distribu- tion associated with the age and metallicity distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Based on the model, we then decomposed each galaxy into a dynam- ically cold disk and a dynamically hot non-disk component, based on circularity distribution of the orbits, and extracted the face-on surface-brightness, age, and metallicity radial profiles for each component separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' We used FCC 177 as an example to illustrate the whole process as shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' The construction of a population-orbit superposition model consists of four steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' First, constructing the model of gravita- tional potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Second, calculating the orbital library under the gravitational potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Third, fitting the luminosity density and kinematic maps by weighting the orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Fourth, fitting the age and metallicity maps by “colorin” the orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' The method has been presented in great detail and carefully validated in Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Here, we just briefly describe some of the key steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Constructing the gravitational potential The gravitational potential combines the contribution of the stars, dark matter halo, and a central black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' To construct the stellar mass distribution, we use the multi- Gaussian expansion (MGE, Cappellari (2002)) to fit the r-band image from FDS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' The galaxies in our sample have a variety of morphological and kinematic properties, which can be classified in three categories: galaxies which are generally flat and with strong rotation indicating an extended disk, galaxies which are generally round but flatter, and with stronger rotation in the inner regions indicating an inner disk, galaxies with no clear evidence of a disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' For the first two categories, we consider an axisymmet- ric solution by adopting a constant positional angle (PA) for the Gaussians to match that of the photometric and kinematic signa- ture of the inner or outer disk component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Photometric PA from the disk region is taken, which is generally consistent with the kinematic PA in that region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' For the third category, we still con- sider axisymmetric solution, but we did not put any extra con- strain on the PA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' thus, the photometric PA directly derived from the whole galaxy image was taken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' The best-fit Gaussian param- eters are listed in the Appendix for all the galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Then we de-project the 2D Gaussian components to 3D by assuming a set of viewing angles (θ, ψ, φ), where θ and φ define the orientation of the line of sight with respect to the principal axes of the galaxy and ψ is chosen to specify the rotation of the galaxy around the line-of-sight in the projected sky-plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' By combining the 3D Gaussian components, we obtained the 3D luminosity density distribution of the galaxy and we multiply a stellar mass-to-light ratio M∗/L to get the 3D mass density distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' The gravitational potential could then be calculated 1 We have fixed the bug in the orbit mirroring as reported in Quen- neville et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' (2022) although our results are unaffected as confirmed in Thater et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' (2022a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' by the classical Chandrasekhar formula (van den Bosch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' In practice, we do not directly use the three viewing angles as free parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' In van den Bosch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' (2008), three intrinsic shape parameters (p, q, u) are used instead of the viewing angles (θ, ψ, φ), where q = Z/X, p = Y/X, and u = X′/X, where X, Y, Z are the intrinsic long, intermediate, and short axis of the galaxy and X′ is the projected major axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' The conversion of the two sets of parameters follows Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' 10 in van den Bosch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Exploring the space of intrinsic shape parameters is more efficient than that of the three viewing angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' For instance, the deprojection of an axisymmetric system would have no constrain on the parameter φ but have a finite axis-ratio between Y and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' In this work, we follow the approach by van den Bosch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' (2008) and allow for some degree of triaxiality of the galaxy by setting a non-unity u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' We use a spherical Navarro-Frenk-White (NFW, Navarro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' (1996)) dark matter halo, with one free parameter dark matter virial mass M200, while concentration C is fixed by the correlation between M200 and C (Dutton & Macciò 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Al- though there is a large scatter in the M200 − C plane for real galaxies, the choice of a fixed C will not significantly affect our results because the two parameters are degenerated and will not be constrained separately with our kinematic data covering out to 2 − 3Re.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' The potential also includes a central black hole char- acterized by a Plummer potential (van den Bosch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' 2008), with the back hole mass, M•, fixed according the M•-σ relation (Kormendy & Ho 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' This choice does not affect our results, because the black hole sphere of influence is mostly not resolved by our kinematic data and we cannot directly constrain the black hole mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' In summary, we have five free "hyper-parameters," for the gravitational potential: the mass-to-light ratio, M∗/L, three pa- rameters on the intrinsic shape of stellar distribution, p, q, and u, and the dark matter virial mass, M200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Computing the orbit library For each model, with a set of hyper-parameters, we calculated an orbit library with tens of thousands orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' The orbits sampling follows the way described in van den Bosch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' We first sampled regular orbits according to a separable tri- axial potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' The orbits are sampled from the three integrals of motion: energy, E, second integral of motion, I2, and third in- tegral of motion, I3 (Binney & Tremaine 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' We sample two sets of 55 × 11 × 11 combinations of (E, I2, I3) as initial condi- tions, which include co-rotating and counter-rotating orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Box orbits are crucial for supporting the triaxial structures, we sampled another set of box orbits by constructing the ini- tial conditions on equipotential surfaces with the energy, E, two spherical angles, θ and φ, which gives another set of 55×11×11 orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' The number of orbits, especially across E, we sample here is larger than that used in previous works (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=', van den Bosch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' 2018a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Jin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' For instance, 21 × 7 × 7 orbits were used in fitting the CALIFA data (Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' 2018a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Because the kinematic data used in this work have larger spatial coverage and higher spatial resolution than those from the previous IFU surveys, more freedom is required by the model to fit the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' To reduce the Poisson noise of the model, we dither every orbit by slightly perturbing the initial conditions to give 53 orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Article number, page 6 of 31 Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Ding et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' : F3D: the environmental effects on the assembly of disks Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Best-fit population-orbit superposition model of FCC 177.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' From top to bottom: Maps of the data, model, and residuals (model−data).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' From left to right: Maps of the surface-brightness, SB, mean velocity, V, velocity dispersion, σ, Gauss-Hermite coefficients h3 and h4, light-weighted age and metallicity of the stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Similar plots for the other sample galaxies are shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='1-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Orbital decomposition of FCC 177.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Probability density distribution p(r, λz) in the left panel, age distribution p(r, t) in the central panel, and metallicity distribution p(r, Z) in the right panel of the stellar orbits in the phase space of time-averaged radius, r, versus circularity, λz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' The probability densities are normalized to unity within the data coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' All the distributions are averages of multiple Best-fit models that fall within the 1σ confidence level of the model hyper-parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' The dashed line marks our orbit-based division into two components: a dynamically cold disk component (λz ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='8) and a dynamically hot non-disk component (λz < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' The shadow regions are beyond the data coverage of age and metallicity maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Similar plots for the other galaxies are shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='1-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Fitting stellar luminosity density and kinematics We fit the luminosity density and stellar kinematics of the galaxy by weighting the orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' The 2D surface-brightness, 3D luminos- ity density deprojected from the MGE, and kinematic maps are used as model constrains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' The kinematic maps include the line- of-sight mean velocity, V, velocity dispersion, σ, Gauss-Hermite coefficients, h3 and h4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' It is worth noting that we did not fit V and σ maps directly, but the Gauss-Hermite coefficients h1, h2, h3, and h4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' We extracted similar luminosity and kinematic maps from the model superposed by orbits, then obtained a solution of the orbit weights by minimizing the χ2 between the data and model using a non-negative least squares (NNLS) method (Law- son & Hanson 1974;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Lawson & Hanson 1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' We used an optimized grid searching process (Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' 2018b) to adjust the free hyper-parameters of the gravitational potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' We started with a model with an initial guess of the hyper-parameters, then we performed an iterative searching pro- cess with intervals of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='05, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='02, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='01, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='01, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='05 for the hyper-parameters, M∗/L, p, q, u, and log M200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' We note that the concentration parameter, C, in the NFW dark matter halo is fixed by the M200−C relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' After the previous sampled models were completed, we selected the best-fit models with χ2 − χ2 min < 200 and sampled the new models around the selected ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' We con- tinued the iterative process until an area of χ2 minimum was found and all models within 3 − σ confidence level around the minimum χ2 were calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' At the end, we calculated a few hundred models for each galaxy, the resulting parameter grid for FCC 177 is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='1 in the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' The models with least-χ2 are selected as the Best-fit model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' 1, we show the Best-fit model of FCC 177, where the model matches the observed kinematic data in great detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Considering the model uncertainties, we defined the 1σ confidence level in a similar way as Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' (2018): δχ2 = χ2 − χ2 min < f � 2 × nGH × Nobs , (1) where nGH = 4 is the number of stellar kinematic moments and Nobs is the number of bins in the kinematic data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' We adopt f = 1 for the galaxies with Nobs < 200, similar to the previous models for the CALIFA data;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' for galaxies with a much larger value for Nobs, we found that the χ2 fluctuation caused by numerical noise Article number, page 7 of 31 SB [log 10 Lofpc?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='] V [km/s] α [kmfs] h3 h4 t [Gyr] Z [ZZ] 50 50 20 4D a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='D 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='1 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='5 O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='D 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='D 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content="D [] EE EEEEE FEEEEF FEEEE FF 25 0 25 50 25 50 25 25 0 25 4 51 25 0 F1 X [arcsec] X [arcsec] X [arcsec] X [arcsec] X [arcsec] X [arcsec] X [arcsec]Probabilitydensity t [Gyr] Z[ZZo] 0-0'0 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='02 4 8 12 0 1 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='0 + 20 40 60 器心 20 40 60 器心 20 40 60 80 rarcsec rarcsec r[arcsec]A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' main is much higher and thus we adopted f = 4, following the results of a bootstrapping analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' 2 Once we fit the stellar kinematics, we obtained the intrin- sic stellar orbit distribution of the galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' We use the circularity, λz, and time-averaged radius, r, to characterize the orbits, where λz is defined as the orbital angular momentum around the z di- rection, normalized by the maximum that is allowed by a circu- lar orbit with the same binding energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Here, λz ∼ 1 represents highly rotating short-axis tube orbits and the λz ∼ 0 represents mostly long-axis tube or box orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' The stellar orbit distribution from the Best-fit model of FCC 177 is shown in the left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Tagging orbits with age and metallicity Next, we fit the age and metallicity maps by tagging age and metallicities to the orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' We took the models within the 1σ con- fidence level selected by the kinematic fitting as given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' For each model, we have the stellar orbit distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' We applied a Voronoi binning scheme to the orbits in the phase-space of λz vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' r and decompose them into ∼ 100 orbital bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Orbits with similar r and λz are included in the same orbital bundle and each bundle has a minimum orbital weight of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' After the Voronoi binning, we assume that each orbital bun- dle k has a single value of age, tk, and metallicity, Zk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Then the age, ti obs, and metallicity, Zi obs, of each observational aperture, i, can be expressed as: ti obs = ΣNbundle k=1 tk f i k/Σk f i k, (2) Zi obs = ΣNbundle k=1 Zk f i k/Σk f i k, k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=', Nbundle, (3) where Nbundle is the total number of orbital bundles and f i k is the luminosity contribution of orbital bundle, k, at an aperture, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' We applied a Bayesian statistical analysis (Python package pymc3)(Salvatier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' 2016) to first solve tk by fitting the ob- served age map and then to solve Zk by fitting the observed metallicity map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' To use the Bayesian theorem to compute the posterior probability of a model, it is necessary to include the prior probability and data likelihood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' For the prior probability of tk, we adopted a bounded normal distribution, where the mean value µ(tk) is randomly sampled around the average of the ob- served ti obs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' We adopted a student’s t-distribution for the data likelihood (Salvatier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' 2016), which allows for some outliers in the data and results in a robust fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' For each tk, we ran a chain with 2000 steps and take the last 500 steps to calculate the mean and 1σ values of tk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' For a few galaxies, we did not get a good match of the age maps with the above fitting process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Therefore, we tried a sec- ond fitting round with µ(tk) in the bounded normal distribution chosen as the tk value obtained from the first fitting round (Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' We obtained a good match of age maps for all the galaxies after the second round of fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' 2 For galaxies with Nobs ∼ 1000, the δχ2 is obtained by a bootstrap- ping process in the following: in a single model with fixed potential and orbit library, we perturb the kinematic data with its errors and fit the model to the perturbed data for many times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' The standard deviation of χ2 obtained from these fittings are taken as the χ2 fluctuation caused by numerical noise of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Unlike the classic statistical analysis for analytic models, numerical noise is dominating the χ2 in our models, and it could be different for data with different spatial resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' The confidence level we adopt is not motivated by robust statistical consid- eration (more discussion on it could see Lipka & Thomas (2021)), but practically it works well in covering the true values in our model test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Surface-brightness, age, and metallicity radial profiles of the whole galaxy, the dynamically cold disk, and the dynamically hot non- disk component for FCC 177.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' We show the profiles of the whole galaxy (left panels) with the blue solid curve, and the profiles of dynamically cold disk and dynamically hot non-disk component (right panels) with the blue solid curve and black dashed curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' The shadowed areas in- dicate the scatter of the profiles of models that fall within the 1σ con- fidence level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' The age and metallicity profiles of the dynamically cold disks are considered reliable and shown in the regions where the dynam- ically cold disk contributes at least 10% of the total surface brightness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Then we fit the metallicity map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' For the prior probability of Zk, we adopt a log-normal distribution, where the mean value µ(log(Zk)) is set by a build-in age-metallicity relation using tk obtained from the fitting of age map and following Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' The student’s t-distribution is adopted again for the data likelihood, and we run a chain with 2000 steps and take the last 500 steps to calculate the mean and 1σ values of Zk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' The Best-fit model of FCC 177 is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' At the end, we obtained models that simultaneously fit well the surface- brightness, stellar kinematics, age, and metallicity maps of all the galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' We note that the age and metallicity maps we used have different binning schemes and have less data coverage than the kinematic data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' For this reason, the age and metallicity of the orbits beyond the data coverage are not well constrained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Orbital decomposition In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' 2, we show the probability density, age, and metallicity distributions of the orbits of the best-fit model of FCC 177 in the phase-space of λz versus r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' We consider only the age and metallicity distributions within the data coverage (r ≲ 45 arcsec) to be reliable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' The models are usually more noisy (than smooth) in the phase-space of λz vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' As we tested with mock data in Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' (2020), not all the substructures in the model are real, but we can trust the model considering the distributions of orbits and of these age and metallicity in a statistical way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Article number, page 8 of 31 Whole Cold [zd/ 6ol Hot 0 2 12 Age [Gyr] 10 8 6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='0 Metallicity [ZIZ o ] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='25 R/Re R/ReY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Ding et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' : F3D: the environmental effects on the assembly of disks Based on the stellar orbit distribution, we coarsely decom- pose the galaxy into two components: the orbits with λz ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='8 are taken as a dynamically-cold component which is usually a disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' All the remaining orbits with λz < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='8 are considered as a dynamically-hot non-disk component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' We note that here the dynamically cold component is defined as in Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' (2018a), whereas the dynamically hot component actually includes their dynamically warm, hot, and counter-rotating orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' 3, we show the radial profiles of surface-brightness, age and metal- licity of the whole galaxy for FCC 177, as well as those pro- files for the dynamically cold and hot components, separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' At r ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='3Re, the cold disk is about 2 Gyr younger and sig- nificantly more metal-rich than the hot component and the age difference has risen to ∼ 4 Gyr at r ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='7Re.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' However, there is a young and metal-rich nuclear star cluster in the galactic cen- ter (r ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='2Re) which contributes to the hot component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' The luminosity-weighted mean age and metallicity of the hot compo- nent are dominated by this young and metal-rich central regions, as a result, the difference of mean age (metallicity) of the cold disk and the hot component becomes smaller than that revealed in the profiles at r ≳ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='3Re.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Within Re, the stars on the cold disk orbits have an luminosity-weighted average age of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='7 Gyr and an luminosity-weighted metallicity of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='25 Z⊙) while the stars on dynamically warmer orbits have an average age of 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='1 Gyr and an average metallicity of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='81 Z⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Results of the orbital decomposition for the sample galaxies We have 20 galaxies with orbit-superposition models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' For 16 of those (15 ETGs and one LTG, FCC 179), we also have age and metallicity maps and subsequently build a population-orbit model as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' In this section, we show the surface-brightness, age, and metallicity radial profiles of the dynamically cold disk component and dynamically hot non-disk component for all 16 galaxies with their orbits tagged with age and metallicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' For the four LTGs without orbits tagged with age and metallicity, we are limited to the surface-brightness profiles of each component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Surface-brightness, age, and metallicity radial profiles As illustrated by the best-fit model of FCC 177, we decomposed each model into a dynamically cold disk (with λz ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='8) and a dynamically hot non-disk component (including all the remain- ing orbits with λz < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Then, we reconstruct the 3D luminos- ity, age and metallicity distributions of each orbital component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' We project each orbital component to be face-on and then ex- tract the radial profiles of surface-brightness, age, and metallicity from the projected maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' We note that the model already takes into the account the information for the 2D maps of the observational data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' When we extract the radial profiles from the face-on projection, the radial profile along different directions are the same for the disk which is axisymmetric, while very similar for the hot non-disk component which could be moderately triaxial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' For each galaxy, we have tens to hundreds of models within the 1σ confidence level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' The aforementioned radial profiles are extracted for each model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' We take the average of these profiles from models within the 1σ confidence level as the mean profile, and their scatter as the 1σ uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' There are significant variations of internal structure from galaxy to galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' We define the luminosity fraction of cold disk as: fcold = λz≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='8 � k fk, (4) where fk is the luminosity fraction of orbital bundle k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' The cold-disk fraction of a galaxy varies with radius, we calculated fcold(r < Re) and fcold(r < 2Re) with orbits within Re and 2Re, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' We classify the galaxies into two categories: 11 of them have a relatively large cold-disk fraction and their cold disks have an extended surface-brightness, whereas the other 9 galaxies have a relatively small cold-disk fraction and their cold disks are con- centrated in the very inner regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Although the two categories are classified by eye, this is fairly consistent with a separation based on the cold-disk fraction fcold(r < 2Re): the galaxies in the first category generally have fcold(r < 2Re) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='1, most galax- ies in the second category have fcold(r < 2Re) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='1, but a few > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' We take this visual classification for practical reasons that we can obtain reliable age and metallicity gradients for the ex- tended cold disks in the following analysis, but not for the con- centrated ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' The age and metallicity observed at any radius is a combi- nation of different components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Our model tests in Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' (2020) show that we cannot recover the age and metallicity of a component at radii where it contributes ≤ 10% to the total luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' For the galaxies with extended disks, the surface- brightness of the cold disk component contributes at least 10% of the total surface-brightness over a radial region extending to at least one Re.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Reliable age and metallicity profiles of the disk are thus obtained over extended radial regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' On the contrary, for the galaxies with small cold-disk fractions and concentrated cold components, the age and metallicity radial profiles of the cold component are only derived in inner radial regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' 4, we show the radial profiles of surface-brightness of the whole galaxy and of the dynamically cold and hot com- ponents, for all the 20 galaxies, as well as the age and metal- licity radial profiles for the 16 galaxies with their orbits tagged with age and metallicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' The galaxies with extended cold disks are shown in the left panels, while those with concentrated cold components are shown in the right panels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' The dynamically cold and hot components are well separated with different surface- brightness, age, and metallicity radial profiles, which will be dis- cussed in the following sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' We note that the surface-brightness profiles of the cold disks do not follow the traditional exponential law: the surface- brightness in the central regions is much lower than the inward extrapolation of the exponential profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' This finding is consis- tent with the results of orbital decomposition of the CALIFA galaxies (Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' 2018) and with those from direct spectrum fitting, where bulge and disk are separated through their stellar populations (Breda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Quantitative description of the radial profiles We quantify the luminosity fraction fcold of the cold disk compo- nent within Re (or 2Re), with respect to the cumulative luminos- ity of the galaxy within that radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' The 1σ uncertainty of fcold is calculated by the scatter of the models within the 1σ confidence level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' We calculate fcold for all the 20 sample galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' For the 16 galaxies with their orbits tagged with age and metallicity, we obtain the mean value and gradient of the radial profiles of age and metallicity of each component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' We calculated Article number, page 9 of 31 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' main Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Surface-brightness, age, and metallicity radial profiles of the galaxy, dynamically cold disk, and dynamically hot non-disk component for the sample galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' We divided the galaxies into two groups: 11 galaxies with a dynamically cold disk extended out to large radii (left panels) and 9 galaxies with a dynamically cold disk concentrated in the inner regions (right panels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' The radial profiles are color-coded by the galaxy stellar mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' The shadowed areas indicates the scatter of the profiles of models that fall within the 1σ confidence level of the model hyper-parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Dashed lines refer to the four LTGs without orbits tagged with age and metallicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' The age and metallicity radial profiles of the dynamically cold disks are considered reliable and shown in the regions where the dynamically cold disk contributes at least 10% of galaxy surface-brightness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Therefore, these profiles are available only for a limited radial range for the galaxies with a concentrated cold disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' the mean age and mean metallicity of the cold disk by ⟨tcold⟩ = 1 fcold λz≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='8 � k tk fk, (5) ⟨Zcold⟩ = 1 fcold λz≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='8 � k Zk fk, (6) where tk is the age of the orbital bundle, k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' The lumonisity- weighted mean age ⟨thot⟩ and metallicity ⟨Zhot⟩ of the hot compo- nent with the orbits λz < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='8 are calculated in a similar manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Note that the mean age is luminosity-weighted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Due to the dif- ferent surface-brightness profiles of the cold and hot component, the difference in mean age of these two components sometimes appears differently from that seen in their age profiles, similar to the case of FCC 177 (as we explained in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' The gradients of age and metallicity are calculated as fol- lows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' We first uniformly interpolated 100 data points along the radial profile within the data coverage to smooth the profile, then we calculated the linear slope (gradient) between adjacent data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' In the end, we took the average of the gradients from the data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' We obtain the uncertainty of the gradient by boot- strapping within the shadowed regions of the profiles as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' All these parameters, together with other basic param- eters obtained from the population-orbit superposition models, are included in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Cold-disk age as a proxy of galaxy infall time into a cluster 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Calibration from simulations It is hard to accurately estimate the infall time into the cluster for each galaxy from observations, although we can statistically infer the likelihood of being ancient or recent infallers from their projected position and line-of-sight velocity in the cluster (Iodice et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' By analyzing the cluster galaxies in the cosmologi- cal simulation Illustris TNG50, we find that the cold-disk age is tightly correlated with the infall time of the galaxy into the clus- ter, as a result of star formation quenching in disks associated with galaxy infall into the cluster (Ding et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=', in prep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' To make direct comparison between the Fornax and TNG50 galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' We defined cold disks of TNG50 galaxies in exactly the same way as here for the Fornax cluster galaxies, with the probability density distribution of stars in the phase-space of cir- cularity λz versus r calculated from the 6D position-velocity in- formation of particles in the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' We note that we use λz and r of the particles’ orbits, not directly the instantaneously values of each particle, following previous work in comparing the stellar orbit distribution of observed and simulated galaxies (Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' 2018a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' The cold-disk fraction and cold-disk age are thus defined exactly the same way as described in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' All galaxies in the 14 clusters with virial mass M200 > 1013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='3 M⊙ in the TNG50 simulations are chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' We define the galaxies’ infall time into the cluster as the time when it first reaches the virial radius r200 of the cluster at that time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' For the pre-processed galaxies, we define their infall time as their first time of falling into the pre-cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Article number, page 10 of 31 FCC255 FCC119 FCC177 FCC182 FCC148 FCC263 FCC153 FCC301 FCC290 FCC143 FCC170 FCC308 FCC083 FCC249 FCC312 FCC161 FCC179 FCC276 FCC147 FCC167Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Ding et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' : F3D: the environmental effects on the assembly of disks Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Infall time into the Fornax cluster of the sample galaxies for which we obtained the age of the dynamically cold disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Left panel: Correlation between the galaxy infall time tinfall and age of the dynamically cold disk tcold for four different mass bins: 8 < log M∗/M⊙ < 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='6 (light cyan), 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='6 < log M∗/M⊙ < 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='2 (cyan), 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='2 < log M∗/M⊙ < 10 (blue), and 10 < log M∗/M⊙ < 12 (dark blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Dashed lines mark the 1σ confidence limits of each correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Circles and crosses correspond to galaxies with extended and concentrated dynamically cold disks, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Each galaxy is plotted by the age of its dynamically cold disk and infall time given by the median of the correlation corresponding to its stellar mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' The error bars corresponds to the 1σ uncertainty of the infall time inferred from the correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Right panel: Distribution of the left-panel galaxies in the phase-space of the projected line-of-sight velocity of the galaxy normalized by line-of-sight velocity dispersion of the cluster VLOS/σLOS versus the projected clustercentric radius of the galaxy normalized by the cluster virial radius Rproj/Rvir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Each galaxy is color-coded according to its infall time except for the 4 LTGs without age and metallicity information shown in gray.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' The symbols are the same as in the left panel and the LTGs are marked by gray symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' The boundaries of the regions A, B, C, D, and E are defined as in Rhee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' We found that the correlation between the cold-disk age and the infall time of the galaxy into the cluster does not strongly depend on the cluster mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' We divided all the galax- ies in the 14 clusters into four mass bins with log10 M∗/M⊙ ∈ (8, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='6), (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='6, 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='2), (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='2, 10), and (10, 11), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' In the left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' 5, we show the running median and ±1σ scatter of the correlation in the four mass bins color-coded by the stellar mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' The 1σ scatter is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='63, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='63, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='94, and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='34 Gyr from the low to high mass bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' The correlation is tighter for the least- massive galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' The scatter becomes significantly larger in the most massive galaxies because a significant fraction of them are already quenched or had little gas left before they fell into the cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' For galaxies with the cold-disk age obtained from the population-orbit superposition model, we can thus use this corre- lation to infer their infall time into the Fornax cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' 5, we obtained the infall time of each galaxy by means of the correlation for the mass bin including its stellar mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' The values inferred from the median and 1σ scatter of the correlation are taken as the median and 1σ uncertainty of the galaxy infall time tinfall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' They are listed in the last column of Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Comparison with galaxy locations in the projected phase-space We cross-checked the infall time inferred from the cold-disk age with the galaxy location in the phase-space VLOS/σLOS vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Rproj/Rvir, as shown in the right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Here, VLOS/σLOS is the projected line-of-sight velocity of the galaxy normalized by line-of-sight velocity dispersion of the cluster and Rproj/Rvir is the projected clustercentric radius of the galaxy normalized by the cluster virial radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' The galaxies are color-coded by the Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Fractions of ancient, intermediate, and recent infallers in the regions of E, D, and B+C we find in our sample and a comparison with that predicated from Rhee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' (2017) in brackets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Region Ancient Intermediate Recent E 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='7% (50%) 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='3% (20%) 0% (25%) D 50% (20%) 0% (30%) 50% (50%) B+C 40% (30%) 20% (20%) 40% (50%) infall time inferred from the cold-disk age in the left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Following Iodice et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' (2019), we define ancient infallers as galaxies that fell into the cluster at 8-12 Gyr ago, intermedi- ate infallers at 4-8 Gyr ago, and recent infallers at 0-4 Gyr ago.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' We divided the phase-space into five regions like in Iodice et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' (2019), with the fraction of ancient infallers decreaseing from regions E and D to B+C (based on Rhee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' (2017)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' About 50% of the galaxies in region E are assumed to be ancient in- fallers in our sample, while six out of seven galaxies in region E are ancient infallers and one is intermediate infaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Galaxies in region D are supposed to be dominated by recent and intermedi- ate infallers, while we have two ancient infallers and two recent infallers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Galaxies in region B+C are suggested to be dominated by recent infallers, while we have two ancient, one intermedi- ate, and two recent infallers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' The fractions of ancient, intermedi- ate and ancient infallers in different regions we obtained and in comparison to that predicted from Rhee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' (2017) are shown in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' Although we have a low amount of statistics at hand, we re- main consistent with Rhee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' (2017) in that the fraction of ancient infallers is highest in region E and then lower in D and B+C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' However, the fraction of ancient infallers in our sample is Article number, page 11 of 31 8< log10M+/M。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content=' < 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='6 Extended B Concentrated 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E5T4oBgHgl3EQfSw_w/content/2301.05532v1.pdf'} +page_content='6 supx∈Ω |x| with center in the +origin such that Ω ⊂ BR. The complements of Ω and BR are denoted by Ωc := Rd \ Ω +Bc +R := Rd \ BR, resp., the open complement of BR is denoted by B+ +R := Rd \ BR (the overbar +over sets denotes their closure in Rd), and the boundary of BR, the sphere, by SR := ∂BR +(cf. Fig. 1). The open complement of Ω is denoted by Ω+ := Rd \ Ω. By ν we denote the +outward-pointing (w.r.t. either Ω or BR) unit normal vector on ∂Ω or SR, respectively. +Trace operators will be denoted by one and the same symbol γ; the concrete meaning (e.g., +3 + +Resonant compactly supported nonlinearities +January 30, 2023 +Ω +SR +uinc +Figure 1: The nonlinear medium Ω is excited by an incident field uinc (d = 2) +traces on the common interface of an interior and exterior domain) will be clear from the +context. +The classical direct problem of radiation and propagation of an electromagnetic field – ac- +tually just one component of it – by/in the penetrable obstacle Ω is governed by a nonlinear +Helmholtz equation with a variable complex-valued wave coefficient: +− ∆u(x) − κ2c(x, u) u = f(x, u) +for (almost) all x ∈ Rd, +(1) +where the wavenumber κ > 0 is fixed. The physical properties of the obstacle Ω are described +by the coefficient c : Rd × C → C (physically the square of the refractive index) and the +right-hand side f : +Rd × C → C. In general, both functions are nonlinear and have the +following properties: +supp(1 − c(·, w)) = Ω +and +supp f(·, w) ⊂ Ω +for all w ∈ C. +(2) +The function 1 − c is often called the contrast function. Basically we assume that c and +f are Carath´eodory functions, i.e. the mapping x �→ c(x, v) is (Lebesgue-)measurable for +all v ∈ C, and the mapping v �→ c(x, v) is continuous for almost all x ∈ Rd. These two +conditions imply that x �→ c(x, v(x)) is measurable for any measurable v. The same applies +to f. +The unknown total field u : Rd → C should have the following structure: +u = +� +urad + uinc +in Ωc, +utrans +in Ω, +(3) +where urad : Ωc → C is the unknown radiated/scattered field, utrans : Ω → C denotes the +unknown transmitted field, and the incident field uinc ∈ H1 +loc(Ω+) is given. The incident +field is usually a (weak) solution of either the homogeneous or inhomogeneous Helmholtz +equation (even in the whole space). Typically it is generated either by concentrated sources +located in a bounded region of Ω+ or by sources at infinity, e.g. travalling waves. +Example 1 (d = 2). The incident plane wave, whose transmission and scattering is inves- +tigated, is given by +uinc(x) := αinc exp(i(Φx1 − Γx2)), x = (x1, x2)⊤ ∈ B+ +R +4 + +Resonant compactly supported nonlinearities +January 30, 2023 +with amplitude αinc and angle of incidence ϕinc, |ϕinc| < π, where Φ := κ sin ϕinc is the +longitudinal wave number and Γ := +√ +κ2 − Φ2 = κ cos ϕinc the transverse wave number. In +polar coordinates is then +uinc(r, ϕ) = αinc exp(i(Φr cos ϕ − Γr sin ϕ)) += αinc exp(iκr(sin ϕinc cos ϕ − cos ϕinc sin ϕ)) += αinc exp(iκr sin(ϕinc − ϕ)), +(r, ϕ) ∈ B+ +R. +The radiated/scattered field urad should satisfy an additional condition, the so-called Som- +merfeld radiation condition: +lim +|x|→∞ |x|(d−1)/2 � +ˆx · ∇urad − iκurad� += 0 +(4) +uniformly for all directions ˆx := x/|x|, where ˆx·∇urad denotes the derivative of urad in radial +direction ˆx, cf. [CK13, eq. (3.7) for d = 3, eq. (3.96) for d = 2]. Physically, the condition (4) +allows only outgoing waves at infinity; mathematically it guaranties the uniqueness of the +solution uscat : B+ +R → C of the following exterior Dirichlet problem +−∆uscat − κ2uscat = 0 +in B+ +R, +uscat = fSR +on SR, +lim +|x|→∞ |x|(d−1)/2 � +ˆx · ∇uscat − iκuscat� += 0, +(5) +where fSR : SR → C is given. We mention that, in the context of classical solutions (i.e. +uscat ∈ C2(B+ +R)) to problem (5), Rellich [Rel43] has shown that the condition (4) can be +weakened to the following integral version: +lim +|x|→∞ +� +SR +��ˆx · ∇uscat − iκuscat��2 ds(x) = 0. +In the context of weak solutions (i.e. uscat ∈ H1 +loc(B+ +R)), an analogous equivalence statement +can be found in [McL00, Thm. 9.6]. +3 The exterior problem in Bc +R +For a given fSR ∈ C(SR) and d = 3, the unique solvability of problem (5) in C2(B+ +R)∩C(Bc +R) +is proved, for example, in [CK13, Thm. 3.21]. In addition, if fSR is smoother, say fSR ∈ +C∞(SR), then the normal derivative of uscat on the boundary SR is a well-defined continuous +function [CK13, Thm. 3.27]. These assertions remain valid in the case d = 2, see [CK13, +Sect. 3.10]. +Therefore, by solving (5) for given fSR ∈ C∞(SR), a mapping can be introduced that takes +the Dirichlet data on SR to the corresponding Neumann data on SR, i.e. +fSR �→ TκfSR := ˆx · ∇uscat�� +SR , +(6) +see, e.g., [CK19, Sect. 3.2]. +5 + +Resonant compactly supported nonlinearities +January 30, 2023 +Furthermore, it is well-known that the mapping Tκ can be extended to a bounded linear +operator Tκ : Hs+1/2(SR) → Hs−1/2(SR) for any |s| ≤ 1/2 [CWGLS12, Thm. 2.31] (we keep +the notation already introduced for this continued operator). This operator is called the +Dirichlet-to-Neumann operator, in short DtN operator, or capacity operator. +Since the problem (5) is considered in a spherical exterior domain, an explicit series represen- +tation of the solution is available using standard separation techniques in polar or spherical +coordinates, respectively. The term-by-term differentiation of this series thus also provides +a series representation of the image of Tκ. +The solution of the problem (5) in the two-dimensionsional case (here with uscat replaced by +u) is given by [Mas87, Proposition 2.1], [KG89, eq. (30)]: +u(x) = u(rˆx) = u(r, ϕ) = +� +n∈Z +H(1) +n (κr) +H(1) +n (κR) +fn(R)Yn(ˆx) = +� +n∈Z +H(1) +n (κr) +H(1) +n (κR) +fn(R)Yn(ϕ), +x = rˆx ∈ Sr, r > R, ϕ ∈ [0, 2π] +(7) +(identifying u(x) with u(r, ϕ) and Yn(ˆx) with Yn(ϕ) for x = rˆx = r(cos ϕ, sin ϕ)⊤), where +(r, ϕ) are the polar coordinates, H(1) +n +are the cylindrical Hankel functions of the first kind of +order n [DLMF22, Sect. 10.2]1, Yn are the circular harmonics defined by +Yn(ϕ) = einϕ +√ +2π +, +n ∈ Z, +fn(R) are the Fourier coefficients of fSR defined by +fn(R) := (fSR(R·), Yn)S1 = +� +S1 +fSR(Rˆx)Yn(ˆx)ds(ˆx) = +� 2π +0 +fSR(R, ϕ)Yn(ϕ)dϕ, +(8) +and ds(ˆx) is the Lebesgue arc length element. +Now we formally differentiate the representation (7) with respect to r to obtain the outward +normal derivative of u: +ˆx · ∇u(x) = ∂u +∂r (rˆx) = κ +� +n∈Z +H(1)′ +n +(κr) +H(1) +n (κR) +fn(R)Yn(ˆx), +x = rˆx ∈ Sr, r > R. +Setting fR := u|SR and letting x in this representation approach the boundary SR, we can +formally define the (extended) DtN operator by +Tκu(x) := 1 +R +� +n∈Z +Zn(κR)un(R)Yn(ˆx), +x = Rˆx ∈ SR, +(9) +where +Zn(ξ) := ξ H(1)′ +n +(ξ) +H(1) +n (ξ) +, +1Instead of (4) [Mas87] considered the ingoing Sommerfeld condition and thus obtained a representation +in terms of the cylindrical Hankel functions of the second kind. Note that H(2) +n (−ξ) = −(−1)nH(1) +n (ξ) +[DLMF22, (10.11.5)]. +6 + +Resonant compactly supported nonlinearities +January 30, 2023 +and un(R) are the Fourier coefficients of u|SR analogously to (8). The admissibility of this +procedure has been proven in many sources in the classical context, for example [CK19, +Sect. 3.5]. For the present case, in the paper [Ern96, Thm. 1] it was shown that the operator +Tκ : Hs+1/2(SR) → Hs−1/2(SR) is bounded for any s ∈ N0. Ernst’s result was extended to +all s ≥ 0 in [HNPX11, Thm. 3.1]. +In the case d = 3, the solution of the problem (5) is given by [KG89, eq. (33)]: +u(x) = u(rˆx) = u(r, ϕ, θ) = +� +n∈N0 +� +|m|≤n +h(1) +n (κr) +h(1) +n (κR) +f m +n (R)Y m +n (ˆx) += +� +n∈N0 +� +|m|≤n +h(1) +n (κr) +h(1) +n (κR) +f m +n (R)Y m +n (ϕ, θ), +x ∈ Sr, r > R, (ϕ, θ) ∈ [0, 2π] × [0, π] +(10) +(identifying u(x) with u(r, ϕ, θ) and Y m +n (ˆx) with Y m +n (ϕ, θ) for x = rˆx = r(cos ϕ sin θ, +sin ϕ sin θ, cos θ)⊤), where (r, ϕ, θ) are the spherical coordinates, h(1) +n are the spherical Hankel +functions of the first kind of order n [DLMF22, Sect. 10.47], Y m +n are the spherical harmonics +defined by +Y m +n (ϕ, θ) = +� +2n + 1 +4π +(n − |m|)! +(n + |m|)! P |m| +n (cos θ)eimϕ, +n ∈ N0, |m| ≤ n, +(identifying Y m +n (ˆx) with Y m +n (ϕ, θ) for ˆx = (cos ϕ sin θ, sin ϕ sin θ, cos θ)⊤), where P m +n are the +associated Legendre functions of the first kind [DLMF22, Sect. 14.21], f m +n (R) are the Fourier +coefficients defined by +f m +n (R) = (fSR(R·), Y m +n )S1 = +� +S1 +fSR(Rˆx)Y m +n (ˆx)ds(ˆx) += +� 2π +0 +� π +0 +fSR(R, ϕ, θ)Y m +n (ϕ, θ) sin θdθdϕ, +(11) +and ds(ˆx) is the Lebesgue surface area element. +Proceeding as in the two-dimensional case, we get +ˆx · ∇u(x) = ∂u +∂r (rˆx) = κ +� +n∈N0 +� +|m|≤n +h(1) +n (κr) +h(1) +n (κR) +f m +n (R)Y m +n (ˆx), +x = rˆx ∈ Sr, r > R. +Setting fR := u|SR and letting r → R, we can define the (extended) DtN operator by +Tκu(x) = 1 +R +� +n∈N0 +� +|m|≤n +zn(κR)um +n (R)Y m +n (ˆx), +x = Rˆx ∈ SR, +(12) +where +zn(ξ) := ξ h(1)′ +n (ξ) +h(1) +n (ξ) +, +7 + +Resonant compactly supported nonlinearities +January 30, 2023 +and um +n (R) are the Fourier coefficients of u|SR analogously to (11). The admissibility of this +procedure is proved in [CK19, Thm. 2.15] or [N´ed01, Thm. 2.6.2], for example. For the +present situation there is a boundedness result for d = 3 analogous to [HNPX11, Thm. 3.1] +in [N´ed01, Thm. 2.6.4]. In summary, the following statement applies to both dimensions. +Theorem 2. The DtN operator Tκ : Hs+1/2(SR) → Hs−1/2(SR) is bounded for any s ≥ 0. +Remark 3. A more refined analysis of the DtN operator in the case s = 0 results in a sharp +estimate of the its norm w.r.t. the wavenumber [BSW16, Thm. 1.4]: Given κ0 > 0, there +exists a constant C > 0 independent of κ such that +∥Tκv∥−1/2,2,SR ≤ Cκ∥v∥1/2,2,SR +for all v ∈ H1 +loc(B+ +R) +and +κ ≥ κ0. +The result from [BSW16, Thm. 1.4] applies to more general domains, for the present situation +it already follows from the proof of Lemma 23 (see the estimates (46), (47) for s = 0, where +the bounds do not depend on N). +At the end of this section we give a collection of some properties of the coefficient functions +in the representations (9), (12) which will be used in some of the subsequent proofs. +Lemma 4. For all ξ > 0, the following holds: +−n ≤ Re Zn(ξ) ≤ −1 +2, +0 < Im Zn(ξ) < ξ +for all |n| ∈ N, +−1 +2 ≤ Re Z0(ξ) < 0, +ξ < Im Z0(ξ), +−(n + 1) ≤ Re zn(ξ) ≤ −1, +0 < Im zn(ξ) ≤ ξ +for all n ∈ N, +Re z0(ξ) = −1, +Im z0(ξ) = ξ. +Proof. For the case d = 2, the estimates can be found in [SW07, eq. (2.34)]. The other +estimates can be found in [N´ed01, Thm. 2.6.1], see also [SW07, eqs. (2.22), (2.23)]. Although +only 0 ≤ Im zn(ξ) is specified in the formulation of the cited theorem, the strict positivity +follows from the positivity of the function qℓ in [N´ed01, eq. (2.6.34)], as has been mentioned +in [MS10]. +Corollary 5. For all ξ > 0, the following holds: +|Zn(ξ)|2 ≤ (1 + n2)(1 + |ξ|2) +for all |n| ∈ N, +|zn(ξ)|2 ≤ (1 + n2)(2 + |ξ|2) +for all n ∈ N0. +Proof. The estimates of the real and imaginary parts of Zn from Lemma 4 immediately +imlpy that +1 +1 + n2|Zn(ξ)|2 = +1 +1 + n2 +� +| Re Zn(ξ)|2 + | Im Zn(ξ)|2� +≤ +1 +1 + n2 +� +n2 + |ξ|2� +≤ 1 + +|ξ|2 +1 + n2 ≤ 1 + |ξ|2, +n ∈ N. +Since H(1) +−n(ξ) = (−1)nH(1) +n (ξ), n ∈ N [DLMF22, eq. (10.4.2)], the estimate is also valid for n +such that −n ∈ N. +8 + +Resonant compactly supported nonlinearities +January 30, 2023 +Analogously we obtain from Lemma 4 that +1 +1 + n2|zn(ξ)|2 = +1 +1 + n2 +� +| Re zn(ξ)|2 + | Im zn(ξ)|2� +≤ +1 +1 + n2 +� +(1 + n)2 + |ξ|2� +≤ 2 + +|ξ|2 +1 + n2 ≤ 2 + |ξ|2. +4 Weak formulations of the interior problem +Now we turn to the consideration of the problem (1)–(4). +In the classical setting it can be formulated as follows: Given uinc ∈ H1 +loc(Ω+), determine the +transmitted field utrans : Ω → C and the radiated/scattered field urad : Ωc → C satisfying +−∆utrans − κ2c(·, utrans) utrans = f(·, utrans) +in Ω, +−∆urad − κ2urad = 0 +in Ω+, +utrans = urad + uinc +on ∂Ω, +ν · ∇utrans = ν · ∇urad + ν · ∇uinc +on ∂Ω +(13) +and the radiation condition (4). Note that the incident field is usually a (weak) solution +of either the homogeneous or inhomogeneous Helmholtz equation in Ω+, i.e. the second +equation in (13) can be replaced by +− ∆u − κ2u = f inc +in Ω+, +(14) +where f inc : Ω+ → C is an eventual source density. For simplicity we do not include the +case of a nontrivial source density in our investigation, but the subsequent theory can be +easily extended by adding an appropriate linear functional, say ℓsrc, on the right-hand side +of the obtained weak formulations (see (15) or (19) later). +In order to give a weak formulation of (13) with the modification (14) in the case f inc = 0, +we introduce the (complex) linear function spaces +H1 +comp(Ω+) := +� +v ∈ H1(Ω+) : supp v is compact +� +, +VRd := {v ∈ L2(Rd) : v|Ω ∈ H1(Ω), v|Ω+ ∈ H1 +loc(Ω+) : γv|Ω = γv|Ω+ on ∂Ω}, +WRd := {v ∈ L2(Rd) : v|Ω ∈ H1(Ω), v|Ω+ ∈ H1 +comp(Ω+) : γv|Ω = γv|Ω+ on ∂Ω} +(note the comment at the beginning of Section 2 on the notation for trace operators) and +multiply the first equation of (13) by the restriction v|Ω of an arbitrary element v ∈ VRd and +(14) by the restriction v|Ω+ of v ∈ VRd, respectively, and integrate py parts: +(∇utrans, ∇v)Ω − (ν · ∇utrans, ∇v)∂Ω − κ2(c(·, utrans)utrans, v)Ω = (f(·, utrans), v)Ω, +(∇u, ∇v)Ω − (ν · ∇u, ∇v)∂Ω+ − κ2(u, v)Ω+ = 0. +9 + +Resonant compactly supported nonlinearities +January 30, 2023 +Here we use the notation, for any domain M ⊂ Rd with boundary ∂M and appropriately +defined functions on M or ∂M, +(∇w, ∇v)M := +� +M +∇w · ∇vdx, +(w, v)M := +� +M +wvdx, +(w, v)∂M := +� +∂M +wvds(x) +(the overbar over functions denotes complex conjugation). Taking into consideration the last +transmission condition in (13), the relationsship ν|Ω = −ν|Ω+, and the fact that the last but +one transmission condition in (13) is included in the definition of the space VRd, we define a +bivariate nonlinear form on VRd × WRd by +aRd(w, v) := (∇w, ∇v)Ω + (∇w, ∇v)Ω+ − κ2(c(·, w)w, v)Rd, +cf., e.g., [Wlo87, Example 21.8]. +Definition 6. Given uinc ∈ H1 +loc(Ω+), a weak solution to the problem (1)–(4) is defined as +an element u ∈ VRd that has the structure (3), satisfies the variational equation +aRd(u, v) = (f(·, u), v)Rd +for all v ∈ WRd +(15) +and the Sommerfeld radiation condition (4). +A second weak formulation can be obtained if we do not replace the second Helmholtz +equation in (13) by (14). Then the first step in the derivation of the weak formulation reads +as +(∇utrans, ∇v)Ω − (ν · ∇utrans, ∇v)∂Ω − κ2(c(·, utrans)utrans, v)Ω = (f(·, utrans), v)Ω, +(∇urad, ∇v)Ω − (ν · ∇urad, ∇v)∂Ω+ − κ2(urad, v)Ω+ = 0. +The last transmission condition in (13) allows to rewrite the first equation as +(∇utrans, ∇v)Ω − (ν · ∇urad, ∇v)∂Ω − κ2(c(·, utrans)utrans, v)Ω += (f(·, utrans), v)Ω + (ν · ∇uinc, ∇v)∂Ω, +leading to the weak formulation +(∇u0, ∇v)Ω+(∇u0, ∇v)Ω+−κ2(c(·, u0)u0, v)Rd = (f(·, u0), v)Rd+(ν·∇uinc, v)∂Ω +for all v ∈ WRd +(16) +with respect to the structure +u0 := +� +urad +in Ωc, +utrans +in Ω, +(17) +where urad ∈ H1 +loc(Ω+), utrans ∈ H1(Ω). +The advantage of this formulation is that it clearly separates the unknown and the known +parts of the fields, so we call this formulation the input-output formulation. The disadvantage +10 + +Resonant compactly supported nonlinearities +January 30, 2023 +is that the natural function space of the solution u0 is not a linear space due to the last but +one transmission condition in (13). +Instead of the problem (1)–(4) we want to solve an equivalent problem in the bounded +domain BR, that is, we define +V := {v ∈ L2(BR) : v|Ω ∈ H1(Ω), v|BR\Ω ∈ H1(BR \ Ω) : γv|Ω = γv|BR\Ω on ∂Ω} +and look for an element u ∈ V such that +−∆utrans − κ2c(·, utrans) u = f(·, utrans) +in Ω, +−∆u − κ2u = 0 +in BR \ Ω, +utrans = urad + uinc +on ∂Ω, +ν · ∇utrans = ν · ∇urad + ν · ∇uinc +on ∂Ω, +ˆx · ∇urad = Tκurad +on SR +(18) +formally holds. Now the weak formulation of problem (18) reads as follows: +Find u ∈ V such that +(∇u, ∇v)Ω + (∇u, ∇v)BR\Ω − κ2(c(·, u)u, v)BR − (Tκu, v)SR += (f(·, u), v)BR − (Tκuinc, v)SR + (ˆx · ∇uinc, v)SR +(19) +for all v ∈ V holds. +Lemma 7. The weak formulations (15) and (19) of the problems (1)–(4) and (18), resp., +are equivalent. +Proof. First let u ∈ V (Rd) be a weak solution to (1)–(4), i.e. it satisfies (15). Then its +restriction to BR belongs to V . +To demonstrate that this restriction satisfies the weak formulation (19), we construct the +radiating solution uBc +R′ of the homogeneous Helmholtz equation outside of a smaller ball BR′ +such that Ω ⊂ BR′ ⊂ BR and uBc +R′ +��� +SR′ = (u − uinc)|SR′. This solution can be constructed +in the form of a series expansion in terms of Hankel functions as explained in the previous +section. +By elliptic regularity (see, e.g., [McL00, Thm. 4.16], [Eva15, Sect. 6.3.1]), the +solution of this problem satisfies the Helmholtz equation in Bc +R′. Moreover, by uniqueness +[N´ed01, Thm. 2.6.5], it coincides with u − uinc = urad in Bc +R′. +Now we choose a finite partition of unity covering BR, denoted by {ϕj}J [Wlo87, Sect. 1.2], +such that its index set J can be decomposed into two disjoint subsets J1, J2 as follows: +BR′ ⊂ int +� � +j∈J1 +supp ϕj +� +, +� +j∈J1 +supp ϕj ⊂ BR, +� +j∈J2 +supp ϕj ⊂ Bc +R′. +For example, we can choose {ϕj}J1 to consist of one element, say ϕ1, namely the usual +mollifier function with support B′, where the open ball B′ (centered at the origin) lies +between BR′ and BR, i.e. BR′ ⊂ B′ = int (supp ϕ1), supp ϕ1 ⊂ BR. Then the second part +consists of a finite open covering of the spherical shell BR \ B′. +11 + +Resonant compactly supported nonlinearities +January 30, 2023 +Then we take, for any v ∈ V , the product v1 := v � +j∈J1 ϕj. This is an element of V , too, +with support in BR, and it can be continued by zero to an element of W(Rd) (keeping the +notation). Hence we can take it as a test function in the weak formulation (15) and obtain +aRd(u, v1) = (f(·, u), v1)Rd. +This is equal to +(∇u, ∇v1)Ω + (∇u, ∇v1)BR\Ω − κ2(c(·, u)u, v1)BR − (Tκu, v1)SR += (f(·, u), v1)BR − (Tκuinc, v1)SR + (ˆx · ∇uinc, v1)SR +due to the properties of the support of v1 (in particular, all terms “living” on SR are equal +to zero). +Since the homogeneous Helmholtz equation is satisfied in � +j∈J2 supp ϕj ⊂ Bc +R′, we can +proceed as follows. We continue the test function v2 := v � +j∈J2 ϕj by zero into the complete +ball BR and have +(f(·, u), v2)BR\Ω = 0 = (−∆u − κ2u, v2)BR\BR′ += (∇u, ∇v2)BR\BR′ − κ2(u, v2)BR\BR′ − (ν · ∇u, v2)∂(BR\BR′) += (∇u, ∇v2)BR\BR′ − κ2(u, v2)BR\BR′ − (ˆx · ∇u, v2)SR. +Now, taking into consideration the properties of the support of v2, we easily obtain the +following relations: +(∇u, ∇v2)BR\BR′ = (∇u, ∇v2)Ω + (∇u, ∇v2)BR\Ω, +(u, v2)BR\BR′ = (c(·, u)u, v2)BR, +(ˆx · ∇u, v2)SR = (ˆx · ∇urad, v2)SR + (ˆx · ∇uinc, v2)SR += (Tκurad, v2)SR + (ˆx · ∇uinc, v2)SR += (Tκu, v2)SR − (Tκuinc, v2)SR + (ˆx · ∇uinc, v2)SR, +where the treatment of the last term makes use of the construction of the Dirichlet-to- +Neumann map Tκ. +Adding both relations and observing that v = v1+v2, we arrive at the variational formulation +(19). +Conversely, let u ∈ V be a solution to (19). To continue it into Bc +R, similar to the first +part of the proof we construct the radiating solution uBc +R of the Helmholtz equation outside +BR such that uBc +R +�� +SR = (u − uinc)|SR and set u := uBc +R + uinc in B+ +R. Hence we have that +Tκu = +∂uBc +R +∂ˆx + Tκuinc. +Now we take an element v ∈ W(Rd). Its restriction to BR is an element of V and thus can +be taken as a test function in (19): +(∇u, ∇v)Ω + (∇u, ∇v)BR\Ω − κ2(c(·, u)u, v)BR − (Tκu, v)SR += (f(·, u), v)BR − (Tκuinc, v)SR + (ˆx · ∇uinc, v)SR. +(20) +12 + +Resonant compactly supported nonlinearities +January 30, 2023 +Since v has a compact support, we can choose a ball B ⊂ Rd centered at the origin such +that BR ∪ supp v ⊂ B. The homogeneous Helmholtz equation is obviously satisfied in the +spherical shell B \ BR: +−∆uBc +R − κ2uBc +R = 0. +We multiply this equation by the complex conjugate of the test function v ∈ V , then integrate +over the shell, and apply the first Green’s formula: +(∇uBc +R, ∇v)B\BR − κ2(uBc +R, v)B\BR − (ν · ∇uBc +R, v)∂(B\BR) = 0. +Now we observe that +(∇uBc +R, ∇v)B\BR = (∇uBc +R, ∇v)B+ +R, +(uBc +R, v)B\BR = (uBc +R, v)B+ +R, +(ν · ∇uBc +R, v)∂(B\BR) = −(ˆx · ∇uBc +R, v)SR = −(Tκu − Tκuinc, v)SR +where the minus sign in the last line results from the change in the orientation of the outer +normal (once w.r.t. the shell, once w.r.t. BR) and the construction of uBc +R. So we arrive at +(∇uBc +R, ∇v)B+ +R − κ2(uBc +R, v)B+ +R + (Tκu, v)SR = (Tκuinc, v)SR. +(21) +Finally, since the incident field satisfies the homogeneous Helmholtz equation in the spherical +shell, too, we see by an analogous argument that the variational equation +(∇uinc, ∇v)B+ +R − κ2(uinc, v)B+ +R = −(ˆx · ∇uinc, v)SR +(22) +holds. +Adding the variational equations (20) – (22), we arrive at the variational formulation (15). +5 Existence and uniqueness of a weak solution +In this section we investigate the existence and uniqueness of the weak solution of the interior +problem (18). We define the sesquilinear form +a(w, v) := (∇w, ∇v)Ω + (∇w, ∇v)BR\Ω − κ2(w, v)BR − (Tκw, v)SR +for all w, v ∈ V, +(23) +the nonlinear form +n(w, v) := κ2(c(·, w) − 1)w, v)BR + (f(·, w), v)BR +− (Tκuinc, v)SR + (ˆx · ∇uinc, v)SR +(24) +and reformulate (19) as follows: Find u ∈ V such that +a(u, v) = n(u, v) +for all v ∈ V. +(25) +On the space V , we use the standard seminorm and norm: +|v|V := +� +∥∇v∥2 +0,2,Ω + ∥∇v∥2 +0,2,BR\Ω +�1/2 +, +∥v∥V := +� +|v|2 +V + ∥v∥2 +0,2,BR +�1/2 . +(26) +13 + +Resonant compactly supported nonlinearities +January 30, 2023 +For κ > 0, the following so-called wavenumber dependent norm on V is also common: +∥v∥V,κ := +� +|v|2 +V + κ2∥v∥2 +0,2,BR +�1/2 . +(27) +It is not difficult to verify that the standard norm and the wavenumber dependent norm are +equivalent on V , i.e. it holds +C−∥v∥V ≤ ∥v∥V,κ ≤ C+∥v∥V +for all v ∈ V, +(28) +where the equivalence constants depend on κ in the following way: C− := min{1; κ} and +C+ := max{1; κ}. We now proceed to examine the linear aspects of the problem (25). +Lemma 8. The sesquilinear form a is bounded on V . +Proof. Applying to each addend in the definition of a the appropriate Cauchy-Bunyakovsky- +Schwarz inequality, we obtain +|a(w, v)| ≤ |w|V |v|V + κ2∥w∥0,2,BR∥v∥0,2,BR ++ ∥Tκw∥−1/2,2,SR∥v∥1/2,2,SR +for all w, v ∈ V. +According to Thm. 2 the DtN operator Tκ is bounded, i.e. there exists a constant CTκ > 0 +such that +∥Tκw∥−1/2,2,SR ≤ CTκ∥w∥1/2,2,SR +for all w ∈ V. +It remains to apply a trace theorem [McL00, Thm. 3.37]: +|a(w, v)| ≤ |w|V |v|V + κ2∥w∥0,2,BR∥v∥0,2,BR + CTκC2 +tr∥w∥1,2,BR\Ω∥v∥1,2,BR\Ω +≤ |w|V |v|V + κ2∥w∥0,2,BR∥v∥0,2,BR + CTκC2 +tr∥w∥V ∥v∥V +≤ min{(max{1, κ2} + CTκC2 +tr)∥w∥V ∥v∥V , (1 + CTκC2 +tr)∥w∥V,κ∥v∥V,κ} +for all w, v ∈ V. +Lemma 9. Given κ0 > 0 and R0 > 0, assume that κ ≥ κ0 (cf. Rem. 3) and R ≥ R0. In +addition, κ0 ≥ 1 is required for d = 2. Then the sesquilinear form a satisfies a G˚arding’s +inequality of the form +Re a(v, v) ≥ ∥v∥2 +V,κ − 2κ2∥v∥2 +0,2,BR +for all v ∈ V. +Proof. From the definitions of a and the wavenumber dependent norm it follows immediately +that +Re a(v, v) = ∥v∥2 +V,κ − 2κ2∥v∥2 +0,2,BR − Re (Tκv, v)SR +≥ ∥v∥2 +V,κ − 2κ2∥v∥2 +0,2,BR + CR−1∥v∥2 +0,2,SR +≥ ∥v∥2 +V,κ − 2κ2∥v∥2 +0,2,BR, +where the first estimate follows from [MS10, Lemma 3.3] with a constant C > 0 depending +soleley on κ0 > 0 and R0 > 0. +14 + +Resonant compactly supported nonlinearities +January 30, 2023 +Next we discuss the solvability and stability of the problem (25) for the case that the right- +hand side is just an antilinear continuous functional ℓ : +V → C. The linear problem of +finding u ∈ V such that +a(u, v) = ℓ(v) +for all v ∈ V +(29) +holds can be formulated equivalently as an operator equation in the dual space V ∗ of V +consisting of all continuous antilinear functionals from V to C. Namely, if we define the +linear operator A : V → V ∗ by +Aw(v) := a(w, v) +for all w, v ∈ V, +(30) +problem (29) is equivalent to solving the operator equation +Au = ℓ +(31) +for u ∈ V . +Note that A is a bounded operator by Lemma 8. +Theorem 10. Under the assumptions of Lemma 9, the problem (31) is uniquely solvable for +any ℓ ∈ V ∗. +Proof. The basic ideas of the proof are taken from the proof of [MS10, Thm 3.8]. Since the +embedding of V into L2(BR) is compact by the compactness theorem of Rellich–Kondrachov +[McL00, Thm. 3.27] together with Tikhonov’s product theorem [KN63, Thm. 4.1], the com- +pact perturbation theorem [McL00, Thm. 2.34] together with Lemma 9 imply that the +Fredholm alternative [McL00, Thm. 2.27] holds for the equation (31). +Hence it is sufficient to demonstrate that the homogeneous adjoint problem (cf. [McL00, +p. 43]) of finding u ∈ V such that a(v, u) = 0 holds for all v ∈ V only allows for the trivial +solution. +So suppose u ∈ V is a solution of the homogeneous adjoint problem. We take v := u and +consider the imaginary part of the resulting equation: +0 = Im a(u, u) = − Im (Tκu, u)SR = Im (Tκu, u)SR. +Then [MS10, Lemma 3.3] implies u = 0 on SR. Then u satisfies the variational equation +(∇u, ∇v)Ω + (∇u, ∇v)BR\Ω − κ2(u, v)BR = 0 +for all v ∈ V, +i.e. it is a weak solution of the homogeneous interior transmission Neumann problem for the +wave equation on BR. On the other hand, u can be extended to the whole space Rd by +zero to an element ˜u ∈ V (Rd), and this element can be interpreted as a weak solution of a +homogeneous full-space transmission problem, for instance in the sense of [TW93, Problem +(P)]. Then it follows from [TW93, Lemma 7.1] that ˜u = 0 und thus u = 0. +Since a Fredholm operator has a closed image [McL00, p. 33], it follows from the Open +Mapping Theorem and Thm. 10 (cf. [McL00, Cor. 2.2]) that the inverse operator A−1 is +bounded, i.e. there exists a constant C(R, κ) > 0 such that +∥u∥V,κ = ∥A−1ℓ∥V,κ ≤ C(R, κ)∥ℓ∥V ∗ +for all ℓ ∈ V ∗. +15 + +Resonant compactly supported nonlinearities +January 30, 2023 +Then it holds +1 +C(R, κ) ≤ ∥ℓ∥V ∗ +∥u∥V,κ += +sup +v∈V \{0} +|ℓ(v)| +∥u∥V,κ∥v∥V,κ += +sup +v∈V \{0} +|a(u, v)| +∥u∥V,κ∥v∥V,κ +. +This estimate proves the following result. +Lemma 11. Under the assumptions of Lemma 9, the sesquilinear form a satisfies an inf-sup +condition: +β(R, κ) := +inf +w∈V \{0} +sup +v∈V \{0} +|a(w, v)| +∥w∥V,κ∥v∥V,κ +> 0. +Now we turn to the nonlinear situation and concretize the assumptions regarding the Cara- +th´eodory functions c and f. +Lemma 12. Let pf ∈ +� +[2, ∞), +d = 2, +[2, 6], +d = 3, and assume there exist nonnegative functions mf, gf ∈ +L∞(Ω) such that +|f(x, ξ)| ≤ mf(x)|ξ|pf−1 + gf(x) +for all (x, ξ) ∈ Ω × C. +Then vf(·, w) ∈ L1(Ω) for all w, v ∈ V . +Proof. Since f is a Carath´eodory function, the composition f(·, w) is measurable and it +sufficies to estimate the integral of |vf(·, w)|. Moreover, it suffices to consider the term +mfv|w|pf−1 in more detail. By H¨older’s inequality for three functions, it holds that +∥vf(·, w)∥0,1,Ω ≤ ∥mf∥0,∞,Ω∥v∥0,pf,Ω∥wpf−1∥0,q,Ω +with 1 +pf ++ 1 +q = 1. +The Lpf-norm of v is bounded thanks to the embedding V |Ω ⊂ Lpf(Ω) for the allowed values +of pf [AF03, Thm. 4.12]. Since |wpf−1|q = |w|p +f, the Lq-norm of wpf−1 is bounded by the +same reasoning. +Lemma 13. Let pc ∈ +� +[2, ∞), +d = 2, +[2, 6], +d = 3, and assume there exist nonnegative functions mc, gc ∈ +L∞(Ω) such that +|c(x, ξ) − 1| ≤ mc(x)|ξ|pc−2 + gc(x) +for all (x, ξ) ∈ Ω × C. +Then zv(c(·, w) − 1) ∈ L1(Ω) for all z, w, v ∈ V . +Proof. Similar to the proof of Lemma 12 it is sufficient to consider the term mczv|w|pc−2 in +more detail. By H¨older’s inequality for four functions, it holds that +∥zv(c(·, w) − 1)∥0,1,Ω ≤ ∥mc∥0,∞,Ω∥z∥0,pc,Ω∥v∥0,pc,Ω∥wpc−2∥0,q,Ω +with 2 +pc ++ 1 +q = 1. +The Lpc-norms of z, v are bounded thanks to the embedding theorem [AF03, Thm. 4.12]. +Since |wpc−2|q = |w|p +c, the Lq-norm of wpc−2 is bounded by the same reasoning. +16 + +Resonant compactly supported nonlinearities +January 30, 2023 +Corollary 14. Under the assumptions of Lemma 12 and Lemma 13, resp., the following +estimates hold for all z, w, v ∈ V : +|(f(·, w), v)Ω| ≤ C +pf +emb∥mf∥0,∞,Ω∥w∥ +pf−1 +1,2,Ω∥v∥1,2,Ω ++ +� +|Ω|d ∥gf∥0,∞,Ω∥v∥0,2,Ω, +(32) +|((c(·, w) − 1)z, v)Ω| ≤ Cpc +emb∥mc∥0,∞,Ω∥w∥pc−2 +1,2,Ω∥z∥1,2,Ω∥v∥1,2,Ω ++ ∥gc∥0,∞,Ω∥z∥0,2,Ω∥v∥0,2,Ω, +(33) +where |Ω|d is the d-volume of Ω. +Proof. Replace v by v in Lemmata 12, 13 to get the first addend of the bounds. The estimate +of the second addend is trivial. +Example 15. An important example for the nonlinearities is +c(x, ξ) := +� +1, +(x, ξ) ∈ Ω+ × C, +ε(L)(x) + α(x)|ξ|2, +(x, ξ) ∈ Ω × C, +with given ε(L), α ∈ L∞(Ω), and f = 0. Here pc = 4, which is within the range of validity of +Lemma 13, and mc = |α|, gc = |ε(L) − 1|. +The estimates from Corollary 14 show that the first two terms on the right-hand side of the +variational equation (25) can be considered as values of nonlinear mappings from V to V ∗, +i.e. we can define +ℓcontr : V → V ∗ +by +⟨ℓcontr(w), v⟩ := κ2(c(·, w) − 1)w, v)Ω, +ℓsrc : V → V ∗ +by +⟨ℓsrc(w), v⟩ := (f(·, w), v)Ω +for all w, v ∈ V. +Furthermore, if uinc ∈ H1 +loc(Ω+) is such that additionally ∆uinc belongs to L2,loc(Ω+) (where +∆uinc is understood in the distributional sense), the last two terms on the right-hand side of +(24) form an antilinear continuous functional on ℓinc ∈ V ∗: +⟨ℓinc, v⟩ := (ˆx · ∇uinc − Tκuinc, v)SR +for all v ∈ V. +This is a consequence of Thm. 2 and the estimates before the trace theorem [KA21, Thm. 6.13]. +Hence +∥ℓinc∥V ∗ ≤ ˜Ctr[∥∆uinc∥0,2,BR\Ω + ∥uinc∥0,2,BR\Ω] + CTκC2 +tr∥uinc∥1,2,BR\Ω, +where ˜Ctr is the norm of the trace operator defined in [KA21, eq. (6.39)]. +However, it is more intuitive to utilize the estimate +∥ℓinc∥V ∗ ≤ Ctr∥ˆx · ∇uinc − Tκuinc∥−1/2,2,SR. +(34) +The reason for this is that the bound can be interpreted as a measure of the deviation of the +function uinc from a radiating solution of the corresponding Helmholtz equation. In other +words: If the function uinc satisfies the boundary value problem (5) with fSR := uinc|SR, then +the functional ℓinc is not present. +17 + +Resonant compactly supported nonlinearities +January 30, 2023 +Consequently, setting +F(w) := ℓcontr(w) + ℓsrc(w) + ℓinc +for all w ∈ V, +we obtain a nonlinear operator F : V → V ∗, and the problem (25) is then equivalent to the +operator equation +Au = F(u) +in V ∗, +and further, by Lemma 11, equivalent to the fixed-point problem +u = A−1F(u) +in V. +(35) +In order to prove the subsequent existence and uniqueness theorem, we specify some addi- +tional properties of the nonlinearities c and f. +Definition 16. The functions c and f are said to generate locally Lipschitz continuous Ne- +mycki operators in V if the following holds: For some parameters pc, pf ∈ +� +[2, ∞), +d = 2, +[2, 6], +d = 3,, +there exist Carath´eodory functions Lc : Ω×C×C → (0, ∞) and Lf : Ω×C×C → (0, ∞) such +that the composition operators Ω × V × V → Lqc(Ω) : (x, w, v) �→ Lc(x, w, v), Ω × V × V → +Lqf(Ω) : (x, w, v) �→ Lf(x, w, v) are bounded for qc, qf > 0 with +3 +pc + 1 +qc = +2 +pf + 1 +qf = 1, and +|c(x, ξ) − c(x, η)| ≤ Lc(x, ξ, η)|ξ − η|, +|f(x, ξ) − f(x, η)| ≤ Lf(x, ξ, η)|ξ − η| +(36) +for all (x, ξ, η) ∈ Ω × C × C. +Remark 17. If the nonlinearities c and f generate locally Lipschitz continuous Nemycki +operators in the sense of the above Definition 16, the assumptions of Lemmata 12, 13 can +be replaced by the requirement that there exist functions wf, wc ∈ V such that f(·, wf) ∈ +Lpf /(pf −1)(Ω) and c(·, wf) ∈ Lpc/(pc−2)(Ω), respectively. +Proof. Indeed, similar to the proofs of the two lemmata mentioned, we have that +∥vf(·, w)∥0,1,Ω ≤ ∥vf(·, wf)∥0,1,Ω + ∥v(f(·, w) − f(·, wf))∥0,1,Ω +≤ ∥vf(·, wf)∥0,1,Ω + ∥vLf(·, w, wf)|w − wf|∥0,1,Ω +≤ ∥v∥0,pf,Ω∥f(·, wf)∥0,˜qf,Ω + ∥v∥0,pf,Ω∥Lf(·, w, wf)∥0,qf,Ω∥w − wf∥0,pf,Ω +≤ +� +∥f(·, wf)∥0,˜qf,Ω + ∥Lf(·, w, wf)∥0,qf,Ω(∥w∥V + ∥wf∥V ) +� +∥v∥V , +∥zvc(·, w)∥0,1,Ω ≤ ∥zvc(·, wc)∥0,1,Ω + ∥zv(c(·, w) − c(·, wc))∥0,1,Ω +≤ ∥zvc(·, wc)∥0,1,Ω + ∥zvLc(·, w, wc)|w − wc|∥0,1,Ω +≤ ∥z∥0,pc,Ω∥v∥0,pc,Ω∥c(·, wc)∥0,˜qc,Ω ++ ∥z∥0,pc,Ω∥v∥0,pc,Ω∥Lc(·, w, wc)∥0,qc,Ω∥w − wc∥0,pc,Ω +≤ [∥c(·, wc)∥0,˜qc,Ω + ∥Lc(·, w, wc)∥0,qc,Ω(∥w∥V + ∥wc∥V )] ∥z∥V ∥v∥V +with +1 +pf + 1 +˜qf = 1 and +2 +pc + 1 +˜qc = 1. +18 + +Resonant compactly supported nonlinearities +January 30, 2023 +Theorem 18. Under the assumptions of Lemma 9, let the functions c and f generate locally +Lipschitz continuous Nemycki operators in V and assume that there exist functions wf, wc ∈ +V such that f(·, wf) ∈ Lpf/(pf −1)(Ω) and c(·, wf) ∈ Lpc/(pc−2)(Ω), respectively. +Furthermore let uinc ∈ H1 +loc(Ω+) be such that additionally ∆uinc ∈ L2,loc(Ω+) holds. +If there exist numbers ̺ > 0 and LF ∈ (0, β(R, κ)) such that the following two conditions +κ2 [∥c(·, wc) − 1∥0,˜qc,Ω + ∥Lc(·, w, wc)∥0,qc,Ω(̺ + ∥wc∥V )] ̺ ++ +� +∥f(·, wf)∥0,˜qf,Ω + ∥Lf(·, w, wf)∥0,qf,Ω(̺ + ∥wf∥V ) +� +(37) ++ Ctr∥ˆx · ∇uinc − Tκuinc∥−1/2,2,SR ≤ ̺β(R, κ), +κ2 [∥Lc(·, w, v)∥0,qc,Ω̺ + ∥c(·, wc) − 1∥0,˜qc,Ω + ∥Lc(·, w, wc)∥0,qc,Ω(̺ + ∥wc∥V )] ++ ∥Lf(·, w, v)∥0,qf,Ω ≤ LF +(38) +are satisfied for all w, v ∈ Kcl +̺ := {v ∈ V : ∥v∥V ≤ ̺}, then the problem (35) has a unique +solution u ∈ Kcl +̺ . +Proof. First we mention that Kcl +̺ is a closed nonempty subset of V . +Next we show that A−1F(Kcl +̺ ) ⊂ Kcl +̺ . To this end we make use of the estimates given in +the proof of Remark 17 and obtain +∥F(w)∥V ∗ ≤ ∥ℓcontr(w)∥V ∗ + ∥ℓsrc(w)∥V ∗ + ∥ℓinc∥V ∗ +≤ κ2 [∥c(·, wc) − 1∥0,˜qc,Ω + ∥Lc(·, w, wc)∥0,qc,Ω(∥w∥V + ∥wc∥V )] ∥w∥V ++ +� +∥f(·, wf)∥0,˜qf,Ω + ∥Lf(·, w, wf)∥0,qf,Ω(∥w∥V + ∥wf∥V ) +� ++ ∥ℓinc∥V ∗ +≤ κ2 [∥c(·, wc) − 1∥0,˜qc,Ω + ∥Lc(·, w, wc)∥0,qc,Ω(̺ + ∥wc∥V )] ̺ ++ +� +∥f(·, wf)∥0,˜qf,Ω + ∥Lf(·, w, wf)∥0,qf,Ω(̺ + ∥wf∥V ) +� ++ Ctr∥ˆx · ∇uinc − Tκuinc∥−1/2,2,SR . +Hence the assumption (37) implies ∥A−1F(w)∥V ≤ ̺. +It remains to show that the mapping A−1F is a contraction. +We start with the consideration of the contrast term. From the elementary decomposition +(c(·, w) − 1)w − (c(·, v) − 1)v = (c(·, w) − c(·, v))w + (c(·, v) − 1)(w − v) +we see that +∥ℓcontr(w) − ℓcontr(v)∥V ∗ +≤ κ2∥Lc(·, w, v)∥0,qc,Ω∥w − v∥V ∥w∥V ++ κ2 [∥c(·, wc) − 1∥0,˜qc,Ω + ∥Lc(·, w, wc)∥0,qc,Ω∥w − wc∥V ] ∥w − v∥V +≤ κ2∥Lc(·, w, v)∥0,qc,Ω∥w − v∥V ̺ ++ κ2 [∥c(·, wc) − 1∥0,˜qc,Ω + ∥Lc(·, w, wc)∥0,qc,Ω(̺ + ∥wc∥V )] ∥w − v∥V +≤ κ2 [∥Lc(·, w, v)∥0,qc,Ω̺ + ∥c(·, wc) − 1∥0,˜qc,Ω + ∥Lc(·, w, wc)∥0,qc,Ω(̺ + ∥wc∥V )] ∥w − v∥V . +The estimate of the source term follows immediately from the properties of f: +∥ℓsrc(w) − ℓsrc(v)∥V ∗ ≤ ∥Lf(·, w, v)∥0,qf,Ω∥w − v∥V . +19 + +Resonant compactly supported nonlinearities +January 30, 2023 +From +∥F(w) − F(v)∥V ∗ ≤ ∥ℓcontr(w) − ℓcontr(v)∥V ∗ + ∥ℓsrc(w) − ℓsrc(v)∥V ∗ +and assumption (38) we thus obtain +∥F(w) − F(v)∥V ∗ ≤ LF∥w − v∥V . +In summary, Banach’s fixed point theorem can be applied (see e.g. [Eva15, Sect. 9.2.1]) and +we conclude that the problem (35) has a unique solution u ∈ Kcl +̺ . +If we introduce the function space +˜V := {v ∈ L2(BR) : v|Ω ∈ H1(Ω), v|BR\Ω ∈ H1(BR \ Ω)} +equipped with the norm +∥v∥ ˜V := +� +∥v∥2 +1,2,Ω + ∥v∥2 +1,2,BR\Ω +�1/2 +for all v ∈ ˜V , +the ball Kcl +̺ appearing in the above theorem can be interpreted as a ball in ˜V of radius ̺ +with center in +u0 := +� +0 +in Ω, +−uinc +in BR \ Ω. +Indeed, for u of the form (3), it holds that +∥u − u0∥2 +˜V = ∥utrans∥2 +1,2,Ω + ∥urad + uinc∥2 +1,2,BR\Ω = ∥u∥2 +V . +This means that the influence of the incident field uinc on the radius ̺ in Thm. 18 depends +only on the deviation of uinc from a radiating field measured by ∥ℓinc∥V ∗, but not directly on +the intensity of uinc. In other words, if the incident field uinc is radiating (i.e., it also satisfies +the Sommerfeld radiation condition (4) and thus ℓinc = 0), the radius ̺ does not depend +on uinc. In particular, uinc can be a strong field, which is important for the occurence of +generation efffects of higher harmonics [AY19]. +Example 19 (Example 15 continued). The identity +c(·, ξ) − c(·, η) = α (|ξ|2 − |η|2) = α (|ξ| + |η|)(|ξ| − |η|) +for all ξ, η ∈ C and the inequality ||ξ| − |η|| ≤ |ξ − η| show that +|c(·, ξ) − c(·, η)| ≤ |α|(|ξ| + |η|)|ξ − η| +holds, hence we can set Lc(·, ξ, η) := |α|(|ξ| + |η|). With pc = qc = 4, c generates a locally +Lipschitz continuous Nemycki operator in V . Furthermore we may choose wc = 0. Then: +∥c(·, wc) − 1∥0,˜qc,Ω = ∥ε(L) − 1∥0,2,Ω, +∥Lc(·, w, v)∥0,qc,Ω = ∥α(|w| + |v|)∥0,4,Ω ≤ ∥αw∥0,4,Ω + ∥αv∥0,4,Ω +≤ ∥α∥0,∞,Ω [∥w∥0,4,Ω + ∥v∥0,4,Ω] ≤ Cemb∥α∥0,∞,Ω [∥w∥V + ∥v∥V ] , +∥Lc(·, w, wc)∥0,qc,Ω = ∥αw∥0,4,Ω ≤ Cemb∥α∥0,∞,Ω∥w∥V . +20 + +Resonant compactly supported nonlinearities +January 30, 2023 +Hence the validity of the following conditions is sufficient for (37), (38): +κ2 � +∥ε(L) − 1∥0,2,Ω + Cemb∥α∥0,∞,Ω̺2� +̺ ++ Ctr∥ˆx · ∇uinc − Tκuinc∥−1/2,2,SR ≤ ̺β(R, κ), +κ2 � +∥ε(L) − 1∥0,2,Ω + 3Cemb∥α∥0,∞,Ω̺2� +≤ LF. +A consideration of these condition shows that there can be different scenarios for which +they can be fulfilled. In particular, one of the smallness requirements concerns the product +∥α∥0,∞,Ω̺3. +Example 20 (saturated Kerr nonlinearity). Another important example for the nonlineari- +ties is [Akh98] +c(x, ξ) := +� +1, +(x, ξ) ∈ Ω+ × C, +ε(L)(x) + α(x)|ξ|2/(1 + γ|ξ|2), +(x, ξ) ∈ Ω × C, +with given ε(L), α ∈ L∞(Ω), saturation parameter γ > 0, and f = 0. Based on the identity +|ξ|2 +1 + γ|ξ|2 − +|η|2 +1 + γ|η|2 = (1 + γ|η|2)|ξ|2 − (1 + γ|ξ|2)|η|2 +(1 + γ|ξ|2)(1 + γ|η|2) += +|ξ|2 − |η|2 +(1 + γ|ξ|2)(1 + γ|η|2) +for all ξ, η ∈ C we obtain +���� +|ξ|2 +1 + γ|ξ|2 − +|η|2 +1 + γ|η|2 +���� = (|ξ| + |η|) ||ξ| − |η|| +(1 + γ|ξ|2)(1 + γ|η|2) ≤ (|ξ| + |η|)|ξ − η|. +Hence on Ω we arrive at the same Lipschitz function as in the previous Example 19, that is +Lc(x, ξ, η) := +� +0, +(x, ξ, η) ∈ Ω+ × C × C, +|α|(|ξ| + |η|), +(x, ξ, η) ∈ Ω × C × C. +Moreover, since +c(x, wc) = c(x, 0) = +� +0, +(x, ξ) ∈ Ω+ × C, +ε(L), +(x, ξ) ∈ Ω × C. +we get the same sufficient conditions. +6 The modified boundary value problem +Since the exact DtN operator is represented as an infinite series (see (9), (12)), it is practically +necessary to truncate this nonlocal operator and consider only finite sums +Tκ,Nu(x) := 1 +R +� +|n|≤N +Zn(κR)un(R)Yn(ˆx), +x = Rˆx ∈ SR ⊂ R2, +(39) +Tκ,Nu(x) = 1 +R +N +� +n=0 +� +|m|≤n +zn(κR)um +n (R)Y m +n (ˆx), +x = Rˆx ∈ SR ⊂ R3 +(40) +21 + +Resonant compactly supported nonlinearities +January 30, 2023 +for some N ∈ N0. The map Tκ,N is called the truncated DtN operator, and N is the truncation +order of the DtN operator. +The replacement of the exact DtN operator Tκ in the problem (18) by the truncated DtN +operator Tκ,N introduces a perturbation, hence we have to answer the question of existence +and uniqueness of a solution to the following problem: +Find uN ∈ V such that +aN(uN, v) = nN(uN, v) +for all v ∈ V. +(41) +holds, where aN and nN are the forms defined by (23), (24) with Tκ replaced by Tκ,N. +The next result is the counterpart to Lemmata 8, 9. Here we formulate a different version +of G˚arding’s inequality compared to the case d = 2 considered in [HNPX11, Thm. 4.4]. +Lemma 21. The sesquilinear form aN +(i) is bounded, i.e. there exists a constant C > 0 independent of N such that +|aN(w, v)| ≤ C∥w∥V ∥v∥V +for all w, v ∈ V, +and +(ii) satisfies a G˚arding’s inequality in the form +Re aN(v, v) ≥ ∥v∥2 +V,κ − 2κ2∥v∥2 +0,2,BR +for all v ∈ V. +Proof. (i) If the proof of [MS10, eq. (3.4a)] is carried out with finitely many terms of the +expansion of Tκ only, the statement follows easily. Alternatively, Lemma 23 with s = 0 can +also be used. +(ii) As in the proof of Lemma 9, the definitions of aN and the wavenumber dependent norm +yield +Re aN(v, v) = ∥v∥2 +V,κ − 2κ2∥v∥2 +0,2,BR − Re (Tκ,Nv, v)SR. +Hence it remains to estimate the last term. In the case d = 2, we have (see (39)) +Tκ,Nv(x) := 1 +R +� +|n|≤N +Zn(κR)vn(R)Yn(ˆx), +x = Rˆx ∈ SR. +Then, using the L2(S1)-orthonormality of the circular harmonics [Zei95, Prop. 3.2.1], we get +−(Tκ,Nv, v)SR = − 1 +R +� +|n|≤N +Zn(κR)(vn(R)Yn, vn(R)Yn)SR += − 1 +R +� +|n|≤N +Zn(κR)|vn(R)|2(Yn, Yn)SR += − +� +|n|≤N +Zn(κR)|vn(R)|2(Yn, Yn)S1 += − +� +|n|≤N +Zn(κR)|vn(R)|2. +22 + +Resonant compactly supported nonlinearities +January 30, 2023 +Hence, by Lemma 4, +− Re (Tκ,Nv, v)SR = +� +|n|≤N +(− Re Zn(κR)) +� +�� +� +≥1/2 +|vn(R)|2 + (− Re Z0(κR)) +� +�� +� +>0 +|v0(R)|2 +≥ 1 +2 +� +|n|≤N +|vn(R)|2 ≥ 0. +The case d = 3 can be treated similarly. From +Tκ,Nv(x) = 1 +R +N +� +n=0 +� +|m|≤n +zn(κR)vm +n (R)Y m +n (ˆx) +(see (40)), we immediately obtain, using the L2(S1)-orthonormality of the spherical harmon- +ics [CK19, Thm. 2.8] that +−(Tκ,Nv, v)SR = − 1 +R +N +� +n=0 +� +|m|≤n +zn(κR)(vm +n (R)Y m +n , vm +n (R)Y m +n )SR += − 1 +R +N +� +n=0 +� +|m|≤n +zn(κR)|vm +n (R)|2(Y m +n , Y m +n )SR += −R +N +� +n=0 +� +|m|≤n +zn(κR)|vm +n (R)|2(Y m +n , Y m +n )S1 += −R +N +� +n=0 +� +|m|≤n +zn(κR)|vm +n (R)|2, +and Lemma 4 implies +− Re (Tκ,Nv, v)SR = R +N +� +n=0 +� +|m|≤n +(− Re zn(κR)) +� +�� +� +≥1 +|vm +n (R)|2 ≥ R +N +� +n=0 +� +|m|≤n +|vm +n (R)|2 ≥ 0. +In both cases we obtain the same G˚arding’s inequality as in the original (untruncated) +problem Lemma 9. +The next result is the variational version of the truncation error estimate. It closely follows +the lines of the proof of [HNPX11, Thm. 3.3], where an estimate of ∥(Tκ − Tκ,N)v∥s−1/2,2,SR, +s ∈ R, was proved in the case d = 2. +Lemma 22. For given w, v ∈ H1/2(SR) it holds that +���((Tκ − Tκ,N)w, v)SR +��� ≤ c(N, w, v)∥w∥1/2,2,SR∥v∥1/2,2,SR, +where c(N, w, v) ≥ 0 and limN→∞ c(N, w, v) = 0. +23 + +Resonant compactly supported nonlinearities +January 30, 2023 +Proof. We start with the two-dimensional situation. So let +w(x) = w(Rˆx) = +� +|n|∈N0 +wn(R)Yn(ˆx), +v(x) = v(Rˆx) = +� +|k|∈N0 +vk(R)Yk(ˆx), +x ∈ SR, +(42) +be series representations of w|SR, v|SR with the Fourier coefficients +wn(R) = (w(R·), Yn)S1 = +� +S1 +w(Rˆx)Yn(ˆx)ds(ˆx), +vk(R) = (v(R·), Yk)S1 = +� +S1 +v(Rˆx)Yk(ˆx)ds(ˆx). +The norm on the Sobolev space Hs(SR), s ≥ 0, can be defined as follows [LM72, Ch. 1, +Rem. 7.6]: +∥v∥2 +s,2,SR := R +� +n∈Z +(1 + n2)s|vn(R)|2. +(43) +Then, by (39), the orthonormality of the circular harmonics [Zei95, Prop. 3.2.1] and (43), +���((Tκ − Tκ,N)w, v)SR +��� = 1 +R +������ +� +|n|,|k|>N +� +Zn(κR)wn(R)Yn(R−1·), vk(R)Yk(R−1·) +� +SR +������ += +������ +� +|n|,|k|>N +Zn(κR) (wn(R)Yn, vk(R)Yk)S1 +������ += +������ +� +|n|>N +Zn(κR)wn(R)vn(R) +������ += +������ +� +|n|>N +Zn(κR) +(1 + n2)1/2(1 + n2)1/4wn(R)(1 + n2)1/4vn(R) +������ +≤ max +|n|>N +���� +Zn(κR) +(1 + n2)1/2 +���� +� +|n|>N +��(1 + n2)1/4wn(R)(1 + n2)1/4vn(R) +�� +≤ max +|n|>N +���� +Zn(κR) +(1 + n2)1/2 +���� + + � +|n|>N +(1 + n2)1/2 |wn(R)|2 + + +1/2 +× + + � +|n|>N +(1 + n2)1/2 |vn(R)|2 + + +1/2 +≤ 1 +R max +|n|>N +���� +Zn(κR) +(1 + n2)1/2 +���� ˜c(N, w, v)∥w∥1/2,2,SR∥v∥1/2,2,SR, +24 + +Resonant compactly supported nonlinearities +January 30, 2023 +where +˜c(N, w, v)2 := +� +|n|>N(1 + n2)1/2|wn(R)|2 +� +|n|∈N0(1 + n2)1/2|wn(R)|2 +� +|n|>N(1 + n2)1/2|vn(R)|2 +� +|n|∈N0(1 + n2)1/2|vn(R)|2. +The coefficient ˜c(N, w, v) tends to zero for N → ∞ thanks to (43), (45).. Corollary 5 implies +the estimate +1 +1 + n2|Zn(κR)|2 ≤ max{|Z0(κR)|2, 1 + |κR|2}, +|n| ∈ N0, +hence we can set +c(N, w, v) := ˜c(N, w, v) +R +max{|Z0(κR)|, (1 + |κR|2)1/2}. +The investigation of the case d = 3 runs similarly. So let +w(x) = w(Rˆx) = +� +n∈N0 +� +|m|≤n +wm +n (R)Y m +n (ˆx), +v(x) = v(Rˆx) = +� +k∈N0 +� +|l|≤k +vl +k(R)Y l +k(ˆx), +x ∈ SR, +(44) +be series representations of w|SR, v|SR with the Fourier coefficients +wm +n (R) = (w(R·), Y m +n )S1 = +� +S1 +w(Rˆx)Y m +n (ˆx)ds(ˆx), +vl +k(R) = (v(R·), Y l +k)S1 = +� +S1 +v(Rˆx)Y l +k(ˆx)ds(ˆx). +The norm on the Sobolev space Hs(SR), s ≥ 0, can be defined as follows [LM72, Ch. 1, +Rem. 7.6]: +∥v∥2 +s,2,SR := R2 � +n∈N0 +� +|m|≤n +(1 + n2)s|vm +n (R)|2. +(45) +Then, by (40), the orthonormality of the spherical harmonics [CK19, Thm. 2.8] and (45), +���((Tκ − Tκ,N)w, v)SR +��� = 1 +R +������ +� +n,k>N +� +|m|≤n,|l|≤k +� +zn(κR)wm +n (R)Y m +n (R−1·), vl +k(R)Y l +k(R−1·) +� +SR +������ += R +������ +� +n,k>N +� +|m|≤n,|l|≤k +zn(κR) +� +wm +n (R)Y m +n , vl +k(R)Y l +k) +� +S1 +������ += R +������ +� +n>N +� +|m|≤n +zn(κR)wm +n (R)vm +n (R) +������ += R +������ +� +n>N +� +|m|≤n +zn(κR) +(1 + n2)1/2(1 + n2)1/4wm +n (R)(1 + n2)1/4vm +n (R) +������ +25 + +Resonant compactly supported nonlinearities +January 30, 2023 +≤ R max +n>N +���� +zn(κR) +(1 + n2)1/2 +���� +� +n>N +� +|m|≤n +��(1 + n2)1/4wm +n (R)(1 + n2)1/4vm +n (R) +�� +≤ R max +n>N +���� +zn(κR) +(1 + n2)1/2 +���� + +� +n>N +� +|m|≤n +(1 + n2)1/2 |wm +n (R)|2 + + +1/2 +× + +� +n>N +� +|m|≤n +(1 + n2)1/2 |vm +n (R)|2 + + +1/2 +≤ 1 +R max +n>N +���� +zn(κR) +(1 + n2)1/2 +���� ˜c(N, w, v)∥w∥1/2,2,SR∥v∥1/2,2,SR, +where +˜c(N, w, v)2 := +� +n>N +� +|m|≤n(1 + n2)1/2 |wm +n (R)|2 +� +|n|∈N0 +� +|m|≤n(1 + n2)1/2 |wm +n (R)|2 +� +n>N +� +|m|≤n(1 + n2)1/2 |vm +n (R)|2 +� +|n|∈N0 +� +|m|≤n(1 + n2)1/2 |vm +n (R)|2. +Thanks to Corollary 5 we can define +c(N, w, v) := ˜c(N, w, v) +R +� +2 + |κR|2�1/2 . +Lemma 23. For s ∈ [0, 1/2) and w ∈ H1−s(BR \ Ω), v ∈ H1+s(BR \ Ω) it holds that +|(Tκ,Nw, v)SR| ≤ Cbl∥w∥1−s,2,BR\Ω∥v∥1+s,2,BR\Ω, +where the constant Cbl ≥ 0 does not depend on N. +Proof. We start with the two-dimensional situation as in the proof of Lemma 22. If w, v +have the representations (42), then, by (39), the orthonormality of the circular harmonics +[Zei95, Prop. 3.2.1] and (43), +|(Tκ,Nw, v)SR| = 1 +R +������ +� +|n|,|k|≤N +� +Zn(κR)wn(R)Yn(R−1·), vk(R)Yk(R−1·) +� +SR +������ += +������ +� +|n|,|k|≤N +Zn(κR) (wn(R)Yn, vk(R)Yk)S1 +������ += +������ +� +|n|≤N +Zn(κR)wn(R)vn(R) +������ += +������ +� +|n|≤N +Zn(κR) +(1 + n2)1/2(1 + n2)(1/2−s)/2wn(R)(1 + n2)(1/2+s)/2vn(R) +������ +≤ max +|n|≤N +���� +Zn(κR) +(1 + n2)1/2 +���� +� +|n|≤N +��(1 + n2)(1/2−s)/2wn(R)(1 + n2)(1/2+s)/2vn(R) +�� +26 + +Resonant compactly supported nonlinearities +January 30, 2023 +≤ max +|n|≤N +���� +Zn(κR) +(1 + n2)1/2 +���� + + � +|n|≤N +(1 + n2)1/2−s |wn(R)|2 + + +1/2 +× + + � +|n|≤N +(1 + n2)1/2+s |vn(R)|2 + + +1/2 +≤ 1 +R max +|n|≤N +���� +Zn(κR) +(1 + n2)1/2 +���� ∥w∥1/2−s,2,SR∥v∥1/2+s,2,SR. +Corollary 5 implies the estimate +1 +1 + n2|Zn(κR)|2 ≤ max{|Z0(κR)|2, 1 + |κR|2}, +|n| ∈ N0, +hence +|(Tκ,Nw, v)SR| ≤ 1 +R max{|Z0(κR)|, (1 + |κR|2)1/2}∥w∥1/2−s,2,SR∥v∥1/2+s,2,SR. +(46) +By the trace theorem [McL00, Thm. 3.38], we finally arrive at +|(Tκ,Nw, v)SR| ≤ C2 +tr +R max{|Z0(κR)|, (1 + |κR|2)1/2}∥w∥1−s,2,BR\Ω∥v∥1+s,2,BR\Ω. +The investigation of the case d = 3 runs similarly. So let w, v have the representations (44), +then, by (40), the orthonormality of the spherical harmonics [CK19, Thm. 2.8] and (45), +|(Tκ,Nw, v)SR| = 1 +R +������ +N +� +n,k=0 +� +|m|≤n,|l|≤k +� +zn(κR)wm +n (R)Y m +n (R−1·), vl +k(R)Y l +k(R−1·) +� +SR +������ += R +������ +N +� +n,k=0 +� +|m|≤n,|l|≤k +zn(κR) +� +wm +n (R)Y m +n , vl +k(R)Y l +k) +� +S1 +������ += R +������ +N +� +n=0 +� +|m|≤n +zn(κR)wm +n (R)vm +n (R) +������ += R +������ +N +� +n=0 +� +|m|≤n +zn(κR) +(1 + n2)1/2(1 + n2)(1/2−s)/2wm +n (R)(1 + n2)(1/2+s)/2vm +n (R) +������ +≤ R max +n∈N0 +���� +zn(κR) +(1 + n2)1/2 +���� +N +� +n=0 +� +|m|≤n +��(1 + n2)(1/2−s)/2wm +n (R)(1 + n2)(1/2+s)/2vm +n (R) +�� +≤ R max +n∈N0 +���� +zn(κR) +(1 + n2)1/2 +���� + + +N +� +n=0 +� +|m|≤n +(1 + n2)1/2−s |wm +n (R)|2 + + +1/2 +× + + +N +� +n=0 +� +|m|≤n +(1 + n2)1/2+s |vm +n (R)|2 + + +1/2 +27 + +Resonant compactly supported nonlinearities +January 30, 2023 +≤ 1 +R max +n∈N0 +���� +zn(κR) +(1 + n2)1/2 +���� ∥w∥1/2−s,2,SR∥v∥1/2+s,2,SR. +Corollary 5 yields +|(Tκ,Nw, v)SR| ≤ 1 +R +� +2 + |κR|2�1/2 ∥w∥1/2−s,2,SR∥v∥1/2+s,2,SR. +(47) +By the trace theorem [McL00, Thm. 3.38], we finally arrive at +|(Tκ,Nw, v)SR| ≤ C2 +tr +R +� +2 + |κR|2�1/2 ∥w∥1−s,2,BR\Ω∥v∥1+s,2,BR\Ω. +Theorem 24. Under the assumptions of Lemma 9, given an antilinear continuous functional +ℓ : V → C, there exists a constant N∗ > 0 such that for N ≥ N∗ the problem +Find uN ∈ V such that +aN(uN, v) = ℓ(v) +for all v ∈ V +(48) +is uniquely solvable. +Proof. First we show that the problem (48) has at most one solution. We start as in the +proof of [HNPX11, Thm. 4.5] and argue by contradiction, i.e. we suppose the following: +∀N∗ ∈ N +∃N = N(N∗) ≥ N∗ +and +uN = uN(N∗) ∈ V +such that +aN(uN, v) = 0 +for all v ∈ V +and ∥uN∥V = 1. +(49) +However, the subsequent discussion differs significantly from the proof of [HNPX11, Thm. 4.5]. +We apply an argument the idea of which goes back to Schatz [Sch74]. +First we assume there exists a solution uN ∈ V of (48) and derive an a priori estimate of +the error ∥u − uN∥V , where u ∈ V is the solution of (29), see Thm. 10. Since aN satisfies a +G˚arding’s inequality (Lemma 21(ii)), we have, making use of (28), +C2 +−∥u − uN∥2 +V − 2κ2∥u − uN∥2 +0,2,BR ≤ Re aN(u − uN, u − uN). +Since +aN(u − uN, v) = aN(u, v) − aN(uN, v) += a(u, v) +� �� � +=ℓ(v) ++aN(u, v) − a(u, v) − aN(uN, v) +� +�� +� +=ℓ(v) += ((Tκ − Tκ,N)u, v)SR , +we obtain +C2 +−∥u − uN∥2 +V − 2κ2∥u − uN∥2 +0,2,BR ≤ η1∥u − uN∥V +(50) +with +η1 := sup +v∈V +Re ((Tκ − Tκ,N)u, v)SR +∥v∥V +. +Now we consider the following auxiliary adjoint problem (cf. [McL00, p. 43]): +28 + +Resonant compactly supported nonlinearities +January 30, 2023 +Find wN ∈ V such that +a(v, wN) = (v, u − uN)BR +for all v ∈ V. +(51) +Since A is a Fredholm operator (see the proof of Thm. 10), the adjoint problem possesses a +unique solution wN ∈ V . Then +∥u − uN∥2 +0,2,SR = a(u − uN, wN) = a(u, wN) − a(uN, wN) += a(u, wN) − aN(uN, wN) +� +�� +� +=ℓ(wN)−ℓ(wN)=0 ++aN(uN, wN) − a(uN, wN) += ((Tκ − Tκ,N)uN, wN)SR. +In particular, this relation shows that ((Tκ − Tκ,N)uN, wN)SR is real. With +η2 := sup +v∈V +((Tκ − Tκ,N)uN, v)SR +∥v∥V +we obtain +∥u − uN∥2 +0,2,BR ≤ η2∥wN∥V ≤ η2C−1 +− C(R, κ)∥u − uN∥V ∗. +The continuous embedding V ⊂ V ∗ yields +∥u − uN∥2 +0,2,BR ≤ η2C−1 +− C(R, κ)Cemb∥u − uN∥V . +Applying this estimate in (50), we get +C2 +−∥u − uN∥2 +V − 2κ2η2C−1 +− C(R, κ)Cemb∥u − uN∥V ≤ η1∥u − uN∥V . +Now, if ∥u − uN∥V ̸= 0, we finally arrive at +C2 +−∥u − uN∥V ≤ η1 + 2κ2η2C−1 +− C(R, κ)Cemb. +(52) +Clearly this inequality is true also for ∥u − uN∥V = 0 so that we can remove this interim +assumption. +Thanks to Lemma 22 we have that +���((Tκ − Tκ,N)u, v)SR +��� ≤ c(N, u, v)∥u∥1/2,2,SR∥v∥1/2,2,SR ≤ c(N, u, c)C2 +tr∥u∥V ∥v∥V , +hence +η1 ≤ c+(N, u)C2 +tr∥u∥V +with +c+(N, u) := sup +v∈V +c(N, u, v), +(53) +where limN→∞ c+(N, u) = 0. Note that, as can be seen from the proof of Lemma 22, the +second fractional factor in the representation of ˜c(N, w, v) can be estimated from above by +one without losing the limit behaviour for N → ∞. Consequently, η1 can be made arbitrarily +small provided N is large enough. +In order to estimate η2 we cannot apply Lemma 22 directly since the second argument in +the factor c(N, uN, v) depends on N, too. Therefore we give a more direct estimate. +29 + +Resonant compactly supported nonlinearities +January 30, 2023 +Namely, let v ∈ V have the representation (42) or (44), respectively. Then we define +VN|SR := +� +span|n|≤N{Yn(R−1·)}, +d = 2, +spann=0...N,|m|≤n{Y m +n (R−1·)}, +d = 3, +and introduce an orthogonal projector +PN : V |SR → VN|SR : v �→ PNv := +�� +|n|≤N vn(R)Yn(R−1·), +d = 2, +�N +n=0 +� +|m|≤n vm +n (R)Y m +n (R−1·), +d = 3. +Then it holds that VN|SR ⊂ ker(TκPN − Tκ,N). +Indeed, if d = 2 and v ∈ VN|SR, then +PNv = v = � +|n|≤N vn(R)Yn(R−1·) and +TκPNv = Tκv = 1 +R +� +|n|≤N +Zn(κR)vn(R)Yn(R−1·) = Tκ,Nv. +An analogous argument applies in the case d = 3. +Now we return to the estimate of η2 and write, for uN ∈ V , +(Tκ − Tκ,N)uN = (Tκ − TκPN)uN + (TκPN − Tκ,N)uN = Tκ(id −PN)uN, +where we have used the above property. The advantage of this approach is that we can apply +a wellknown estimate of the projection error. The proof of this estimate runs similarly to +the proof of Lemma 22 but only without the coefficients Zn or zn, respectively: +��((id −PN)w, v)SR +�� = +������ +� +|n|,|k|>N +� +wn(R)Yn(R−1·), vk(R)Yk(R−1·) +� +SR +������ += R +������ +� +|n|,|k|>N +(wn(R)Yn, vk(R)Yk)S1 +������ += R +������ +� +|n|>N +wn(R)vn(R) +������ += R +������ +� +|n|>N +1 +(1 + n2)1/2(1 + n2)1/4wn(R)(1 + n2)1/4vn(R) +������ +≤ max +|n|>N +R +(1 + n2)1/2 +� +|n|>N +��(1 + n2)1/4wn(R)(1 + n2)1/4vn(R) +�� +≤ +R +(1 + N2)1/2 + + � +|n|>N +(1 + n2)1/2 |wn(R)|2 + + +1/2 +× + + � +|n|>N +(1 + n2)1/2 |vn(R)|2 + + +1/2 +≤ +1 +(1 + N2)1/2∥w∥1/2,2,SR∥v∥1/2,2,SR. +30 + +Resonant compactly supported nonlinearities +January 30, 2023 +The same estimate holds true for d = 3. Then we get, by Remark 3 (or Lemma 23), +���((Tκ − Tκ,N)uN, v)SR +��� = +��(Tκ(id −PN)uN, v)SR +�� +≤ +Cκ +(1 + N2)1/2∥uN∥1/2,2,SR∥v∥1/2,2,SR +≤ +CC2 +trκ +(1 + N2)1/2∥uN∥V ∥v∥V , +thus +η2 ≤ +CC2 +trκ +(1 + N2)1/2∥uN∥V . +Using this estimate and (53) in (52), we obtain +C2 +−∥u − uN∥V ≤ c+(N, u)C2 +tr∥u∥V + 2κ2C−1 +− C(R, κ)Cemb +CC2 +trκ +(1 + N2)1/2∥uN∥V . +(54) +Now we appply this estimate to the solutions uN of the homogeneous truncated problems in +(49). By Thm. 10, the homogeneous linear interior problem (29) (i.e. ℓ = 0) has the solution +u = 0, and the above estimate implies +C2 +−∥uN∥V ≤ 2κ2C−1 +− C(R, κ)Cemb +CC2 +trκ +(1 + N2)1/2∥uN∥V , +which is a contradiction to ∥uN∥V = 1 for all N. +Although the proof of Thm. 24 allows an analogous conclusion as in Lemma 11 that the +truncated bilinear form aN satisfies an inf-sup condition, such a conclusion is not fully +satisfactory since the question remains whether and how the inf-sup constant depends on N +or not. However, at least for sufficiently large N, a positive answer can given. +Lemma 25. Under the assumptions of Lemma 9, there exists a number N∗ ∈ N such that +βN∗(R, κ) := +inf +w∈V \{0} +sup +v∈V \{0} +|aN(w, v)| +∥w∥V,κ∥v∥V,κ +> 0 +is independent of N ≥ N∗. +In the proof a formula is given that expresses βN∗(R, κ) in terms of β(R, κ). +Proof. We return to the proof of Thm. 24 and mention that the estimate (54) is valid for +solutions u, uN of the general linear problems (29) (or, equally, (31)) and (48), respectively. +By the triangle inequality, +∥uN∥V ≤ ∥u∥V + ∥u − uN∥V +≤ ∥u∥V + c+(N, u)C−2 +− C2 +tr∥u∥V + 2κ2C−3 +− C(R, κ)Cemb +CC2 +trκ +(1 + N2)1/2∥uN∥V . +31 + +Resonant compactly supported nonlinearities +January 30, 2023 +If N∗ is sufficiently large such that +κ2C−3 +− C(R, κ)Cemb +CC2 +trκ +(1 + N2)1/2 ≤ 1 +4 +and +c+(N, u)C−2 +− C2 +tr ≤ 1 +for all N ≥ N∗, +then, by Lemma 11, +∥uN∥V ≤ 4∥u∥V ≤ 4 +C− +∥u∥V,κ ≤ ∥ℓ∥V ∗. +That is, the sesquilinear form aN satisfies an inf-sup condition +βN∗(R, κ) := +inf +w∈V \{0} +sup +v∈V \{0} +|aN(w, v)| +∥w∥V,κ∥v∥V,κ +> 0 +with βN∗(R, κ) := C−β(R, κ) +4C+ +independent of N ≥ N∗. +Analogously to (30) we introduce the truncated linear operator AN : V → V ∗ by +ANw(v) := aN(w, v) +for all w, v ∈ V. +By Lemma 21, AN is a bounded operator, and Lemma 25 implies that AN has a bounded +inverse: +∥w∥V,κ ≤ βN∗(R, κ)−1∥ANw∥∗ +for all w ∈ V. +Furthermore, we define a nonlinear operator FN : V → V ∗ by +FN(w)(v) := ℓcontr(w) + ℓsrc(w) + ℓinc +N +for all w ∈ V, +where +⟨ℓinc +N , v⟩ := (ˆx · ∇uinc − Tκ,Nuinc, v)SR. +The problem (41) is then equivalent to the operator equation +ANu = FN(u) +in V ∗, +and further to the fixed-point problem +u = A−1 +N FN(u) +in V. +(55) +Theorem 26. Under the assumptions of Lemma 9, let the functions c and f generate locally +Lipschitz continuous Nemycki operators in V and assume that there exist functions wf, wc ∈ +V such that f(·, wf) ∈ Lpf/(pf −1)(Ω) and c(·, wf) ∈ Lpc/(pc−2)(Ω), respectively. +Furthermore let uinc ∈ H1 +loc(Ω+) be such that additionally ∆uinc ∈ L2,loc(Ω+) holds. +If there exist numbers ̺ > 0 and LF ∈ (0, βN∗(R, κ)) (where N∗ and βN∗(R, κ) are from +Lemma 25) such that the following two conditions +κ2 [∥c(·, wc) − 1∥0,˜qc,Ω + ∥Lc(·, w, wc)∥0,qc,Ω(̺ + ∥wc∥V )] ̺ ++ +� +∥f(·, wf)∥0,˜qf,Ω + ∥Lf(·, w, wf)∥0,qf,Ω(̺ + ∥wf∥V ) +� ++ Ctr∥ˆx · ∇uinc − Tκ,Nuinc∥−1/2,2,SR ≤ ̺βN∗(R, κ), +κ2 [∥Lc(·, w, v)∥0,qc,Ω̺ + ∥c(·, wc) − 1∥0,˜qc,Ω + ∥Lc(·, w, wc)∥0,qc,Ω(̺ + ∥wc∥V )] ++ ∥Lf(·, w, v)∥0,qf,Ω ≤ LF +are satisfied for all w, v ∈ Kcl +̺ , then the problem (35) has a unique solution uN ∈ Kcl +̺ for all +N ≥ N∗. +Proof. 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Cam- +bridge University Press, Cambridge, 2000. +[MS10] +J.M. Melenk and M. Sauter. Convergence analysis for finite element discretiza- +tions of the Helmholtz equation with Dirichlet-to-Neumann boundary condi- +tions. Math. Comp., 79(272):1871–1914, 2010. +[MV22] +R. Maier and B. Verf¨urth. Multiscale scattering in nonlinear Kerr-type media. +Math. Comp., 91(336):1655–1685, 2022. +[N´ed01] +J.C. N´ed´elec. +Acoustic and Electromagnetic Equations. Integral Representa- +tions for Harmonic Problems, volume 144 of Applied Mathematical Sciences. +Springer-Verlag, New York, 2001. +[Rel43] +F. Rellich. ¨Uber das asymptotische Verhalten der L¨osungen von ∆u+λu = 0 in +unendlichen Gebieten. Jahresbericht der Deutschen Mathematiker-Vereinigung, +53:57–65, 1943. +[Sch74] +A.H. Schatz. An observation concerning Ritz-Galerkin methods with indefinite +bilinear forms. Math. Comp., 28(128):959–962, 1974. +[SW07] +J. Shen and L.-L. Wang. Analysis of a spectral-Galerkin approximation to the +Helmholtz equation in exterior domains. SIAM J. Numer. Anal., 45(5):1954– +1978, 2007. +[TW93] +R.H. Torres and G.V. Welland. +The Helmholtz equation and transmission +problems with Lipschitz interfaces. Indiana Univ. Math. J., 42(4):1457–1485, +1993. +[Wlo87] +J. Wloka. Partial Differential Equations. Cambridge University Press, Cam- +bridge, 1987. +[Zei95] +E. Zeidler. Applied functional analysis. Applications to mathematical physics. +Springer-Verlag, New York-Berlin-Heidelberg, 1995. Applied Mathematical Sci- +ences, vol. 108. +34 + diff --git a/AdFKT4oBgHgl3EQfVy5o/content/tmp_files/load_file.txt b/AdFKT4oBgHgl3EQfVy5o/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e6c3865523402e87abf1b69b46b4ec183c758429 --- /dev/null +++ b/AdFKT4oBgHgl3EQfVy5o/content/tmp_files/load_file.txt @@ -0,0 +1,1179 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf,len=1178 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='11789v1 [math-ph] 27 Jan 2023 A radiation and propagation problem for a Helmholtz equation with a compactly supported nonlinearity Lutz Angermann∗ January 30, 2023 The present work describes some extensions of an approach, originally devel- oped by V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Yatsyk and the author, for the theoretical and numerical analysis of scattering and radiation effects on infinite plates with cubically polarized lay- ers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' The new aspects lie on the transition to more generally shaped, two- or three-dimensional objects, which no longer necessarily have to be represented in terms a Cartesian product of real intervals, to more general nonlinearities (in- cluding saturation) and the possibility of an efficient numerical approximation of the electromagnetic fields and derived quantities (such as energy, transmission coefficient, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' The paper advocates an approach that consists in transform- ing the original full-space problem for a nonlinear Helmholtz equation (as the simplest model) into an equivalent boundary-value problem on a bounded do- main by means of a nonlocal Dirichlet-to-Neumann (DtN) operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' It is shown that the transformed problem is equivalent to the original one and can be solved uniquely under suitable conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Morever, the impact of the truncation of the DtN operator on the resulting solution is investigated, so that the way to the numerical solution by appropriate finite element methods is available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Keywords: Scattering, radiation, nonlinear Helmholtz equation, nonlinearly polarizable medium, DtN operator, truncation AMS Subject Classification (2022): 35 J 05 35 Q 60 78 A 45 1 Introduction The present work deals with the mathematical modeling of the response of a penetrable two- or three-dimensional object (obstacle), represented by a bounded domain, to the excitation ∗Dept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' of Mathematics, Clausthal University of Technology, Erzstr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 1, D-38678 Clausthal-Zellerfeld, Ger- many, lutz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='angermann@tu-clausthal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='de 1 Resonant compactly supported nonlinearities January 30, 2023 by an external electromagnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' A special aspect of the paper is that, in contrast to many other, thematically comparable works, nonlinear constitutive laws of this object are in the foreground.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' A standard example are the so-called Kerr nonlinearities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' It is physically known, but also only little investigated mathematically that sufficiently strong incident fields, under certain conditions, cause effects such as frequency multiplication, which cannot occur in the linear models frequently considered in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' On the other hand, such effects are interest- ing in applications, which is why a targeted exploitation, for example from a numerical or optimization point of view, first requires thorough theoretical investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' A relatively simple mathematical model for this is a nonlinear Helmholtz equation, which results from the transition from the time-space formulation of Maxwell’s equations to the frequency-space formulation together with further simplifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Although some interesting nonlinear effects cannot be modeled by means of a single scalar equation alone, its inves- tigation is of own importance, for example from the aspect of variable coefficients, and on the other hand its understanding is also the basis for further development, for example for systems of nonlinear Helmholtz equations, see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=', [AY19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' The latter is also the reason why we consider a splitted nonlinearity and not concentrate the nonlinearity in one term as is obvious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' The Helmholtz equation with nonlinearities has only recently become the focus of mathe- matical investigations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' However, problems are mainly dealt with in which the nonlinearities are globally smooth, while here a formulation as a transmission problem is used that allows less smooth transitions at the object boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' In addition, we allow more general nonlin- earities than the Kerr nonlinearities mentioned, in particular saturation effects can be taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Starting from a physically oriented problem description as a full-space problem, we derive a weak formulation on a bounded domains using the well-known technique of DtN operators, and show its equivalence to the weakly formulated original problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Since the influence of the external field only occurs indirectly in the weak formulation, we also give a second variant of the weak formulation that better clarifies this influence and which we call the input-output formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Since the DtN operators are non-local, their practical application (numerics) causes prob- lems, which is why a well-known truncation technique is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' This raises the problem of proving the well-posedness of the reduced problem and establishing a connection (error estimate) of the solution of the reduced problem to the original problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Although these questions in the linear case have been discussed in the literature for a relatively long time, they even for the linear case seemed to have been treated only selectively and sometimes only very vaguely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' The latter concerns in particular the question of the independence of the stability constant from the truncation parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' In this work, both stability and er- ror estimates are given for the two- and three-dimensional case, whereby a formula-based relationship between the discrete and the continuous stability constant is established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Another difference to many existing, especially older works is that the present paper works with variational (weak) formulations but not with integral equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Unfortunately, the complete tracking of the dependence of the occurring parameters on the wave number (so- called wavenumber-independent bounds) has not yet been included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' It has already been mentioned that, for the linear situation, in connection with scattering problems or with problems that are formulated from the very beginning in bounded domains 2 Resonant compactly supported nonlinearities January 30, 2023 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=', with impedance boundary conditions), there is an extensive and multi-threaded body of literature that is beyond the scope of this article to list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Transmission problems of the type considered here are rarely found in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Nevertheless, without claiming completeness, a few works should be mentioned here that had an influence on the present results and whose bibliographies may be of help.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' A frequently cited work that deals with linear scattering problems in two dimensions and also served as the motivation for the present work is [HNPX11], which, however, does not discuss the dependence of the stability constant on the truncation parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' A number of later works by other authors quote this work, but sometimes assume results that cannot be found in the original.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' The work that comes closest to our intentions is [Koy07], where the exterior Dirichlet boundary-value problem for the linear Helmholtz equation is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' In this paper, no separate, parameter-uniform stability estimate of the truncated problem is given, but the truncation error is included in the error estimate of a finite element approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' A similar work is [Koy09], but in which another boundary condition at the boundary of the auxiliary domain is considered, the so-called modified DtN condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Among the more recent papers, works by Mandel [Man19], Chen, Ev´equoz & Weth [CEW21], and Maier & Verf¨urth [MV22] should be mentioned, especially because of the cited sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' In his cumulative habilitation thesis, which contains further references, Mandel examines ex- istence and uniqueness questions for solutions of systems of nonlinear Helmholtz equations in the full-space case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Scattering or transmission problems are not considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' consider the scattering problem with quite high regularity assumptions to the superlinear nonlinearities, but without truncation approaches and not in the context of variational solu- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Maier & Verf¨urth, who focus mainly on multiscale aspects for a nonlinear Helmholtz equation over a bounded domain with impedance boundary conditions, give an instructive review of the literature on nonlinear Helmholtz equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' The structure of the present work is based on the program outlined above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' After the problem formulation in Section 2, the exterior auxiliary problem required for truncation is discussed, after which the weak formulation and equivalence statement follow in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Section 5 is dedicated to the existence and uniqueness of the weak solution, where in particular the assumptions on the nonlinear terms are discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' The final section then deals with the properties of the truncated problem – uniform (with respect to the truncation parameter) well-posedness and estimate of the truncation error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 2 Problem formulation Let Ω ⊂ Rd be a bounded domain with a Lipschitz boundary ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' It represents a medium with a nonlinear behaviour with respect to electromagnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Since Ω is bounded, we can choose an open Euclidean d-ball BR ⊂ Rd of radius R > supx∈Ω |x| with center in the origin such that Ω ⊂ BR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' The complements of Ω and BR are denoted by Ωc := Rd \\ Ω Bc R := Rd \\ BR, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=', the open complement of BR is denoted by B+ R := Rd \\ BR (the overbar over sets denotes their closure in Rd), and the boundary of BR, the sphere, by SR := ∂BR (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' The open complement of Ω is denoted by Ω+ := Rd \\ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' By ν we denote the outward-pointing (w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' either Ω or BR) unit normal vector on ∂Ω or SR, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Trace operators will be denoted by one and the same symbol γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' the concrete meaning (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=', 3 Resonant compactly supported nonlinearities January 30, 2023 Ω SR uinc Figure 1: The nonlinear medium Ω is excited by an incident field uinc (d = 2) traces on the common interface of an interior and exterior domain) will be clear from the context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' The classical direct problem of radiation and propagation of an electromagnetic field – ac- tually just one component of it – by/in the penetrable obstacle Ω is governed by a nonlinear Helmholtz equation with a variable complex-valued wave coefficient: − ∆u(x) − κ2c(x, u) u = f(x, u) for (almost) all x ∈ Rd, (1) where the wavenumber κ > 0 is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' The physical properties of the obstacle Ω are described by the coefficient c : Rd × C → C (physically the square of the refractive index) and the right-hand side f : Rd × C → C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' In general, both functions are nonlinear and have the following properties: supp(1 − c(·, w)) = Ω and supp f(·, w) ⊂ Ω for all w ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' (2) The function 1 − c is often called the contrast function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Basically we assume that c and f are Carath´eodory functions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' the mapping x �→ c(x, v) is (Lebesgue-)measurable for all v ∈ C, and the mapping v �→ c(x, v) is continuous for almost all x ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' These two conditions imply that x �→ c(x, v(x)) is measurable for any measurable v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' The same applies to f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' The unknown total field u : Rd → C should have the following structure: u = � urad + uinc in Ωc, utrans in Ω, (3) where urad : Ωc → C is the unknown radiated/scattered field, utrans : Ω → C denotes the unknown transmitted field, and the incident field uinc ∈ H1 loc(Ω+) is given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' The incident field is usually a (weak) solution of either the homogeneous or inhomogeneous Helmholtz equation (even in the whole space).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Typically it is generated either by concentrated sources located in a bounded region of Ω+ or by sources at infinity, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' travalling waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Example 1 (d = 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' The incident plane wave, whose transmission and scattering is inves- tigated, is given by uinc(x) := αinc exp(i(Φx1 − Γx2)), x = (x1, x2)⊤ ∈ B+ R 4 Resonant compactly supported nonlinearities January 30, 2023 with amplitude αinc and angle of incidence ϕinc, |ϕinc| < π, where Φ := κ sin ϕinc is the longitudinal wave number and Γ := √ κ2 − Φ2 = κ cos ϕinc the transverse wave number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' In polar coordinates is then uinc(r, ϕ) = αinc exp(i(Φr cos ϕ − Γr sin ϕ)) = αinc exp(iκr(sin ϕinc cos ϕ − cos ϕinc sin ϕ)) = αinc exp(iκr sin(ϕinc − ϕ)), (r, ϕ) ∈ B+ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' The radiated/scattered field urad should satisfy an additional condition, the so-called Som- merfeld radiation condition: lim |x|→∞ |x|(d−1)/2 � ˆx · ∇urad − iκurad� = 0 (4) uniformly for all directions ˆx := x/|x|, where ˆx·∇urad denotes the derivative of urad in radial direction ˆx, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' [CK13, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='7) for d = 3, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='96) for d = 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Physically, the condition (4) allows only outgoing waves at infinity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' mathematically it guaranties the uniqueness of the solution uscat : B+ R → C of the following exterior Dirichlet problem −∆uscat − κ2uscat = 0 in B+ R, uscat = fSR on SR, lim |x|→∞ |x|(d−1)/2 � ˆx · ∇uscat − iκuscat� = 0, (5) where fSR : SR → C is given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' We mention that, in the context of classical solutions (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' uscat ∈ C2(B+ R)) to problem (5), Rellich [Rel43] has shown that the condition (4) can be weakened to the following integral version: lim |x|→∞ � SR ��ˆx · ∇uscat − iκuscat��2 ds(x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' In the context of weak solutions (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' uscat ∈ H1 loc(B+ R)), an analogous equivalence statement can be found in [McL00, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 3 The exterior problem in Bc R For a given fSR ∈ C(SR) and d = 3, the unique solvability of problem (5) in C2(B+ R)∩C(Bc R) is proved, for example, in [CK13, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' In addition, if fSR is smoother, say fSR ∈ C∞(SR), then the normal derivative of uscat on the boundary SR is a well-defined continuous function [CK13, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' These assertions remain valid in the case d = 2, see [CK13, Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Therefore, by solving (5) for given fSR ∈ C∞(SR), a mapping can be introduced that takes the Dirichlet data on SR to the corresponding Neumann data on SR, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' fSR �→ TκfSR := ˆx · ∇uscat�� SR , (6) see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=', [CK19, Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 5 Resonant compactly supported nonlinearities January 30, 2023 Furthermore, it is well-known that the mapping Tκ can be extended to a bounded linear operator Tκ : Hs+1/2(SR) → Hs−1/2(SR) for any |s| ≤ 1/2 [CWGLS12, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='31] (we keep the notation already introduced for this continued operator).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' This operator is called the Dirichlet-to-Neumann operator, in short DtN operator, or capacity operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Since the problem (5) is considered in a spherical exterior domain, an explicit series represen- tation of the solution is available using standard separation techniques in polar or spherical coordinates, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' The term-by-term differentiation of this series thus also provides a series representation of the image of Tκ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' The solution of the problem (5) in the two-dimensionsional case (here with uscat replaced by u) is given by [Mas87, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='1], [KG89, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' (30)]: u(x) = u(rˆx) = u(r, ϕ) = � n∈Z H(1) n (κr) H(1) n (κR) fn(R)Yn(ˆx) = � n∈Z H(1) n (κr) H(1) n (κR) fn(R)Yn(ϕ), x = rˆx ∈ Sr, r > R, ϕ ∈ [0, 2π] (7) (identifying u(x) with u(r, ϕ) and Yn(ˆx) with Yn(ϕ) for x = rˆx = r(cos ϕ, sin ϕ)⊤), where (r, ϕ) are the polar coordinates, H(1) n are the cylindrical Hankel functions of the first kind of order n [DLMF22, Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='2]1, Yn are the circular harmonics defined by Yn(ϕ) = einϕ √ 2π , n ∈ Z, fn(R) are the Fourier coefficients of fSR defined by fn(R) := (fSR(R·), Yn)S1 = � S1 fSR(Rˆx)Yn(ˆx)ds(ˆx) = � 2π 0 fSR(R, ϕ)Yn(ϕ)dϕ, (8) and ds(ˆx) is the Lebesgue arc length element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Now we formally differentiate the representation (7) with respect to r to obtain the outward normal derivative of u: ˆx · ∇u(x) = ∂u ∂r (rˆx) = κ � n∈Z H(1)′ n (κr) H(1) n (κR) fn(R)Yn(ˆx), x = rˆx ∈ Sr, r > R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Setting fR := u|SR and letting x in this representation approach the boundary SR, we can formally define the (extended) DtN operator by Tκu(x) := 1 R � n∈Z Zn(κR)un(R)Yn(ˆx), x = Rˆx ∈ SR, (9) where Zn(ξ) := ξ H(1)′ n (ξ) H(1) n (ξ) , 1Instead of (4) [Mas87] considered the ingoing Sommerfeld condition and thus obtained a representation in terms of the cylindrical Hankel functions of the second kind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Note that H(2) n (−ξ) = −(−1)nH(1) n (ξ) [DLMF22, (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='5)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 6 Resonant compactly supported nonlinearities January 30, 2023 and un(R) are the Fourier coefficients of u|SR analogously to (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' The admissibility of this procedure has been proven in many sources in the classical context, for example [CK19, Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' For the present case, in the paper [Ern96, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 1] it was shown that the operator Tκ : Hs+1/2(SR) → Hs−1/2(SR) is bounded for any s ∈ N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Ernst’s result was extended to all s ≥ 0 in [HNPX11, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' In the case d = 3, the solution of the problem (5) is given by [KG89, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' (33)]: u(x) = u(rˆx) = u(r, ϕ, θ) = � n∈N0 � |m|≤n h(1) n (κr) h(1) n (κR) f m n (R)Y m n (ˆx) = � n∈N0 � |m|≤n h(1) n (κr) h(1) n (κR) f m n (R)Y m n (ϕ, θ), x ∈ Sr, r > R, (ϕ, θ) ∈ [0, 2π] × [0, π] (10) (identifying u(x) with u(r, ϕ, θ) and Y m n (ˆx) with Y m n (ϕ, θ) for x = rˆx = r(cos ϕ sin θ, sin ϕ sin θ, cos θ)⊤), where (r, ϕ, θ) are the spherical coordinates, h(1) n are the spherical Hankel functions of the first kind of order n [DLMF22, Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='47], Y m n are the spherical harmonics defined by Y m n (ϕ, θ) = � 2n + 1 4π (n − |m|)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' (n + |m|)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' P |m| n (cos θ)eimϕ, n ∈ N0, |m| ≤ n, (identifying Y m n (ˆx) with Y m n (ϕ, θ) for ˆx = (cos ϕ sin θ, sin ϕ sin θ, cos θ)⊤), where P m n are the associated Legendre functions of the first kind [DLMF22, Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='21], f m n (R) are the Fourier coefficients defined by f m n (R) = (fSR(R·), Y m n )S1 = � S1 fSR(Rˆx)Y m n (ˆx)ds(ˆx) = � 2π 0 � π 0 fSR(R, ϕ, θ)Y m n (ϕ, θ) sin θdθdϕ, (11) and ds(ˆx) is the Lebesgue surface area element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Proceeding as in the two-dimensional case, we get ˆx · ∇u(x) = ∂u ∂r (rˆx) = κ � n∈N0 � |m|≤n h(1) n (κr) h(1) n (κR) f m n (R)Y m n (ˆx), x = rˆx ∈ Sr, r > R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Setting fR := u|SR and letting r → R, we can define the (extended) DtN operator by Tκu(x) = 1 R � n∈N0 � |m|≤n zn(κR)um n (R)Y m n (ˆx), x = Rˆx ∈ SR, (12) where zn(ξ) := ξ h(1)′ n (ξ) h(1) n (ξ) , 7 Resonant compactly supported nonlinearities January 30, 2023 and um n (R) are the Fourier coefficients of u|SR analogously to (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' The admissibility of this procedure is proved in [CK19, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='15] or [N´ed01, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='2], for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' For the present situation there is a boundedness result for d = 3 analogous to [HNPX11, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='1] in [N´ed01, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' In summary, the following statement applies to both dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' The DtN operator Tκ : Hs+1/2(SR) → Hs−1/2(SR) is bounded for any s ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' A more refined analysis of the DtN operator in the case s = 0 results in a sharp estimate of the its norm w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' the wavenumber [BSW16, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='4]: Given κ0 > 0, there exists a constant C > 0 independent of κ such that ∥Tκv∥−1/2,2,SR ≤ Cκ∥v∥1/2,2,SR for all v ∈ H1 loc(B+ R) and κ ≥ κ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' The result from [BSW16, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='4] applies to more general domains, for the present situation it already follows from the proof of Lemma 23 (see the estimates (46), (47) for s = 0, where the bounds do not depend on N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' At the end of this section we give a collection of some properties of the coefficient functions in the representations (9), (12) which will be used in some of the subsequent proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' For all ξ > 0, the following holds: −n ≤ Re Zn(ξ) ≤ −1 2, 0 < Im Zn(ξ) < ξ for all |n| ∈ N, −1 2 ≤ Re Z0(ξ) < 0, ξ < Im Z0(ξ), −(n + 1) ≤ Re zn(ξ) ≤ −1, 0 < Im zn(ξ) ≤ ξ for all n ∈ N, Re z0(ξ) = −1, Im z0(ξ) = ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' For the case d = 2, the estimates can be found in [SW07, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='34)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' The other estimates can be found in [N´ed01, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='1], see also [SW07, eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='22), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='23)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Although only 0 ≤ Im zn(ξ) is specified in the formulation of the cited theorem, the strict positivity follows from the positivity of the function qℓ in [N´ed01, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='34)], as has been mentioned in [MS10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' For all ξ > 0, the following holds: |Zn(ξ)|2 ≤ (1 + n2)(1 + |ξ|2) for all |n| ∈ N, |zn(ξ)|2 ≤ (1 + n2)(2 + |ξ|2) for all n ∈ N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' The estimates of the real and imaginary parts of Zn from Lemma 4 immediately imlpy that 1 1 + n2|Zn(ξ)|2 = 1 1 + n2 � | Re Zn(ξ)|2 + | Im Zn(ξ)|2� ≤ 1 1 + n2 � n2 + |ξ|2� ≤ 1 + |ξ|2 1 + n2 ≤ 1 + |ξ|2, n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Since H(1) −n(ξ) = (−1)nH(1) n (ξ), n ∈ N [DLMF22, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='2)], the estimate is also valid for n such that −n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 8 Resonant compactly supported nonlinearities January 30, 2023 Analogously we obtain from Lemma 4 that 1 1 + n2|zn(ξ)|2 = 1 1 + n2 � | Re zn(ξ)|2 + | Im zn(ξ)|2� ≤ 1 1 + n2 � (1 + n)2 + |ξ|2� ≤ 2 + |ξ|2 1 + n2 ≤ 2 + |ξ|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 4 Weak formulations of the interior problem Now we turn to the consideration of the problem (1)–(4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' In the classical setting it can be formulated as follows: Given uinc ∈ H1 loc(Ω+), determine the transmitted field utrans : Ω → C and the radiated/scattered field urad : Ωc → C satisfying −∆utrans − κ2c(·, utrans) utrans = f(·, utrans) in Ω, −∆urad − κ2urad = 0 in Ω+, utrans = urad + uinc on ∂Ω, ν · ∇utrans = ν · ∇urad + ν · ∇uinc on ∂Ω (13) and the radiation condition (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Note that the incident field is usually a (weak) solution of either the homogeneous or inhomogeneous Helmholtz equation in Ω+, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' the second equation in (13) can be replaced by − ∆u − κ2u = f inc in Ω+, (14) where f inc : Ω+ → C is an eventual source density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' For simplicity we do not include the case of a nontrivial source density in our investigation, but the subsequent theory can be easily extended by adding an appropriate linear functional, say ℓsrc, on the right-hand side of the obtained weak formulations (see (15) or (19) later).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' In order to give a weak formulation of (13) with the modification (14) in the case f inc = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' we introduce the (complex) linear function spaces H1 comp(Ω+) := � v ∈ H1(Ω+) : supp v is compact � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' VRd := {v ∈ L2(Rd) : v|Ω ∈ H1(Ω),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' v|Ω+ ∈ H1 loc(Ω+) : γv|Ω = γv|Ω+ on ∂Ω},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' WRd := {v ∈ L2(Rd) : v|Ω ∈ H1(Ω),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' v|Ω+ ∈ H1 comp(Ω+) : γv|Ω = γv|Ω+ on ∂Ω} (note the comment at the beginning of Section 2 on the notation for trace operators) and multiply the first equation of (13) by the restriction v|Ω of an arbitrary element v ∈ VRd and (14) by the restriction v|Ω+ of v ∈ VRd,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' respectively,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' and integrate py parts: (∇utrans,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' ∇v)Ω − (ν · ∇utrans,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' ∇v)∂Ω − κ2(c(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' utrans)utrans,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' v)Ω = (f(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' utrans),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' v)Ω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' (∇u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' ∇v)Ω − (ν · ∇u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' ∇v)∂Ω+ − κ2(u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' v)Ω+ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 9 Resonant compactly supported nonlinearities January 30, 2023 Here we use the notation, for any domain M ⊂ Rd with boundary ∂M and appropriately defined functions on M or ∂M, (∇w, ∇v)M := � M ∇w · ∇vdx, (w, v)M := � M wvdx, (w, v)∂M := � ∂M wvds(x) (the overbar over functions denotes complex conjugation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Taking into consideration the last transmission condition in (13), the relationsship ν|Ω = −ν|Ω+, and the fact that the last but one transmission condition in (13) is included in the definition of the space VRd, we define a bivariate nonlinear form on VRd × WRd by aRd(w, v) := (∇w, ∇v)Ω + (∇w, ∇v)Ω+ − κ2(c(·, w)w, v)Rd, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=', e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=', [Wlo87, Example 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Given uinc ∈ H1 loc(Ω+), a weak solution to the problem (1)–(4) is defined as an element u ∈ VRd that has the structure (3), satisfies the variational equation aRd(u, v) = (f(·, u), v)Rd for all v ∈ WRd (15) and the Sommerfeld radiation condition (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' A second weak formulation can be obtained if we do not replace the second Helmholtz equation in (13) by (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Then the first step in the derivation of the weak formulation reads as (∇utrans, ∇v)Ω − (ν · ∇utrans, ∇v)∂Ω − κ2(c(·, utrans)utrans, v)Ω = (f(·, utrans), v)Ω, (∇urad, ∇v)Ω − (ν · ∇urad, ∇v)∂Ω+ − κ2(urad, v)Ω+ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' The last transmission condition in (13) allows to rewrite the first equation as (∇utrans, ∇v)Ω − (ν · ∇urad, ∇v)∂Ω − κ2(c(·, utrans)utrans, v)Ω = (f(·, utrans), v)Ω + (ν · ∇uinc, ∇v)∂Ω, leading to the weak formulation (∇u0, ∇v)Ω+(∇u0, ∇v)Ω+−κ2(c(·, u0)u0, v)Rd = (f(·, u0), v)Rd+(ν·∇uinc, v)∂Ω for all v ∈ WRd (16) with respect to the structure u0 := � urad in Ωc, utrans in Ω, (17) where urad ∈ H1 loc(Ω+), utrans ∈ H1(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' The advantage of this formulation is that it clearly separates the unknown and the known parts of the fields, so we call this formulation the input-output formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' The disadvantage 10 Resonant compactly supported nonlinearities January 30, 2023 is that the natural function space of the solution u0 is not a linear space due to the last but one transmission condition in (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Instead of the problem (1)–(4) we want to solve an equivalent problem in the bounded domain BR, that is, we define V := {v ∈ L2(BR) : v|Ω ∈ H1(Ω), v|BR\\Ω ∈ H1(BR \\ Ω) : γv|Ω = γv|BR\\Ω on ∂Ω} and look for an element u ∈ V such that −∆utrans − κ2c(·, utrans) u = f(·, utrans) in Ω, −∆u − κ2u = 0 in BR \\ Ω, utrans = urad + uinc on ∂Ω, ν · ∇utrans = ν · ∇urad + ν · ∇uinc on ∂Ω, ˆx · ∇urad = Tκurad on SR (18) formally holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Now the weak formulation of problem (18) reads as follows: Find u ∈ V such that (∇u, ∇v)Ω + (∇u, ∇v)BR\\Ω − κ2(c(·, u)u, v)BR − (Tκu, v)SR = (f(·, u), v)BR − (Tκuinc, v)SR + (ˆx · ∇uinc, v)SR (19) for all v ∈ V holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' The weak formulations (15) and (19) of the problems (1)–(4) and (18), resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=', are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' First let u ∈ V (Rd) be a weak solution to (1)–(4), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' it satisfies (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Then its restriction to BR belongs to V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' To demonstrate that this restriction satisfies the weak formulation (19), we construct the radiating solution uBc R′ of the homogeneous Helmholtz equation outside of a smaller ball BR′ such that Ω ⊂ BR′ ⊂ BR and uBc R′ ��� SR′ = (u − uinc)|SR′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' This solution can be constructed in the form of a series expansion in terms of Hankel functions as explained in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' By elliptic regularity (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=', [McL00, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='16], [Eva15, Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='1]), the solution of this problem satisfies the Helmholtz equation in Bc R′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Moreover, by uniqueness [N´ed01, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='5], it coincides with u − uinc = urad in Bc R′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Now we choose a finite partition of unity covering BR, denoted by {ϕj}J [Wlo87, Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='2], such that its index set J can be decomposed into two disjoint subsets J1, J2 as follows: BR′ ⊂ int � � j∈J1 supp ϕj � , � j∈J1 supp ϕj ⊂ BR, � j∈J2 supp ϕj ⊂ Bc R′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' For example, we can choose {ϕj}J1 to consist of one element, say ϕ1, namely the usual mollifier function with support B′, where the open ball B′ (centered at the origin) lies between BR′ and BR, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' BR′ ⊂ B′ = int (supp ϕ1), supp ϕ1 ⊂ BR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Then the second part consists of a finite open covering of the spherical shell BR \\ B′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 11 Resonant compactly supported nonlinearities January 30, 2023 Then we take, for any v ∈ V , the product v1 := v � j∈J1 ϕj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' This is an element of V , too, with support in BR, and it can be continued by zero to an element of W(Rd) (keeping the notation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Hence we can take it as a test function in the weak formulation (15) and obtain aRd(u, v1) = (f(·, u), v1)Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' This is equal to (∇u, ∇v1)Ω + (∇u, ∇v1)BR\\Ω − κ2(c(·, u)u, v1)BR − (Tκu, v1)SR = (f(·, u), v1)BR − (Tκuinc, v1)SR + (ˆx · ∇uinc, v1)SR due to the properties of the support of v1 (in particular, all terms “living” on SR are equal to zero).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Since the homogeneous Helmholtz equation is satisfied in � j∈J2 supp ϕj ⊂ Bc R′, we can proceed as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' We continue the test function v2 := v � j∈J2 ϕj by zero into the complete ball BR and have (f(·, u), v2)BR\\Ω = 0 = (−∆u − κ2u, v2)BR\\BR′ = (∇u, ∇v2)BR\\BR′ − κ2(u, v2)BR\\BR′ − (ν · ∇u, v2)∂(BR\\BR′) = (∇u, ∇v2)BR\\BR′ − κ2(u, v2)BR\\BR′ − (ˆx · ∇u, v2)SR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Now, taking into consideration the properties of the support of v2, we easily obtain the following relations: (∇u, ∇v2)BR\\BR′ = (∇u, ∇v2)Ω + (∇u, ∇v2)BR\\Ω, (u, v2)BR\\BR′ = (c(·, u)u, v2)BR, (ˆx · ∇u, v2)SR = (ˆx · ∇urad, v2)SR + (ˆx · ∇uinc, v2)SR = (Tκurad, v2)SR + (ˆx · ∇uinc, v2)SR = (Tκu, v2)SR − (Tκuinc, v2)SR + (ˆx · ∇uinc, v2)SR, where the treatment of the last term makes use of the construction of the Dirichlet-to- Neumann map Tκ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Adding both relations and observing that v = v1+v2, we arrive at the variational formulation (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Conversely, let u ∈ V be a solution to (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' To continue it into Bc R, similar to the first part of the proof we construct the radiating solution uBc R of the Helmholtz equation outside BR such that uBc R �� SR = (u − uinc)|SR and set u := uBc R + uinc in B+ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Hence we have that Tκu = ∂uBc R ∂ˆx + Tκuinc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Now we take an element v ∈ W(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Its restriction to BR is an element of V and thus can be taken as a test function in (19): (∇u, ∇v)Ω + (∇u, ∇v)BR\\Ω − κ2(c(·, u)u, v)BR − (Tκu, v)SR = (f(·, u), v)BR − (Tκuinc, v)SR + (ˆx · ∇uinc, v)SR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' (20) 12 Resonant compactly supported nonlinearities January 30, 2023 Since v has a compact support, we can choose a ball B ⊂ Rd centered at the origin such that BR ∪ supp v ⊂ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' The homogeneous Helmholtz equation is obviously satisfied in the spherical shell B \\ BR: −∆uBc R − κ2uBc R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' We multiply this equation by the complex conjugate of the test function v ∈ V , then integrate over the shell, and apply the first Green’s formula: (∇uBc R, ∇v)B\\BR − κ2(uBc R, v)B\\BR − (ν · ∇uBc R, v)∂(B\\BR) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Now we observe that (∇uBc R, ∇v)B\\BR = (∇uBc R, ∇v)B+ R, (uBc R, v)B\\BR = (uBc R, v)B+ R, (ν · ∇uBc R, v)∂(B\\BR) = −(ˆx · ∇uBc R, v)SR = −(Tκu − Tκuinc, v)SR where the minus sign in the last line results from the change in the orientation of the outer normal (once w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' the shell, once w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' BR) and the construction of uBc R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' So we arrive at (∇uBc R, ∇v)B+ R − κ2(uBc R, v)B+ R + (Tκu, v)SR = (Tκuinc, v)SR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' (21) Finally, since the incident field satisfies the homogeneous Helmholtz equation in the spherical shell, too, we see by an analogous argument that the variational equation (∇uinc, ∇v)B+ R − κ2(uinc, v)B+ R = −(ˆx · ∇uinc, v)SR (22) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Adding the variational equations (20) – (22), we arrive at the variational formulation (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 5 Existence and uniqueness of a weak solution In this section we investigate the existence and uniqueness of the weak solution of the interior problem (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' We define the sesquilinear form a(w, v) := (∇w, ∇v)Ω + (∇w, ∇v)BR\\Ω − κ2(w, v)BR − (Tκw, v)SR for all w, v ∈ V, (23) the nonlinear form n(w, v) := κ2(c(·, w) − 1)w, v)BR + (f(·, w), v)BR − (Tκuinc, v)SR + (ˆx · ∇uinc, v)SR (24) and reformulate (19) as follows: Find u ∈ V such that a(u, v) = n(u, v) for all v ∈ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' (25) On the space V , we use the standard seminorm and norm: |v|V := � ∥∇v∥2 0,2,Ω + ∥∇v∥2 0,2,BR\\Ω �1/2 , ∥v∥V := � |v|2 V + ∥v∥2 0,2,BR �1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' (26) 13 Resonant compactly supported nonlinearities January 30, 2023 For κ > 0, the following so-called wavenumber dependent norm on V is also common: ∥v∥V,κ := � |v|2 V + κ2∥v∥2 0,2,BR �1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' (27) It is not difficult to verify that the standard norm and the wavenumber dependent norm are equivalent on V , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' it holds C−∥v∥V ≤ ∥v∥V,κ ≤ C+∥v∥V for all v ∈ V, (28) where the equivalence constants depend on κ in the following way: C− := min{1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' κ} and C+ := max{1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' κ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' We now proceed to examine the linear aspects of the problem (25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' The sesquilinear form a is bounded on V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Applying to each addend in the definition of a the appropriate Cauchy-Bunyakovsky- Schwarz inequality, we obtain |a(w, v)| ≤ |w|V |v|V + κ2∥w∥0,2,BR∥v∥0,2,BR + ∥Tκw∥−1/2,2,SR∥v∥1/2,2,SR for all w, v ∈ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' According to Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 2 the DtN operator Tκ is bounded, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' there exists a constant CTκ > 0 such that ∥Tκw∥−1/2,2,SR ≤ CTκ∥w∥1/2,2,SR for all w ∈ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' It remains to apply a trace theorem [McL00, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='37]: |a(w, v)| ≤ |w|V |v|V + κ2∥w∥0,2,BR∥v∥0,2,BR + CTκC2 tr∥w∥1,2,BR\\Ω∥v∥1,2,BR\\Ω ≤ |w|V |v|V + κ2∥w∥0,2,BR∥v∥0,2,BR + CTκC2 tr∥w∥V ∥v∥V ≤ min{(max{1, κ2} + CTκC2 tr)∥w∥V ∥v∥V , (1 + CTκC2 tr)∥w∥V,κ∥v∥V,κ} for all w, v ∈ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Given κ0 > 0 and R0 > 0, assume that κ ≥ κ0 (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Rem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 3) and R ≥ R0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' In addition, κ0 ≥ 1 is required for d = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Then the sesquilinear form a satisfies a G˚arding’s inequality of the form Re a(v, v) ≥ ∥v∥2 V,κ − 2κ2∥v∥2 0,2,BR for all v ∈ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' From the definitions of a and the wavenumber dependent norm it follows immediately that Re a(v, v) = ∥v∥2 V,κ − 2κ2∥v∥2 0,2,BR − Re (Tκv, v)SR ≥ ∥v∥2 V,κ − 2κ2∥v∥2 0,2,BR + CR−1∥v∥2 0,2,SR ≥ ∥v∥2 V,κ − 2κ2∥v∥2 0,2,BR, where the first estimate follows from [MS10, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='3] with a constant C > 0 depending soleley on κ0 > 0 and R0 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 14 Resonant compactly supported nonlinearities January 30, 2023 Next we discuss the solvability and stability of the problem (25) for the case that the right- hand side is just an antilinear continuous functional ℓ : V → C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' The linear problem of finding u ∈ V such that a(u, v) = ℓ(v) for all v ∈ V (29) holds can be formulated equivalently as an operator equation in the dual space V ∗ of V consisting of all continuous antilinear functionals from V to C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Namely, if we define the linear operator A : V → V ∗ by Aw(v) := a(w, v) for all w, v ∈ V, (30) problem (29) is equivalent to solving the operator equation Au = ℓ (31) for u ∈ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Note that A is a bounded operator by Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Under the assumptions of Lemma 9, the problem (31) is uniquely solvable for any ℓ ∈ V ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' The basic ideas of the proof are taken from the proof of [MS10, Thm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Since the embedding of V into L2(BR) is compact by the compactness theorem of Rellich–Kondrachov [McL00, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='27] together with Tikhonov’s product theorem [KN63, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='1], the com- pact perturbation theorem [McL00, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='34] together with Lemma 9 imply that the Fredholm alternative [McL00, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='27] holds for the equation (31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Hence it is sufficient to demonstrate that the homogeneous adjoint problem (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' [McL00, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 43]) of finding u ∈ V such that a(v, u) = 0 holds for all v ∈ V only allows for the trivial solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' So suppose u ∈ V is a solution of the homogeneous adjoint problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' We take v := u and consider the imaginary part of the resulting equation: 0 = Im a(u, u) = − Im (Tκu, u)SR = Im (Tκu, u)SR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Then [MS10, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='3] implies u = 0 on SR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Then u satisfies the variational equation (∇u, ∇v)Ω + (∇u, ∇v)BR\\Ω − κ2(u, v)BR = 0 for all v ∈ V, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' it is a weak solution of the homogeneous interior transmission Neumann problem for the wave equation on BR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' On the other hand, u can be extended to the whole space Rd by zero to an element ˜u ∈ V (Rd), and this element can be interpreted as a weak solution of a homogeneous full-space transmission problem, for instance in the sense of [TW93, Problem (P)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Then it follows from [TW93, Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='1] that ˜u = 0 und thus u = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Since a Fredholm operator has a closed image [McL00, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 33], it follows from the Open Mapping Theorem and Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 10 (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' [McL00, Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='2]) that the inverse operator A−1 is bounded, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' there exists a constant C(R, κ) > 0 such that ∥u∥V,κ = ∥A−1ℓ∥V,κ ≤ C(R, κ)∥ℓ∥V ∗ for all ℓ ∈ V ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 15 Resonant compactly supported nonlinearities January 30, 2023 Then it holds 1 C(R, κ) ≤ ∥ℓ∥V ∗ ∥u∥V,κ = sup v∈V \\{0} |ℓ(v)| ∥u∥V,κ∥v∥V,κ = sup v∈V \\{0} |a(u, v)| ∥u∥V,κ∥v∥V,κ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' This estimate proves the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Lemma 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Under the assumptions of Lemma 9, the sesquilinear form a satisfies an inf-sup condition: β(R, κ) := inf w∈V \\{0} sup v∈V \\{0} |a(w, v)| ∥w∥V,κ∥v∥V,κ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Now we turn to the nonlinear situation and concretize the assumptions regarding the Cara- th´eodory functions c and f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Lemma 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Let pf ∈ � [2, ∞), d = 2, [2, 6], d = 3, and assume there exist nonnegative functions mf, gf ∈ L∞(Ω) such that |f(x, ξ)| ≤ mf(x)|ξ|pf−1 + gf(x) for all (x, ξ) ∈ Ω × C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Then vf(·, w) ∈ L1(Ω) for all w, v ∈ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Since f is a Carath´eodory function, the composition f(·, w) is measurable and it sufficies to estimate the integral of |vf(·, w)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Moreover, it suffices to consider the term mfv|w|pf−1 in more detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' By H¨older’s inequality for three functions, it holds that ∥vf(·, w)∥0,1,Ω ≤ ∥mf∥0,∞,Ω∥v∥0,pf,Ω∥wpf−1∥0,q,Ω with 1 pf + 1 q = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' The Lpf-norm of v is bounded thanks to the embedding V |Ω ⊂ Lpf(Ω) for the allowed values of pf [AF03, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Since |wpf−1|q = |w|p f, the Lq-norm of wpf−1 is bounded by the same reasoning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Lemma 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Let pc ∈ � [2, ∞), d = 2, [2, 6], d = 3, and assume there exist nonnegative functions mc, gc ∈ L∞(Ω) such that |c(x, ξ) − 1| ≤ mc(x)|ξ|pc−2 + gc(x) for all (x, ξ) ∈ Ω × C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Then zv(c(·, w) − 1) ∈ L1(Ω) for all z, w, v ∈ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Similar to the proof of Lemma 12 it is sufficient to consider the term mczv|w|pc−2 in more detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' By H¨older’s inequality for four functions, it holds that ∥zv(c(·, w) − 1)∥0,1,Ω ≤ ∥mc∥0,∞,Ω∥z∥0,pc,Ω∥v∥0,pc,Ω∥wpc−2∥0,q,Ω with 2 pc + 1 q = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' The Lpc-norms of z, v are bounded thanks to the embedding theorem [AF03, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Since |wpc−2|q = |w|p c, the Lq-norm of wpc−2 is bounded by the same reasoning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 16 Resonant compactly supported nonlinearities January 30, 2023 Corollary 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Under the assumptions of Lemma 12 and Lemma 13, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=', the following estimates hold for all z, w, v ∈ V : |(f(·, w), v)Ω| ≤ C pf emb∥mf∥0,∞,Ω∥w∥ pf−1 1,2,Ω∥v∥1,2,Ω + � |Ω|d ∥gf∥0,∞,Ω∥v∥0,2,Ω, (32) |((c(·, w) − 1)z, v)Ω| ≤ Cpc emb∥mc∥0,∞,Ω∥w∥pc−2 1,2,Ω∥z∥1,2,Ω∥v∥1,2,Ω + ∥gc∥0,∞,Ω∥z∥0,2,Ω∥v∥0,2,Ω, (33) where |Ω|d is the d-volume of Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Replace v by v in Lemmata 12, 13 to get the first addend of the bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' The estimate of the second addend is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Example 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' An important example for the nonlinearities is c(x, ξ) := � 1, (x, ξ) ∈ Ω+ × C, ε(L)(x) + α(x)|ξ|2, (x, ξ) ∈ Ω × C, with given ε(L), α ∈ L∞(Ω), and f = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Here pc = 4, which is within the range of validity of Lemma 13, and mc = |α|, gc = |ε(L) − 1|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' The estimates from Corollary 14 show that the first two terms on the right-hand side of the variational equation (25) can be considered as values of nonlinear mappings from V to V ∗, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' we can define ℓcontr : V → V ∗ by ⟨ℓcontr(w), v⟩ := κ2(c(·, w) − 1)w, v)Ω, ℓsrc : V → V ∗ by ⟨ℓsrc(w), v⟩ := (f(·, w), v)Ω for all w, v ∈ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Furthermore, if uinc ∈ H1 loc(Ω+) is such that additionally ∆uinc belongs to L2,loc(Ω+) (where ∆uinc is understood in the distributional sense), the last two terms on the right-hand side of (24) form an antilinear continuous functional on ℓinc ∈ V ∗: ⟨ℓinc, v⟩ := (ˆx · ∇uinc − Tκuinc, v)SR for all v ∈ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' This is a consequence of Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 2 and the estimates before the trace theorem [KA21, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Hence ∥ℓinc∥V ∗ ≤ ˜Ctr[∥∆uinc∥0,2,BR\\Ω + ∥uinc∥0,2,BR\\Ω] + CTκC2 tr∥uinc∥1,2,BR\\Ω, where ˜Ctr is the norm of the trace operator defined in [KA21, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='39)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' However, it is more intuitive to utilize the estimate ∥ℓinc∥V ∗ ≤ Ctr∥ˆx · ∇uinc − Tκuinc∥−1/2,2,SR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' (34) The reason for this is that the bound can be interpreted as a measure of the deviation of the function uinc from a radiating solution of the corresponding Helmholtz equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' In other words: If the function uinc satisfies the boundary value problem (5) with fSR := uinc|SR, then the functional ℓinc is not present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 17 Resonant compactly supported nonlinearities January 30, 2023 Consequently, setting F(w) := ℓcontr(w) + ℓsrc(w) + ℓinc for all w ∈ V, we obtain a nonlinear operator F : V → V ∗, and the problem (25) is then equivalent to the operator equation Au = F(u) in V ∗, and further, by Lemma 11, equivalent to the fixed-point problem u = A−1F(u) in V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' (35) In order to prove the subsequent existence and uniqueness theorem, we specify some addi- tional properties of the nonlinearities c and f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Definition 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' The functions c and f are said to generate locally Lipschitz continuous Ne- mycki operators in V if the following holds: For some parameters pc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' pf ∈ � [2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' ∞),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' d = 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' [2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 6],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' d = 3,,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' there exist Carath´eodory functions Lc : Ω×C×C → (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' ∞) and Lf : Ω×C×C → (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' ∞) such that the composition operators Ω × V × V → Lqc(Ω) : (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' v) �→ Lc(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' v),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Ω × V × V → Lqf(Ω) : (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' v) �→ Lf(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' v) are bounded for qc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' qf > 0 with 3 pc + 1 qc = 2 pf + 1 qf = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' and |c(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' ξ) − c(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' η)| ≤ Lc(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' η)|ξ − η|,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' |f(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' ξ) − f(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' η)| ≤ Lf(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' η)|ξ − η| (36) for all (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' η) ∈ Ω × C × C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Remark 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' If the nonlinearities c and f generate locally Lipschitz continuous Nemycki operators in the sense of the above Definition 16, the assumptions of Lemmata 12, 13 can be replaced by the requirement that there exist functions wf, wc ∈ V such that f(·, wf) ∈ Lpf /(pf −1)(Ω) and c(·, wf) ∈ Lpc/(pc−2)(Ω), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Indeed,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' similar to the proofs of the two lemmata mentioned,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' we have that ∥vf(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' w)∥0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='Ω ≤ ∥vf(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' wf)∥0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='Ω + ∥v(f(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' w) − f(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' wf))∥0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='Ω ≤ ∥vf(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' wf)∥0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='Ω + ∥vLf(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' wf)|w − wf|∥0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='Ω ≤ ∥v∥0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='pf,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='Ω∥f(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' wf)∥0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='˜qf,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='Ω + ∥v∥0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='pf,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='Ω∥Lf(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' wf)∥0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='qf,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='Ω∥w − wf∥0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='pf,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='Ω ≤ � ∥f(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' wf)∥0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='˜qf,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='Ω + ∥Lf(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' wf)∥0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='qf,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='Ω(∥w∥V + ∥wf∥V ) � ∥v∥V ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' ∥zvc(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' w)∥0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='Ω ≤ ∥zvc(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' wc)∥0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='Ω + ∥zv(c(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' w) − c(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' wc))∥0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='Ω ≤ ∥zvc(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' wc)∥0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='Ω + ∥zvLc(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' wc)|w − wc|∥0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='Ω ≤ ∥z∥0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='pc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='Ω∥v∥0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='pc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='Ω∥c(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' wc)∥0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='˜qc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='Ω + ∥z∥0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='pc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='Ω∥v∥0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='pc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='Ω∥Lc(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' wc)∥0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='qc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='Ω∥w − wc∥0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='pc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='Ω ≤ [∥c(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' wc)∥0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='˜qc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='Ω + ∥Lc(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' wc)∥0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='qc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='Ω(∥w∥V + ∥wc∥V )] ∥z∥V ∥v∥V with 1 pf + 1 ˜qf = 1 and 2 pc + 1 ˜qc = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 18 Resonant compactly supported nonlinearities January 30, 2023 Theorem 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Under the assumptions of Lemma 9, let the functions c and f generate locally Lipschitz continuous Nemycki operators in V and assume that there exist functions wf, wc ∈ V such that f(·, wf) ∈ Lpf/(pf −1)(Ω) and c(·, wf) ∈ Lpc/(pc−2)(Ω), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Furthermore let uinc ∈ H1 loc(Ω+) be such that additionally ∆uinc ∈ L2,loc(Ω+) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' If there exist numbers ̺ > 0 and LF ∈ (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' β(R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' κ)) such that the following two conditions κ2 [∥c(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' wc) − 1∥0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='˜qc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='Ω + ∥Lc(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' wc)∥0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='qc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='Ω(̺ + ∥wc∥V )] ̺ + � ∥f(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' wf)∥0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='˜qf,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='Ω + ∥Lf(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' wf)∥0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='qf,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='Ω(̺ + ∥wf∥V ) � (37) + Ctr∥ˆx · ∇uinc − Tκuinc∥−1/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='SR ≤ ̺β(R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' κ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' κ2 [∥Lc(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' v)∥0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='qc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='Ω̺ + ∥c(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' wc) − 1∥0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='˜qc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='Ω + ∥Lc(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' wc)∥0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='qc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='Ω(̺ + ∥wc∥V )] + ∥Lf(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' v)∥0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='qf,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='Ω ≤ LF (38) are satisfied for all w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' v ∈ Kcl ̺ := {v ∈ V : ∥v∥V ≤ ̺},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' then the problem (35) has a unique solution u ∈ Kcl ̺ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' First we mention that Kcl ̺ is a closed nonempty subset of V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Next we show that A−1F(Kcl ̺ ) ⊂ Kcl ̺ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' To this end we make use of the estimates given in the proof of Remark 17 and obtain ∥F(w)∥V ∗ ≤ ∥ℓcontr(w)∥V ∗ + ∥ℓsrc(w)∥V ∗ + ∥ℓinc∥V ∗ ≤ κ2 [∥c(·, wc) − 1∥0,˜qc,Ω + ∥Lc(·, w, wc)∥0,qc,Ω(∥w∥V + ∥wc∥V )] ∥w∥V + � ∥f(·, wf)∥0,˜qf,Ω + ∥Lf(·, w, wf)∥0,qf,Ω(∥w∥V + ∥wf∥V ) � + ∥ℓinc∥V ∗ ≤ κ2 [∥c(·, wc) − 1∥0,˜qc,Ω + ∥Lc(·, w, wc)∥0,qc,Ω(̺ + ∥wc∥V )] ̺ + � ∥f(·, wf)∥0,˜qf,Ω + ∥Lf(·, w, wf)∥0,qf,Ω(̺ + ∥wf∥V ) � + Ctr∥ˆx · ∇uinc − Tκuinc∥−1/2,2,SR .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Hence the assumption (37) implies ∥A−1F(w)∥V ≤ ̺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' It remains to show that the mapping A−1F is a contraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' We start with the consideration of the contrast term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' From the elementary decomposition (c(·, w) − 1)w − (c(·, v) − 1)v = (c(·, w) − c(·, v))w + (c(·, v) − 1)(w − v) we see that ∥ℓcontr(w) − ℓcontr(v)∥V ∗ ≤ κ2∥Lc(·, w, v)∥0,qc,Ω∥w − v∥V ∥w∥V + κ2 [∥c(·, wc) − 1∥0,˜qc,Ω + ∥Lc(·, w, wc)∥0,qc,Ω∥w − wc∥V ] ∥w − v∥V ≤ κ2∥Lc(·, w, v)∥0,qc,Ω∥w − v∥V ̺ + κ2 [∥c(·, wc) − 1∥0,˜qc,Ω + ∥Lc(·, w, wc)∥0,qc,Ω(̺ + ∥wc∥V )] ∥w − v∥V ≤ κ2 [∥Lc(·, w, v)∥0,qc,Ω̺ + ∥c(·, wc) − 1∥0,˜qc,Ω + ∥Lc(·, w, wc)∥0,qc,Ω(̺ + ∥wc∥V )] ∥w − v∥V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' The estimate of the source term follows immediately from the properties of f: ∥ℓsrc(w) − ℓsrc(v)∥V ∗ ≤ ∥Lf(·, w, v)∥0,qf,Ω∥w − v∥V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 19 Resonant compactly supported nonlinearities January 30, 2023 From ∥F(w) − F(v)∥V ∗ ≤ ∥ℓcontr(w) − ℓcontr(v)∥V ∗ + ∥ℓsrc(w) − ℓsrc(v)∥V ∗ and assumption (38) we thus obtain ∥F(w) − F(v)∥V ∗ ≤ LF∥w − v∥V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' In summary, Banach’s fixed point theorem can be applied (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' [Eva15, Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='1]) and we conclude that the problem (35) has a unique solution u ∈ Kcl ̺ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' If we introduce the function space ˜V := {v ∈ L2(BR) : v|Ω ∈ H1(Ω), v|BR\\Ω ∈ H1(BR \\ Ω)} equipped with the norm ∥v∥ ˜V := � ∥v∥2 1,2,Ω + ∥v∥2 1,2,BR\\Ω �1/2 for all v ∈ ˜V , the ball Kcl ̺ appearing in the above theorem can be interpreted as a ball in ˜V of radius ̺ with center in u0 := � 0 in Ω, −uinc in BR \\ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Indeed, for u of the form (3), it holds that ∥u − u0∥2 ˜V = ∥utrans∥2 1,2,Ω + ∥urad + uinc∥2 1,2,BR\\Ω = ∥u∥2 V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' This means that the influence of the incident field uinc on the radius ̺ in Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 18 depends only on the deviation of uinc from a radiating field measured by ∥ℓinc∥V ∗, but not directly on the intensity of uinc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' In other words, if the incident field uinc is radiating (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=', it also satisfies the Sommerfeld radiation condition (4) and thus ℓinc = 0), the radius ̺ does not depend on uinc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' In particular, uinc can be a strong field, which is important for the occurence of generation efffects of higher harmonics [AY19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Example 19 (Example 15 continued).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' The identity c(·, ξ) − c(·, η) = α (|ξ|2 − |η|2) = α (|ξ| + |η|)(|ξ| − |η|) for all ξ, η ∈ C and the inequality ||ξ| − |η|| ≤ |ξ − η| show that |c(·, ξ) − c(·, η)| ≤ |α|(|ξ| + |η|)|ξ − η| holds, hence we can set Lc(·, ξ, η) := |α|(|ξ| + |η|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' With pc = qc = 4, c generates a locally Lipschitz continuous Nemycki operator in V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Furthermore we may choose wc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Then: ∥c(·, wc) − 1∥0,˜qc,Ω = ∥ε(L) − 1∥0,2,Ω, ∥Lc(·, w, v)∥0,qc,Ω = ∥α(|w| + |v|)∥0,4,Ω ≤ ∥αw∥0,4,Ω + ∥αv∥0,4,Ω ≤ ∥α∥0,∞,Ω [∥w∥0,4,Ω + ∥v∥0,4,Ω] ≤ Cemb∥α∥0,∞,Ω [∥w∥V + ∥v∥V ] , ∥Lc(·, w, wc)∥0,qc,Ω = ∥αw∥0,4,Ω ≤ Cemb∥α∥0,∞,Ω∥w∥V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 20 Resonant compactly supported nonlinearities January 30, 2023 Hence the validity of the following conditions is sufficient for (37), (38): κ2 � ∥ε(L) − 1∥0,2,Ω + Cemb∥α∥0,∞,Ω̺2� ̺ + Ctr∥ˆx · ∇uinc − Tκuinc∥−1/2,2,SR ≤ ̺β(R, κ), κ2 � ∥ε(L) − 1∥0,2,Ω + 3Cemb∥α∥0,∞,Ω̺2� ≤ LF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' A consideration of these condition shows that there can be different scenarios for which they can be fulfilled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' In particular, one of the smallness requirements concerns the product ∥α∥0,∞,Ω̺3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Example 20 (saturated Kerr nonlinearity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Another important example for the nonlineari- ties is [Akh98] c(x, ξ) := � 1, (x, ξ) ∈ Ω+ × C, ε(L)(x) + α(x)|ξ|2/(1 + γ|ξ|2), (x, ξ) ∈ Ω × C, with given ε(L), α ∈ L∞(Ω), saturation parameter γ > 0, and f = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Based on the identity |ξ|2 1 + γ|ξ|2 − |η|2 1 + γ|η|2 = (1 + γ|η|2)|ξ|2 − (1 + γ|ξ|2)|η|2 (1 + γ|ξ|2)(1 + γ|η|2) = |ξ|2 − |η|2 (1 + γ|ξ|2)(1 + γ|η|2) for all ξ, η ∈ C we obtain ���� |ξ|2 1 + γ|ξ|2 − |η|2 1 + γ|η|2 ���� = (|ξ| + |η|) ||ξ| − |η|| (1 + γ|ξ|2)(1 + γ|η|2) ≤ (|ξ| + |η|)|ξ − η|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Hence on Ω we arrive at the same Lipschitz function as in the previous Example 19, that is Lc(x, ξ, η) := � 0, (x, ξ, η) ∈ Ω+ × C × C, |α|(|ξ| + |η|), (x, ξ, η) ∈ Ω × C × C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Moreover, since c(x, wc) = c(x, 0) = � 0, (x, ξ) ∈ Ω+ × C, ε(L), (x, ξ) ∈ Ω × C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' we get the same sufficient conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 6 The modified boundary value problem Since the exact DtN operator is represented as an infinite series (see (9), (12)), it is practically necessary to truncate this nonlocal operator and consider only finite sums Tκ,Nu(x) := 1 R � |n|≤N Zn(κR)un(R)Yn(ˆx), x = Rˆx ∈ SR ⊂ R2, (39) Tκ,Nu(x) = 1 R N � n=0 � |m|≤n zn(κR)um n (R)Y m n (ˆx), x = Rˆx ∈ SR ⊂ R3 (40) 21 Resonant compactly supported nonlinearities January 30, 2023 for some N ∈ N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' The map Tκ,N is called the truncated DtN operator, and N is the truncation order of the DtN operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' The replacement of the exact DtN operator Tκ in the problem (18) by the truncated DtN operator Tκ,N introduces a perturbation, hence we have to answer the question of existence and uniqueness of a solution to the following problem: Find uN ∈ V such that aN(uN, v) = nN(uN, v) for all v ∈ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' (41) holds, where aN and nN are the forms defined by (23), (24) with Tκ replaced by Tκ,N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' The next result is the counterpart to Lemmata 8, 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Here we formulate a different version of G˚arding’s inequality compared to the case d = 2 considered in [HNPX11, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Lemma 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' The sesquilinear form aN (i) is bounded, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' there exists a constant C > 0 independent of N such that |aN(w, v)| ≤ C∥w∥V ∥v∥V for all w, v ∈ V, and (ii) satisfies a G˚arding’s inequality in the form Re aN(v, v) ≥ ∥v∥2 V,κ − 2κ2∥v∥2 0,2,BR for all v ∈ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' (i) If the proof of [MS10, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='4a)] is carried out with finitely many terms of the expansion of Tκ only, the statement follows easily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Alternatively, Lemma 23 with s = 0 can also be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' (ii) As in the proof of Lemma 9, the definitions of aN and the wavenumber dependent norm yield Re aN(v, v) = ∥v∥2 V,κ − 2κ2∥v∥2 0,2,BR − Re (Tκ,Nv, v)SR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Hence it remains to estimate the last term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' In the case d = 2, we have (see (39)) Tκ,Nv(x) := 1 R � |n|≤N Zn(κR)vn(R)Yn(ˆx), x = Rˆx ∈ SR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Then, using the L2(S1)-orthonormality of the circular harmonics [Zei95, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='1], we get −(Tκ,Nv, v)SR = − 1 R � |n|≤N Zn(κR)(vn(R)Yn, vn(R)Yn)SR = − 1 R � |n|≤N Zn(κR)|vn(R)|2(Yn, Yn)SR = − � |n|≤N Zn(κR)|vn(R)|2(Yn, Yn)S1 = − � |n|≤N Zn(κR)|vn(R)|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 22 Resonant compactly supported nonlinearities January 30, 2023 Hence, by Lemma 4, − Re (Tκ,Nv, v)SR = � |n|≤N (− Re Zn(κR)) � �� � ≥1/2 |vn(R)|2 + (− Re Z0(κR)) � �� � >0 |v0(R)|2 ≥ 1 2 � |n|≤N |vn(R)|2 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' The case d = 3 can be treated similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' From Tκ,Nv(x) = 1 R N � n=0 � |m|≤n zn(κR)vm n (R)Y m n (ˆx) (see (40)), we immediately obtain, using the L2(S1)-orthonormality of the spherical harmon- ics [CK19, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='8] that −(Tκ,Nv, v)SR = − 1 R N � n=0 � |m|≤n zn(κR)(vm n (R)Y m n , vm n (R)Y m n )SR = − 1 R N � n=0 � |m|≤n zn(κR)|vm n (R)|2(Y m n , Y m n )SR = −R N � n=0 � |m|≤n zn(κR)|vm n (R)|2(Y m n , Y m n )S1 = −R N � n=0 � |m|≤n zn(κR)|vm n (R)|2, and Lemma 4 implies − Re (Tκ,Nv, v)SR = R N � n=0 � |m|≤n (− Re zn(κR)) � �� � ≥1 |vm n (R)|2 ≥ R N � n=0 � |m|≤n |vm n (R)|2 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' In both cases we obtain the same G˚arding’s inequality as in the original (untruncated) problem Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' The next result is the variational version of the truncation error estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' It closely follows the lines of the proof of [HNPX11, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='3], where an estimate of ∥(Tκ − Tκ,N)v∥s−1/2,2,SR, s ∈ R, was proved in the case d = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Lemma 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' For given w, v ∈ H1/2(SR) it holds that ���((Tκ − Tκ,N)w, v)SR ��� ≤ c(N, w, v)∥w∥1/2,2,SR∥v∥1/2,2,SR, where c(N, w, v) ≥ 0 and limN→∞ c(N, w, v) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 23 Resonant compactly supported nonlinearities January 30, 2023 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' We start with the two-dimensional situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' So let w(x) = w(Rˆx) = � |n|∈N0 wn(R)Yn(ˆx), v(x) = v(Rˆx) = � |k|∈N0 vk(R)Yk(ˆx), x ∈ SR, (42) be series representations of w|SR, v|SR with the Fourier coefficients wn(R) = (w(R·), Yn)S1 = � S1 w(Rˆx)Yn(ˆx)ds(ˆx), vk(R) = (v(R·), Yk)S1 = � S1 v(Rˆx)Yk(ˆx)ds(ˆx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' The norm on the Sobolev space Hs(SR), s ≥ 0, can be defined as follows [LM72, Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 1, Rem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='6]: ∥v∥2 s,2,SR := R � n∈Z (1 + n2)s|vn(R)|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' (43) Then, by (39), the orthonormality of the circular harmonics [Zei95, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='1] and (43),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' ���((Tκ − Tκ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='N)w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' v)SR ��� = 1 R ������ � |n|,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='|k|>N � Zn(κR)wn(R)Yn(R−1·),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' vk(R)Yk(R−1·) � SR ������ = ������ � |n|,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='|k|>N Zn(κR) (wn(R)Yn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' vk(R)Yk)S1 ������ = ������ � |n|>N Zn(κR)wn(R)vn(R) ������ = ������ � |n|>N Zn(κR) (1 + n2)1/2(1 + n2)1/4wn(R)(1 + n2)1/4vn(R) ������ ≤ max |n|>N ���� Zn(κR) (1 + n2)1/2 ���� � |n|>N ��(1 + n2)1/4wn(R)(1 + n2)1/4vn(R) �� ≤ max |n|>N ���� Zn(κR) (1 + n2)1/2 ���� \uf8eb \uf8ed � |n|>N (1 + n2)1/2 |wn(R)|2 \uf8f6 \uf8f8 1/2 × \uf8eb \uf8ed � |n|>N (1 + n2)1/2 |vn(R)|2 \uf8f6 \uf8f8 1/2 ≤ 1 R max |n|>N ���� Zn(κR) (1 + n2)1/2 ���� ˜c(N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' v)∥w∥1/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='SR∥v∥1/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='SR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 24 Resonant compactly supported nonlinearities January 30,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 2023 where ˜c(N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' v)2 := � |n|>N(1 + n2)1/2|wn(R)|2 � |n|∈N0(1 + n2)1/2|wn(R)|2 � |n|>N(1 + n2)1/2|vn(R)|2 � |n|∈N0(1 + n2)1/2|vn(R)|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' The coefficient ˜c(N, w, v) tends to zero for N → ∞ thanks to (43), (45).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='. Corollary 5 implies the estimate 1 1 + n2|Zn(κR)|2 ≤ max{|Z0(κR)|2, 1 + |κR|2}, |n| ∈ N0, hence we can set c(N, w, v) := ˜c(N, w, v) R max{|Z0(κR)|, (1 + |κR|2)1/2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' The investigation of the case d = 3 runs similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' So let w(x) = w(Rˆx) = � n∈N0 � |m|≤n wm n (R)Y m n (ˆx), v(x) = v(Rˆx) = � k∈N0 � |l|≤k vl k(R)Y l k(ˆx), x ∈ SR, (44) be series representations of w|SR, v|SR with the Fourier coefficients wm n (R) = (w(R·), Y m n )S1 = � S1 w(Rˆx)Y m n (ˆx)ds(ˆx), vl k(R) = (v(R·), Y l k)S1 = � S1 v(Rˆx)Y l k(ˆx)ds(ˆx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' The norm on the Sobolev space Hs(SR), s ≥ 0, can be defined as follows [LM72, Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 1, Rem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='6]: ∥v∥2 s,2,SR := R2 � n∈N0 � |m|≤n (1 + n2)s|vm n (R)|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' (45) Then, by (40), the orthonormality of the spherical harmonics [CK19, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='8] and (45),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' ���((Tκ − Tκ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='N)w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' v)SR ��� = 1 R ������ � n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='k>N � |m|≤n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='|l|≤k � zn(κR)wm n (R)Y m n (R−1·),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' vl k(R)Y l k(R−1·) � SR ������ = R ������ � n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='k>N � |m|≤n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='|l|≤k zn(κR) � wm n (R)Y m n ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' vl k(R)Y l k) � S1 ������ = R ������ � n>N � |m|≤n zn(κR)wm n (R)vm n (R) ������ = R ������ � n>N � |m|≤n zn(κR) (1 + n2)1/2(1 + n2)1/4wm n (R)(1 + n2)1/4vm n (R) ������ 25 Resonant compactly supported nonlinearities January 30,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 2023 ≤ R max n>N ���� zn(κR) (1 + n2)1/2 ���� � n>N � |m|≤n ��(1 + n2)1/4wm n (R)(1 + n2)1/4vm n (R) �� ≤ R max n>N ���� zn(κR) (1 + n2)1/2 ���� \uf8eb \uf8ed� n>N � |m|≤n (1 + n2)1/2 |wm n (R)|2 \uf8f6 \uf8f8 1/2 × \uf8eb \uf8ed� n>N � |m|≤n (1 + n2)1/2 |vm n (R)|2 \uf8f6 \uf8f8 1/2 ≤ 1 R max n>N ���� zn(κR) (1 + n2)1/2 ���� ˜c(N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' v)∥w∥1/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='SR∥v∥1/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='SR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' where ˜c(N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' v)2 := � n>N � |m|≤n(1 + n2)1/2 |wm n (R)|2 � |n|∈N0 � |m|≤n(1 + n2)1/2 |wm n (R)|2 � n>N � |m|≤n(1 + n2)1/2 |vm n (R)|2 � |n|∈N0 � |m|≤n(1 + n2)1/2 |vm n (R)|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Thanks to Corollary 5 we can define c(N, w, v) := ˜c(N, w, v) R � 2 + |κR|2�1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Lemma 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' For s ∈ [0, 1/2) and w ∈ H1−s(BR \\ Ω), v ∈ H1+s(BR \\ Ω) it holds that |(Tκ,Nw, v)SR| ≤ Cbl∥w∥1−s,2,BR\\Ω∥v∥1+s,2,BR\\Ω, where the constant Cbl ≥ 0 does not depend on N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' We start with the two-dimensional situation as in the proof of Lemma 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' If w, v have the representations (42), then, by (39), the orthonormality of the circular harmonics [Zei95, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='1] and (43),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' |(Tκ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='Nw,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' v)SR| = 1 R ������ � |n|,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='|k|≤N � Zn(κR)wn(R)Yn(R−1·),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' vk(R)Yk(R−1·) � SR ������ = ������ � |n|,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='|k|≤N Zn(κR) (wn(R)Yn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' vk(R)Yk)S1 ������ = ������ � |n|≤N Zn(κR)wn(R)vn(R) ������ = ������ � |n|≤N Zn(κR) (1 + n2)1/2(1 + n2)(1/2−s)/2wn(R)(1 + n2)(1/2+s)/2vn(R) ������ ≤ max |n|≤N ���� Zn(κR) (1 + n2)1/2 ���� � |n|≤N ��(1 + n2)(1/2−s)/2wn(R)(1 + n2)(1/2+s)/2vn(R) �� 26 Resonant compactly supported nonlinearities January 30,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 2023 ≤ max |n|≤N ���� Zn(κR) (1 + n2)1/2 ���� \uf8eb \uf8ed � |n|≤N (1 + n2)1/2−s |wn(R)|2 \uf8f6 \uf8f8 1/2 × \uf8eb \uf8ed � |n|≤N (1 + n2)1/2+s |vn(R)|2 \uf8f6 \uf8f8 1/2 ≤ 1 R max |n|≤N ���� Zn(κR) (1 + n2)1/2 ���� ∥w∥1/2−s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='SR∥v∥1/2+s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='SR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Corollary 5 implies the estimate 1 1 + n2|Zn(κR)|2 ≤ max{|Z0(κR)|2, 1 + |κR|2}, |n| ∈ N0, hence |(Tκ,Nw, v)SR| ≤ 1 R max{|Z0(κR)|, (1 + |κR|2)1/2}∥w∥1/2−s,2,SR∥v∥1/2+s,2,SR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' (46) By the trace theorem [McL00, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='38], we finally arrive at |(Tκ,Nw, v)SR| ≤ C2 tr R max{|Z0(κR)|, (1 + |κR|2)1/2}∥w∥1−s,2,BR\\Ω∥v∥1+s,2,BR\\Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' The investigation of the case d = 3 runs similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' So let w, v have the representations (44), then, by (40), the orthonormality of the spherical harmonics [CK19, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='8] and (45),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' |(Tκ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='Nw,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' v)SR| = 1 R ������ N � n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='k=0 � |m|≤n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='|l|≤k � zn(κR)wm n (R)Y m n (R−1·),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' vl k(R)Y l k(R−1·) � SR ������ = R ������ N � n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='k=0 � |m|≤n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='|l|≤k zn(κR) � wm n (R)Y m n ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' vl ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='k(R)Y l ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='k) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='S1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='= R ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='N ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='� ' metadata={'source': 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+page_content='≤ R max ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='n∈N0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='zn(κR) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='(1 + n2)1/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='\uf8eb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='\uf8ed ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='N ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='n=0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='|m|≤n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='(1 + n2)1/2−s |wm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='n (R)|2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='\uf8f6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='\uf8f8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='1/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='\uf8eb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='\uf8ed ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='N ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='n=0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='|m|≤n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='(1 + n2)1/2+s |vm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='n (R)|2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='\uf8f6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='\uf8f8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='1/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='27 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='Resonant compactly supported nonlinearities ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='January 30,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 2023 ≤ 1 R max n∈N0 ���� zn(κR) (1 + n2)1/2 ���� ∥w∥1/2−s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='SR∥v∥1/2+s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='SR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Corollary 5 yields |(Tκ,Nw, v)SR| ≤ 1 R � 2 + |κR|2�1/2 ∥w∥1/2−s,2,SR∥v∥1/2+s,2,SR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' (47) By the trace theorem [McL00, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='38], we finally arrive at |(Tκ,Nw, v)SR| ≤ C2 tr R � 2 + |κR|2�1/2 ∥w∥1−s,2,BR\\Ω∥v∥1+s,2,BR\\Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Theorem 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Under the assumptions of Lemma 9, given an antilinear continuous functional ℓ : V → C, there exists a constant N∗ > 0 such that for N ≥ N∗ the problem Find uN ∈ V such that aN(uN, v) = ℓ(v) for all v ∈ V (48) is uniquely solvable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' First we show that the problem (48) has at most one solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' We start as in the proof of [HNPX11, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='5] and argue by contradiction, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' we suppose the following: ∀N∗ ∈ N ∃N = N(N∗) ≥ N∗ and uN = uN(N∗) ∈ V such that aN(uN, v) = 0 for all v ∈ V and ∥uN∥V = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' (49) However, the subsequent discussion differs significantly from the proof of [HNPX11, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' We apply an argument the idea of which goes back to Schatz [Sch74].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' First we assume there exists a solution uN ∈ V of (48) and derive an a priori estimate of the error ∥u − uN∥V , where u ∈ V is the solution of (29), see Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Since aN satisfies a G˚arding’s inequality (Lemma 21(ii)), we have, making use of (28), C2 −∥u − uN∥2 V − 2κ2∥u − uN∥2 0,2,BR ≤ Re aN(u − uN, u − uN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Since aN(u − uN, v) = aN(u, v) − aN(uN, v) = a(u, v) � �� � =ℓ(v) +aN(u, v) − a(u, v) − aN(uN, v) � �� � =ℓ(v) = ((Tκ − Tκ,N)u, v)SR , we obtain C2 −∥u − uN∥2 V − 2κ2∥u − uN∥2 0,2,BR ≤ η1∥u − uN∥V (50) with η1 := sup v∈V Re ((Tκ − Tκ,N)u, v)SR ∥v∥V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Now we consider the following auxiliary adjoint problem (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' [McL00, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 43]): 28 Resonant compactly supported nonlinearities January 30, 2023 Find wN ∈ V such that a(v, wN) = (v, u − uN)BR for all v ∈ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' (51) Since A is a Fredholm operator (see the proof of Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 10), the adjoint problem possesses a unique solution wN ∈ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Then ∥u − uN∥2 0,2,SR = a(u − uN, wN) = a(u, wN) − a(uN, wN) = a(u, wN) − aN(uN, wN) � �� � =ℓ(wN)−ℓ(wN)=0 +aN(uN, wN) − a(uN, wN) = ((Tκ − Tκ,N)uN, wN)SR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' In particular, this relation shows that ((Tκ − Tκ,N)uN, wN)SR is real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' With η2 := sup v∈V ((Tκ − Tκ,N)uN, v)SR ∥v∥V we obtain ∥u − uN∥2 0,2,BR ≤ η2∥wN∥V ≤ η2C−1 − C(R, κ)∥u − uN∥V ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' The continuous embedding V ⊂ V ∗ yields ∥u − uN∥2 0,2,BR ≤ η2C−1 − C(R, κ)Cemb∥u − uN∥V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Applying this estimate in (50), we get C2 −∥u − uN∥2 V − 2κ2η2C−1 − C(R, κ)Cemb∥u − uN∥V ≤ η1∥u − uN∥V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Now, if ∥u − uN∥V ̸= 0, we finally arrive at C2 −∥u − uN∥V ≤ η1 + 2κ2η2C−1 − C(R, κ)Cemb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' (52) Clearly this inequality is true also for ∥u − uN∥V = 0 so that we can remove this interim assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Thanks to Lemma 22 we have that ���((Tκ − Tκ,N)u, v)SR ��� ≤ c(N, u, v)∥u∥1/2,2,SR∥v∥1/2,2,SR ≤ c(N, u, c)C2 tr∥u∥V ∥v∥V , hence η1 ≤ c+(N, u)C2 tr∥u∥V with c+(N, u) := sup v∈V c(N, u, v), (53) where limN→∞ c+(N, u) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Note that, as can be seen from the proof of Lemma 22, the second fractional factor in the representation of ˜c(N, w, v) can be estimated from above by one without losing the limit behaviour for N → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Consequently, η1 can be made arbitrarily small provided N is large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' In order to estimate η2 we cannot apply Lemma 22 directly since the second argument in the factor c(N, uN, v) depends on N, too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Therefore we give a more direct estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 29 Resonant compactly supported nonlinearities January 30, 2023 Namely, let v ∈ V have the representation (42) or (44), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Then we define VN|SR := � span|n|≤N{Yn(R−1·)}, d = 2, spann=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='N,|m|≤n{Y m n (R−1·)}, d = 3, and introduce an orthogonal projector PN : V |SR → VN|SR : v �→ PNv := �� |n|≤N vn(R)Yn(R−1·), d = 2, �N n=0 � |m|≤n vm n (R)Y m n (R−1·), d = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Then it holds that VN|SR ⊂ ker(TκPN − Tκ,N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Indeed, if d = 2 and v ∈ VN|SR, then PNv = v = � |n|≤N vn(R)Yn(R−1·) and TκPNv = Tκv = 1 R � |n|≤N Zn(κR)vn(R)Yn(R−1·) = Tκ,Nv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' An analogous argument applies in the case d = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Now we return to the estimate of η2 and write, for uN ∈ V , (Tκ − Tκ,N)uN = (Tκ − TκPN)uN + (TκPN − Tκ,N)uN = Tκ(id −PN)uN, where we have used the above property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' The advantage of this approach is that we can apply a wellknown estimate of the projection error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' The proof of this estimate runs similarly to the proof of Lemma 22 but only without the coefficients Zn or zn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' respectively: ��((id −PN)w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' v)SR �� = ������ � |n|,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='|k|>N � wn(R)Yn(R−1·),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' vk(R)Yk(R−1·) � SR ������ = R ������ � |n|,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='|k|>N (wn(R)Yn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' vk(R)Yk)S1 ������ = R ������ � |n|>N wn(R)vn(R) ������ = R ������ � |n|>N 1 (1 + n2)1/2(1 + n2)1/4wn(R)(1 + n2)1/4vn(R) ������ ≤ max |n|>N R (1 + n2)1/2 � |n|>N ��(1 + n2)1/4wn(R)(1 + n2)1/4vn(R) �� ≤ R (1 + N2)1/2 \uf8eb \uf8ed � |n|>N (1 + n2)1/2 |wn(R)|2 \uf8f6 \uf8f8 1/2 × \uf8eb \uf8ed � |n|>N (1 + n2)1/2 |vn(R)|2 \uf8f6 \uf8f8 1/2 ≤ 1 (1 + N2)1/2∥w∥1/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='SR∥v∥1/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='SR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 30 Resonant compactly supported nonlinearities January 30, 2023 The same estimate holds true for d = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Then we get, by Remark 3 (or Lemma 23), ���((Tκ − Tκ,N)uN, v)SR ��� = ��(Tκ(id −PN)uN, v)SR �� ≤ Cκ (1 + N2)1/2∥uN∥1/2,2,SR∥v∥1/2,2,SR ≤ CC2 trκ (1 + N2)1/2∥uN∥V ∥v∥V , thus η2 ≤ CC2 trκ (1 + N2)1/2∥uN∥V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Using this estimate and (53) in (52), we obtain C2 −∥u − uN∥V ≤ c+(N, u)C2 tr∥u∥V + 2κ2C−1 − C(R, κ)Cemb CC2 trκ (1 + N2)1/2∥uN∥V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' (54) Now we appply this estimate to the solutions uN of the homogeneous truncated problems in (49).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' By Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 10, the homogeneous linear interior problem (29) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' ℓ = 0) has the solution u = 0, and the above estimate implies C2 −∥uN∥V ≤ 2κ2C−1 − C(R, κ)Cemb CC2 trκ (1 + N2)1/2∥uN∥V , which is a contradiction to ∥uN∥V = 1 for all N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Although the proof of Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 24 allows an analogous conclusion as in Lemma 11 that the truncated bilinear form aN satisfies an inf-sup condition, such a conclusion is not fully satisfactory since the question remains whether and how the inf-sup constant depends on N or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' However, at least for sufficiently large N, a positive answer can given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Lemma 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Under the assumptions of Lemma 9, there exists a number N∗ ∈ N such that βN∗(R, κ) := inf w∈V \\{0} sup v∈V \\{0} |aN(w, v)| ∥w∥V,κ∥v∥V,κ > 0 is independent of N ≥ N∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' In the proof a formula is given that expresses βN∗(R, κ) in terms of β(R, κ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' We return to the proof of Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 24 and mention that the estimate (54) is valid for solutions u, uN of the general linear problems (29) (or, equally, (31)) and (48), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' By the triangle inequality, ∥uN∥V ≤ ∥u∥V + ∥u − uN∥V ≤ ∥u∥V + c+(N, u)C−2 − C2 tr∥u∥V + 2κ2C−3 − C(R, κ)Cemb CC2 trκ (1 + N2)1/2∥uN∥V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 31 Resonant compactly supported nonlinearities January 30, 2023 If N∗ is sufficiently large such that κ2C−3 − C(R, κ)Cemb CC2 trκ (1 + N2)1/2 ≤ 1 4 and c+(N, u)C−2 − C2 tr ≤ 1 for all N ≥ N∗, then, by Lemma 11, ∥uN∥V ≤ 4∥u∥V ≤ 4 C− ∥u∥V,κ ≤ ∥ℓ∥V ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' That is, the sesquilinear form aN satisfies an inf-sup condition βN∗(R, κ) := inf w∈V \\{0} sup v∈V \\{0} |aN(w, v)| ∥w∥V,κ∥v∥V,κ > 0 with βN∗(R, κ) := C−β(R, κ) 4C+ independent of N ≥ N∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Analogously to (30) we introduce the truncated linear operator AN : V → V ∗ by ANw(v) := aN(w, v) for all w, v ∈ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' By Lemma 21, AN is a bounded operator, and Lemma 25 implies that AN has a bounded inverse: ∥w∥V,κ ≤ βN∗(R, κ)−1∥ANw∥∗ for all w ∈ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Furthermore, we define a nonlinear operator FN : V → V ∗ by FN(w)(v) := ℓcontr(w) + ℓsrc(w) + ℓinc N for all w ∈ V, where ⟨ℓinc N , v⟩ := (ˆx · ∇uinc − Tκ,Nuinc, v)SR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' The problem (41) is then equivalent to the operator equation ANu = FN(u) in V ∗, and further to the fixed-point problem u = A−1 N FN(u) in V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' (55) Theorem 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Under the assumptions of Lemma 9, let the functions c and f generate locally Lipschitz continuous Nemycki operators in V and assume that there exist functions wf, wc ∈ V such that f(·, wf) ∈ Lpf/(pf −1)(Ω) and c(·, wf) ∈ Lpc/(pc−2)(Ω), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Furthermore let uinc ∈ H1 loc(Ω+) be such that additionally ∆uinc ∈ L2,loc(Ω+) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' If there exist numbers ̺ > 0 and LF ∈ (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' βN∗(R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' κ)) (where N∗ and βN∗(R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' κ) are from Lemma 25) such that the following two conditions κ2 [∥c(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' wc) − 1∥0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='˜qc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='Ω + ∥Lc(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' wc)∥0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='qc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='Ω(̺ + ∥wc∥V )] ̺ + � ∥f(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' wf)∥0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='˜qf,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='Ω + ∥Lf(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' wf)∥0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='qf,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='Ω(̺ + ∥wf∥V ) � + Ctr∥ˆx · ∇uinc − Tκ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='Nuinc∥−1/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='SR ≤ ̺βN∗(R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' κ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' κ2 [∥Lc(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' v)∥0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='qc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='Ω̺ + ∥c(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' wc) − 1∥0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='˜qc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='Ω + ∥Lc(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' wc)∥0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='qc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='Ω(̺ + ∥wc∥V )] + ∥Lf(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' v)∥0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='qf,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='Ω ≤ LF are satisfied for all w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' v ∈ Kcl ̺ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' then the problem (35) has a unique solution uN ∈ Kcl ̺ for all N ≥ N∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' Analogously to the proof of Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' 32 Resonant compactly supported nonlinearities January 30, 2023 7 Conclusion A mathematical model together with an investigation of existence and uniqueness of its solution for radiation and propagation effects on compactly supported cubic nonlinearities is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' The full-space problem is reduced to an equivalent truncated local problem, whereby in particular the dependence of the solution on the truncation parameter (with regard to stability and errors) is studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=' The results form the basis for the use of numerical methods, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFKT4oBgHgl3EQfVy5o/content/2301.11789v1.pdf'} +page_content=', FEM, for the approximate solution of the original problem with controllable accuracy.' metadata={'source': 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b/B9A0T4oBgHgl3EQfAP-Y/content/tmp_files/2301.01960v1.pdf.txt @@ -0,0 +1,2078 @@ +Applying the Quantum Error-correcting Codes for Fault-tolerant Blind Quantum +Computation +Qiang Zhao +Department of Computer Science +and Engineering +The Chinese University of Hong Kong +Hong Kong, China +Email:qiangzhao@cuhk.edu.hk +Qiong Li +School of Computer Science +and Technology +Harbin Institute of Technology +Harbin, China +Email: qiongli@hit.edu.cn +John C.S. Lui +Department of Computer Science +and Engineering +The Chinese University of Hong Kong +Hong Kong, China +Email:cslui@cse.cuhk.edu.hk +Abstract—The Blind Quantum Computation (BQC) is a dele- +gated protocol, which allows a client to rent a remote quantum +server to implement desired quantum computations, while +keeping her inputs, outputs and algorithms privacy. However, +the qubit errors during the quantum computation are realistic +issues that are needed to consider. In this paper, we propose +a fault-tolerant blind quantum computation protocol with +quantum error-correcting codes to avoid the accumulation and +propagation of qubit errors during the computing. Meanwhile, +we also present the ϵ-blindness in our protocol. To improve the +error correction performance, the concatenated codes are used +in our protocol. We further present the resources consumption +of photon pulses by the optimal level concatenation codes. The +simulation results show that our scheme not only can improve +the preparation efficiency but also reduce quantum resources, +which shows a significant improvement to a realistic fault- +tolerant BQC. +1. Introduction +Quantum cloud computing is an innovative research at +the interface of maths, computer science and physics, which +promises revolutionary improvements in communications +and computations, dealing with tasks that are impossible +for traditional computers. Quantum cloud computing is +a distributed system composed of quantum computer as +server and simple quantum devices as clients that exchange +quantum information and computing instructions through +quantum channel and classical channel. It will be conve- +nient for clients to rent quantum computing resources to +achieve desired quantum computation. However, classical +communication network protocols can not ensure the secu- +rity of quantum information and computation. One possible +solution to address this problem is via the Blind Quantum +Computation (BQC) protocol supported by quantum cloud +computing, which enables a client(say Alice) with limited +quantum technology to delegate a computing task to a +remote quantum server(say Bob) while ensuring computing +information privacy [1], [2], [3]. In recent years, a number +of BQC protocols have been proposed [4], [5], [6], [7], +[8], [9]. Among these protocols, Universal Blind Quantum +Computation (UBQC) is of particular interest as it not only +can guarantee that Alice’s inputs, outputs and quantum algo- +rithms are unknown to server while delegating computation, +but also only requires Alice to prepare the single photon +states. +As we all know, qubits and quantum gates are easily +affected by environmental and devices noise, which will +lead qubits and quantum gates to rotate errors. It is usually +called the decoherence effect. Since noise will result in more +qubit and quantum gate errors as quantum computational +scales up, it is very formidable challenge to realize a large +scale quantum computation. However, Noisy Intermediate- +Scale Quantum (NISQ) technology will be available in +near future, which can help quantum computers with 50- +100 qubits to perform tasks that surpasses classical com- +puters [10]. UBQC is composed of quantum preparation, +quantum measurement, one-way quantum communication +and two-way classical communication [11]. In the noisy +intermediate-scale UBQC, due to the influence of noise, it +is inevitable to make errors of qubits and quantum gates +in each of stage [9], [12]. Hence, UBQC will have an +interesting challenge how to deal with qubit errors. Besides, +the quantum gates should be also fault-tolerant to prevent +the accumulation and propagation of errors in the subsequent +computation. Therefore, it is necessary for UBQC to utilize +quantum error-correcting codes [13] or other quantum codes +[14], [15] with error correcting capability as logical qubits +to implement a fault-tolerant blind quantum computation. +Since quantum error correction imposes a heavy overhead +cost in number of qubits and number of gates, it is a +formidable challenge to trade off between fault tolerance +and heavy resource overhead (the number of required pulses) +because a large number of ancilla qubits are used to correct +erroneous data qubits to ensure the required fault tolerance. +In this context, this work gives an alternative protocol to +implement a fault tolerant blind quantum computation in +NISQ, and optimize quantum resource consumption, in term +of the number of photon pulses. Moreover, our quantum +error-correcting scheme can be also applied to other graphs +of BQC and perform fault tolerant quantum computation. +arXiv:2301.01960v1 [quant-ph] 5 Jan 2023 + +The main contributions of this paper are as follows: +• +We present a remote blind quantum error-correcting +codes preparation protocol on cluster states to cor- +rect qubit errors and improve the successful prob- +ability of required qubits. The encoded circuit is +transformed to cluster states instead of brickwork +states, i.e. multi-particle entangled graph states, +and then delegate server to prepare quantum error- +correcting codes in the framework of measurement- +based quantum computation. +• +Based on the above preparation, we propose a fault- +tolerant blind quantum computation protocol with +quantum error-correcting codes to avoid the accu- +mulation and the propagation of qubit errors and +improve the performance in a delegated quantum +computation. Since these quantum error-correcting +codes are unknown logical qubits to server, each of +them is considered as logical unit to take place of +original qubit on brickwork state to perform a fault- +tolerant computation. Meanwhile, we also demon- +strate the ϵ-blindness of our protocol in case of +malicious server. +• +To achieve a higher fault-tolerant performance, we +propose to utilize the concatenated stabilizer codes, +that is to concatenate the same or different stabi- +lizer codes in a multi-level encoding manner, to +prepare high-quality encoded logical qubits to do +fault-tolerant computation. Moreover, we deduce the +lower bound of quantum resource consumption in +each level of the concatenation codes, i.e. the num- +ber of required pulses. +• +To reduce quantum resource overhead in a fault- +tolerant quantum computation, we build an opti- +mization model of quantum resource consumption, +which describes the ratio of the number of pulses +in the concatenation codes to the non-coding case +under the same probability of successful preparation. +When it reaches minimum, we can achieve the opti- +mal number of levels in the concatenated stabilizer +codes. +The rest of this paper is organized as follows: In Section +II, the background and technical preliminaries are intro- +duced. In Section III, we firstly present a remote blind quan- +tum error-correcting codes preparation protocol on cluster +state to correct qubit errors, and then propose a fault-tolerant +blind quantum computation with quantum error-correcting +codes to do delegated computing. Furthermore, we also give +a proof of the blindness of our fault-tolerant protocol. In +Section IV, the multi-level quantum error-correcting codes +are concatenated to reduce the error rate of the generated +logical qubits. A lower bound of the number of required +pulses is deduced with different levels in fault-tolerant blind +quantum computation. In Section V, we present simula- +tion results to further validate the theoretical model of the +number of required pulses with concatenated codes, and +give an optimal level number of the concatenation codes.In +Section VI, we surveys the related works. In Section VII, +we conclude our work and provide the direction of future +work. +2. Background +BQC protocols can be generally divided into two cate- +gories, they are measurement-based BQC and circuit-based +BQC. In 2005, Childs proposed the first BQC protocol [1], +which utilized quantum circuit model and the quantum one- +time pad to realize a secure quantum computation. However +the client was required to have quantum memory and the +ability to perform quantum SWAP gate. Since then, some +other circuit-based BQC protocols [2], [16] were proposed, +but they all required client to possess quantum memory, +quantum measurement or limited quantum computing. In +2009, Broadbent, Fitzsimons and Kashefi proposed the Uni- +versal Blind Quantum Computation (UBQC) protocol [3], +which is the first measurement-based BQC. This protocol +only requires client to prepare the single photons, and other +operations can be delegated to server, which can greatly +reduce client’s burden. After then, it has been also experi- +mentally demonstrated. Based on UBQC [17], some other +measurement-based BQC protocols [7], [18], [19], [20], +[21], [22] have been devised to be as practical as possible +in recent years. To verify the blindness and correctness of +practical BQC, some protocols [18], [19] were also proposed +in noisy channels or existing any malicious attacker. In order +to make Alice as classical as possible, multiple-server BQC +protocols [20], [21] were proposed based on the shared +entanglement states between servers. For tolerating more +faults or noise, some fault-tolerant BQC protocols [7], [22] +were presented based on the idea of quantum error correc- +tion. With development of quantum technology, UBQC is +gradually moving from theoretical researches to practical +applications. +UBQC is within the framework of a measurement-based +quantum computation (MBQC) [3], [23]. The underlying +resource of MBQC is the brickwork states, which is multi- +particle entangled graph states constructed by prepared +qubits. A delegated quantum computation can be imple- +mented by measurements on brickwork states. Let us have a +client (say Alice ) and a server (say Bob) with full-fledged +quantum computer. In preparation stage, Alice prepares +a series of single qubits |+θ⟩ with random polarization +θ ∈ {0, π/4, ..., 7π/4} and sends to Bob through one-way +quantum channel, and Bob uses Controlled Z (CZ) gates +to act on these received qubits to build brickwork state. In +interactive measurement stage, Alice carefully designs the +quantum circuit based on the computational task, and trans- +forms a sequence of ordered quantum gates in the circuit +into the measurement angles, and then sends to Bob through +two-way classical channel. Bob performs measurement op- +erations for each qubit on brickwork state, and returns +back the results to Alice. The measurement procedure is +repeated in turn on brickwork state until the desired quantum +computation result is obtained. The schematic diagram of +UBQC is shown in Fig.1. + +Note that for each qubit on brickwork state, Alice can +calculate the measurement angle φ′ +x,y according to her +desired angle φx,y and the previous measurement result +[3]. In order to ensure the privacy of computation, the +measurement angle φx,y needs to be further processed as +δx,y = φ′ +x,y + θx,y + πrx,y, rx,y∈R {0, 1}. Then, the real +measurement angle δx,y is sent to Bob to perform the fault- +tolerant measurement for each encoded logical qubit |ψx,y⟩L +on encoded brickwork state. The measurement result sx,y +is returned to Alice. If the random bit chosen by Alice, +rx,y = 1, Alice flips the measurement result sx,y; otherwise +she does nothing. +Laser +(Weak Coherent Pulses) + Classical-Computer +(Data Processing) +Quantum Resources +(Measurement-Based +Quantum Computation) +Classical-Computer +(Data Processing) +Measurement Angles +Measurement Results +Quantum Algorithm +Quantum Graph States +Quantum Channel +Classical Channel +Alice +Bob +Measurement +Angles +Measurement +Results +Measurement +Results +Measurement +Bases +Figure 1. The schematic diagram of UBQC. +2.1. Remote blind qubit state preparation protocol +with two decoy states +In UBQC, the qubits could be encoded in the polar- +ization of a single photon generated by a realistic single +photon source. However, it is inevitable to send two or more +identically polarized photons instead of one in any physi- +cal implementation, which will destroy the perfect privacy +of client. Hence, a Remote Blind qubit State Preparation +(RBSP) protocol was proposed to prepare quantum states, +which can arbitrarily approach to the perfect qubits [4]. +Moreover, this protocol can allow us to achieve ϵ-blind +UBQC for any security parameter ϵ > 0. Nevertheless, the +client’s overhead, which is the number of pulses needed +for preparing a qubit, is usually very large, and this is true +especially for a long-distance communication. Therefore, a +modified RBSP with two decoy states was presented by +Zhao and Li [5], [6] to optimize the number of required +pulses. They showed that two decoy states protocol can +achieve a smaller number of pulses than one decoy state. +The RBSP with two decoy states is summarized as follows: +(1) Alice generates N weak coherent pulses which con- +tain signal and two decoy states, and sends them to Bob. +(2) For each received state, Bob reports to Alice which +pulses were received. Based on the reported statistic values, +Alice calculates the gains of the signal state and the two +decoy states, and determines to continue the protocol if they +are within the specified thresholds. +(3) Bob discards the decoy states declared by Alice, and +separates the remaining signal states into S groups (in fact +S is the computational scale). Each group needs to perform +the interlaced 1-D cluster computation subroutine(I1DC) [4] +so that the desired qubit |+θ⟩ can be prepared. +(4) With Alice’s knowledge about the angles of signal +states in the I1DC procedure and outcomes reported by Bob, +Alice can compute the polarization angle θ of the prepared +qubit. +Based on RBSP with two decoy states [6], one can esti- +mate the lower bound of Alice’s overhead, i.e. the required +total number of pulses N. +Theorem 1. In RBSP with two decoy states, the average +photon number of signal states is denoted as µ, and the +average photon number of two decoy states is denoted +as v1, v2. The gain of signal states is Qµ, which is the +probability of the detecting event of the signal pulses. +The proportion of single photon in the signal states is +described as p1, and its lower bound is pL,v1,v2 +1 +. The +probabilities of signal and two decoy states chosen by +the client are defined as pµ, pv1, pv2, respectively, with +pµ+pv1+pv2 = 1. For a UBQC protocol of computation +size S (where S ≫ Mµ/Mmulti, Mµ is the total number +of signal states, Mmulti is the number of multiphotons +signal states.), the lower bound of the required total +number of pulses, denoted as N L,v1,v2, can be expressed +as follows. +N ≥ N L,v1,v2 = +S +pµQµ +ln (ϵ/S) +ln +� +1 − pL,v1,v2 +1 +�. +(1) +2.2. Quantum error-correcting codes +In quantum computation, the errors are inevitable be- +cause of the decoherence effect of quantum state. In order +to correct qubit error, we need to establish a error model for +quantum computation. In [24], We conclude that the evolu- +tion of the qubit can be expressed as a linear combination +of four possibilities: (1) no error occurs, (2) the bit flip +|0⟩ ↔ |1⟩, (3) the relation phase of |0⟩ and |1⟩, (4) both a +bit flip and a phase flip occur. Then, the error super-operator +E is diagonal in Pauli basis. The error model can be taken +the form +E (|ψ⟩ ⟨ψ|) = +� +Ei∈E +p (Ei)Ei |ψ⟩ ⟨ψ| E† +i , +(2) +where +all +error +Ei +are +Pauli +operators +Ei += +⊗n +j=1Xa +j Zb +j, a, b ∈ {0, 1}, and p (Ei) is the probability for +the error Ei to occur. In Eq.2, we have the normalization +condition +E† +i Ei += +I, ∀Ei, +and +the +trace-preserving +constraint +� +Ei∈E +p (Ei) = 1. For correcting qubit errors, we +need to diagnose which of these four possibilities actually +occurred, and then correct the error by applying the Pauli +basis. + +In the procedure for determining error syndrome, we +must make sure that quantum state can not be destroyed +for subsequent quantum computation, and its information is +private. Therefore, we need to use a suitable quantum code +to handle these qubits, which may occur errors. A popular +quantum code is the [[n, k, d]] stabilizer code [24], [25], +[26], which can encode k qubits into n. The parameter d +is the distance of the code. A code with distance d can +correct ⌊d/2⌋ simultaneous errors from the error set E. +The code space is the eigenspace of the generators of the +code stabilizer S, which is a set of (n − k) independent +commuting operators g. Each codeword |ψm⟩ obeys the +eigenvalue equations |ψm⟩ = g |ψm⟩ , ∀m = 0, ..., 2k − 1. +The stabilizer code can be also described by the generator +matrix G, which has 2n columns and n − k rows. The +generator matrix is denoted as G = (XG|ZG). Each row +in G encodes a generator g of the stabilizer. The column +index of XG and ZG labels the qubits. The positions of the +1’s in XG indicate the qubits that are acted on by X in +the listed generators, and the 1’s in ZG indicated the qubits +acted on by Z. If a 1 appears in same position in both XG +and ZG, then the product Y = ZX acts on that qubit. +|0⟩L = +1 +2 +√ +2(|0000000⟩ + |0001111⟩ + |0110011⟩ ++ |0111100⟩ + |1010101⟩ + |1011010⟩ ++ |1100110⟩ + |1101001⟩) +|1⟩L = +1 +2 +√ +2(|1111111⟩ + |1110000⟩ + |1001100⟩ ++ |1000011⟩ + |0101010⟩ + |0100101⟩ ++ |0011001⟩ + |0010110⟩) +(3) +In the quantum error correction, the diagnosis of error is +often named as the error syndrome measurement, which +is increasingly difficult as the number of logical qubits +increases in an encoded block. Hence, we use a common +stabilizer code with fewer qubits, i.e. the 7-qubit Steane’s +code [[7,1,3]], which can encode one qubit in seven, and +corrects one-qubit error. The encoded logical qubit basis is +denoted as {|0⟩L, |1⟩L}, and see Eq.3. +In fact, the code is also a special case of the CSS +code, which is an extension of the classical Hamming code +in quantum error correction. The generator matrix of the +[[7,1,3]] is shown as follows [25]. +G[[7,1,3]] = (XG|ZG)[[7,1,3]]= +� +� +� +� +� +� +0 +0 +0 +1 +1 +1 +1 +0 +1 +1 +0 +0 +1 +1 +1 +0 +1 +0 +1 +0 +1 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +���������� +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +1 +1 +1 +1 +0 +1 +1 +0 +0 +1 +1 +1 +0 +1 +0 +1 +0 +1 +� +� +� +� +� +� +(4) +The encoding circuit of quantum error-correcting codes +can be designed according to the generator matrix. In +Fig. 2(a), the circuit is used to encode an unknown logical +qubit [24]. The CNOT gates of the circuit are based on +XG. In Fig. 2(b), two ancilla states are prepared to perform +H +H +H +0 +0 +0 +0 +0 +0 +0 +1 + + + +H +H +M +M +7 +Z +X +Data +Ancilla +(a) +(b) +0 +1 +L +L + + + +Figure 2. The encoding and correction circuits of the [[7,1,3]] code [24]. +(a) An unknown logical qubit and 6 ancilla qubits can be used to encode +into [[7,1,3]] code. (b) Two 7-qubit Steane states are used to correct error +qubits for an 7-qubit data block. Each CNOT gate in the diagram represents +7 CNOT gates performed in parallel. +error syndrome measurements of bit flip and phase flip. +The measurment results of ancilla qubits are multiplied by +the row vectors of G[[7,1,3]] to obtain the parity bits, which +can diagnose the error syndrome. Then, we can use Pauli +operator to correct the qubit errors in the encoded data block. +2.3. Problem statement +In the quantum computation, a basic qubit unit is often +stored on the polarization of a single photon or the spin +of a single electron. A qubit has superposition property, +which can be described by two-dimension Hilbert space. +An quantum algorithm is described by the quantum circuit +which consists of a sequence of quantum gates. In UBQC, +we know that the quantum gates are firstly transformed +into bricks, and then server performs measurements on each +qubit of brickwork state to implement client’s computation +based on MBQC. Obviously, quantum gates can be realized +by performing measurements on qubits. In the practical +UBQC system, we only need to consider the impact of +qubits. Due to the influence of noise from environment and +device, qubits are prone to polarization errors. Hence, it is +necessary to integrate quantum error correction idea into +UBQC to ensure the correctness of a delegated computation. +In UBQC, the main processes of a quantum computation +are preparation and measurement. In preparation, we need to +modify the above RBSP protocol to prepare quantum error- +correcting codes instead of single qubits. In order to prepare +quantum error-correcting codes, we firstly design a quantum +encoding circuit, and then transform it into graph states. +Based on MBQC model, we can delegate server to prepare +quantum error-correcting codes on graph states. However, +both good encoding circuits and good graph states require +a large number of data qubits and ancilla qubits, which are +directly determine the size of computation. The challenge is +to achieve better error correction performance with limited +quantum resources. +In measurement, client sends measurement angles to +server, and guide him to perform measurement-based com- +putations on brickwork state. To prevent the accumulation +and propagation of qubits in computing, it is necessary to +perform fault-tolerant quantum computation. The quantum + +error-correcting codes can be considered as logical unites to +take the place of original qubits on brickwork state. Quan- +tum gates must be fault tolerant in the quantum computing +circuit. The measurements of logical qubits on brickwork +state are also required to be fault tolerant. Since the qubits, +quantum gates and measurements are different from previ- +ous states, the security of the fault-tolerant blind quantum +computation are needed to be re-demonstrated based on +original protocol. The challenge is the ϵ-blindness of a +practical fault-tolerant protocol. The good fault tolerance +requires a lot of redundant qubits to resist the spread of +qubit errors. Hence, other challenge is to trade off between +quantum resource consumption and fault tolerance. +3. Fault-tolerant blind quantum computation +with quantum error-correcting codes +To effectively remove qubit errors, the encoded quantum +gates need to ensure that a failure in executing any encoded +gate can be only propagated to a small number of qubits in +each encoded data block, so that the aggregated error rate +does not exceed the designed fault tolerant threshold. In ad- +dition, any error-correction scheme can introduce errors on +the encoded qubits, so one must be very careful in designing +error-correction procedures that do not introduce too many +additional errors into the encoded data. Therefore, for a +quantum circuit, in order to perform a fault-tolerant quantum +computing, the fault-tolerant preparation of quantum error- +correcting codes, fault-tolerant error correction operating, +fault-tolerant quantum gates application and fault-tolerant +measurements are needed to prevent the accumulation and +propagation of errors in quantum computing [24], and this +is shown in Firgure.3. +FT +preparation +FT +preparation +FT error +Correction +FT error +correct +FT +CNOT +FT error +Correction +FT error +Correction +FT H +FT error +Correction +FT error +Correction +FT +measurement +FT +measurement +6 +0 + +   +6 +0 + +   +L + +L + +H +Figure 3. The process of fault-tolerant quantum computation +In UBQC, a fault-tolerant computing should be per- +formed as in Fig.3. For example, Alice delegated Bob to per- +form fault-tolerant preparation of quantum error-correcting +codes. After then, these encoded logical qubits are used to +create encoded brickwork state to realize fault-tolerant quan- +tum computing. In fault-tolerant preparation [3], thousands +of qubits are used to prepare quantum error-correcting codes. +On one hand, for delegated preparation, quantum gates +of the encoding circuit are equivalently transformed into +brickwork state or other graph state, and then implemented +by measurement-based quantum computation. The quantum +gates between non-neighbouring qubits can not be directly +implemented on brickwork state, so we need many SWAP +gates to ensure that any quantum gate acts on adjacent +qubits. On the other hand, the CNOT gate between non- +adjacent qubits is frequently used in fault-tolerant encoding +circuit, so a large number of SWAP gates are needed to +transformed into brickwork state, which requires a lot of +auxiliary quantum resources. Hence, it is very difficult to +achieve a desired fault-tolerant quantum computation with +current quantum technology. If the high-frequency CNOT +gate of the encoding circuit can be implemented at high +successful probability, the error rate of the prepared quantum +error-correcting code will be very low. Consequently, with +limited quantum resources, a fault-tolerant blind quantum +computation can be implemented by in a compromised +way using a combination of quantum error-correcting code +preparation and fault-tolerant measurements in UBQC. +3.1. Delegated +preparation +of +quantum +error- +correcting codes +In a quantum computation, we know that arbitrary quan- +tum gate can be realized on cluster state. The essence of +a quantum circuit is a series of ordered quantum gates. +Hence, each quantum circuit can be implemented on cluster +state [27]. The quantum error-correcting code circuit can +be also realized on cluster state to generate the encoded +logical qubits. These codes can be considered as logical +units to take the place of original qubits on brickwork state +to perform a fault-tolerant computing. Hence, we propose a +remote quantum error-correcting code preparation protocol +on cluster state, as shown in Protocol.1 +Protocol 1: A remote quantum error-correcting +code preparation on cluster state +Input: data weak coherent pulses (signal states and +two decoy states) with polarization ρσ, +σ ∈R {kπ/4 : 0 ≤ k ≤ 7}; ancilla pulses +with polarization |+⟩. +Output: quantum error-correcting codes +{|+θi⟩L}S +1 . +1 Alice sends data and ancilla pulses to Bob. +2 for i=1 to S; do +3 +Bob prepares qubit |+θi⟩ based on RBSP with +two decoy states, and uses it and a group of +ancilla qubits {|+⟩} to build cluster state. +4 +for x=1 to m, y=1 to n; do +5 +Realization of the quantum encoding circuit +on cluster state. +6 +if qxy is white then qxy is measured with +{|0⟩, |1⟩} ; +7 +if qxy is green then qxy is measured with +M(0); +8 +if qxy is red then qxy is measured with +M(π/2); +9 +end +10 +Bob prepares the encoding logical qubit +{|+θi⟩L}. +11 end +12 return {|+θi⟩L}. +In the following, we will give the [[7,1,3]] circuit as +an example to prepare quantum error-correcting codes ac- + +Figure 4. Realization of the [[7,1,3]] encoding circuit on cluster state. +Circles in green, red and white represent cluster qubits measured in the +eigenbasis of X, Y , Z, respectively. +cording to Prtocol.1. In Fig.2(a), we note that the depth of +[[7,1,3]] circuit is low, and the frequently used gate is only +CNOT. If the CNOT gate can be implemented at a high +successful rate, the successful rate of preparation will also be +high. According to quantum computation model on cluster +states [27], the quantum gates of [[7,1,3]] circuit can be +transformed into cluster state, and sequentially measure each +qubit on cluster state to prepare quantum error-correcting +codes. +Based on Protocol.1, Alice firstly sends data photon +pulses and ancilla photon pulses to Bob. Bob prepares data +qubit based RBSP with two decoy states [6], and then utilize +a series of ancilla qubits and prepared qubit to build the +initial cluster state. According to [[7,1,3]] circuit in Fig.2, +Alice can transform each quantum gate into measurement +angles on cluster state, as shown Fig.4. After then, Bob +uses the computational basis {|0⟩, |1⟩} to eliminate the +redundant qubits according to Alice’s requirements. The +remaining qubits on cluster state are used to prepare quan- +tum error-correcting code based on Alice’s measurement +basis M(δ), which is defined by orthogonal projections on +|±δ⟩ = (|0⟩ ± eiδ|1⟩)/ +√ +2. The parameter δ ∈ [0, 2π] is +called the measurement angle. For δ = 0 or π/2, one obtains +the X or Y Pauli measurement. Obviously, the measurement +will be understood as destructive measurements. The mea- +surement outcome at qubit i will be denoted by si ∈ Z2. We +take the specific convention that si = 0 if the state collapses +to |+δ⟩ under the corresponding measurement, and si = 1 +if to |−δ⟩. +In the process of eliminating redundant qubits, the struc- +ture information of the underlying cluster state can be +achieved to Bob, which may bring about the leakage of +quantum gates used. However, Alice only needs to ensure +that the encoding logical qubit |+θi⟩L is unknown to Bob +in preparation, and quantum gates of the encoding circuit +can be public to Bob. Three kinds of measurement ba- +sises are needed in preparation, including the computational +basis {|0⟩, |1⟩}, measurement basises M(0) and M(π/2), +which are corresponding to the eigenstates of Z, X, Y in +Pauli gates, respectively. These measurement basises are +independent of the polarization angle θi, which means that +the information of θi is not leaked to Bob in measurement +computing. Hence, cluster state can be used to prepare +quantum error-correcting codes, which are also the unknown +encoding logical qubits to Bob. +3.2. Fault-tolerant computation of quantum error- +correcting codes +Based on the RBSP with two decoy states, the polariza- +tion angle of prepared qubit θ is ϵ-blind, and Bob can not get +any information about it. The prepared quantum state as data +qubit is combined with a series of ancilla qubits to prepare +the required quantum error-correcting code |+θ⟩L according +to Protocol.1. In the whole preparation process, Bob are +unable to obtain information about θ. Hence, these prepared +quantum error-correcting codes are unknown logical qubits +to Bob. Every encoded logical qubit can be considered as +logical unit to take the place of original qubit on brickwork +state. The new brickwork state can be used to do fault- +tolerant quantum computation. Based on the original UBQC, +we propose a fault-tolerant blind quantum computation with +quantum error-correcting codes , as shown in protocol.2. +Protocol 2: Fault-tolerant blind quantum compu- +tation with quantum error-correcting codes +Input: data weak coherent pulses with polarization +ρσ, σ ∈R {kπ/4 : 0 ≤ k ≤ 7}, and ancilla +pulses with polarization |+⟩. +Output: measurement result (sx,y)nm +1,1 . +1 Alice sends data and ancilla pulses to Bob. +2 for i=1 to S; do +3 +Bob uses data pulse ρσ to prepare the required +qubit |+θi⟩, and uses it and a group of ancilla +qubits {|+⟩} to prepare the desired encoded +logical qubit |+θi⟩L. +4 end +5 Bob uses these encoded logical qubits {|+θi⟩L}S +1 +to build the encoded brickwork state. +6 for x=1 to n; y=1 to m; do +7 +Alice calculates angle φ′ +x,y, the beginning +values sX +0,y = sZ +0,y = 0; and then calculates +measurement angle +δx,y = φ′ +x,y + θx,y + πrx,y,rx,y ∈R {0, 1}. +8 +Bob measures encoded logical qubit |ψx,y⟩L in +the base {|+δx,y⟩L, |−δx,y⟩L}, return +measurement result sx,y ∈ {0, 1}. +9 +if rx,y = 1 then flip sx,y; +10 +if rx,y = 0 then continue; +11 end +12 return sx,y. +In the following, for a simple example, as shown in +Fig.5(a), we use our Protocol.2 to illustrate the delegated +fault-tolerant quantum computation. Firstly, Alice sends +weak coherent pulses to Bob, which includes data states +and ancilla states. The polarization angles of data pulses are +selected randomly from {kπ/4 : 0 ≤ k ≤ 7}, and the ancilla +is |+⟩. Secondly, Bob uses the data pulse ρσ to prepare the +required qubit |+θi⟩ based on RBSP with two decoy states +[6], and uses it and a group of ancilla qubits {|+⟩} to build +cluster state to prepare the desired encoded logical qubit +|+θi⟩L, as shown in Fig.4. Bob uses these encoded logical + +qubits {|+θi⟩L}S +1 to build the encoded brickwork state, as +shown in Fig.5(c). Finally, the interactive measurements +are performed between Alice and Bob. Alice transforms +the quantum circuit to fault-tolerant quantum circuit, as +shown in Fig.5, and then calculates angle φ′ +x,y based on +the desired angle φx,y and previous measurement outcomes. +Let sX +x,y = ⊕i∈Dx,ysi be the parity of all measurement +outcomes for qubits in Xx,y, where Dx,y ⊆ [x − 1] × [m] is +a set of X-dependencies. Similarly, sZ +x,y = ⊕i∈D′x,ysi is the +parity of all the measurement outcomes for qubits in Zx,y, +where D′ +x,y ⊆ [x − 1] × [m] is a set of Z-dependencies. +We assume that the dependency sets Xx,y and Zx,y are +obtained via UBQC [3]. Then, the actual measurement angle +is φ′ +x,y = (−1)sX +x,yφx,y + sZ +x,yπ. The measurement angles +{δx,y} can be calculated according to the original UBQC +[3], i.e. δx,y = φ′ +x,y + θx,y + πrx,y, where rx,y is randomly +chosen in {0, 1}, as shown in Fig.5(c). Afterward, the fault- +tolerant measurements on the encoded brickwork state are +used to realize the fault-tolerant quantum computing. This +procedure is repeated in turn on the encoded brickwork state +until the desired quantum computation result is obtained. +Not that arbitrary quantum rotation gate and Controlled- +NOT (CNOT) gate can be used to build an universal gate +group [28]. Hence, their own fault-tolerant logical gates can +be also composed into the universal logical gate group, +which can be used to implement arbitrary fault-tolerant +quantum gate, as shown in Fig.5(d). +In Protocol.2, we note that Alice only use cluster state +to prepare quantum error-correcting codes according to Pro- +tocol.1. In contrast to brickwork state, cluster state can re- +duce quantum resource consumption. Although the encoded +circuit in preparation may be leaked to Bob, the prepared +quantum error-correcting codes are unknown to Bob. Hence, +these codes can be used as logical unit to build encoded +brickwork state to do fault-tolerant quantum computation. +The fault-tolerant quantum gates are needed to transfer +on the encoded brickwork state. For each encoded logical +qubit on brickwork state, the measurements are performed +simultaneously. +3.3. Security Analysis +If the ideal joint state shared by the client and the server +can be described by the state πideal +AB , then a malicious server +can not achieve anything about client’s information. For any +UBQC protocol, the ideal joint state πideal +AB +is evolved into +one of the family of states, i.e. F(πideal +AB ). Due to the security +holds for any action of the server, any state of the family is +equally blind. In order to analyse the security in a realistic +implementation, we consider the settings where the client +sends general states ρθi instead of the perfect state |+θi⟩. +We can introduce the definition of ϵ blindness [4]: +Definition 1. A UBQC protocol with imperfect preparation +is ϵ-blind, if the trace distance between ideal state πθi +AB +and realistic state πρθi +AB is less than ϵ: +min +π +θi +AB∈F +1 +2 +���π{ρθi} +AB +− π{θi} +AB +��� ≤ ε. +(5) +U1 +U2 +U3 +1,1 + +1,2 + +1,3 + +1,4 + +2,6 + +2,1 + +2,2 + +2,3 + +2,4 + +3,6 + +3,1 + +3,2 + +3,3 + +3,4 + +3,7 + +4,1 + +4,2 + +4,3 + +4,4 + +2,7 + +2,8 + +3,8 + +2,5 + +3,5 + +U1 +U2 +U3 +1,5 + +1,6 + +1,7 + +1,8 + +4,8 + +4,7 + +4,6 + +4,5 + +FT +U1 +FT +U2 +FT +U3 +Fault- +tolerant +U3 +0 +0 +  +Z +R + +L +  +X +R + +L +  +Z +R + +L + +' +Z +R + +L + +' +X +R + +L + +' +Z +R + +L + + + +' + +' + +' + +0 +0 +0 +0 +0 +4 + +4 + +4 + + +L +L +L +L +L +L +L +L +L +L +L +L +L +L +L +L +L +L +L +L +L +L +L +L +L +L +L +L +L +L +L +L +L +L +L +L +L +L +L +L +L +L +L +L +L +L +L +L +L +L +L +L +L +L +L +L +(a) +(b) +(d) +(c) +Figure 5. An example illustration of fault-tolerant blind quantum compu- +tation with quantum error-correcting codes. (a) A simple quantum circuit +including three quantum gates U1, U2, U3, each thin line represents a +qubit. (b) The fault-tolerant quantum circuit, every thick box represents +a fault-tolerant quantum gate, and each thick line represents a encoded +logical qubit with 7 qubits, i.e. [[7,1,3]] code. (c) Implementation of the +fault-tolerant quantum computation on brickwork state, and each angle δ +in circles presents a measurement basis M(δ), each orange thick circle +with subscript L represents an encoded logical qubit, each gray thick box +with subscript L represents an output logical qubit. (d) Two brick models +built with the encoded logical qubits, i.e. arbitrary rotation logical gate and +CNOT logical gate, which can be used to form an universal logical gate +group. +Based on Protocol.2, we can delegated a remote Bob +to do a fault-tolerant quantum computation. If the Bob is +malicious or dishonest, does our protocol still ensure the +privacy of Alice’s information ? To solve this problem, we +will demonstrate the security of protocol.2 to ensure that +Bob does not get any computational information except the +scale of encoded brickwork state. In fault-tolerant comput- +ing, Bob only has some information about the measurement +bases and the encoded states. Therefore, one needs to show +that Alice’s information is independent of Bob. + +Theorem 2. The fault-tolerant protocol.2 is ϵ-blind while +leaking at most (n, m), which is the dimension of +computational scale, provided that the prepared encoded +logical qubit |+θ⟩L is ϵ-blind. +Proof: Let (n, m) be the dimension of the brick- +work state. According to definition of brickwork state [3], +it ensures that Bob does not get any information on the +underlying computation except n and m. +On the one hand, one needs to prove that Bob’s mea- +surement bases are ϵ-independent of Alice’s computing +information. Alice has two secret bit strings in the fault- +tolerant quantum computing. One is the polarization angle +θ of quantum error-correcting code, while the other is +the actual measurement base φ′, which is determined by +computational tasks and previous measurement results. The +measurement angle sent to Bob can be calculated according +to δ = φ′+θ+πr, r ∈R {0, 1}, and then it is used to perform +fault-tolerant measurements on the encoded brickwork state, +where r is a random bit string determined by Alice, which +is unknown to Bob. Since the polarization angle θ is also +ϵ-uniform in {kπ/4|k = 0, 1, 2, ..., 7}, so θ + πr is also +ϵ-uniform random. Hence, Bob’s measurement angle δ is +ϵ-independent of the computational information ψ′. +On the other hand, the information from encoded quan- +tum states is ϵ-blind to Bob. Since the bit rx,y is random, +each encoded logical qubit for Bob will have one of the +following two possibilities: +1)If rx,y = 0 and δx,y = φ′ +x,y + θx,y, then |ψx,y⟩L = +1 +√ +2 +� +|0⟩L + ei(δx,y−φ′ +x,y)|1⟩L +� +. +2)If rx,y = 1 and δx,y = φ′ +x,y + θx,y + π, then |ψx,y⟩L = +1 +√ +2 +� +|0⟩L − ei(δx,y−φ′ +x,y)|1⟩L +� +. +Without loss of generality, if the angle δ is fixed, θ +depends on φ′, but the random bit rx,y is unknown to Bob. +Hence, the encoded logical qubit for Bob is a mixed state +of two cases, which is ϵ-independent of Alice’s computing +information φ′. □ +In summary, according to Protocol.2, we know that +Alice can delegate Bob to prepare ϵ-blind quantum error- +correcting codes to correct errors, meanwhile these codes +can be used as encoded logical qubits to do subsequent +fault-tolerant quantum computation. In addition, Protocol.2 +only requires Alice to send weak coherent pulses to prepare +quantum error-correcting codes, which can decouple Alice +dependency on quantum computing and quantum memory. +The number of ancilla qubits increases only by a constant +factor with the increasing computational scale, rather than a +linear increase in the original UBQC [3]. Hence, Protocol.2 +have advantages from Alice’s perspective of saving resource +consumption and getting rid of quantum dependence. +4. Concatenation codes and resource require- +ments +There is a beautiful construction based on concatenated +codes which can be used to reduce the effective error rate +achieved by the computation even further. The idea is to +recursively apply the scheme described above for simulating +a circuit using an encoded circuit, constructing a hierarchy +of quantum circuits Level=1 (the initial encoded circuit), +Level=2,3 and so on. In the first stage of this construction, +each qubit in the original circuit is encoded in a quantum +code whose qubits are themselves encoded in a quantum +code, whose own qubits are encoded yet again, and so on, +as illustrated in Fig.6. Hence, in order to improve error +correction performance of qubits, we study the application +of concatenation codes in UBQC as shown in Protocol.3. +Protocol 3: Fault-tolerant blind quantum compu- +tation with concatenation codes +Input: Data pulses and ancilla pulses +Output: Measurement results (sx,y)nm +1,1 +1 for i=1 to S; do +2 +Bob uses data pulse ρσ to prepare the required +qubit |+θi,0⟩ based on RBSP with two decoy +states +3 +if l >= k then +4 +Output(|+θi,l⟩L) +5 +else +6 +Bob uses |+θi,l−1⟩L and a group of ancilla +qubits {|+⟩} to build cluster state, and +then prepare the desired encoded logical +qubit |+θi,l⟩L based on the lth-level +concatenated circuit; l++. +7 +end +8 end +9 Bob uses these encoded logical qubits +� +|+θi,k⟩L +�S +1 +to build the encoded brickwork state. +10 The interactive measurement process, refer to +Protocol.2 to perform. +11 return sx,y. +we use two-level concatenated [[7,1,3]] codes to reduce +error rate, and each [[7,1,3]] code encodes a single qubit +using a block of 7 qubits. When examine one of the 7 +qubits in this block with higher resolution, we first note +that it is itself an encoded sub-block. With this method, +the complexity of quantum error correction does not grow +so sharply as we increase the error-correcting capacity of +quantum code. Note that the [[7,1,3]] code can correct one +error. If the probability of error per actual physical qubit +at the lowest level of the code is e0, and these errors are +uncorrected, and the correction is fault-tolerant, then the +probability of a correction failure in [[7,1,3]] code is of +order e2 +0. If we concatenate the code to construct a new +block, then an error occurs in the block only if two of the +sub-blocks of size 7 fail, which occurs with a probability of +order e4 +0. +Let the error probability of each block at level i be +denoted as ei. At each level of the concatenated code, the +block fails if there are errors in at least two contained +sub-blocks. For the i-level concatenated code, the failure +probability can be estimated according to Eq.6. + +FT +Preparation + +H +H +H +H +H +H +H +H +H +H +H +H +H +H +H +FT +Correction + +FT +Preparation + +FT +Correction + +FT +Syndrome Measurement + +FT +Syndrome Measurement + +Ancilla State +0 L +0 L +0 L +0 +0 +0 +Ancilla State +Ancilla Qubits Pool +Ancilla Qubits Pool +2nd Level +1st Level +Figure 6. +The two-level concatenated [[7,1,3]] codes. When inspected +at higher resolution, each qubit or quantum gate in the block is itself an +encoded sub-block. Each block comprises fault-tolerant syndrome measure- +ment, fault-tolerant correction and fault-tolerant quantum gate(a transversal +application of quantum gate). +ei = +7 +� +k≥2 +� +7 +k +� +ek +i−1 +≈ +� +7 +2 +� +e2 +i−1 + o +� +e2 +i−1 +� +≈ (21e0)2i +21 ++ o +� +e2i +0 +� +(6) +In order to significantly reduce the probability of error +significantly, the condition e0 < 1/21 must be satisfied in +the concatenation codes (the concatenation codes hold if and +only if (21e0)2i ≪ 1). Further, we can make the error rate +arbitrarily small by adding sufficient levels of concatenation. +If the concatenated [[7,1,3]] code circuit is delegated to Bob +to prepare quantum error-correcting code on the cluster state +in Protocol.2, then the number of required ancilla pulses +will increase with the number of levels. Since a required +data qubit is used to prepare an encoded logical qubit using +[[7,1,3]] encoding circuit, the number of required data qubits +is consistent with the RBSP with two decoy states, which +remaining unchanged. +1-level coding +2-level coding +3-level coding +A data qubit +Figure 7. The encoding structure for a date qubit in 3-level concatenation +code. +In Protocol.3, the weak coherent pulses are sent to Bob, +which contain data pulses and ancilla pulses. According +to RBSP with two decoy states, data pulses are used to +prepare the required data qubits {|+θi⟩}S +1 , and each data +qubit is encoded into a required logical qubit. Hence, the +number of data pulses is equal to RBSP with two decoy +states, denoted as N d [7]. The number of ancilla pulses +depends on encoding circuit on cluster state, as shown in +Fig.4. Note that the required number of ancilla qubits is a +constant when preparing an encoded logical qubit, denoted +as C. If the transmittance of the quantum channel is T, and +the computational scale is S, then the number of ancilla +pulses N a = CS/T. Hence, the total number of required +pulses in Protocol.3 is the sum of data and ancilla pulses, +i.e.N = N d + N a. +The number of required ancilla pulses consists of the +preparations, syndrome measurements and corrections. Note +that, the required ancilla qubits in the syndrome measure- +ments and corrections can be reused in our fault-tolerant +protocol, therefore, this part of the resource consumption of +ancilla qubits can be ignored. One only needs to analyze the +preparation consumption of ancilla qubits. In Protocol.3, we +use cluster state to prepare the encoded logical qubits. Based +on circuit model, we know that the block size of ancilla +qubits to prepare an encoded logical qubit is a constant +C. According to the encoding structure for a data qubits +in Fig.7, the block size of ancilla qubits is the sum of +the required number at all levels. If we use an n-level +concatenation code to prepare an encoded logical qubit, the +number of ancilla qubits is denoted as N a +n, and its value can +be calculated as follows: +N a +n = +n +� +i=1 +7i−1C = (7n − 1) C +6 +. +(7) +The number of required data pulses is consistent with RBSP +with two decoy states. According to Eq.1, we have N d ≥ +N L,v1,v2. The required total number of pulses in the n-level +concatenated [[7,1,3]] code, i.e. Nn, is estimated as follows: +Nn = N d + N a +n +≥ N L,v1,v2 + (7n − 1) CS +6T +≈ S +T +� +� +ln (ϵ/S) +pµµ ln +� +1 − pL,v1,v2 +1 +� + (7n − 1) C +6 +� +� +(8) +where S is the computational scale, ϵ is security parameter +of the blind quantum computation, T is the transmittance +of the quantum channel between Alice and Bob, µ,v1, v2 +are the average number of signal photons and two decoy +photons, respectively, the proportion of single photon in +the signal states is described as p1, and its lower bound +is pL,v1,v2 +1 +, the probability of signal pulses chosen by the +client is defined as pµ. +In the RBSP protocol with two decoy states, for the +computation size S, if an error occurs in prepared qubits, +the whole preparation protocol will fail. We assume that the +preparation protocol is a repeatable Bernoulli’s experiment. +If the error probability of each prepared qubit is e0, the +successful probability of preparation in each experiment +will be (1 − e0)S. After repeating k times, the successful +probability of preparation will be 1 − (1 − (1 − e0)S)k. + +00In the n-level concatenated code, the successful probability +of each preparation is (1 − en)S. If the successful proba- +bility is the same in both coded and non-coded cases, i.e. +1−(1−(1−e0)S)k = (1−en)S, then the repetition number +k can be calculated as follows: +k = ln[1 − (1 − en)S]/ ln[1 − (1 − e0)S]. +(9) +where en can be estimated according to Eq.6. In other +words, if Alice wants to use the non-coding protocol to +achieve the same successful probability as encoding, the +number of required data pulses is k times than that of the +original case, i.e. kN d. Let R describe the ratio of the +number of pulses for concatenation codes to that of the non- +coding case at the same successful probability. The ratio R is +a resource consumption function with the number of levels +n, which can be used to estimate the optimal number of +concatenated levels. The estimation of resource consumption +ratio R(n) is shown in Eq.10. +R (n) = Nn +kN d = N d+N a +n +kN d +≤ N L,v1,v2 + N a +n +kN L,v1,v2 += +ln +� +1 − (1 − e0)S� +ln +� +1 − (1 − en)S� +�(7n − 1) CS +6TN L,v1,v2 +1 +� +(10) +where N L,v1,v2 is the lower bound of the number of required +pulses in the RBSP with two decoy states, which can be +estimated according to Eq.1. When the computation scale +S is fixed, the ratio R (n) ∼ (7/2)n according to Eq.6 +and Eq.10. Hence, the ratio R has an exponential growth +trend for a large number of levels n. For a high-level +concatenation code, a large number of qubits are used for +repeated encoding. In our concatenation code, we not only +need to consider improving the successful preparation prob- +ability, but also reducing the resource consumption. Hence, +we only take into account the optimal level in the low- +level concatenation code (0 ≤ n ≤ 4). When the resource +consumption ratio R(n) reaches its minimum, the proportion +of the number of the encoding pulses in the non-coding is +the lowest. At this point, only a small number of pulses for +concatenation code can be used to achieve a high successful +probability, in which case the corresponding level n is the +optimal one. +According to Eq.8, the number of required data pulses +is unchanged with the increasing number of levels accord- +ing to, and the number of required ancilla pulses grows +exponentially with the increasing number of levels in the +concatenation code. Furthermore, the optimal level n can +be estimated according to the resource consumption ratio +R(n). +5. Performance evaluation +In simulation, our computer uses an the Intel(R) +Core(TM) i7-6700HQ CPU with 12.0GB RAM and Win +10 pro OS to complete the experiment through MATLAB +software. The overall transmittance T (including fiber trans- +mittance and detection efficiency) is calculated as follows: +T = ts · ηs · 10−αL/10 +(11) +TABLE 1. THE SIMULATION PARAMETERS FOR OUR PROTOCOL +α +tS +ηS +µ +v1 +v2 +pµ +pv1 +pv2 +S +ϵ +e +0.2 0.45 0.1 0.6 0.125 +0 +0.9 0.05 0.05 1000 1010 0.01 +The parameter α is the loss coefficient measured in +dB/km, L is the length of the fiber in km, ts is denoted as the +internal transmittance of optical components in Bob’s side, +and ηs is detector efficiency in Bob’s side. The mean photon +number of signal states and two decoy states are represented +as µ, v1, v2, respectively. The chosen probability of signal +states and two decoy states are denoted as pµ, pv1, pv2, +respectively. The computation scale is S. The required blind- +ness of the UBQC is denoted as ϵ. The error probability +of each qubit is denoted as e0. Based on Fig.2 and 4, we +can calculate the number of ancilla qubits, C = 1774. The +relative parameters are set in Table.1(refer to the data in [29] +and [30]). If we use [[7,1,3]] to encode n-level concatenation +code in our protocol, the quantum resource consumption is +shown as follows. +Suppose Alice has a laser transmitter with frequency +f = 1MHZ, and Bob has full-fledged quantum computer. +According to the setting in Table.1, we can derive the +efficiency of prepared qubits in our protocols, that is the +number of qubits generated per second. Combined with +Eq.8, we can further estimate the upper bounder of the +efficiency in the concatenation code. +E = S · f/Nn +≤ S · f/ +� +N L,v1,v2 + N a +n +� +(12) +100 +101 +102 +Communication Distance(L) +106 +107 +108 +109 +1010 +1011 +1012 +1013 +The Total Number of Required Pulses(N) +Asymptotic Case +Non-Coding Case +Our Coding Case +Figure 8. The total number of required pulses N with the same probability +of successful preparation vs. the communication distance between Alice and +Bob. The green and red lines show the simulation results of our Protocol.1 +with coding and non-coding case, respectively. The black line shows the +simulation results in the asymptotic case (an infinite data-size and near- +perfect qubits preparation). +In Fig. 8, we can see that the total number of required +pulse for the coding case in Protocol.1 is less than for the + +0 +10 +20 +30 +40 +50 +60 +70 +80 +90 +100 +Communication Distance L (km) +0 +100 +200 +300 +400 +500 +600 +700 +Efficiency E (qubits/s) +Asymptotic Case +Non-Coding Case +Our Coding Case +Figure 9. +The preparation efficiency E with the same probability of +successful preparation vs. the communication distance L. The green, red +and black lines show efficiency curves of our protocol.1 with coding, non- +coding, asymptotic case respectively. +non-coding case in RBSP with two decoy states under the +same probability of successful preparation. Since the initial +error probability of each qubit is e0, the error probability +of each quantum error-correcting code prepared is e2 +0, and +the probability of successful preparation with size S is +(1 − e2 +0)S. The error probability of the encoded logical +qubits prepared is much lower that of RBSP with two decoy +states. In order to obtain the same probability of successful +preparation, non-coding case needs to be repeated k times. +As the communication distance increases, the value of N +grows rapidly, which indicates that the channel loss and +qubit error rate have a great influence on N. For long +distance communication, the advantages of our Protocol.1 +are more outstanding than non-coding case, which means +that the required number of pulses N is closer to asymptotic +case(an infinite data-size and without qubit error [6], [29]). +In Fig. 9, we note that the efficiency of qubits prepared +gradually decreases with increasing communication distance +until it tends to 0. Compared with non-coding case, the +efficiency of qubits for coding case in our Protocol.1 is +closer to the asymptotic limit. Obviously, as the error rate +decreases from e0 to e2 +0, more pulses as ancilla qubits are +used to prepare quantum error-correcting codes, which will +lead to a sharp drop in efficiency. +According to Eq.(10), we know that the resource con- +sumption ratio R(n) is the proportion of the number of +pulses for concatenated coding in that of the non-coding +case. In order to reach the optimal level, the partial derivative +∂R (n) /∂n = 0 to solve the extreme value. Nevertheless, +we note that the partial derivative is so complicated that it +is hard to solve. Hence, we use the simulation experiment +to estimate the optimal level n, as shown in Fig.10. We note +that the optimal n is about 2, and the resource consumption +ration R(n) reaches the minimum value. It means the 2-level +concatenation code can utilize relatively fewer quantum re- +sources than other levels to achieve a better error-correcting +performance. +In Fig.11, the number of required pulses N in each +case is increasing trend with the communication distance L, +which indicates that both the channel loss and qubit error +1 +1.5 +2 +2.5 +3 +3.5 +4 +The Number of Concatenated Levels(n) +0 +0.005 +0.01 +0.015 +0.02 +0.025 + The Ratio of Resoure Consumptio(R) +Asymptotic Case +25km Case +50km Case +100km Case +Figure 10. The ratio of resource consumption R(n) vs. the number of +levels in the concatenation code. The blue, green and red lines show the +simulation results of resource consumption ratio as distance is 25km, 50km +and 100km, respectively. The black line shows simulation results in the +asymptotic case. +101 +102 +Communication Distance(L) +106 +108 +1010 +1012 +1014 +1016 +The Total Number of Required Pulses(N) +Asymptotic Case +Non-Coding Case +1-Level Concatenated Code +2-Level Concatenated Code +3-Level Concatenated Code +4-Level Concatenated Code +Figure 11. The total number of required pulses N with the same probability +of successful preparation vs. the communication distance between Alice +and Bob. The green, red, yellow, cyan and blue lines show the simulation +results of our protocol with the 1,2,3,4-level concatenation code and non- +encoding case, respectively. The black line shows the simulation results in +the asymptotic case. +rate have a great influence on N. Compared with the non- +coding case at the same successful probability, the number +of required pulses N for encoding is closer to asymptotic +case(an infinite data-size and near-perfect qubits prepara- +tion). The reason is that the error probability of the encoded +logical qubits prepared by the concatenation circuit is much +lower than the non-coding case. In order to obtain the same +successful probability, it is necessary for RBSP with two +decoy states to be repeated k times, which results in the +waste of a vast number of pulses. Further, we can see that the +number of required pulse in the 2-level concatenation code +is less than other levels case at same successful probability, +which shows the 2-level concatenation code can achieve +a better error-correcting performance in the finite quantum +resources case. +In Fig.6, the preparation efficiency E in the asymptotic +case is much higher than in the real case. Since the fluctu- +ation of finite data and imperfect preparation of required +qubits are inevitable in real UBQC, a large number of +pulses as ancilla qubits are used to deal with the impacts + +0 +10 +20 +30 +40 +50 +60 +70 +80 +90 +100 +Communication Distance L (km) +0 +100 +200 +300 +400 +500 +600 +700 +Efficiency E(qubits/s) +Asymptotic Case +Non-Coding Case +1-Level Concatenated Code +2-Level Concatenated Code +3-Level Concatenated Code +4-Level Concatenated Code +Figure 12. +The preparation efficiency E with the same probability of +successful preparation vs. the communication distance L. The green, red, +yellow, cyan and blue lines show the simulation results of our Protocol.3 +with 1,2,3,4-level concatenation code and non-coding case, respectively. +The black line shows simulation results in the asymptotic case. +0 +10 +20 +30 +40 +50 +60 +70 +80 +90 +100 +Communication Distance L (km) +0 +0.05 +0.1 +0.15 +0.2 +0.25 +0.3 +0.35 +0.4 +Efficiency E(qubits/s) +Non-Coding Case +1-Level Concatenated Code +2-Level Concatenated Code +3-Level Concatenated Code +4-Level Concatenated Code +Figure 13. +The preparation efficiency E with the same probability of +successful preparation vs. the communication distance L. The figure is +a detail view to show simulation results with 1,2,3,4-level concatenation +code and non-coding case in our Protocol.3. It is the partial enlargement +of Fig.12. +of fluctuation and qubit errors, which leads to a drop in +efficiency. In order to better show our simulation results, we +give a partial enlargement of Fig.12, as shown in Fig.13. +Not that the preparation efficiency E has a sharp decline +trend as the communication distance increases. The error +probability of quantum error-correcting code prepared is +e1 +06, the probability of successful preparation with size S is +(1 − e1 +06)S. with the same probability, we demonstrate the +simulation results of the efficiency for the non-coding case +and the concatenation codes at different levels. Although the +high-level concatenation code can obtain a lower error rate, +it also requires massive quantum resource consumption, and +results in a very lower efficiency, which is not applicable in +the reality. In UBQC with limited quantum resource, we +can see that the 2-level concatenation code can achieve the +optimal preparation efficiency in Fig.13. Hence, in the 2- +level concatenation code, our Protocol.3 can be used not +only to improve the preparation efficiency, but also to save +quantum resources. It is also very significant to improve the +error-correcting performance in fault-tolerant blind quantum +computation. +6. Related works +In original fault-tolerant UBQC [3], fault-tolerant quan- +tum computing is performed on the top of brickwork state. +Thus, many SWAP gates are used to process those CNOT +gates acting on non-adjacent qubits, which results in a +linear increase in size of brickwork state with the increasing +computational scale. The required number of prepared qubits +is also a linear function multiple of the original UBQC pro- +tocol [3]. In our protocol.2, the number of ancilla qubits is +more constant than the original UBQC protocol. Compared +to Chien’s two fault-tolerant protocols [12], our protocol +only requires Alice to send weak coherent pulses to pre- +pare quantum error-corrrecting codes, which can free Alice +dependence on quantum computing and quantum memory. +In addition, since different quantum gates have different +sizes, the used quantum gates can be guessed by Bob when +eliminating redundant qubits. However, our protocol.2 only +prepares quantum error-correcting codes on cluster state, +which can reduce quantum resource consumption while +ensuring that Bob is unknown to the prepared state, and then +performs the fault-tolerant quantum computing on encoded +brickwork state, which can guarantee our protocol is ϵ-blind. +However, the current quantum technology still struggles +with noises and imperfect measurement, which results in a +high error rate in the fault-tolerant UBQC. To this end, the +concatenated stabilizer codes can be considered to reduce +qubit requirements by tailoring circuits to suppress the +dominant effect of qubit errors [31], [32]. Chamberland, +Jochym and Laflamme et al. concatenated the 7-qubit Steane +code with the 15-qubit Reed-Muller code to implement +universal fault-tolerant quantum computation without magic +state distillation [33]. Paul Webster et al. presented a gen- +eral framework for universal fault-tolerant logic with no-go +theorem and stabilizer codes [34], which can be applied to +a wide range of stabilizer code families, including concate- +nated codes and conventional topological stabilizer codes. +A fault-tolerant quantum computation with concatenated +quantum codes was proposed by Chamberland, Noh and +Preskill et al. [32], which can reduce the consumption of +qubits by tailoring the quantum error-correcting codes to +suppress the dominant phase-flip errors. To avoid coher- +ent errors, Yingkai Ouyang proposed rotated concatenated +stabilizer codes [35], namely, concatenating an [[n,k,d]] +stabilizer outer code with constant-excitation inner codes. +It was shown that when the stabilizer outer code is fault- +tolerant, the concatenated codes are immune from coherent +phase errors. The concatenation codes above are useful for +improving the fault-tolerant threshold and reducing quantum +resource consumption in a realistic UBQC system. +In recent years, many encoding methods [14], [36], [37], +[38] were proposed to deal with this problem, such as +concatenation code [38], RHG Lattice code [14] and surface +code [37]. However, these remain daunting for the complex +encoding structures and large resource consumption. In this +paper, we propose a general fault-tolerant blind quantum +computation with concatenation codes to reduce the error +rate of logical qubits, thereby improving the fault-tolerant + +threshold of UBQC. Besides, we optimize the number of +required pulses in the concatenated code to reduce quantum +resource consumption. In fault-tolerant UBQC with limited +quantum resources, the 2-level concatenated code can obtain +the optimal performance. +7. Conclusions +In the paper, we propose a method to realize quantum +error-correcting codes on cluster state. A fault-tolerant blind +quantum computation with quantum error-correcting codes +is proposed to address errors in the quantum computation. +In order to improve the error-correcting performance, the +stabilizer codes are used for multi-level concatenating to +improve the fault-tolerant threshold of qubits. To reduce +quantum resource consumption, we also analyse the number +of required pulses in each level concatenation code. Due to +a large number of quantum resources are required in the +high-level concatenation codes, the low-level concatenated +codes (0 ≤ n ≤ 4) are only considered in this paper. The +optimal level in the concatenation codes is about 2, and +the resource consumption ratio reaches the minimum value, +which means that the number of required pulses in the 2- +level concatenation codes is less than other levels under +the same probability of successful preparation. From the +perspective of reducing quantum resources, the simulation +results show that the 2-level concatenated code can achieve +a better performance than other levels with the increase of +communication distance. Our encoded scheme can also be +used to other blind quantum computation protocols with +different graphs. In order to further improve the practical +performance of UBQC, we will continue to design the con- +catenated codes between other quantum codes and stabilizer +code in the future. Meanwhile, we will also attempt to use +topological codes with high fault tolerance to do a delegated +computing in the case of limited quantum resources. +Acknowledgments +This work is supported by Space Science and Tech- +nology Advance Research Joint Funds(Grant Number: +6141B06110105), National Natural Science Foundation of +China(Grant Number: 61771168) and National Natural Sci- +ence Foundation of China (grant number: 62071151). +References +[1] +A. M. Childs, “Secure assisted quantum computation,” Quantum Info. +Comput., vol. 5, no. 6, pp. 456–466, 2005. +[2] +P. Arrighi and L. Salvail, “Blind quantum computation,” International +Journal of Quantum Information, vol. 4, no. 05, pp. 883–898, 2006. +[3] +A. Broadbent, J. Fitzsimons, and E. Kashefi, “Universal blind +quantum computation,” in Foundations of Computer Science, 2009. +FOCS’09. 50th Annual IEEE Symposium on. +IEEE, 2009, Confer- +ence Proceedings, pp. 517–526. +[4] +V. Dunjko, E. Kashefi, and A. 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Guo, “The concatenation of the +gottesman-kitaev-preskill code with the xzzx surface code,” arXiv +preprint arXiv:2207.04383, 2022. + diff --git a/B9A0T4oBgHgl3EQfAP-Y/content/tmp_files/load_file.txt b/B9A0T4oBgHgl3EQfAP-Y/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..6bec5fa995dd218ebb44e938acd325ae483218d7 --- /dev/null +++ b/B9A0T4oBgHgl3EQfAP-Y/content/tmp_files/load_file.txt @@ -0,0 +1,933 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf,len=932 +page_content='Applying the Quantum Error-correcting Codes for Fault-tolerant Blind Quantum Computation Qiang Zhao Department of Computer Science and Engineering The Chinese University of Hong Kong Hong Kong, China Email:qiangzhao@cuhk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='hk Qiong Li School of Computer Science and Technology Harbin Institute of Technology Harbin, China Email: qiongli@hit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='cn John C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Lui Department of Computer Science and Engineering The Chinese University of Hong Kong Hong Kong, China Email:cslui@cse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='cuhk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='hk Abstract—The Blind Quantum Computation (BQC) is a dele- gated protocol, which allows a client to rent a remote quantum server to implement desired quantum computations, while keeping her inputs, outputs and algorithms privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' However, the qubit errors during the quantum computation are realistic issues that are needed to consider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' In this paper, we propose a fault-tolerant blind quantum computation protocol with quantum error-correcting codes to avoid the accumulation and propagation of qubit errors during the computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Meanwhile, we also present the ϵ-blindness in our protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' To improve the error correction performance, the concatenated codes are used in our protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' We further present the resources consumption of photon pulses by the optimal level concatenation codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' The simulation results show that our scheme not only can improve the preparation efficiency but also reduce quantum resources, which shows a significant improvement to a realistic fault- tolerant BQC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Introduction Quantum cloud computing is an innovative research at the interface of maths, computer science and physics, which promises revolutionary improvements in communications and computations, dealing with tasks that are impossible for traditional computers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Quantum cloud computing is a distributed system composed of quantum computer as server and simple quantum devices as clients that exchange quantum information and computing instructions through quantum channel and classical channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' It will be conve- nient for clients to rent quantum computing resources to achieve desired quantum computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' However, classical communication network protocols can not ensure the secu- rity of quantum information and computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' One possible solution to address this problem is via the Blind Quantum Computation (BQC) protocol supported by quantum cloud computing, which enables a client(say Alice) with limited quantum technology to delegate a computing task to a remote quantum server(say Bob) while ensuring computing information privacy [1], [2], [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' In recent years, a number of BQC protocols have been proposed [4], [5], [6], [7], [8], [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Among these protocols, Universal Blind Quantum Computation (UBQC) is of particular interest as it not only can guarantee that Alice’s inputs, outputs and quantum algo- rithms are unknown to server while delegating computation, but also only requires Alice to prepare the single photon states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' As we all know, qubits and quantum gates are easily affected by environmental and devices noise, which will lead qubits and quantum gates to rotate errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' It is usually called the decoherence effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Since noise will result in more qubit and quantum gate errors as quantum computational scales up, it is very formidable challenge to realize a large scale quantum computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' However, Noisy Intermediate- Scale Quantum (NISQ) technology will be available in near future, which can help quantum computers with 50- 100 qubits to perform tasks that surpasses classical com- puters [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' UBQC is composed of quantum preparation, quantum measurement, one-way quantum communication and two-way classical communication [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' In the noisy intermediate-scale UBQC, due to the influence of noise, it is inevitable to make errors of qubits and quantum gates in each of stage [9], [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Hence, UBQC will have an interesting challenge how to deal with qubit errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Besides, the quantum gates should be also fault-tolerant to prevent the accumulation and propagation of errors in the subsequent computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Therefore, it is necessary for UBQC to utilize quantum error-correcting codes [13] or other quantum codes [14], [15] with error correcting capability as logical qubits to implement a fault-tolerant blind quantum computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Since quantum error correction imposes a heavy overhead cost in number of qubits and number of gates, it is a formidable challenge to trade off between fault tolerance and heavy resource overhead (the number of required pulses) because a large number of ancilla qubits are used to correct erroneous data qubits to ensure the required fault tolerance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' In this context, this work gives an alternative protocol to implement a fault tolerant blind quantum computation in NISQ, and optimize quantum resource consumption, in term of the number of photon pulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Moreover, our quantum error-correcting scheme can be also applied to other graphs of BQC and perform fault tolerant quantum computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='01960v1 [quant-ph] 5 Jan 2023 The main contributions of this paper are as follows: We present a remote blind quantum error-correcting codes preparation protocol on cluster states to cor- rect qubit errors and improve the successful prob- ability of required qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' The encoded circuit is transformed to cluster states instead of brickwork states, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' multi-particle entangled graph states, and then delegate server to prepare quantum error- correcting codes in the framework of measurement- based quantum computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Based on the above preparation, we propose a fault- tolerant blind quantum computation protocol with quantum error-correcting codes to avoid the accu- mulation and the propagation of qubit errors and improve the performance in a delegated quantum computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Since these quantum error-correcting codes are unknown logical qubits to server, each of them is considered as logical unit to take place of original qubit on brickwork state to perform a fault- tolerant computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Meanwhile, we also demon- strate the ϵ-blindness of our protocol in case of malicious server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' To achieve a higher fault-tolerant performance, we propose to utilize the concatenated stabilizer codes, that is to concatenate the same or different stabi- lizer codes in a multi-level encoding manner, to prepare high-quality encoded logical qubits to do fault-tolerant computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Moreover, we deduce the lower bound of quantum resource consumption in each level of the concatenation codes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' the num- ber of required pulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' To reduce quantum resource overhead in a fault- tolerant quantum computation, we build an opti- mization model of quantum resource consumption, which describes the ratio of the number of pulses in the concatenation codes to the non-coding case under the same probability of successful preparation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' When it reaches minimum, we can achieve the opti- mal number of levels in the concatenated stabilizer codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' The rest of this paper is organized as follows: In Section II, the background and technical preliminaries are intro- duced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' In Section III, we firstly present a remote blind quan- tum error-correcting codes preparation protocol on cluster state to correct qubit errors, and then propose a fault-tolerant blind quantum computation with quantum error-correcting codes to do delegated computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Furthermore, we also give a proof of the blindness of our fault-tolerant protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' In Section IV, the multi-level quantum error-correcting codes are concatenated to reduce the error rate of the generated logical qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' A lower bound of the number of required pulses is deduced with different levels in fault-tolerant blind quantum computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' In Section V, we present simula- tion results to further validate the theoretical model of the number of required pulses with concatenated codes, and give an optimal level number of the concatenation codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='In Section VI, we surveys the related works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' In Section VII, we conclude our work and provide the direction of future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Background BQC protocols can be generally divided into two cate- gories, they are measurement-based BQC and circuit-based BQC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' In 2005, Childs proposed the first BQC protocol [1], which utilized quantum circuit model and the quantum one- time pad to realize a secure quantum computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' However the client was required to have quantum memory and the ability to perform quantum SWAP gate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Since then, some other circuit-based BQC protocols [2], [16] were proposed, but they all required client to possess quantum memory, quantum measurement or limited quantum computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' In 2009, Broadbent, Fitzsimons and Kashefi proposed the Uni- versal Blind Quantum Computation (UBQC) protocol [3], which is the first measurement-based BQC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' This protocol only requires client to prepare the single photons, and other operations can be delegated to server, which can greatly reduce client’s burden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' After then, it has been also experi- mentally demonstrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Based on UBQC [17], some other measurement-based BQC protocols [7], [18], [19], [20], [21], [22] have been devised to be as practical as possible in recent years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' To verify the blindness and correctness of practical BQC, some protocols [18], [19] were also proposed in noisy channels or existing any malicious attacker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' In order to make Alice as classical as possible, multiple-server BQC protocols [20], [21] were proposed based on the shared entanglement states between servers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' For tolerating more faults or noise, some fault-tolerant BQC protocols [7], [22] were presented based on the idea of quantum error correc- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' With development of quantum technology, UBQC is gradually moving from theoretical researches to practical applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' UBQC is within the framework of a measurement-based quantum computation (MBQC) [3], [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' The underlying resource of MBQC is the brickwork states, which is multi- particle entangled graph states constructed by prepared qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' A delegated quantum computation can be imple- mented by measurements on brickwork states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Let us have a client (say Alice ) and a server (say Bob) with full-fledged quantum computer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' In preparation stage, Alice prepares a series of single qubits |+θ⟩ with random polarization θ ∈ {0, π/4, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=', 7π/4} and sends to Bob through one-way quantum channel, and Bob uses Controlled Z (CZ) gates to act on these received qubits to build brickwork state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' In interactive measurement stage, Alice carefully designs the quantum circuit based on the computational task, and trans- forms a sequence of ordered quantum gates in the circuit into the measurement angles, and then sends to Bob through two-way classical channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Bob performs measurement op- erations for each qubit on brickwork state, and returns back the results to Alice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' The measurement procedure is repeated in turn on brickwork state until the desired quantum computation result is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' The schematic diagram of UBQC is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Note that for each qubit on brickwork state, Alice can calculate the measurement angle φ′ x,y according to her desired angle φx,y and the previous measurement result [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' In order to ensure the privacy of computation, the measurement angle φx,y needs to be further processed as δx,y = φ′ x,y + θx,y + πrx,y, rx,y∈R {0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Then, the real measurement angle δx,y is sent to Bob to perform the fault- tolerant measurement for each encoded logical qubit |ψx,y⟩L on encoded brickwork state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' The measurement result sx,y is returned to Alice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' If the random bit chosen by Alice, rx,y = 1, Alice flips the measurement result sx,y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' otherwise she does nothing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Laser (Weak Coherent Pulses) Classical-Computer (Data Processing) Quantum Resources (Measurement-Based Quantum Computation) Classical-Computer (Data Processing) Measurement Angles Measurement Results Quantum Algorithm Quantum Graph States Quantum Channel Classical Channel Alice Bob Measurement Angles Measurement Results Measurement Results Measurement Bases Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' The schematic diagram of UBQC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Remote blind qubit state preparation protocol with two decoy states In UBQC, the qubits could be encoded in the polar- ization of a single photon generated by a realistic single photon source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' However, it is inevitable to send two or more identically polarized photons instead of one in any physi- cal implementation, which will destroy the perfect privacy of client.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Hence, a Remote Blind qubit State Preparation (RBSP) protocol was proposed to prepare quantum states, which can arbitrarily approach to the perfect qubits [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Moreover, this protocol can allow us to achieve ϵ-blind UBQC for any security parameter ϵ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Nevertheless, the client’s overhead, which is the number of pulses needed for preparing a qubit, is usually very large, and this is true especially for a long-distance communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Therefore, a modified RBSP with two decoy states was presented by Zhao and Li [5], [6] to optimize the number of required pulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' They showed that two decoy states protocol can achieve a smaller number of pulses than one decoy state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' The RBSP with two decoy states is summarized as follows: (1) Alice generates N weak coherent pulses which con- tain signal and two decoy states, and sends them to Bob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' (2) For each received state, Bob reports to Alice which pulses were received.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Based on the reported statistic values, Alice calculates the gains of the signal state and the two decoy states, and determines to continue the protocol if they are within the specified thresholds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' (3) Bob discards the decoy states declared by Alice, and separates the remaining signal states into S groups (in fact S is the computational scale).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Each group needs to perform the interlaced 1-D cluster computation subroutine(I1DC) [4] so that the desired qubit |+θ⟩ can be prepared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' (4) With Alice’s knowledge about the angles of signal states in the I1DC procedure and outcomes reported by Bob, Alice can compute the polarization angle θ of the prepared qubit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Based on RBSP with two decoy states [6], one can esti- mate the lower bound of Alice’s overhead, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' the required total number of pulses N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' In RBSP with two decoy states, the average photon number of signal states is denoted as µ, and the average photon number of two decoy states is denoted as v1, v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' The gain of signal states is Qµ, which is the probability of the detecting event of the signal pulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' The proportion of single photon in the signal states is described as p1, and its lower bound is pL,v1,v2 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' The probabilities of signal and two decoy states chosen by the client are defined as pµ, pv1, pv2, respectively, with pµ+pv1+pv2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' For a UBQC protocol of computation size S (where S ≫ Mµ/Mmulti, Mµ is the total number of signal states, Mmulti is the number of multiphotons signal states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' ), the lower bound of the required total number of pulses, denoted as N L,v1,v2, can be expressed as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' N ≥ N L,v1,v2 = S pµQµ ln (ϵ/S) ln � 1 − pL,v1,v2 1 �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' (1) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Quantum error-correcting codes In quantum computation, the errors are inevitable be- cause of the decoherence effect of quantum state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' In order to correct qubit error, we need to establish a error model for quantum computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' In [24], We conclude that the evolu- tion of the qubit can be expressed as a linear combination of four possibilities: (1) no error occurs, (2) the bit flip |0⟩ ↔ |1⟩, (3) the relation phase of |0⟩ and |1⟩, (4) both a bit flip and a phase flip occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Then, the error super-operator E is diagonal in Pauli basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' The error model can be taken the form E (|ψ⟩ ⟨ψ|) = � Ei∈E p (Ei)Ei |ψ⟩ ⟨ψ| E† i , (2) where all error Ei are Pauli operators Ei = ⊗n j=1Xa j Zb j, a, b ∈ {0, 1}, and p (Ei) is the probability for the error Ei to occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='2, we have the normalization condition E† i Ei = I, ∀Ei, and the trace-preserving constraint � Ei∈E p (Ei) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' For correcting qubit errors, we need to diagnose which of these four possibilities actually occurred, and then correct the error by applying the Pauli basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' In the procedure for determining error syndrome, we must make sure that quantum state can not be destroyed for subsequent quantum computation, and its information is private.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Therefore, we need to use a suitable quantum code to handle these qubits, which may occur errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' A popular quantum code is the [[n, k, d]] stabilizer code [24], [25], [26], which can encode k qubits into n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' The parameter d is the distance of the code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' A code with distance d can correct ⌊d/2⌋ simultaneous errors from the error set E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' The code space is the eigenspace of the generators of the code stabilizer S, which is a set of (n − k) independent commuting operators g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Each codeword |ψm⟩ obeys the eigenvalue equations |ψm⟩ = g |ψm⟩ , ∀m = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=', 2k − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' The stabilizer code can be also described by the generator matrix G, which has 2n columns and n − k rows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' The generator matrix is denoted as G = (XG|ZG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Each row in G encodes a generator g of the stabilizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' The column index of XG and ZG labels the qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' The positions of the 1’s in XG indicate the qubits that are acted on by X in the listed generators, and the 1’s in ZG indicated the qubits acted on by Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' If a 1 appears in same position in both XG and ZG, then the product Y = ZX acts on that qubit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' |0⟩L = 1 2 √ 2(|0000000⟩ + |0001111⟩ + |0110011⟩ + |0111100⟩ + |1010101⟩ + |1011010⟩ + |1100110⟩ + |1101001⟩) |1⟩L = 1 2 √ 2(|1111111⟩ + |1110000⟩ + |1001100⟩ + |1000011⟩ + |0101010⟩ + |0100101⟩ + |0011001⟩ + |0010110⟩) (3) In the quantum error correction, the diagnosis of error is often named as the error syndrome measurement, which is increasingly difficult as the number of logical qubits increases in an encoded block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Hence, we use a common stabilizer code with fewer qubits, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' the 7-qubit Steane’s code [[7,1,3]], which can encode one qubit in seven, and corrects one-qubit error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' The encoded logical qubit basis is denoted as {|0⟩L, |1⟩L}, and see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' In fact, the code is also a special case of the CSS code, which is an extension of the classical Hamming code in quantum error correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' The generator matrix of the [[7,1,3]] is shown as follows [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' G[[7,1,3]] = (XG|ZG)[[7,1,3]]= � � � � � � 0 0 0 1 1 1 1 0 1 1 0 0 1 1 1 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ���������� 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 1 1 0 0 1 1 1 0 1 0 1 0 1 � � � � � � (4) The encoding circuit of quantum error-correcting codes can be designed according to the generator matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' 2(a), the circuit is used to encode an unknown logical qubit [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' The CNOT gates of the circuit are based on XG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' 2(b), two ancilla states are prepared to perform H H H 0 0 0 0 0 0 0 1 \uf061 \uf062 \uf02b H H M M 7 Z X Data Ancilla (a) (b) 0 1 L L \uf061 \uf062 \uf02b Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' The encoding and correction circuits of the [[7,1,3]] code [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' (a) An unknown logical qubit and 6 ancilla qubits can be used to encode into [[7,1,3]] code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' (b) Two 7-qubit Steane states are used to correct error qubits for an 7-qubit data block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Each CNOT gate in the diagram represents 7 CNOT gates performed in parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' error syndrome measurements of bit flip and phase flip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' The measurment results of ancilla qubits are multiplied by the row vectors of G[[7,1,3]] to obtain the parity bits, which can diagnose the error syndrome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Then, we can use Pauli operator to correct the qubit errors in the encoded data block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Problem statement In the quantum computation, a basic qubit unit is often stored on the polarization of a single photon or the spin of a single electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' A qubit has superposition property, which can be described by two-dimension Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' An quantum algorithm is described by the quantum circuit which consists of a sequence of quantum gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' In UBQC, we know that the quantum gates are firstly transformed into bricks, and then server performs measurements on each qubit of brickwork state to implement client’s computation based on MBQC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Obviously, quantum gates can be realized by performing measurements on qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' In the practical UBQC system, we only need to consider the impact of qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Due to the influence of noise from environment and device, qubits are prone to polarization errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Hence, it is necessary to integrate quantum error correction idea into UBQC to ensure the correctness of a delegated computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' In UBQC, the main processes of a quantum computation are preparation and measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' In preparation, we need to modify the above RBSP protocol to prepare quantum error- correcting codes instead of single qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' In order to prepare quantum error-correcting codes, we firstly design a quantum encoding circuit, and then transform it into graph states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Based on MBQC model, we can delegate server to prepare quantum error-correcting codes on graph states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' However, both good encoding circuits and good graph states require a large number of data qubits and ancilla qubits, which are directly determine the size of computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' The challenge is to achieve better error correction performance with limited quantum resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' In measurement, client sends measurement angles to server, and guide him to perform measurement-based com- putations on brickwork state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' To prevent the accumulation and propagation of qubits in computing, it is necessary to perform fault-tolerant quantum computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' The quantum error-correcting codes can be considered as logical unites to take the place of original qubits on brickwork state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Quan- tum gates must be fault tolerant in the quantum computing circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' The measurements of logical qubits on brickwork state are also required to be fault tolerant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Since the qubits, quantum gates and measurements are different from previ- ous states, the security of the fault-tolerant blind quantum computation are needed to be re-demonstrated based on original protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' The challenge is the ϵ-blindness of a practical fault-tolerant protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' The good fault tolerance requires a lot of redundant qubits to resist the spread of qubit errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Hence, other challenge is to trade off between quantum resource consumption and fault tolerance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Fault-tolerant blind quantum computation with quantum error-correcting codes To effectively remove qubit errors, the encoded quantum gates need to ensure that a failure in executing any encoded gate can be only propagated to a small number of qubits in each encoded data block, so that the aggregated error rate does not exceed the designed fault tolerant threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' In ad- dition, any error-correction scheme can introduce errors on the encoded qubits, so one must be very careful in designing error-correction procedures that do not introduce too many additional errors into the encoded data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Therefore, for a quantum circuit, in order to perform a fault-tolerant quantum computing, the fault-tolerant preparation of quantum error- correcting codes, fault-tolerant error correction operating, fault-tolerant quantum gates application and fault-tolerant measurements are needed to prevent the accumulation and propagation of errors in quantum computing [24], and this is shown in Firgure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' FT preparation FT preparation FT error Correction FT error correct FT CNOT FT error Correction FT error Correction FT H FT error Correction FT error Correction FT measurement FT measurement 6 0 \uf071 \uf0c4 \uf0c4 \uf02b 6 0 \uf071 \uf0c4 \uf0c4 \uf02b L \uf071\uf02b L \uf071\uf02b H Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' The process of fault-tolerant quantum computation In UBQC, a fault-tolerant computing should be per- formed as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' For example, Alice delegated Bob to per- form fault-tolerant preparation of quantum error-correcting codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' After then, these encoded logical qubits are used to create encoded brickwork state to realize fault-tolerant quan- tum computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' In fault-tolerant preparation [3], thousands of qubits are used to prepare quantum error-correcting codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' On one hand, for delegated preparation, quantum gates of the encoding circuit are equivalently transformed into brickwork state or other graph state, and then implemented by measurement-based quantum computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' The quantum gates between non-neighbouring qubits can not be directly implemented on brickwork state, so we need many SWAP gates to ensure that any quantum gate acts on adjacent qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' On the other hand, the CNOT gate between non- adjacent qubits is frequently used in fault-tolerant encoding circuit, so a large number of SWAP gates are needed to transformed into brickwork state, which requires a lot of auxiliary quantum resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Hence, it is very difficult to achieve a desired fault-tolerant quantum computation with current quantum technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' If the high-frequency CNOT gate of the encoding circuit can be implemented at high successful probability, the error rate of the prepared quantum error-correcting code will be very low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Consequently, with limited quantum resources, a fault-tolerant blind quantum computation can be implemented by in a compromised way using a combination of quantum error-correcting code preparation and fault-tolerant measurements in UBQC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Delegated preparation of quantum error- correcting codes In a quantum computation, we know that arbitrary quan- tum gate can be realized on cluster state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' The essence of a quantum circuit is a series of ordered quantum gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Hence, each quantum circuit can be implemented on cluster state [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' The quantum error-correcting code circuit can be also realized on cluster state to generate the encoded logical qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' These codes can be considered as logical units to take the place of original qubits on brickwork state to perform a fault-tolerant computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Hence, we propose a remote quantum error-correcting code preparation protocol on cluster state, as shown in Protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='1 Protocol 1: A remote quantum error-correcting code preparation on cluster state Input: data weak coherent pulses (signal states and two decoy states) with polarization ρσ, σ ∈R {kπ/4 : 0 ≤ k ≤ 7};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' ancilla pulses with polarization |+⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Output: quantum error-correcting codes {|+θi⟩L}S 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' 1 Alice sends data and ancilla pulses to Bob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' 2 for i=1 to S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' do 3 Bob prepares qubit |+θi⟩ based on RBSP with two decoy states, and uses it and a group of ancilla qubits {|+⟩} to build cluster state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' 4 for x=1 to m, y=1 to n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' do 5 Realization of the quantum encoding circuit on cluster state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' 6 if qxy is white then qxy is measured with {|0⟩, |1⟩} ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' 7 if qxy is green then qxy is measured with M(0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' 8 if qxy is red then qxy is measured with M(π/2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' 9 end 10 Bob prepares the encoding logical qubit {|+θi⟩L}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' 11 end 12 return {|+θi⟩L}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' In the following, we will give the [[7,1,3]] circuit as an example to prepare quantum error-correcting codes ac- Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Realization of the [[7,1,3]] encoding circuit on cluster state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Circles in green, red and white represent cluster qubits measured in the eigenbasis of X, Y , Z, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' cording to Prtocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='2(a), we note that the depth of [[7,1,3]] circuit is low, and the frequently used gate is only CNOT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' If the CNOT gate can be implemented at a high successful rate, the successful rate of preparation will also be high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' According to quantum computation model on cluster states [27], the quantum gates of [[7,1,3]] circuit can be transformed into cluster state, and sequentially measure each qubit on cluster state to prepare quantum error-correcting codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Based on Protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='1, Alice firstly sends data photon pulses and ancilla photon pulses to Bob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Bob prepares data qubit based RBSP with two decoy states [6], and then utilize a series of ancilla qubits and prepared qubit to build the initial cluster state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' According to [[7,1,3]] circuit in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='2, Alice can transform each quantum gate into measurement angles on cluster state, as shown Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' After then, Bob uses the computational basis {|0⟩, |1⟩} to eliminate the redundant qubits according to Alice’s requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' The remaining qubits on cluster state are used to prepare quan- tum error-correcting code based on Alice’s measurement basis M(δ), which is defined by orthogonal projections on |±δ⟩ = (|0⟩ ± eiδ|1⟩)/ √ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' The parameter δ ∈ [0, 2π] is called the measurement angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' For δ = 0 or π/2, one obtains the X or Y Pauli measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Obviously, the measurement will be understood as destructive measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' The mea- surement outcome at qubit i will be denoted by si ∈ Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' We take the specific convention that si = 0 if the state collapses to |+δ⟩ under the corresponding measurement, and si = 1 if to |−δ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' In the process of eliminating redundant qubits, the struc- ture information of the underlying cluster state can be achieved to Bob, which may bring about the leakage of quantum gates used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' However, Alice only needs to ensure that the encoding logical qubit |+θi⟩L is unknown to Bob in preparation, and quantum gates of the encoding circuit can be public to Bob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Three kinds of measurement ba- sises are needed in preparation, including the computational basis {|0⟩, |1⟩}, measurement basises M(0) and M(π/2), which are corresponding to the eigenstates of Z, X, Y in Pauli gates, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' These measurement basises are independent of the polarization angle θi, which means that the information of θi is not leaked to Bob in measurement computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Hence, cluster state can be used to prepare quantum error-correcting codes, which are also the unknown encoding logical qubits to Bob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Fault-tolerant computation of quantum error- correcting codes Based on the RBSP with two decoy states, the polariza- tion angle of prepared qubit θ is ϵ-blind, and Bob can not get any information about it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' The prepared quantum state as data qubit is combined with a series of ancilla qubits to prepare the required quantum error-correcting code |+θ⟩L according to Protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' In the whole preparation process, Bob are unable to obtain information about θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Hence, these prepared quantum error-correcting codes are unknown logical qubits to Bob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Every encoded logical qubit can be considered as logical unit to take the place of original qubit on brickwork state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' The new brickwork state can be used to do fault- tolerant quantum computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Based on the original UBQC, we propose a fault-tolerant blind quantum computation with quantum error-correcting codes , as shown in protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Protocol 2: Fault-tolerant blind quantum compu- tation with quantum error-correcting codes Input: data weak coherent pulses with polarization ρσ, σ ∈R {kπ/4 : 0 ≤ k ≤ 7}, and ancilla pulses with polarization |+⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Output: measurement result (sx,y)nm 1,1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' 1 Alice sends data and ancilla pulses to Bob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' 2 for i=1 to S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' do 3 Bob uses data pulse ρσ to prepare the required qubit |+θi⟩, and uses it and a group of ancilla qubits {|+⟩} to prepare the desired encoded logical qubit |+θi⟩L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' 4 end 5 Bob uses these encoded logical qubits {|+θi⟩L}S 1 to build the encoded brickwork state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' 6 for x=1 to n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' y=1 to m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' do 7 Alice calculates angle φ′ x,y, the beginning values sX 0,y = sZ 0,y = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' and then calculates measurement angle δx,y = φ′ x,y + θx,y + πrx,y,rx,y ∈R {0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' 8 Bob measures encoded logical qubit |ψx,y⟩L in the base {|+δx,y⟩L, |−δx,y⟩L}, return measurement result sx,y ∈ {0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' 9 if rx,y = 1 then flip sx,y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' 10 if rx,y = 0 then continue;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' 11 end 12 return sx,y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' In the following, for a simple example, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='5(a), we use our Protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='2 to illustrate the delegated fault-tolerant quantum computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Firstly, Alice sends weak coherent pulses to Bob, which includes data states and ancilla states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' The polarization angles of data pulses are selected randomly from {kπ/4 : 0 ≤ k ≤ 7}, and the ancilla is |+⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Secondly, Bob uses the data pulse ρσ to prepare the required qubit |+θi⟩ based on RBSP with two decoy states [6], and uses it and a group of ancilla qubits {|+⟩} to build cluster state to prepare the desired encoded logical qubit |+θi⟩L, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Bob uses these encoded logical qubits {|+θi⟩L}S 1 to build the encoded brickwork state, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='5(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Finally, the interactive measurements are performed between Alice and Bob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Alice transforms the quantum circuit to fault-tolerant quantum circuit, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='5, and then calculates angle φ′ x,y based on the desired angle φx,y and previous measurement outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Let sX x,y = ⊕i∈Dx,ysi be the parity of all measurement outcomes for qubits in Xx,y, where Dx,y ⊆ [x − 1] × [m] is a set of X-dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Similarly, sZ x,y = ⊕i∈D′x,ysi is the parity of all the measurement outcomes for qubits in Zx,y, where D′ x,y ⊆ [x − 1] × [m] is a set of Z-dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' We assume that the dependency sets Xx,y and Zx,y are obtained via UBQC [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Then, the actual measurement angle is φ′ x,y = (−1)sX x,yφx,y + sZ x,yπ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' The measurement angles {δx,y} can be calculated according to the original UBQC [3], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' δx,y = φ′ x,y + θx,y + πrx,y, where rx,y is randomly chosen in {0, 1}, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='5(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Afterward, the fault- tolerant measurements on the encoded brickwork state are used to realize the fault-tolerant quantum computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' This procedure is repeated in turn on the encoded brickwork state until the desired quantum computation result is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Not that arbitrary quantum rotation gate and Controlled- NOT (CNOT) gate can be used to build an universal gate group [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Hence, their own fault-tolerant logical gates can be also composed into the universal logical gate group, which can be used to implement arbitrary fault-tolerant quantum gate, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='5(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' In Protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='2, we note that Alice only use cluster state to prepare quantum error-correcting codes according to Pro- tocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' In contrast to brickwork state, cluster state can re- duce quantum resource consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Although the encoded circuit in preparation may be leaked to Bob, the prepared quantum error-correcting codes are unknown to Bob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Hence, these codes can be used as logical unit to build encoded brickwork state to do fault-tolerant quantum computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' The fault-tolerant quantum gates are needed to transfer on the encoded brickwork state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' For each encoded logical qubit on brickwork state, the measurements are performed simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Security Analysis If the ideal joint state shared by the client and the server can be described by the state πideal AB , then a malicious server can not achieve anything about client’s information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' For any UBQC protocol, the ideal joint state πideal AB is evolved into one of the family of states, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' F(πideal AB ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Due to the security holds for any action of the server, any state of the family is equally blind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' In order to analyse the security in a realistic implementation, we consider the settings where the client sends general states ρθi instead of the perfect state |+θi⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' We can introduce the definition of ϵ blindness [4]: Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' A UBQC protocol with imperfect preparation is ϵ-blind, if the trace distance between ideal state πθi AB and realistic state πρθi AB is less than ϵ: min π θi AB∈F 1 2 ���π{ρθi} AB − π{θi} AB ��� ≤ ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' (5) U1 U2 U3 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='1 \uf064 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='2 \uf064 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='3 \uf064 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='4 \uf064 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='6 \uf064 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='1 \uf064 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='2 \uf064 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='3 \uf064 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='4 \uf064 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='6 \uf064 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='1 \uf064 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='2 \uf064 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='3 \uf064 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='4 \uf064 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='7 \uf064 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='1 \uf064 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='2 \uf064 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='3 \uf064 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='4 \uf064 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='7 \uf064 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='8 \uf064 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='8 \uf064 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='5 \uf064 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='5 \uf064 U1 U2 U3 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='5 \uf064 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='6 \uf064 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='7 \uf064 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='8 \uf064 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='8 \uf064 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='7 \uf064 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='6 \uf064 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content="5 \uf064 FT U1 FT U2 FT U3 Fault- tolerant U3 0 0 \uf028 \uf029 Z R \uf061 L \uf028 \uf029 X R \uf062 L \uf028 \uf029 Z R \uf067 L \uf028 \uf029' Z R \uf061 L \uf028 \uf029' X R \uf062 L \uf028 \uf029' Z R \uf067 L \uf061 \uf062 \uf067 ' \uf061 ' \uf062 ' \uf067 0 0 0 0 0 4 \uf070 4 \uf070 4 \uf070 \uf02d L L L L L L L L L L L L L L L L L L L L L L L L L L L L L L L L L L L L L L L L L L L L L L L L L L L L L L L L (a) (b) (d) (c) Figure 5." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' An example illustration of fault-tolerant blind quantum compu- tation with quantum error-correcting codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' (a) A simple quantum circuit including three quantum gates U1, U2, U3, each thin line represents a qubit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' (b) The fault-tolerant quantum circuit, every thick box represents a fault-tolerant quantum gate, and each thick line represents a encoded logical qubit with 7 qubits, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' [[7,1,3]] code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' (c) Implementation of the fault-tolerant quantum computation on brickwork state, and each angle δ in circles presents a measurement basis M(δ), each orange thick circle with subscript L represents an encoded logical qubit, each gray thick box with subscript L represents an output logical qubit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' (d) Two brick models built with the encoded logical qubits, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' arbitrary rotation logical gate and CNOT logical gate, which can be used to form an universal logical gate group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Based on Protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='2, we can delegated a remote Bob to do a fault-tolerant quantum computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' If the Bob is malicious or dishonest, does our protocol still ensure the privacy of Alice’s information ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' To solve this problem, we will demonstrate the security of protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='2 to ensure that Bob does not get any computational information except the scale of encoded brickwork state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' In fault-tolerant comput- ing, Bob only has some information about the measurement bases and the encoded states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Therefore, one needs to show that Alice’s information is independent of Bob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' The fault-tolerant protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='2 is ϵ-blind while leaking at most (n, m), which is the dimension of computational scale, provided that the prepared encoded logical qubit |+θ⟩L is ϵ-blind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Proof: Let (n, m) be the dimension of the brick- work state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' According to definition of brickwork state [3], it ensures that Bob does not get any information on the underlying computation except n and m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' On the one hand, one needs to prove that Bob’s mea- surement bases are ϵ-independent of Alice’s computing information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Alice has two secret bit strings in the fault- tolerant quantum computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' One is the polarization angle θ of quantum error-correcting code, while the other is the actual measurement base φ′, which is determined by computational tasks and previous measurement results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' The measurement angle sent to Bob can be calculated according to δ = φ′+θ+πr, r ∈R {0, 1}, and then it is used to perform fault-tolerant measurements on the encoded brickwork state, where r is a random bit string determined by Alice, which is unknown to Bob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Since the polarization angle θ is also ϵ-uniform in {kπ/4|k = 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=', 7}, so θ + πr is also ϵ-uniform random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Hence, Bob’s measurement angle δ is ϵ-independent of the computational information ψ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' On the other hand, the information from encoded quan- tum states is ϵ-blind to Bob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Since the bit rx,y is random, each encoded logical qubit for Bob will have one of the following two possibilities: 1)If rx,y = 0 and δx,y = φ′ x,y + θx,y, then |ψx,y⟩L = 1 √ 2 � |0⟩L + ei(δx,y−φ′ x,y)|1⟩L � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' 2)If rx,y = 1 and δx,y = φ′ x,y + θx,y + π, then |ψx,y⟩L = 1 √ 2 � |0⟩L − ei(δx,y−φ′ x,y)|1⟩L � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Without loss of generality, if the angle δ is fixed, θ depends on φ′, but the random bit rx,y is unknown to Bob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Hence, the encoded logical qubit for Bob is a mixed state of two cases, which is ϵ-independent of Alice’s computing information φ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' □ In summary, according to Protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='2, we know that Alice can delegate Bob to prepare ϵ-blind quantum error- correcting codes to correct errors, meanwhile these codes can be used as encoded logical qubits to do subsequent fault-tolerant quantum computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' In addition, Protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='2 only requires Alice to send weak coherent pulses to prepare quantum error-correcting codes, which can decouple Alice dependency on quantum computing and quantum memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' The number of ancilla qubits increases only by a constant factor with the increasing computational scale, rather than a linear increase in the original UBQC [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Hence, Protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='2 have advantages from Alice’s perspective of saving resource consumption and getting rid of quantum dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Concatenation codes and resource require- ments There is a beautiful construction based on concatenated codes which can be used to reduce the effective error rate achieved by the computation even further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' The idea is to recursively apply the scheme described above for simulating a circuit using an encoded circuit, constructing a hierarchy of quantum circuits Level=1 (the initial encoded circuit), Level=2,3 and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' In the first stage of this construction, each qubit in the original circuit is encoded in a quantum code whose qubits are themselves encoded in a quantum code, whose own qubits are encoded yet again, and so on, as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Hence, in order to improve error correction performance of qubits, we study the application of concatenation codes in UBQC as shown in Protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Protocol 3: Fault-tolerant blind quantum compu- tation with concatenation codes Input: Data pulses and ancilla pulses Output: Measurement results (sx,y)nm 1,1 1 for i=1 to S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' do 2 Bob uses data pulse ρσ to prepare the required qubit |+θi,0⟩ based on RBSP with two decoy states 3 if l >= k then 4 Output(|+θi,l⟩L) 5 else 6 Bob uses |+θi,l−1⟩L and a group of ancilla qubits {|+⟩} to build cluster state, and then prepare the desired encoded logical qubit |+θi,l⟩L based on the lth-level concatenated circuit;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' l++.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' 7 end 8 end 9 Bob uses these encoded logical qubits � |+θi,k⟩L �S 1 to build the encoded brickwork state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' 10 The interactive measurement process, refer to Protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='2 to perform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' 11 return sx,y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' we use two-level concatenated [[7,1,3]] codes to reduce error rate, and each [[7,1,3]] code encodes a single qubit using a block of 7 qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' When examine one of the 7 qubits in this block with higher resolution, we first note that it is itself an encoded sub-block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' With this method, the complexity of quantum error correction does not grow so sharply as we increase the error-correcting capacity of quantum code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Note that the [[7,1,3]] code can correct one error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' If the probability of error per actual physical qubit at the lowest level of the code is e0, and these errors are uncorrected, and the correction is fault-tolerant, then the probability of a correction failure in [[7,1,3]] code is of order e2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' If we concatenate the code to construct a new block, then an error occurs in the block only if two of the sub-blocks of size 7 fail, which occurs with a probability of order e4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Let the error probability of each block at level i be denoted as ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' At each level of the concatenated code, the block fails if there are errors in at least two contained sub-blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' For the i-level concatenated code, the failure probability can be estimated according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' FT Preparation \uf02d H H H H H H H H H H H H H H H FT Correction \uf02d FT Preparation \uf02d FT Correction \uf02d FT Syndrome Measurement \uf02d FT Syndrome Measurement \uf02d Ancilla State 0 L 0 L 0 L 0 0 0 Ancilla State Ancilla Qubits Pool Ancilla Qubits Pool 2nd Level 1st Level Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' The two-level concatenated [[7,1,3]] codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' When inspected at higher resolution, each qubit or quantum gate in the block is itself an encoded sub-block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Each block comprises fault-tolerant syndrome measure- ment, fault-tolerant correction and fault-tolerant quantum gate(a transversal application of quantum gate).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' ei = 7 � k≥2 � 7 k � ek i−1 ≈ � 7 2 � e2 i−1 + o � e2 i−1 � ≈ (21e0)2i 21 + o � e2i 0 � (6) In order to significantly reduce the probability of error significantly, the condition e0 < 1/21 must be satisfied in the concatenation codes (the concatenation codes hold if and only if (21e0)2i ≪ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Further, we can make the error rate arbitrarily small by adding sufficient levels of concatenation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' If the concatenated [[7,1,3]] code circuit is delegated to Bob to prepare quantum error-correcting code on the cluster state in Protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='2, then the number of required ancilla pulses will increase with the number of levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Since a required data qubit is used to prepare an encoded logical qubit using [[7,1,3]] encoding circuit, the number of required data qubits is consistent with the RBSP with two decoy states, which remaining unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' 1-level coding 2-level coding 3-level coding A data qubit Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' The encoding structure for a date qubit in 3-level concatenation code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' In Protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='3, the weak coherent pulses are sent to Bob, which contain data pulses and ancilla pulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' According to RBSP with two decoy states, data pulses are used to prepare the required data qubits {|+θi⟩}S 1 , and each data qubit is encoded into a required logical qubit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Hence, the number of data pulses is equal to RBSP with two decoy states, denoted as N d [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' The number of ancilla pulses depends on encoding circuit on cluster state, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Note that the required number of ancilla qubits is a constant when preparing an encoded logical qubit, denoted as C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' If the transmittance of the quantum channel is T, and the computational scale is S, then the number of ancilla pulses N a = CS/T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Hence, the total number of required pulses in Protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='3 is the sum of data and ancilla pulses, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='N = N d + N a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' The number of required ancilla pulses consists of the preparations, syndrome measurements and corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Note that, the required ancilla qubits in the syndrome measure- ments and corrections can be reused in our fault-tolerant protocol, therefore, this part of the resource consumption of ancilla qubits can be ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' One only needs to analyze the preparation consumption of ancilla qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' In Protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='3, we use cluster state to prepare the encoded logical qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Based on circuit model, we know that the block size of ancilla qubits to prepare an encoded logical qubit is a constant C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' According to the encoding structure for a data qubits in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='7, the block size of ancilla qubits is the sum of the required number at all levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' If we use an n-level concatenation code to prepare an encoded logical qubit, the number of ancilla qubits is denoted as N a n, and its value can be calculated as follows: N a n = n � i=1 7i−1C = (7n − 1) C 6 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' (7) The number of required data pulses is consistent with RBSP with two decoy states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' According to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='1, we have N d ≥ N L,v1,v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' The required total number of pulses in the n-level concatenated [[7,1,3]] code, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Nn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' is estimated as follows: Nn = N d + N a n ≥ N L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='v1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='v2 + (7n − 1) CS 6T ≈ S T � � ln (ϵ/S) pµµ ln � 1 − pL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='v1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='v2 1 � + (7n − 1) C 6 � � (8) where S is the computational scale,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' ϵ is security parameter of the blind quantum computation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' T is the transmittance of the quantum channel between Alice and Bob,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' µ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='v1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' v2 are the average number of signal photons and two decoy photons,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' respectively,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' the proportion of single photon in the signal states is described as p1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' and its lower bound is pL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='v1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='v2 1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' the probability of signal pulses chosen by the client is defined as pµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' In the RBSP protocol with two decoy states, for the computation size S, if an error occurs in prepared qubits, the whole preparation protocol will fail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' We assume that the preparation protocol is a repeatable Bernoulli’s experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' If the error probability of each prepared qubit is e0, the successful probability of preparation in each experiment will be (1 − e0)S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' After repeating k times, the successful probability of preparation will be 1 − (1 − (1 − e0)S)k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' 00In the n-level concatenated code, the successful probability of each preparation is (1 − en)S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' If the successful proba- bility is the same in both coded and non-coded cases, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' 1−(1−(1−e0)S)k = (1−en)S, then the repetition number k can be calculated as follows: k = ln[1 − (1 − en)S]/ ln[1 − (1 − e0)S].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' (9) where en can be estimated according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' In other words, if Alice wants to use the non-coding protocol to achieve the same successful probability as encoding, the number of required data pulses is k times than that of the original case, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' kN d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Let R describe the ratio of the number of pulses for concatenation codes to that of the non- coding case at the same successful probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' The ratio R is a resource consumption function with the number of levels n, which can be used to estimate the optimal number of concatenated levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' The estimation of resource consumption ratio R(n) is shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' R (n) = Nn kN d = N d+N a n kN d ≤ N L,v1,v2 + N a n kN L,v1,v2 = ln � 1 − (1 − e0)S� ln � 1 − (1 − en)S� �(7n − 1) CS 6TN L,v1,v2 +1 � (10) where N L,v1,v2 is the lower bound of the number of required pulses in the RBSP with two decoy states, which can be estimated according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' When the computation scale S is fixed, the ratio R (n) ∼ (7/2)n according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='6 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Hence, the ratio R has an exponential growth trend for a large number of levels n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' For a high-level concatenation code, a large number of qubits are used for repeated encoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' In our concatenation code, we not only need to consider improving the successful preparation prob- ability, but also reducing the resource consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Hence, we only take into account the optimal level in the low- level concatenation code (0 ≤ n ≤ 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' When the resource consumption ratio R(n) reaches its minimum, the proportion of the number of the encoding pulses in the non-coding is the lowest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' At this point, only a small number of pulses for concatenation code can be used to achieve a high successful probability, in which case the corresponding level n is the optimal one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' According to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='8, the number of required data pulses is unchanged with the increasing number of levels accord- ing to, and the number of required ancilla pulses grows exponentially with the increasing number of levels in the concatenation code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Furthermore, the optimal level n can be estimated according to the resource consumption ratio R(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Performance evaluation In simulation, our computer uses an the Intel(R) Core(TM) i7-6700HQ CPU with 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='0GB RAM and Win 10 pro OS to complete the experiment through MATLAB software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' The overall transmittance T (including fiber trans- mittance and detection efficiency) is calculated as follows: T = ts · ηs · 10−αL/10 (11) TABLE 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' THE SIMULATION PARAMETERS FOR OUR PROTOCOL α tS ηS µ v1 v2 pµ pv1 pv2 S ϵ e 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='125 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='05 1000 1010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='01 The parameter α is the loss coefficient measured in dB/km, L is the length of the fiber in km, ts is denoted as the internal transmittance of optical components in Bob’s side, and ηs is detector efficiency in Bob’s side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' The mean photon number of signal states and two decoy states are represented as µ, v1, v2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' The chosen probability of signal states and two decoy states are denoted as pµ, pv1, pv2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' The computation scale is S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' The required blind- ness of the UBQC is denoted as ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' The error probability of each qubit is denoted as e0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Based on Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='2 and 4, we can calculate the number of ancilla qubits, C = 1774.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' The relative parameters are set in Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='1(refer to the data in [29] and [30]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' If we use [[7,1,3]] to encode n-level concatenation code in our protocol, the quantum resource consumption is shown as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Suppose Alice has a laser transmitter with frequency f = 1MHZ, and Bob has full-fledged quantum computer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' According to the setting in Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='1, we can derive the efficiency of prepared qubits in our protocols, that is the number of qubits generated per second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Combined with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='8, we can further estimate the upper bounder of the efficiency in the concatenation code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' E = S · f/Nn ≤ S · f/ � N L,v1,v2 + N a n � (12) 100 101 102 Communication Distance(L) 106 107 108 109 1010 1011 1012 1013 The Total Number of Required Pulses(N) Asymptotic Case Non-Coding Case Our Coding Case Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' The total number of required pulses N with the same probability of successful preparation vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' the communication distance between Alice and Bob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' The green and red lines show the simulation results of our Protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='1 with coding and non-coding case, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' The black line shows the simulation results in the asymptotic case (an infinite data-size and near- perfect qubits preparation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' 8, we can see that the total number of required pulse for the coding case in Protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='1 is less than for the 0 10 20 30 40 50 60 70 80 90 100 Communication Distance L (km) 0 100 200 300 400 500 600 700 Efficiency E (qubits/s) Asymptotic Case Non-Coding Case Our Coding Case Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' The preparation efficiency E with the same probability of successful preparation vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' the communication distance L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' The green, red and black lines show efficiency curves of our protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='1 with coding, non- coding, asymptotic case respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' non-coding case in RBSP with two decoy states under the same probability of successful preparation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Since the initial error probability of each qubit is e0, the error probability of each quantum error-correcting code prepared is e2 0, and the probability of successful preparation with size S is (1 − e2 0)S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' The error probability of the encoded logical qubits prepared is much lower that of RBSP with two decoy states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' In order to obtain the same probability of successful preparation, non-coding case needs to be repeated k times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' As the communication distance increases, the value of N grows rapidly, which indicates that the channel loss and qubit error rate have a great influence on N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' For long distance communication, the advantages of our Protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='1 are more outstanding than non-coding case, which means that the required number of pulses N is closer to asymptotic case(an infinite data-size and without qubit error [6], [29]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' 9, we note that the efficiency of qubits prepared gradually decreases with increasing communication distance until it tends to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Compared with non-coding case, the efficiency of qubits for coding case in our Protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='1 is closer to the asymptotic limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Obviously, as the error rate decreases from e0 to e2 0, more pulses as ancilla qubits are used to prepare quantum error-correcting codes, which will lead to a sharp drop in efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' According to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' (10), we know that the resource con- sumption ratio R(n) is the proportion of the number of pulses for concatenated coding in that of the non-coding case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' In order to reach the optimal level, the partial derivative ∂R (n) /∂n = 0 to solve the extreme value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Nevertheless, we note that the partial derivative is so complicated that it is hard to solve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Hence, we use the simulation experiment to estimate the optimal level n, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' We note that the optimal n is about 2, and the resource consumption ration R(n) reaches the minimum value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' It means the 2-level concatenation code can utilize relatively fewer quantum re- sources than other levels to achieve a better error-correcting performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='11, the number of required pulses N in each case is increasing trend with the communication distance L, which indicates that both the channel loss and qubit error 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='5 4 The Number of Concatenated Levels(n) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='025 The Ratio of Resoure Consumptio(R) Asymptotic Case 25km Case 50km Case 100km Case Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' The ratio of resource consumption R(n) vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' the number of levels in the concatenation code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' The blue, green and red lines show the simulation results of resource consumption ratio as distance is 25km, 50km and 100km, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' The black line shows simulation results in the asymptotic case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' 101 102 Communication Distance(L) 106 108 1010 1012 1014 1016 The Total Number of Required Pulses(N) Asymptotic Case Non-Coding Case 1-Level Concatenated Code 2-Level Concatenated Code 3-Level Concatenated Code 4-Level Concatenated Code Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' The total number of required pulses N with the same probability of successful preparation vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' the communication distance between Alice and Bob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' The green, red, yellow, cyan and blue lines show the simulation results of our protocol with the 1,2,3,4-level concatenation code and non- encoding case, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' The black line shows the simulation results in the asymptotic case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' rate have a great influence on N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Compared with the non- coding case at the same successful probability, the number of required pulses N for encoding is closer to asymptotic case(an infinite data-size and near-perfect qubits prepara- tion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' The reason is that the error probability of the encoded logical qubits prepared by the concatenation circuit is much lower than the non-coding case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' In order to obtain the same successful probability, it is necessary for RBSP with two decoy states to be repeated k times, which results in the waste of a vast number of pulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Further, we can see that the number of required pulse in the 2-level concatenation code is less than other levels case at same successful probability, which shows the 2-level concatenation code can achieve a better error-correcting performance in the finite quantum resources case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='6, the preparation efficiency E in the asymptotic case is much higher than in the real case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Since the fluctu- ation of finite data and imperfect preparation of required qubits are inevitable in real UBQC, a large number of pulses as ancilla qubits are used to deal with the impacts 0 10 20 30 40 50 60 70 80 90 100 Communication Distance L (km) 0 100 200 300 400 500 600 700 Efficiency E(qubits/s) Asymptotic Case Non-Coding Case 1-Level Concatenated Code 2-Level Concatenated Code 3-Level Concatenated Code 4-Level Concatenated Code Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' The preparation efficiency E with the same probability of successful preparation vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' the communication distance L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' The green, red, yellow, cyan and blue lines show the simulation results of our Protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='3 with 1,2,3,4-level concatenation code and non-coding case, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' The black line shows simulation results in the asymptotic case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' 0 10 20 30 40 50 60 70 80 90 100 Communication Distance L (km) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='4 Efficiency E(qubits/s) Non-Coding Case 1-Level Concatenated Code 2-Level Concatenated Code 3-Level Concatenated Code 4-Level Concatenated Code Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' The preparation efficiency E with the same probability of successful preparation vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' the communication distance L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' The figure is a detail view to show simulation results with 1,2,3,4-level concatenation code and non-coding case in our Protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' It is the partial enlargement of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' of fluctuation and qubit errors, which leads to a drop in efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' In order to better show our simulation results, we give a partial enlargement of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='12, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Not that the preparation efficiency E has a sharp decline trend as the communication distance increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' The error probability of quantum error-correcting code prepared is e1 06, the probability of successful preparation with size S is (1 − e1 06)S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' with the same probability, we demonstrate the simulation results of the efficiency for the non-coding case and the concatenation codes at different levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Although the high-level concatenation code can obtain a lower error rate, it also requires massive quantum resource consumption, and results in a very lower efficiency, which is not applicable in the reality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' In UBQC with limited quantum resource, we can see that the 2-level concatenation code can achieve the optimal preparation efficiency in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Hence, in the 2- level concatenation code, our Protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='3 can be used not only to improve the preparation efficiency, but also to save quantum resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' It is also very significant to improve the error-correcting performance in fault-tolerant blind quantum computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Related works In original fault-tolerant UBQC [3], fault-tolerant quan- tum computing is performed on the top of brickwork state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Thus, many SWAP gates are used to process those CNOT gates acting on non-adjacent qubits, which results in a linear increase in size of brickwork state with the increasing computational scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' The required number of prepared qubits is also a linear function multiple of the original UBQC pro- tocol [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' In our protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='2, the number of ancilla qubits is more constant than the original UBQC protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Compared to Chien’s two fault-tolerant protocols [12], our protocol only requires Alice to send weak coherent pulses to pre- pare quantum error-corrrecting codes, which can free Alice dependence on quantum computing and quantum memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' In addition, since different quantum gates have different sizes, the used quantum gates can be guessed by Bob when eliminating redundant qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' However, our protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content='2 only prepares quantum error-correcting codes on cluster state, which can reduce quantum resource consumption while ensuring that Bob is unknown to the prepared state, and then performs the fault-tolerant quantum computing on encoded brickwork state, which can guarantee our protocol is ϵ-blind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' However, the current quantum technology still struggles with noises and imperfect measurement, which results in a high error rate in the fault-tolerant UBQC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' To this end, the concatenated stabilizer codes can be considered to reduce qubit requirements by tailoring circuits to suppress the dominant effect of qubit errors [31], [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Chamberland, Jochym and Laflamme et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' concatenated the 7-qubit Steane code with the 15-qubit Reed-Muller code to implement universal fault-tolerant quantum computation without magic state distillation [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Paul Webster et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' presented a gen- eral framework for universal fault-tolerant logic with no-go theorem and stabilizer codes [34], which can be applied to a wide range of stabilizer code families, including concate- nated codes and conventional topological stabilizer codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' A fault-tolerant quantum computation with concatenated quantum codes was proposed by Chamberland, Noh and Preskill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' [32], which can reduce the consumption of qubits by tailoring the quantum error-correcting codes to suppress the dominant phase-flip errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' To avoid coher- ent errors, Yingkai Ouyang proposed rotated concatenated stabilizer codes [35], namely, concatenating an [[n,k,d]] stabilizer outer code with constant-excitation inner codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' It was shown that when the stabilizer outer code is fault- tolerant, the concatenated codes are immune from coherent phase errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' The concatenation codes above are useful for improving the fault-tolerant threshold and reducing quantum resource consumption in a realistic UBQC system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' In recent years, many encoding methods [14], [36], [37], [38] were proposed to deal with this problem, such as concatenation code [38], RHG Lattice code [14] and surface code [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' However, these remain daunting for the complex encoding structures and large resource consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' In this paper, we propose a general fault-tolerant blind quantum computation with concatenation codes to reduce the error rate of logical qubits, thereby improving the fault-tolerant threshold of UBQC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Besides, we optimize the number of required pulses in the concatenated code to reduce quantum resource consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' In fault-tolerant UBQC with limited quantum resources, the 2-level concatenated code can obtain the optimal performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Conclusions In the paper, we propose a method to realize quantum error-correcting codes on cluster state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' A fault-tolerant blind quantum computation with quantum error-correcting codes is proposed to address errors in the quantum computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' In order to improve the error-correcting performance, the stabilizer codes are used for multi-level concatenating to improve the fault-tolerant threshold of qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' To reduce quantum resource consumption, we also analyse the number of required pulses in each level concatenation code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Due to a large number of quantum resources are required in the high-level concatenation codes, the low-level concatenated codes (0 ≤ n ≤ 4) are only considered in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' The optimal level in the concatenation codes is about 2, and the resource consumption ratio reaches the minimum value, which means that the number of required pulses in the 2- level concatenation codes is less than other levels under the same probability of successful preparation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' From the perspective of reducing quantum resources, the simulation results show that the 2-level concatenated code can achieve a better performance than other levels with the increase of communication distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Our encoded scheme can also be used to other blind quantum computation protocols with different graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' In order to further improve the practical performance of UBQC, we will continue to design the con- catenated codes between other quantum codes and stabilizer code in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Meanwhile, we will also attempt to use topological codes with high fault tolerance to do a delegated computing in the case of limited quantum resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Acknowledgments This work is supported by Space Science and Tech- nology Advance Research Joint Funds(Grant Number: 6141B06110105), National Natural Science Foundation of China(Grant Number: 61771168) and National Natural Sci- ence Foundation of China (grant number: 62071151).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' References [1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Childs, “Secure assisted quantum computation,” Quantum Info.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9A0T4oBgHgl3EQfAP-Y/content/2301.01960v1.pdf'} +page_content=', vol.' metadata={'source': 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[cond-mat.mtrl-sci] 12 Jan 2023 +Bismuth layer properties in the ultrathin Bi–FeNi multilayer films +probed by spectroscopic ellipsometry +N. N. Kovaleva,1, ∗ D. Chvostova,2 O. Pacherova,2 A. V. Muratov,1 +L. Fekete,2 I. A. Sherstnev,1 K. I. Kugel,3, 4 F. A. Pudonin,1 and A. Dejneka2 +1Lebedev Physical Institute, Russian Academy of Sciences, Leninsky prospect 53, 119991 Moscow, Russia +2Institute of Physics, Academy of Sciences of the Czech Republic, Na Slovance 2, 18221 Prague, Czech Republic +3Institute for Theoretical and Applied Electrodynamics, +Russian Academy of Sciences, 125412 Moscow, Russia +4National Research University Higher School of Economics, 101000 Moscow, Russia +(Dated: January 13, 2023) +Using wide-band (0.5–6.5 eV) spectroscopic ellipsometry we study ultrathin [Bi(0.6–2.5 nm)– +FeNi(0.8,1.2 nm)]N multilayer films grown by rf sputtering deposition, where the FeNi layer has +a nanoisland structure and its morphology and magnetic properties change with decreasing the +nominal layer thickness. From the multilayer model simulations of the ellipsometric angles, Ψ(ω) +and ∆(ω), the complex (pseudo)dielectric function spectra of the Bi layer were extracted. +The +obtained results demonstrate that the Bi layer can possess the surface metallic conductivity, which +is strongly affected by the morphology and magnetic properties of the nanoisland FeNi layer in the +GMR-type Bi–FeNi multilayer structures. +The spin-orbit coupling (SOC) is a relativistic effect +important for the electronic structure of heavy atoms +and solids formed by them. This leads to characteristic +surface metallic states arising from the loss of the inver- +sion symmetry at the surface (Rashba effect) [1]. Bis- +muth (Bi) is a rather heavy element with strong SOC in +the atomic 6p levels (where p3/2 − p1/2 splitting is about +1.5 eV), which facilitates the application of quasi-two- +dimensional (2D) Bi layers in spintronics as spin sources +or filters, as well as in multilayer structures exhibiting +the giant magnetoresistance (GMR) effect. The scaling +of Bi integrated units to smaller dimensions is still going +on toward the thickness of 5 nm and beyond, where 2D +Bi (bismuthene) exhibits extraordinary electronic prop- +erties [2]. For implementing the full potential of GMR +applications by a rational nanostructure design, the in- +formation on the electron band structure of 2D Bi layers +is important. +In bulk Bi, +which crystallizes in the rhombohe- +dral symmetry (space group R¯3m, unit cell parame- +ters a = b = 4.547 ˚A, c = 11.8616˚A, α = β = 90◦, γ = 120◦) +with two atoms per unit cell, five bands accommodate ten +valence electrons, which dictates an insulating behavior +generically. However, the bands close to the Fermi level, +namely, at the T and L points of the Brillouin zone, can +be significantly affected by the strong SOC [3]. As a con- +sequence, three conduction minima at the L points lie at +about 40 meV lower than the single valence-band maxi- +mum at the T point. This indirect band overlap implies +the semimetallic behavior of bulk Bi with electron trans- +port properties dictated by quite small electron effective +mass along a certain axis and unusually long mean free +path. Due to the crystal structure inversion symmetry, +∗Electronic address: kovalevann@lebedev.ru +Analyzer +FIG. 1: Schematics of (a) the ultrathin [Bi–FeNi)]N multi- +layer films investigated in the present study by spectroscopic +ellipsometry and (b) the magnetization configuration in the +nanoisland FeNi layer. +Larger islands exhibit an in-plane +magnetization configuration, while smaller islands can have +an out-of-plane one [9, 14] (for more details, see the text). +(c) SEM image of the nanoisland FeNi layer on the Sitall +substrate obtained by using a JEOL JSM-7001F facility. Re- +produced with permission from Phys. Lett. A 410, 127546 +(2021) [13]. Copyright 2021. Elsevier. +the SOC does not lead to any lifting of the spin degener- +acy in the 6p bands, each having two possible spin states +per k point in the Brillouin zone, while the loss of sym- +metry at the surface or interface can transform Bi from +a semimetal (SM) to a metal when the electron or hole +bands cross the Fermi level. +In the thin-film limit, this effect can be at conflict with +the quantum confinement effect (or size-effect), leading + +(p) +I hw +(C) +(6) +EGW! +B! +19silsnA +E +Fiapf 2onlce2 +to complicated electronic properties. +Quantum effects +can be observed in thin films whose thickness is compa- +rable to the effective wavelength of charge carriers, and +their mean free path exceeds the film thickness. These +conditions should transform Bi from a SM to a semicon- +ductor (SC) at a critical film thickness of about 300 ˚A +[4]. +Due to a very low charge carrier density ranging +from about 1017 cm−3 to 1018 cm−3 and small relative +effective masses of the charge carriers from 0.005 to 0.1, +the optical excitation of charge carriers starts to be rel- +evant only in the far infrared (below 0.1 eV) [5], where +a confinement-induced energy gap in thin Bi films could +manifest itself in the optical experiments. However, the +surface metallic states may hinder the SM-SC transi- +tion in ultrathin Bi films. +The existence of the sur- +face metallic states in ultrathin Bi(001) films was con- +firmed by the broadband terahertz time-domain spec- +troscopy study [6]. It was shown that the surface charge +carrier density, plasma frequency, and scattering rate +dramatically increase with a decrease in the film thick- +ness, reaching n = 3.1 × 1019 cm−3, ωp = 4.0 × 102 THz +(1.65 eV), and γD = 4.8 × 102 THz (2.0 eV), respectively, +in the thinnest investigated 2.8 nm Bi film [6], where the +estimated optical conductivity dc limit σ1(ω→0) = ω2 +p/γD += 2300 Ω−1·cm−1. +Recently, we have demonstrated that the electronic +properties of the free and localized Ta charge carriers +in (Ta–FeNi)N multilayer films (MLFs) can be studied +by spectroscopic ellipsometry (SE) [7, 8]. Here, we ex- +plore the elaborated SE approach to gain insights into +the electron band structure and surface electronic prop- +erties of ultrathin Bi layers in real GMR-type (Bi–FeNi)N +MLF structures, incorporating nanoisland FeNi layers +(see the scheme of the (Bi–FeNi)N MLFs investigated +in the present study by SE in Fig. 1(a)). The morphol- +ogy and magnetic properties of a single-layer nanoisland +FeNi film grown on the Sitall substrate were studied ear- +lier [9–11]. Below the structural percolation transition +at the nominal thickness of 1.5 – 1.8 nm [12], the FeNi +layer is discontinuous and consists of inhomogeneously +distributed FM nanoislands having lateral sizes of 5 – +30 nm and possessing giant magnetic moments of 103– +105 µB (where µB is the Bohr magneton). +As an ex- +ample, Fig. 1(c) shows the scanning electron microscopy +(SEM) image of the nanoisland FeNi film grown on the +Sitall substrate [13]. As schematically shown in Fig. 1(b), +the larger islands (which appear closer to the perco- +lation transition) have the in-plane magnetization con- +figuration, while the smaller islands (existing quite far +from the percolation transition) have the out-of-plane one +[9, 14]. Here, collective superferromagnetic (SFM) states +– FM or AFM – may develop in the self-assembled local +arrangements of FM nanoislands at comparatively high +temperatures [9, 15]. However, small and well separated +FeNi nanoislands are weakly interacting via the magnetic +dipole forces and exhibit superparamagnetic (SPM) be- +havior at high temperatures, which is associated with +strongly fluctuating giant magnetic moments [9]. +(a) +(b) +FIG. 2: Ellipsometric angles, Ψ(ω) and ∆(ω), measured for +the Al2O3(2.1 nm)/[Bi–FeNi(h)]16/Sitall MLF samples hav- +ing the FeNi layer thickness h (a) 1.2 nm and (b) 0.8 nm at an +angle of incidence of 70◦ (symbols) and the fitting results by +the Drude-Lorentz model (Eq. (1)) (displayed for the Bi layer +thickness of 2.5, 2.0, 1.4, and 0.6 nm by solid black, green, +blue, and red curves, respectively). +The (Bi – FeNi)N MLFs were grown by rf sputtering +deposition from 99.95 % pure Bi and Fe21Ni79 targets on +glass Sitall (TiO2) substrates. Before the deposition, the +vacuum chamber was annealed at 200◦C, so that the pre- +pared base pressure in the chamber was below 2 × 10−6 +Torr. During the deposition, the background Ar pres- +sure was 6×10−4 Torr. The actual temperature of the +substrates was about 80◦C. We used the Sitall substrates +with typical sizes of 15×5×0.6 mm3. The Bi and FeNi +layer nominal thickness was controlled by the deposition +time determined by the film deposition rate. +For ex- +ample, the determined FeNi layer deposition rate was +about 0.67 ˚A per second. To protect the grown MLFs +from the oxidation under ambient conditions, the as de- +posited MLFs were covered in situ by the 2.1 nm-thick +Al2O3 layer. In the prepared (Bi–FeNi)N MLF samples, +the FeNi layer nominal thickness was 0.8 and 1.2 nm, the +thickness of the Bi layer was 0.6, 1.4, 2.0, and 2.5 nm, +and the number of Bi/FeNi bilayers was N = 16. In our +recent study, (Ta–FeNi) MLFs grown by rf sputtering de- +position onto Sitall-glass substrates, including ultrathin +0.52 nm-thick FeNi layers, were characterized by scan- +ning/transmission electron microscopy (STEM) (for de- +tails see [8]). Here, the grown Bi–FeNi MLF samples were + +3 +TABLE I: Parameters of the Drude and low-energy Lorentz +bands for the Bi layer in the MLFs [Bi(2.5, 2.0, 1.4 nm)– +FeNi(0.8, 1.2 nm)]16, resulting from the model simulations of +the complex dielectric response (Eq. (1)) (for details see sup- +plementary material). +FeNi +Bi +AD +γD +Aj +Ej +γj +(nm) +(nm) +(eV) +(eV) +(eV) +2.5 +25±4 +1.4±0.1 +96±18 +0.32 +0.83 +1.2 +2.0 +23±3 +2.2±0.1 +97±9 +0.40 +1.08 +1.4 +64±0.3 +0.75±0.17 +19±9 +0.81 +1.32 +2.5 +97±1 +0.459 +1.271 +0.8 +2.0 +98±1 +0.481 +1.354 +1.4 +103±1 +0.429 +1.628 +characterized by the atomic-force microscopy (AFM), X- +ray diffraction, and X-ray reflectivity (see supplementary +material to this article). The X-ray reflectivity measure- +ments confirm a good periodicity and relatively small in- +terface roughness in the grown Bi–FeNi MLF structures, +as well as good agreement with the nominal thickness of +the Bi and FeNi layers. The X-ray diffraction suggests +orientation of the Bi layers along the (012) plane, where +the interlayer distance is 3.28 ˚A. Thus, the prepared Bi– +FeNi MLFs having the Bi layer thickness of 0.6, 1.4, 2.0, +and 2.5 nm correspond to about 2, 4, 6, and 8 Bi(012) +monolayers. +The ellipsometric angles Ψ(ω) and ∆(ω) +were measured for the prepared Al2O3/(Bi-FeNi)16/Sitall +MLF samples at room temperature at three or four angles +of incidence of 60◦, 65◦, 70◦, and 75◦ in a wide photon +energy range of 0.5–6.5 eV with a J.A. Woollam VUV- +VASE spectroscopic ellipsometer (Fig. 2(a,b) illustrates +the Ψ(ω) and ∆(ω) measured at 70◦). The complex di- +electric function ˜ε(ω) = ε1(ω)+iε2(ω) of each Bi or FeNi +layer was modeled by the Drude term and the sum of +Lorentz oscillators to account for the contributions of free +charge carriers and interband optical transitions, respec- +tively +˜ε(E ≡ ℏω) = ǫ∞ − +AD +E2 + iEγD ++ +� +j +AjγjEj +E2 +j − E2 − iEγj +,(1) +where ǫ∞ is the high frequency dielectric constant. The +fitted Drude parameters were AD (related to the plasma +frequency ωp via AD = ǫ∞ℏω2 +p) and scattering rate γD. +The adjustable Lorentz oscillator parameters were Ej, +γj, and Aj of the peak energy, full width at half maxi- +mum, and ε2 peak height, respectively. The ellipsomet- +ric angles, Ψ(ω) and ∆(ω), measured at different angles +of incidence were fitted simultaneously in the framework +of the multilayer model Al2O3/[Bi(2.5,2.0, 1.4, 0.6 nm)– +FeNi(0.8, 1.2 nm)]16/Sitall, where, in addition, the sur- +face roughness was taken into account by the standard +effective medium approximation (EMA) based on the +Bruggeman model (50% Al2O3 – 50% vacuum), using +the J.A. Woollam VASE software [16]. In the simulation +of the ellipsometric angles, Ψ(ω) and ∆(ω), the Bi lay- +ers in each MLF structure were described by the disper- +FIG. 3: +The complex dielectric function spectra, ε2(ω) +and ε1(ω), of the Bi layer in the [Bi(2.5, 2.0, 1.4, 0.6 nm)– +FeNi(h)]16 MLFs for the FeNi layer thickness h (a) 1.2 nm +and (b) 0.8 nm are shown by solid black, green, blue, and +dashed red curves, respectively. +sion models (Eq. (1)), including three Lorentz oscillators +and the Drude term where necessary. The discontinu- +ous nanoisland FeNi layers were modeled by the effective +dielectric function in EMA, which describes the optical +properties of a complex composite by an effective homo- +geneous medium. In the utilized multilayer model, the +spectra of the complex dielectric function of the blank +Sitall substrate obtained from our previous SE studies +[17, 18] were substituted. The Bi and FeNi layer thick- +nesses were fitted to their respective nominal values. The +good quality of the fit obtained for the measured angle +of incidence of 70◦ is demonstrated by Fig. 2(a,b), where +we plot the recorded ellipsometric angles Ψ(ω) and ∆(ω) +and the fitting results. +The details of the used model +and the resulting Drude-Lorentz parameters along with +the fit quality check are given in supplementary mate- +rial to this article. The simulation in the framework of +the multilayer model for the Al2O3/(Bi–FeNi)16/Sitall +MLFs, where the Bi–FeNi interface roughness is explic- +itly included, does not essentially improve the fit (see the + +4 +FIG. 4: +The Bi intralayer optical conductivity, σ1(ω) = +ε2(ω)ω[cm−1]/60, in the [Bi–FeNi(h)]16 MLFs shown by solid +black, green, and blue curves, for the FeNi layer thickness +h (a–c) 1.2 nm and (d–f) 0.8 nm, respectively. +The contri- +butions from the low-energy Lorentz oscillator and the Drude +term are indicated by the cyan and yellow shaded area, respec- +tively. The summary contribution of the Drude and Lorentz +bands is shown by dotted lines. +analysis presented in [7]), suggesting that the Bi–FeNi in- +terface roughness is essentially incorporated in the EMA +dielectric function of the nanoisland FeNi layers. +From the multilayer model simulations, the dielectric +function spectra of the Bi and FeNi layers were ob- +tained (see supplementary material for details). +Here, +we are particularly interested in the dielectric func- +tion spectra, +ε1(ω) and ε2(ω), +of the Bi layer in +the studied [Bi(2.5, 2.0, 1.4, 0.6 nm)–FeNi(1.2 nm)]16 and +[Bi(2.5, 2.0, 1.4, 0.6 nm)–FeNi(0.8 nm)]16 MLFs, and the +spectra obtained from the best-fit simulations of the Ψ(ω) +and ∆(ω) are shown in Fig. 3(a,b). One can notice dif- +ferent trends in the behavior of the corresponding ε1(ω) +and ε2(ω) spectra. Thus, the ε1(ω) spectra in Fig. 3(b) +exhibit a clearly pronounced minimum at about 0.8 eV, +whereas ε1(ω) spectra in Fig. 3(a) display more negative +values falling down to –50. Accordingly, the ε2(ω) spec- +tra in Fig. 3(a) demonstrate a steeper rise at the lowest +probed photon energies. Moreover, the dielectric func- +tion spectra of the 0.6 nm-thick Bi layers in the studied +[Bi(0.6 nm)–FeNi(0.8, 1.2 nm)]16 MLFs display apparent +trends toward more pronounced metallic behavior at the +lowest probed photon energies, the ε1(ω) exhibits a sharp +downturn to negative values, and ε2(ω) dramatically in- +creases (see Fig. 3(a,b) and supplementary material to +this article). +In Fig. 4(a–f) we present the Bi intralayer optical +conductivity, σ1(ω) = ε2(ω)ω[cm−1]/60, in the stud- +ied [Bi(2.5, 2.0, 1.4 nm)–FeNi(0.8,1.2 nm)]16 MLFs. Here, +the dispersion analysis representation resulting from the +multilayer model simulations using Eq. (1) is explicitly +demonstrated. +On one hand, the simulation results +for the three [Bi(2.5, 2.0, 1.4 nm)–FeNi(0.8 nm)]16 MLF +structures, including the 0.8 nm-thick nanoisland FeNi +layer, indicate that the Bi layer low-energy response is +dominated by a pronounced Lorentz band peaking at +0.43 – 0.48 eV having the ε2 peak height of 97 – 103 (see +Table I). The low-energy interband transition with a +high oscillator strength is observed in the dense Bi lay- +ers at the energy of 0.8 eV having the ε2 peak height of +about 120 [19, 20]. The low-energy peak at 0.8 eV is also +seen in the averaged over anisotropy low-energy dielec- +tric function response of single crystals [21, 22]. +This +strong low-energy optical transition is associated with +interband transitions with the onset near the Γ point, +Γ+ +6 − Γ− +6 and Γ+ +45 − Γ− +6 [23], and with interband tran- +sitions near the T point T− +6 −T− +45 [24]. +Therefore, we +can conclude that the low-energy optical response of the +Bi layer in the [Bi(2.5, 2.0, 1.4 nm)–FeNi(0.8 nm)]16 MLFs +(see Fig. 4(d–f)) is dominated by the Bi semimetallic- +like electron band structure. +On the other hand, our +model simulations imply that the optical conductivity of +the Bi layer in the [Bi(2.5, 2.0, 1.4 nm)–FeNi(1.2 nm)]16 +MLFs, including the 1.2 nm-thick nanoisland FeNi layer, +has competing contributions from the low-energy Lorentz +band and from the Drude term (see Fig. 4(a–c) and Table +I). For the 2.5 and 2.0 nm thick Bi layers, the estimated +Drude parameters are similar to those characterizing the +surface metallic states arising due to the Rashba effect +in ultrathin Bi(001) films [6] (σ1(ω→0) = 2300 Ω−1·cm−1 +and γD =2.0 eV). However, it was shown that the surface +layer in the ultrathin Bi(012) films can possess a pseu- +docubic Bi{012}-oriented allotrope with the even num- +ber of layers (represented by black phosphorus-like puck- +ered layers) [25]. We have found that with decreasing +the Bi layer thickness from 2.0 to 1.4 nm (corresponding +to about six and four Bi(012) monolayers, respectively), +the Drude dc limit σ1(ω→0) significantly increases from +about 3100±400 to 8600±40Ω−1·cm−1, and the scatter- +ing rate γD decreases from 2.2±0.2 to 0.8±0.2eV. At the +same time, the low-energy Lorentz band becomes signif- +icantly suppressed (see Fig. 4(a–c)). The observed evo- +lution of the competing Drude and Lorentz parts can be +attributed to the progressive increase in the contribution +of the Bi surface metallic states. However, note the dif- +ference from the results obtained for the 40–2.8nm-thick +Bi(001) single-layer films [6], where the ωp and γD in- +crease with a decrease of the film thickness. We suppose +that here a GMR-like case plays an important role. In- +deed, in the GMR-type MLF structures, FM coupling +is found for 2.5 nm and 1.3 nm-thick spacer layers, and +AFM coupling is found for a 2 nm-thick spacer layer [26]. +The existence of AFM or FM GMR-type correlations be- +tween the discontinuous nanoisland FeNi layers could oc- +cur in the SFM regime [9, 11, 15]. +Therefore, in the + +5 +studied GMR-type [Bi(2.5, 2.0, 1.4 nm)–FeNi(1.2 nm)]16 +MLFs, the magnetic interaction between the neighboring +FeNi layers, which is responsible for the spin-dependent +scattering in the magnetic layer, oscillates from FM via +AFM to FM ones. The spin-dependent scattering at the +Bi/FeNi interface necessarily affects the scattering of the +Bi surface metallic charge carriers (γD), which should +decrease for the FM coupling and increase for the AFM +coupling between the neighdoring FeNi layers. The de- +crease of the γD of the surface metallic charge carriers in +the FM regime will naturally lead to the increase in the +optical dc conductivity limit (see Fig. 4(c)). According +to the results of the present multilayer model simula- +tions for the [Bi(2.5, 2.0, 1.4 nm)–FeNi(0.8, nm)]16 MLFs, +including the 0.8 nm-thick nanoisland FeNi layers, the Bi +layer exhibits semimetallic bulk-like electron band struc- +ture, however, the Drude surface metallic conductivity +(the Drude term) is implicit in Fig. 4(d–f). Here, with de- +creasing the FeNi layer thickness, strong SPM-type fluc- +tuations of giant magnetic moments of the FeNi nanois- +lands become important [9, 11]. In our recent study [8], +we reported that this leads to localization phenomena in +the GMR-type MLFs, introduced by an additional strong +magnetic disorder and long-range many-body interac- +tions between giant magnetic moments of FeNi nanois- +lands. +Therefore, the lack of evidence on the surface +metallic states for the Bi layer in the [Bi(2.5, 2.0, 1.4 nm)– +FeNi(0.8, nm)]16 MLFs can be referred to strong SPM- +type fluctuations of giant magnetic moments of FM FeNi +nanoislands, leading to strong scattering and localization +of scarce free charge carriers in the Bi surface layer. In ad- +dition, we found that the dielectric function spectra of the +two Bi{012} monolayers demonstrate pronounced metal- +licity properties in the [Bi(0.6 nm)–FeNi(0.8, 1.2 nm)]16 +MLFs. The origin of the discovered semimetal-to-metall +crossover needs to be further investigated. In particu- +lar, the impact of lattice mismatch at the interface on +the electron band structure of 2D bismuthene [2] and +the AFM mechanism of a giant SOC-splitting [27] in the +presence of AFM spin textures at the interface should be +considered. +In +conclusion, +using +the +advances +of +the +spec- +troscopic +ellipsometry +approach, +we +extracted +the +(pseudo)dielectric function spectra of the ultrathin Bi +layers incorporating from two to eight Bi(012) mono- +layers in the [Bi(0.6, 1.4, 2.0, 2.5, nm)–FeNi(0.8, 1.2 nm)] +multilayer structures grown by rf sputtering deposition. +We found that the Bi(012) layers inside the studied +multilayer film structures can possess the surface metal- +lic conductivity, which is strongly influenced by the +morphology and magnetic properties of the nanoisland +FeNi layer. +The obtained results may be useful for +implementing the full potential of the GMR applications +based on quasi-2D Bi layers. +See the supplementary material for the atomic force +microscopy (AFM), X-ray reflectivity (XRR), and X-ray +diffraction (XRD) characterization of the MLFs and for +details of the spectroscopic ellipsometry study. +This work was partially supported by the Czech Sci- +ence Foundation (Project No. 20-21864S), and European +Structural and Investment Funds and the Czech Ministry +of Education, Youth, and Sports (Project No. SOLID21, +CZ.02.1.01/0.0/0.0/16−019/0000760). +The theoretical +analysis performed by K. 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' Kovaleva,1, ∗ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' Chvostova,2 O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' Pacherova,2 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' Muratov,1 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' Fekete,2 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' Sherstnev,1 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' Kugel,3, 4 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' Pudonin,1 and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' Dejneka2 1Lebedev Physical Institute, Russian Academy of Sciences, Leninsky prospect 53, 119991 Moscow, Russia 2Institute of Physics, Academy of Sciences of the Czech Republic, Na Slovance 2, 18221 Prague, Czech Republic 3Institute for Theoretical and Applied Electrodynamics, Russian Academy of Sciences, 125412 Moscow, Russia 4National Research University Higher School of Economics, 101000 Moscow, Russia (Dated: January 13, 2023) Using wide-band (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='5–6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='5 eV) spectroscopic ellipsometry we study ultrathin [Bi(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='6–2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='5 nm)– FeNi(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='8,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='2 nm)]N multilayer films grown by rf sputtering deposition, where the FeNi layer has a nanoisland structure and its morphology and magnetic properties change with decreasing the nominal layer thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' From the multilayer model simulations of the ellipsometric angles, Ψ(ω) and ∆(ω), the complex (pseudo)dielectric function spectra of the Bi layer were extracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' The obtained results demonstrate that the Bi layer can possess the surface metallic conductivity, which is strongly affected by the morphology and magnetic properties of the nanoisland FeNi layer in the GMR-type Bi–FeNi multilayer structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' The spin-orbit coupling (SOC) is a relativistic effect important for the electronic structure of heavy atoms and solids formed by them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' This leads to characteristic surface metallic states arising from the loss of the inver- sion symmetry at the surface (Rashba effect) [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' Bis- muth (Bi) is a rather heavy element with strong SOC in the atomic 6p levels (where p3/2 − p1/2 splitting is about 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='5 eV), which facilitates the application of quasi-two- dimensional (2D) Bi layers in spintronics as spin sources or filters, as well as in multilayer structures exhibiting the giant magnetoresistance (GMR) effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' The scaling of Bi integrated units to smaller dimensions is still going on toward the thickness of 5 nm and beyond, where 2D Bi (bismuthene) exhibits extraordinary electronic prop- erties [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' For implementing the full potential of GMR applications by a rational nanostructure design, the in- formation on the electron band structure of 2D Bi layers is important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' In bulk Bi, which crystallizes in the rhombohe- dral symmetry (space group R¯3m, unit cell parame- ters a = b = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='547 ˚A, c = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='8616˚A, α = β = 90◦, γ = 120◦) with two atoms per unit cell, five bands accommodate ten valence electrons, which dictates an insulating behavior generically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' However, the bands close to the Fermi level, namely, at the T and L points of the Brillouin zone, can be significantly affected by the strong SOC [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' As a con- sequence, three conduction minima at the L points lie at about 40 meV lower than the single valence-band maxi- mum at the T point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' This indirect band overlap implies the semimetallic behavior of bulk Bi with electron trans- port properties dictated by quite small electron effective mass along a certain axis and unusually long mean free path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' Due to the crystal structure inversion symmetry, ∗Electronic address: kovalevann@lebedev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='ru Analyzer FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' 1: Schematics of (a) the ultrathin [Bi–FeNi)]N multi- layer films investigated in the present study by spectroscopic ellipsometry and (b) the magnetization configuration in the nanoisland FeNi layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' Larger islands exhibit an in-plane magnetization configuration, while smaller islands can have an out-of-plane one [9, 14] (for more details, see the text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' (c) SEM image of the nanoisland FeNi layer on the Sitall substrate obtained by using a JEOL JSM-7001F facility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' Re- produced with permission from Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' A 410, 127546 (2021) [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' Copyright 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' Elsevier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' the SOC does not lead to any lifting of the spin degener- acy in the 6p bands, each having two possible spin states per k point in the Brillouin zone, while the loss of sym- metry at the surface or interface can transform Bi from a semimetal (SM) to a metal when the electron or hole bands cross the Fermi level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' In the thin-film limit, this effect can be at conflict with the quantum confinement effect (or size-effect), leading (p) I hw (C) (6) EGW!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' B!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' 19silsnA E Fiapf 2onlce2 to complicated electronic properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' Quantum effects can be observed in thin films whose thickness is compa- rable to the effective wavelength of charge carriers, and their mean free path exceeds the film thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' These conditions should transform Bi from a SM to a semicon- ductor (SC) at a critical film thickness of about 300 ˚A [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' Due to a very low charge carrier density ranging from about 1017 cm−3 to 1018 cm−3 and small relative effective masses of the charge carriers from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='005 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='1, the optical excitation of charge carriers starts to be rel- evant only in the far infrared (below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='1 eV) [5], where a confinement-induced energy gap in thin Bi films could manifest itself in the optical experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' However, the surface metallic states may hinder the SM-SC transi- tion in ultrathin Bi films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' The existence of the sur- face metallic states in ultrathin Bi(001) films was con- firmed by the broadband terahertz time-domain spec- troscopy study [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' It was shown that the surface charge carrier density, plasma frequency, and scattering rate dramatically increase with a decrease in the film thick- ness, reaching n = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='1 × 1019 cm−3, ωp = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='0 × 102 THz (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='65 eV), and γD = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='8 × 102 THz (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='0 eV), respectively, in the thinnest investigated 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='8 nm Bi film [6], where the estimated optical conductivity dc limit σ1(ω→0) = ω2 p/γD = 2300 Ω−1·cm−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' Recently, we have demonstrated that the electronic properties of the free and localized Ta charge carriers in (Ta–FeNi)N multilayer films (MLFs) can be studied by spectroscopic ellipsometry (SE) [7, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' Here, we ex- plore the elaborated SE approach to gain insights into the electron band structure and surface electronic prop- erties of ultrathin Bi layers in real GMR-type (Bi–FeNi)N MLF structures, incorporating nanoisland FeNi layers (see the scheme of the (Bi–FeNi)N MLFs investigated in the present study by SE in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' 1(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' The morphol- ogy and magnetic properties of a single-layer nanoisland FeNi film grown on the Sitall substrate were studied ear- lier [9–11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' Below the structural percolation transition at the nominal thickness of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='5 – 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='8 nm [12], the FeNi layer is discontinuous and consists of inhomogeneously distributed FM nanoislands having lateral sizes of 5 – 30 nm and possessing giant magnetic moments of 103– 105 µB (where µB is the Bohr magneton).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' As an ex- ample, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' 1(c) shows the scanning electron microscopy (SEM) image of the nanoisland FeNi film grown on the Sitall substrate [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' As schematically shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' 1(b), the larger islands (which appear closer to the perco- lation transition) have the in-plane magnetization con- figuration, while the smaller islands (existing quite far from the percolation transition) have the out-of-plane one [9, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' Here, collective superferromagnetic (SFM) states – FM or AFM – may develop in the self-assembled local arrangements of FM nanoislands at comparatively high temperatures [9, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' However, small and well separated FeNi nanoislands are weakly interacting via the magnetic dipole forces and exhibit superparamagnetic (SPM) be- havior at high temperatures, which is associated with strongly fluctuating giant magnetic moments [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' (a) (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' 2: Ellipsometric angles, Ψ(ω) and ∆(ω), measured for the Al2O3(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='1 nm)/[Bi–FeNi(h)]16/Sitall MLF samples hav- ing the FeNi layer thickness h (a) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='2 nm and (b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='8 nm at an angle of incidence of 70◦ (symbols) and the fitting results by the Drude-Lorentz model (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' (1)) (displayed for the Bi layer thickness of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='5, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='4, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='6 nm by solid black, green, blue, and red curves, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' The (Bi – FeNi)N MLFs were grown by rf sputtering deposition from 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='95 % pure Bi and Fe21Ni79 targets on glass Sitall (TiO2) substrates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' Before the deposition, the vacuum chamber was annealed at 200◦C, so that the pre- pared base pressure in the chamber was below 2 × 10−6 Torr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' During the deposition, the background Ar pres- sure was 6×10−4 Torr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' The actual temperature of the substrates was about 80◦C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' We used the Sitall substrates with typical sizes of 15×5×0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='6 mm3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' The Bi and FeNi layer nominal thickness was controlled by the deposition time determined by the film deposition rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' For ex- ample, the determined FeNi layer deposition rate was about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='67 ˚A per second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' To protect the grown MLFs from the oxidation under ambient conditions, the as de- posited MLFs were covered in situ by the 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='1 nm-thick Al2O3 layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' In the prepared (Bi–FeNi)N MLF samples, the FeNi layer nominal thickness was 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='8 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='2 nm, the thickness of the Bi layer was 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='6, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='4, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='0, and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='5 nm, and the number of Bi/FeNi bilayers was N = 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' In our recent study, (Ta–FeNi) MLFs grown by rf sputtering de- position onto Sitall-glass substrates, including ultrathin 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='52 nm-thick FeNi layers, were characterized by scan- ning/transmission electron microscopy (STEM) (for de- tails see [8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' Here, the grown Bi–FeNi MLF samples were 3 TABLE I: Parameters of the Drude and low-energy Lorentz bands for the Bi layer in the MLFs [Bi(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='5, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='4 nm)– FeNi(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='8, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='2 nm)]16, resulting from the model simulations of the complex dielectric response (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' (1)) (for details see sup- plementary material).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' FeNi Bi AD γD Aj Ej γj (nm) (nm) (eV) (eV) (eV) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='5 25±4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='4±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='1 96±18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='83 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='0 23±3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='2±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='1 97±9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='40 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='08 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='4 64±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='75±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='17 19±9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='81 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='32 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='5 97±1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='459 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='271 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='0 98±1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='481 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='354 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='4 103±1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='429 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='628 characterized by the atomic-force microscopy (AFM), X- ray diffraction, and X-ray reflectivity (see supplementary material to this article).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' The X-ray reflectivity measure- ments confirm a good periodicity and relatively small in- terface roughness in the grown Bi–FeNi MLF structures, as well as good agreement with the nominal thickness of the Bi and FeNi layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' The X-ray diffraction suggests orientation of the Bi layers along the (012) plane, where the interlayer distance is 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='28 ˚A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' Thus, the prepared Bi– FeNi MLFs having the Bi layer thickness of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='6, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='4, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='0, and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='5 nm correspond to about 2, 4, 6, and 8 Bi(012) monolayers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' The ellipsometric angles Ψ(ω) and ∆(ω) were measured for the prepared Al2O3/(Bi-FeNi)16/Sitall MLF samples at room temperature at three or four angles of incidence of 60◦, 65◦, 70◦, and 75◦ in a wide photon energy range of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='5–6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='5 eV with a J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' Woollam VUV- VASE spectroscopic ellipsometer (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' 2(a,b) illustrates the Ψ(ω) and ∆(ω) measured at 70◦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' The complex di- electric function ˜ε(ω) = ε1(ω)+iε2(ω) of each Bi or FeNi layer was modeled by the Drude term and the sum of Lorentz oscillators to account for the contributions of free charge carriers and interband optical transitions, respec- tively ˜ε(E ≡ ℏω) = ǫ∞ − AD E2 + iEγD + � j AjγjEj E2 j − E2 − iEγj ,(1) where ǫ∞ is the high frequency dielectric constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' The fitted Drude parameters were AD (related to the plasma frequency ωp via AD = ǫ∞ℏω2 p) and scattering rate γD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' The adjustable Lorentz oscillator parameters were Ej, γj, and Aj of the peak energy, full width at half maxi- mum, and ε2 peak height, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' The ellipsomet- ric angles, Ψ(ω) and ∆(ω), measured at different angles of incidence were fitted simultaneously in the framework of the multilayer model Al2O3/[Bi(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='5,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='4, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='6 nm)– FeNi(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='8, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='2 nm)]16/Sitall, where, in addition, the sur- face roughness was taken into account by the standard effective medium approximation (EMA) based on the Bruggeman model (50% Al2O3 – 50% vacuum), using the J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' Woollam VASE software [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' In the simulation of the ellipsometric angles, Ψ(ω) and ∆(ω), the Bi lay- ers in each MLF structure were described by the disper- FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' 3: The complex dielectric function spectra, ε2(ω) and ε1(ω), of the Bi layer in the [Bi(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='5, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='4, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='6 nm)– FeNi(h)]16 MLFs for the FeNi layer thickness h (a) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='2 nm and (b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='8 nm are shown by solid black, green, blue, and dashed red curves, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' sion models (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' (1)), including three Lorentz oscillators and the Drude term where necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' The discontinu- ous nanoisland FeNi layers were modeled by the effective dielectric function in EMA, which describes the optical properties of a complex composite by an effective homo- geneous medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' In the utilized multilayer model, the spectra of the complex dielectric function of the blank Sitall substrate obtained from our previous SE studies [17, 18] were substituted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' The Bi and FeNi layer thick- nesses were fitted to their respective nominal values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' The good quality of the fit obtained for the measured angle of incidence of 70◦ is demonstrated by Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' 2(a,b), where we plot the recorded ellipsometric angles Ψ(ω) and ∆(ω) and the fitting results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' The details of the used model and the resulting Drude-Lorentz parameters along with the fit quality check are given in supplementary mate- rial to this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' The simulation in the framework of the multilayer model for the Al2O3/(Bi–FeNi)16/Sitall MLFs, where the Bi–FeNi interface roughness is explic- itly included, does not essentially improve the fit (see the 4 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' 4: The Bi intralayer optical conductivity, σ1(ω) = ε2(ω)ω[cm−1]/60, in the [Bi–FeNi(h)]16 MLFs shown by solid black, green, and blue curves, for the FeNi layer thickness h (a–c) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='2 nm and (d–f) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='8 nm, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' The contri- butions from the low-energy Lorentz oscillator and the Drude term are indicated by the cyan and yellow shaded area, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' The summary contribution of the Drude and Lorentz bands is shown by dotted lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' analysis presented in [7]), suggesting that the Bi–FeNi in- terface roughness is essentially incorporated in the EMA dielectric function of the nanoisland FeNi layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' From the multilayer model simulations, the dielectric function spectra of the Bi and FeNi layers were ob- tained (see supplementary material for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' Here, we are particularly interested in the dielectric func- tion spectra, ε1(ω) and ε2(ω), of the Bi layer in the studied [Bi(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='5, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='4, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='6 nm)–FeNi(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='2 nm)]16 and [Bi(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='5, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='4, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='6 nm)–FeNi(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='8 nm)]16 MLFs, and the spectra obtained from the best-fit simulations of the Ψ(ω) and ∆(ω) are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' 3(a,b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' One can notice dif- ferent trends in the behavior of the corresponding ε1(ω) and ε2(ω) spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' Thus, the ε1(ω) spectra in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' 3(b) exhibit a clearly pronounced minimum at about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='8 eV, whereas ε1(ω) spectra in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' 3(a) display more negative values falling down to –50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' Accordingly, the ε2(ω) spec- tra in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' 3(a) demonstrate a steeper rise at the lowest probed photon energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' Moreover, the dielectric func- tion spectra of the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='6 nm-thick Bi layers in the studied [Bi(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='6 nm)–FeNi(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='8, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='2 nm)]16 MLFs display apparent trends toward more pronounced metallic behavior at the lowest probed photon energies, the ε1(ω) exhibits a sharp downturn to negative values, and ε2(ω) dramatically in- creases (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' 3(a,b) and supplementary material to this article).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' 4(a–f) we present the Bi intralayer optical conductivity, σ1(ω) = ε2(ω)ω[cm−1]/60, in the stud- ied [Bi(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='5, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='4 nm)–FeNi(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='8,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='2 nm)]16 MLFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' Here, the dispersion analysis representation resulting from the multilayer model simulations using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' (1) is explicitly demonstrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' On one hand, the simulation results for the three [Bi(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='5, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='4 nm)–FeNi(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='8 nm)]16 MLF structures, including the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='8 nm-thick nanoisland FeNi layer, indicate that the Bi layer low-energy response is dominated by a pronounced Lorentz band peaking at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='43 – 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='48 eV having the ε2 peak height of 97 – 103 (see Table I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' The low-energy interband transition with a high oscillator strength is observed in the dense Bi lay- ers at the energy of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='8 eV having the ε2 peak height of about 120 [19, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' The low-energy peak at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='8 eV is also seen in the averaged over anisotropy low-energy dielec- tric function response of single crystals [21, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' This strong low-energy optical transition is associated with interband transitions with the onset near the Γ point, Γ+ 6 − Γ− 6 and Γ+ 45 − Γ− 6 [23], and with interband tran- sitions near the T point T− 6 −T− 45 [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' Therefore, we can conclude that the low-energy optical response of the Bi layer in the [Bi(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='5, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='4 nm)–FeNi(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='8 nm)]16 MLFs (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' 4(d–f)) is dominated by the Bi semimetallic- like electron band structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' On the other hand, our model simulations imply that the optical conductivity of the Bi layer in the [Bi(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='5, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='4 nm)–FeNi(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='2 nm)]16 MLFs, including the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='2 nm-thick nanoisland FeNi layer, has competing contributions from the low-energy Lorentz band and from the Drude term (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' 4(a–c) and Table I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' For the 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='5 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='0 nm thick Bi layers, the estimated Drude parameters are similar to those characterizing the surface metallic states arising due to the Rashba effect in ultrathin Bi(001) films [6] (σ1(ω→0) = 2300 Ω−1·cm−1 and γD =2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='0 eV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' However, it was shown that the surface layer in the ultrathin Bi(012) films can possess a pseu- docubic Bi{012}-oriented allotrope with the even num- ber of layers (represented by black phosphorus-like puck- ered layers) [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' We have found that with decreasing the Bi layer thickness from 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='0 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='4 nm (corresponding to about six and four Bi(012) monolayers, respectively), the Drude dc limit σ1(ω→0) significantly increases from about 3100±400 to 8600±40Ω−1·cm−1, and the scatter- ing rate γD decreases from 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='2±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='2 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='8±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='2eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' At the same time, the low-energy Lorentz band becomes signif- icantly suppressed (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' 4(a–c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' The observed evo- lution of the competing Drude and Lorentz parts can be attributed to the progressive increase in the contribution of the Bi surface metallic states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' However, note the dif- ference from the results obtained for the 40–2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='8nm-thick Bi(001) single-layer films [6], where the ωp and γD in- crease with a decrease of the film thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' We suppose that here a GMR-like case plays an important role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' In- deed, in the GMR-type MLF structures, FM coupling is found for 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='5 nm and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='3 nm-thick spacer layers, and AFM coupling is found for a 2 nm-thick spacer layer [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' The existence of AFM or FM GMR-type correlations be- tween the discontinuous nanoisland FeNi layers could oc- cur in the SFM regime [9, 11, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' Therefore, in the 5 studied GMR-type [Bi(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='5, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='4 nm)–FeNi(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='2 nm)]16 MLFs, the magnetic interaction between the neighboring FeNi layers, which is responsible for the spin-dependent scattering in the magnetic layer, oscillates from FM via AFM to FM ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' The spin-dependent scattering at the Bi/FeNi interface necessarily affects the scattering of the Bi surface metallic charge carriers (γD), which should decrease for the FM coupling and increase for the AFM coupling between the neighdoring FeNi layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' The de- crease of the γD of the surface metallic charge carriers in the FM regime will naturally lead to the increase in the optical dc conductivity limit (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' 4(c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' According to the results of the present multilayer model simula- tions for the [Bi(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='5, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='4 nm)–FeNi(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='8, nm)]16 MLFs, including the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='8 nm-thick nanoisland FeNi layers, the Bi layer exhibits semimetallic bulk-like electron band struc- ture, however, the Drude surface metallic conductivity (the Drude term) is implicit in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' 4(d–f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' Here, with de- creasing the FeNi layer thickness, strong SPM-type fluc- tuations of giant magnetic moments of the FeNi nanois- lands become important [9, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' In our recent study [8], we reported that this leads to localization phenomena in the GMR-type MLFs, introduced by an additional strong magnetic disorder and long-range many-body interac- tions between giant magnetic moments of FeNi nanois- lands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' Therefore, the lack of evidence on the surface metallic states for the Bi layer in the [Bi(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='5, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='4 nm)– FeNi(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='8, nm)]16 MLFs can be referred to strong SPM- type fluctuations of giant magnetic moments of FM FeNi nanoislands, leading to strong scattering and localization of scarce free charge carriers in the Bi surface layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' In ad- dition, we found that the dielectric function spectra of the two Bi{012} monolayers demonstrate pronounced metal- licity properties in the [Bi(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='6 nm)–FeNi(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='8, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='2 nm)]16 MLFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' The origin of the discovered semimetal-to-metall crossover needs to be further investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' In particu- lar, the impact of lattice mismatch at the interface on the electron band structure of 2D bismuthene [2] and the AFM mechanism of a giant SOC-splitting [27] in the presence of AFM spin textures at the interface should be considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' In conclusion, using the advances of the spec- troscopic ellipsometry approach, we extracted the (pseudo)dielectric function spectra of the ultrathin Bi layers incorporating from two to eight Bi(012) mono- layers in the [Bi(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='6, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='4, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='0, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='5, nm)–FeNi(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='8, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='2 nm)] multilayer structures grown by rf sputtering deposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' We found that the Bi(012) layers inside the studied multilayer film structures can possess the surface metal- lic conductivity, which is strongly influenced by the morphology and magnetic properties of the nanoisland FeNi layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' The obtained results may be useful for implementing the full potential of the GMR applications based on quasi-2D Bi layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' See the supplementary material for the atomic force microscopy (AFM), X-ray reflectivity (XRR), and X-ray diffraction (XRD) characterization of the MLFs and for details of the spectroscopic ellipsometry study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' This work was partially supported by the Czech Sci- ence Foundation (Project No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' 20-21864S), and European Structural and Investment Funds and the Czech Ministry of Education, Youth, and Sports (Project No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' SOLID21, CZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='01/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='0/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='0/16−019/0000760).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' The theoretical analysis performed by K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' Kugel was supported by the Russian Science Foundation, project No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' 21-12-0254 (https://rscf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content='ru/en/project/21-12-00254/).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' The authors have no conflicts to disclose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' The data that support the findings of this study are available within this article (and its supplementary ma- terial).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' [1] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' 100, 251605 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' [7] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' Kovaleva, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' Chvostova, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' Pacherova, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' Fekete, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' Kugel, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' Pudonin, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' Dejneka, Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' 111, 183104 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' [8] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' Kovaleva, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' Kusmartsev, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' Mekhiya, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' Trunkin, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' Chvostova, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' Davydov, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' Ovesh- nikov, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' Pacherova, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' Sherstnev, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' Kusmartseva, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' Kugel, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' Dejneka, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' Pudonin, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' Luo, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' Aronzon, Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' Rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' 10, 21172 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' [9] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' Stupakov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' Bagdinov, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' Prokhorov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' Bagdinova, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' Demikhov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' Dejneka, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' Kugel, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' Gorbatsevich, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' Pudonin, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' Kovaleva, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' Nanomater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=', 3190260 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' [10] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' Boltaev, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' Pudonin, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' Sherstnev, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' Egorov, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' Condens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' Matter 30, 295804 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' [11] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' Kovaleva, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' Bagdinov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' Stupakov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' Dejneka, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' Demikhov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' Gorbatsevich, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' Pudonin, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' Kugel, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' Kusmartsev, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' Nanopart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' 20, 109 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' [12] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' Boltaev, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' Pudonin, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' Sherstnev, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' Egorov, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' 125, 465 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' [13] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' Noskova, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' Pudonin, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' Sherstnev, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' Eroshenko, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' Egorov, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' Shadrin, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQffA0H/content/2301.05103v1.pdf'} +page_content=' Lett.' 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100644 index 0000000000000000000000000000000000000000..ccc71c79c391aa9498aa2c7ef75b43f3d3b60b90 --- /dev/null +++ b/HNE2T4oBgHgl3EQf-wn9/content/tmp_files/2301.04243v1.pdf.txt @@ -0,0 +1,946 @@ +Robust Human Identity Anonymization using Pose Estimation +Hengyuan Zhang∗, Jing-Yan Liao∗, David Paz, Henrik I. Christensen +Abstract— Many outdoor autonomous mobile platforms re- +quire more human identity anonymized data to power their +data-driven algorithms. The human identity anonymization +should be robust so that less manual intervention is needed, +which remains a challenge for current face detection and +anonymization systems. In this paper, we propose to use +the skeleton generated from the state-of-the-art human pose +estimation model to help localize human heads. We develop +criteria to evaluate the performance and compare it with the +face detection approach. We demonstrate that the proposed +algorithm can reduce missed faces and thus better protect +the identity information for the pedestrians. We also develop +a confidence-based fusion method to further improve the +performance. +I. INTRODUCTION +Outdoor mobile robots have the huge potential to benefit +human life. Applications such as autonomous driving or +delivery are coming to our life. A critical step before they +can be deployed at scale is the pedestrian motion prediction +problem, given that human motion is highly uncertain and +multi-modal. To address this problem, recent algorithms rely +on data-driven approaches. However, collecting these data +poses risks of leaking identity information. Areas having +identity information, such as human faces and license plates +are required to be anonymized by recent regulations. Recent +released datasets follow this approach to blur faces and +license plates. +Face identity anonymization consists of two parts, face +detection and applying an anonymization algorithm to the +detected region. Face detection can be achieved by applying +a standard object detector trained on a face specific dataset. +Face anonymization methods can then be applied on facial +regions; examples include Gaussian blur and more recent +methods such as DeepPrivacy [1]. In this work, we are +majorly concerned about the face detection phase, where we +localize the identity sensitive region. +Over these years of research and development, face de- +tection has evolved from using hand-crafted features, such +as Haar, to deep-learning ones. Although there are various +techniques to tackle this problem, the recent trend in face +detection algorithms is treating it as a sub-problem in the +object detection field [2]. At the time of writing this paper, +∗These members contributed equally to this publication. +Affiliation: Contextual Robotics Institute, University of California, San +Diego, 9500 Gilman Dr, La Jolla, CA 92093 +©2022 IEEE. Personal use of this material is permitted. Permission from +IEEE must be obtained for all other uses, in any current or future media, +including reprinting/republishing this material for advertising or promotional +purposes, creating new collective works, for resale or redistribution to +servers or lists, or reuse of any copyrighted component of this work in +other works. +the state-of-the-art face detection method is YOLO5Face [3], +which is based on the YOLOv5 [4] object detector. +Nevertheless, the robustness associated with these methods +is subject to their ability to detect facial features across +various orientations and small regions; this can drastically +vary across various image resolutions. Instead, we propose +to infer the human head from the skeleton generated by pose +estimation algorithm. Given that human keypoints across the +entire body will cover a much larger region than the face +alone, this allows us to anonymize identity information more +robustly. +This paper is organized as follows. We review the prior +work in face detection, face anonymization, and keypoint +detection in section II. Our proposed method is introduced +in detail in section III. We evaluate the proposed method and +discuss the metrics in section IV. Finally, we conclude with +a number of key takeaways in section V. +Our major contributions are: +1. We propose to leverage keypoint detectors to infer +head location, which increases the identity sensitive region +detection range and robustness, thus better protecting the +identity of pedestrians. +2. We propose a metric to evaluate the proposed method +and compare it fairly with the face detectors in the +anonymization context. +3. Furthermore, we show that a confidence-based fusion +method can further improve the performance. +II. RELATED WORK +Face Detection: As the first step of the anonymization +system, face detection can directly impact the performance +and robustness. There are several approaches for face de- +tection. Before deep learning based methods were utilized, +handcrafted features such as Haar [5] were used to detect +faces. Recently, after the benchmark dataset WiderFace [6] +was released, face detection grew rapidly and focused on +challenges such as multi-scale. To tackle these challenges, +many works leverage the knowledge from general object +detectors in the face detection task. FDNet [7], as an example +of two-stage detectors, employed multi-scale training, multi- +scale testing with light-head Faster RCNN [8] and redesigned +anchors to help multi-scale performance. For single-stage +methods, TinaFace [2], which is based on RetinaNet [9], +introduced the Inception Module to enhance capability of +multi-scale detection, and used DIoU [10] as regression loss +for small face detection. The state of the art, YOLO5Face [3], +modified the YOLOv5 [4] bottleneck part in the architecture +to increase robustness for large and small faces. Based on +SSD [11], PyramidBox [12] utilized contextual information +arXiv:2301.04243v1 [cs.CV] 10 Jan 2023 + +Original Image +Boxes fused by confidence +Face boxes +Head boxes +OpenPifPaf skeleton +Zoom in view +YOLO5Face +OpenPifPaf +Infer head +Fusion +Fig. 1. +We infer head bounding box from the skeleton predicted by a pose estimation algorithm (OpenPifPaf), and detect face bounding boxes use a face +detector (YOLO5Face). These head and face boxes are fused to produce the final output. (Results are presented in a zoom in view of the original image) +with the proposed Pyramid Anchors (PA), the Low-level +Feature Pyramid Network (LFPN), and the Context-sensitive +Prediction Module (CPM) to predict multi-scale faces. +Face Anonymization: In identity anonymization systems, +once faces are detected, an anonymization algorithm can +be applied to the detected region. Even though this is not +the focus of our paper, we briefly discuss the prior work +to provide the necessary context. This can be done easily +by blocking the region with random pixels, blurring, or +pixelation. However, recent datasets targeting outdoor tasks +not only need to anonymize the identity, but also need to +reduce its impact on pedestrian detection. Recent approaches +use learning to address this problem. Instead of trying to hide +the identity by destroying the details, the methods focus on +changing the appearance, either adding distortion or gener- +ating a completely new face. For example, DeepPrivacy [1] +uses a generative adversarial network to produce faces that +matches the background context but also looks completely +different from the original face. CIAGAN [13] further brings +in control of features of anonymization as well as temporal +consistency in a video sequence. +Pose Estimation: There are two approaches in pose esti- +mation: top-down and bottom-up methods. The former meth- +ods could be perceived as a person detector with a single- +person pose estimator, while the latter predicts body joints +individually and associates them afterwards. By leveraging +a large amount of data and methods from human detection, +top-down methods can achieve competitive results in human +pose estimation. For example, Mask R-CNN [14] extends +instance segmentation to pose estimation by predicting a one- +hot map for each joint. However, top-down methods are not +suitable for our case because partial occlusions can lead to +missed detections in the human detection stage. This is the +result from its two-stage detecting nature. For instance, the +performance of human detection could be unstable if only +partial observations are presented in the image frame, making +it difficult to detect skeletons properly, thus posing challenges +to localize head poses. In contrast, this can be addressed +by bottom-up methods in most cases. Moreover, bottom-up +methods are also more suitable for real-time applications as +they are faster [15]. The pioneer work, DeepCut [16], detects +keypoints individually and associates detected joints with an +integer linear program, which is computationally expensive. +Later works accelerate prediction time with greedy decoders, +such as OpenPose [17], PersonLab [18], and our method of +choice, OpenPifPaf [19]. OpenPifPaf handles pose in various +scale by predicting the joint location and size. Although top- +down methods generally handle scale variance better than +bottom-up methods, we find that OpenPifPaf satisfies our +requirements as the human faces that are too small don’t +need to be detected for anonymization. +III. METHOD +To overcome the limitations of the face-detection based +strategies, we propose to use human body pose estimation +methods to infer head position. The pipeline is shown in +Fig. 1. In this section, we introduce the keypoint detection +and association method we use, OpenPifPaf. We then outline +the process of estimating head position from the keypoints +generated by OpenPifPaf. Finally, we discuss the tracking +and fusion method that jointly leverages face and head +detections. +A. OpenPifPaf +Our method is based on the human body pose estimation +model, OpenPifPaf. OpenPifPaf consists of three parts, back- +bone feature extraction, Composite Intensity Fields (CIF) and +Composite Association Fields (CAF). + +■口 +口口 +口The backbone stage is extract feature from image with +ResNet [20] or ShuffleNetV2 [21], and the output feature +would be shared by CIF and CAF. The CIF is a 1x1 convo- +lution that predicts semantic keypoints, which is identical to +the Part Intensity Field (PIF) introduced in [22]. For human +subjects, the keypoints correspond to body parts such as +joints. For a specific part, the network outputs the confidence +score, its two dimensional vector from each pixel and a scale. +Given that the feature maps estimated by the CIF prediction +head are coarse, a convolution is applied over the coarse +targets using a bivariate Gaussian kernel, which yields a high +resolution confidence map. +The CAF part shares the same backbone; however, this +head predicts a confidence, two vectors that point towards +the two parts this association connects and two spreads for +spatial precision of the regression and two joint sizes. +The output of OpenPifPaf is a set of keypoint locations +and the confidence. Such keypoints are meaningful parts of +a human, such as left shoulder, right eye and nose. Given +the predefined connectivity, we can link the key points and +produce the skeleton of a human. +We note that OpenPifPaf generalizes well on our data +without fine tuning. The detection range of it is much better +than the YOLO5Face detector. It is also more robust on the +same range, which raises our interests to examine its use for +face detection and identity anonymization. +B. Head Prediction +Given the human body pose predictions generated by +OpenPifPaf, which consist of skeletons for the COCO [23] +dataset, we focus on inferring head positions for anonymiza- +tion purposes. The actual procedure differs depending on +whether facial keypoints are predicted or not. +When there are facial keypoints available from OpenPif- +Paf, we can readily estimate the head center from these +facial keypoints. Then the bounding box dimensions (height +and width) of the head are inferred from torso length, +which is given by the distance from the shoulder keypoint +to the hip keypoint. We assume a fixed ratio between the +head dimension to the torso length in terms of pixels. This +actually varies from person to person but serves as a good +approximation for anonymization purpose. +When facial keypoints are not available but shoulder and +hip keypoints are available, the head position needs to be +inferred. This is done simply by centering a bounding box +horizontally between the shoulder keypoints, and vertically +on top of the shoulder keypoints plus a neck length. Again, +the bounding box dimensions are inferred from the torso +length by assuming a fix ratio between them. We also assume +the neck length has a fixed ratio with respect to the dimension +of the head and experimentally find it using our data. +C. Head and Face Fusion +Given the head boxes predicted by OpenPifPaf, it is +intuitive to consider fusion of these head boxes with the face +boxes generated by the YOLO5Face. +Since our goal is to reduce missed faces as much as +possible to better protect identity, we design our approach to +be more tolerant to false positives. Thus the simplest fusion +strategy involves keeping both results. However, we can also +remove face boxes that are covered by the head boxes since +the region will be anonymized. +A more risky strategy is to remove a head bounding box +if there exists a face prediction within. This can potentially +increase the probability of exposing identity information; +however, this strategy is justified by the fact that usually +when a face box is associated with a head box, the subject +in question is closer to us and we are confident about the +face location. +Finally, we can filter the bounding box by its confidence. +We keep the bounding box with a higher confidence score +if for a head bounding box there exists a face prediction +within. The confidence score for face bounding box comes +directly from the output of YOLO5Face while the confidence +score for a head bounding box is the average of the keypoint +confidence from OpenPifPaf. +D. Head and Face Tracking +Given the predicted head bounding boxes from OpenPif- +Paf, we leverage a tracking pipeline to reduce false positives +but also to incorporate temporal consistency. A Kalman Filter +[24] was applied with association based on center-distance. +We modify the SORT [25] tracker for our application. +Initially we associated detections with existing tracks based +on IOU and GIoU [26]; however, this often generated poor +associations as faces are often too small and may not overlap +between adjacent image frames. Thus, we choose an l2 center +distance based association which gives better performance. +IV. EXPERIMENTS +We evaluate the proposed algorithm and present the results +in this section. To this end, we label two video sequences, +one for parameter tuning, which consist of 257 frames, and +another 544 frames in different scene for testing. Then, +we define a metric that allows fair comparison. Finally, we +present the results of our algorithm and its comparison with +the face detection algorithms. It is noteworthy that the data +was recorded during COVID and masks can be observed but +the neural networks were not fine-tuned for this specific case. +A. Metric and Labeling +A fair comparison of our approach against the face de- +tectors is challenging. Standard evaluations use intersection +over union (IOU). But considering that we focus on identity +anonymization, this can be achieved by blurring most of the +face and there will be no consequences of including other +parts of the head. For example, two examples with an IoU +score of 0.5 are shown in Fig. 2. While both achieve the +same score, the example on the right evidently better protects +the sensitive region compared to the example on the left. +Even though additional regions are also included, applying +anonymization to the region has lower risk. + +Fig. 2. +Green solid line boxes are face labels and blue dashed line boxes +are detections. Left and right detection both have 0.5 IoU with the face +label. +Fig. 3. +The green solid line box is a face label and the red dotted line box +is a head label. +Another challenging point is the nature of the algorithms. +Face detectors capture the faces and not the head as depicted +by the green bounding boxes, The bounding boxes may +change largely when the head facing angle changes. Since +our approach mostly captures the bounding boxes of head, +which includes hair and is more invariant to the facing +direction, evaluating them using a standard IoU metric would +lead to inconsistent comparisons. +Therefore we use a two label strategy, as shown in Fig. 3. +We label both faces (green solid line) and heads (red dotted +line). We then denote the bounding box for a detection by +D, a face label by F, and a head label as H. Moreover, two +criteria for evaluation are introduced. First, the face criterion +is defined as +D ∩ F +F +> α, +which implies that at least a portion α of the face label should +be included in D; thus most of the identity information can +be anonymized. +Second, the head criterion is defined by +D ∩ H +D +> β, +which implies that at least a portion β of the detection should +be included in the head label; thus the background portion +within predicted bounding box is bounded. +We collected two video sequences in dense pedestrian +walkways in a university area. For our labeled video se- +quences, both heads and faces of the pedestrians are labeled. +Face labels are considered mostly the front of the head with +skin, not including the ears and hair. Head labels include +the hair. We provide labels up to the distance where human +annotator believes that the face is potentially recognizable. +The video sequence for parameter tuning consists of 259 +frames, with 1018 face labels and 1726 head labels. The +video sequence for testing consists of 543 frames, with 1949 +face labels and 3819 head labels. +B. Comparison with Face Detection +Based on the criteria proposed in the previous section, +we evaluate the proposed OpenPifPaf based algorithm, the +face detector and the fusion method on our testing video +sequence. +The evaluation process are as follows. First, we associate +each face label with a head label, which is based on the +assumption that each face label should be within a head +label. Then for the pedestrians, we have a group of front +facing individuals that have both face labels and head labels, +and another group only have head labels (back of the +pedestrians). Second, we associate the detections with the +head labels using the Hungarian algorithm [27]. Finally, +we evaluate the predicted boxes with the labels using the +proposed criteria. Unless otherwise specified, we set α and +β to 0.5. +For the targets that have a face label and a head label, +we check if the associated detection box satisfies both of the +criteria, face only, head only or none. We denote the result +as match both, face, head or none. For the targets that has +only a head label, we only evaluate the head criterion for +the associated detection. We denote the result as match head +or none. If a detected box is not associated with any of the +labels or doesn’t satisfy any of the criteria with the associated +labels, it is counted as a false positive case. +As shown in the Table I, OpenPifPaf predicted detec- +tions produce significantly more face matches compared to +YOLO5Face; however, they also generate considerably more +false positives. The results suggest that OpenPifPaf predicted +boxes are doing better in terms of finding face regions. +Furthermore, given that we set a higher cost to missing a face +than generating false positives, we consider the performance +of the OpenPifPaf based method acceptable. +The fusion method further improves the performance. If +we keep both the face and the head boxes, the approach +produces the lowest number of missed faces while also gen- +erating the highest number of false positives. Applying other +fusion methods when a predicted face from YOLO5Face is +in the predicted head from OpenPifPaf allows us to greatly +reduce false positives while the number of missed faces +only drops slightly. Among these fusion methods, fusion +by confidence and the keep head method produce similar +results, both are slightly better than the keep face method. +Post processing shows that the similarity of the two fusion +methods is because OpenPifPaf in general gives higher +confidence scores compared to the scores from YOLO5Face, +even when the predicted boxes are less accurate. To make the +confidence fusion more effective, an extension to our work is +to tune the confidence scores from the two neural networks +using the same dataset. +Another advantage is that the OpenPifPaf’s predicted +boxes miss fewer facial regions when the faces are large. We + +Label +Face and Head +Head +FP count +FPS +Task +Match +Both +Face +Head +None +Head +None +Detection +YOLO5Face +1637 +2 +26 +247 +252 +1455 +551 +18.1 +OpenPifPaf Head +1772 +5 +28 +107 +1417 +290 +1540 +5.4 +Detection Fusion +keep head +1838 +5 +18 +51 +1463 +244 +2545 +4.04 +keep face +1829 +4 +28 +51 +1465 +242 +2543 +keep both +1840 +3 +18 +51 +1465 +242 +3784 +by confidence +1839 +4 +18 +51 +1463 +244 +2545 +Tracking +YOLO5Face +1468 +5 +58 +381 +197 +1510 +473 +18.1 +OpenPifPaf Head +1654 +9 +38 +211 +1300 +407 +1288 +5.4 +TABLE I +THE RESULT OF MATCHING PREDICTED OPENPIFPAF HEAD BOXES AND YOLO5FACE FACE BOXES WITH THE GROUNDTRUTH HEAD LABELS AND +FACE LABELS (DETAILS IN SECTION IV-B). +Label +Face and Head +Head +FP count +Task +Match +Both +Face +Head +None +Head +None +Unfiltered +Detection +YOLO5Face +1637 +2 +26 +247 +252 +1455 +551 +OpenPifPaf Head +1772 +5 +28 +107 +1417 +290 +1540 +Detection Fusion +by confidence +1839 +4 +18 +51 +1463 +244 +2545 +Filtered +Detection +YOLO5Face +1634 +2 +26 +234 +249 +1445 +557 +OpenPifPaf +1728 +5 +26 +137 +1368 +326 +644 +Detection Fusion +by confidence +1818 +4 +18 +56 +1443 +251 +1634 +TABLE II +COMPARISON OF RESULTS WITH BOXES FILTERED BY SIZE FOR BOTH HEAD LABELS AND HEAD PREDICTION. +Filter Size (pixel) +Percent of Face Label Missed +0 +5 +10 +15 +15 +20 +25 +30 +35 +40 +YOLO5Face +OpenPifPaf Head +Confidence Fusion +Fig. 4. +Percent of face missed for different methods when filter size +changes. +verify this by removing head labels whose maximum dimen- +sion is smaller than a certain threshold. Then we pick those +face labels that has a head label associated with it and check +how much are these labels not detected by the predicted +boxes. We consider it missed when the face criterion is not +satisfied. As shown in Fig. 4, The fusion results significantly +reduce the rate of missed faces for various bounding box +size thresholds and OpenPifPaf predicted boxes does better +for smaller boxes. The fusion method achieves zero missing +rate for heads whose maximum dimension are larger than 40 +pixels. +In terms of the high false positive count, we would like to +argue that this does not reflect the true performance of the +algorithm. +This is supported by our observation in the data. For +example, as shown in Fig. 5. OpenPifPaf based method +Fig. 5. +OpenPifPaf based method can begin to find head boxes earlier than +it is recognizable. +was able to infer head boxes (cyan bounding boxes) even +when the pedestrians are very far away. These faces are so +blurry that we did not find it necessary to provide labels +for them, since we only provide labels up to the distance +where human annotator believes that the face is potentially +recognizable. These predictions are counted as false positive +because we don’t provide labels for them and it does no +harm to anonymize them any way. This is not the general +case where we consider a false positive in a random place, +which might cause loss of information in other region if we +anonymize it. +We also show this by the following experiment. As shown +in Table II, we compare the result with or without filtering the +label and predicted head boxes by its maximum dimension. +We set the threshold to 15, which is considered a very small + +口 +口head size in an image of dimension 1920x1440. But with +the threshold, we see that the correct cases dropped slightly +while the false positive cases dropped significantly. It verifies +our observation that OpenPifPaf was able to detect heads +beyond the recognition range, where there are no labels. +For this reason, the resulting high false positive count of +OpenPifPaf based results does not imply that it is not a good +algorithm for anonymization. +Another interesting findings from the Table I is that +the tracking pipeline decreases performance. And the per- +formance decreases more for the YOLO5Face. From our +experience working with the labels, this is mostly caused +by the head movement when human are walking. Our head +movement is not linear, but coupled with ups and downs +for each step. This does not align with the linear velocity +assumption for the Kalman Filter. Given that the YOLO5Face +predicts smaller boxes, it is more vulnerable to the error +cause by this. Another reason is the nonlinear motion from +the data collection platform. We collected the data using two +cameras mounted on top of a vehicle. Thus the target motion +is also coupled with the motion of ego vehicle. +We +compare +the +OpenPifPaf +based +method +with +YOLO5Face in challenging cases where the faces are oc- +cluded, truncated or hidden. We present some results in Fig. +6. In Fig. 6(a), the person on the right was detected by both +methods while the person of the left was only detected by the +proposed method. This suggests that the OpenPifPaf based +method can handle face occlusion better than YOLO5Face. +But it also requires a significant part of the body to present. +This is verified by Fig. 6(b)(c) where multiple faces are +missed by both methods. Similarly, the OpenPifPaf based +method is not good at handling truncated faces, as shown +in Fig. 6(d)(e), since part of the body is also outside of +the image. YOLO5Face fails at Fig. 6(d) but succeeds in +Fig. 6(e). Fig. 6(f) presents an example where the face was +considered hidden but both methods were able to make the +detection. It is also worth noting that in Fig. 6(b), there is a +face with mask only detected by OpenPifPaf based method. +This is one of the benefits of using the proposed method, +being more robust to facial coverings and decorations. +C. Result Change for Different Threshold +In the following experiments, we would like to show +how different face criterion threshold α and head criterion +threshold β would influence the result. +During these experiments, we set one threshold to 0.5 and +increase the other threshold from 0.1 to 0.9 by a step size +0.1. For all pedestrians that have both face label and label, +we classify them into match both, face, head or none and +count the total number for each category. The results are +presented in figures. +For face criterion threshold α, the higher it is, the more +facial region we require the predicted bounding box to cover. +It can be seen from Fig. 7 that when α is higher than 0.5, +the performance of YOLO5Face drops quickly. This is not +the case for OpenPifPaf predicted boxes as shown in Fig. +8. The result is very natural as OpenPifPaf predicted larger +(a) +(b) +(c) +(d) +(e) +(f) +Fig. 6. +Example results of YOLO5Face and the OpenPifPaf based method +in challenging scenarios such as occluded (a,b,c), truncated (d,e) and hidden +(f) faces. Bounding boxes represent face labels (green), head labels (red), +YOLO5Face detections (cyan) and the OpenPifPaf based method detections +(blue). +threshold +count +0 +500 +1000 +1500 +2000 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +Both +Face +Head +None +Fig. 7. +YOLO5Face result on different face label threshold. +boxes, which is likely to cover the face box even when the +matching criterion becomes more strict. This implies that +using OpenPifPaf predicted box would cover most of the +face information. Thus better protect the identity. +For head criterion, YOLO5Face is less influenced by the +change of its threshold β as shown in Fig. 10. Comparatively, +OpenPifPaf predicted boxes drops performance significantly +when the criterion becomes stricter. But the overall accuracy +is still high when the threshold is set to 0.6. +D. Inference Speed +Lastly, we would like to discuss the performance trade- +off for our anonymization pipeline. We report frame rate +for various methods testing on single 2080ti. As shown in +the Table I, the YOLO5Face model runs at 18.1 frames per +second (FPS), which is considered real-time in autonomous +driving settings where many datasets are recorded at 10Hz. + +COVID-19 +VACCINATION +SITE司threshold +count +0 +500 +1000 +1500 +2000 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +Both +Face +Head +None +Fig. 8. +OpenPifPaf predicted head result on different face label threshold. +threshold +count +0 +500 +1000 +1500 +2000 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +Both +Face +Head +None +Fig. 9. +Fusion by confidence result on different face label threshold. +When combined with Openpifpaf, the proposed pipeline runs +at 4 FPS, which no longer satisfies the real-time requirement. +While the proposed method performs better at the expense +of computational costs, we see its value in offline use cases +and potential to reduce its run time. As an offline method, the +proposed algorithm with higher accuracy can reduce the risk +of identity leak in large-scale datasets. It will also require less +post-examination that can save human efforts. Moreover, our +proposed algorithm is a plug-and-play pipeline, which means +as the state-of-the-art model in pose estimation evolves, the +system could be faster and more accurate. +V. CONCLUSION +We proposed to use the skeleton from pose estima- +tion algorithm to infer head bounding boxes for identity +anonymization. Based on this, we further fuse the output +with face detector output to get better results. We show that +the proposed methods can significantly reduce the failure +cases compare to the face detectors alone. Thus we believe +this can be apply to a face anonymization system to better +protect the identity. For future work, we will further improve +the head prediction localization accuracy and develop more +sophisticated fusion method. +REFERENCES +[1] H˚akon Hukkel˚as, Rudolf Mester, and Frank Lindseth. Deepprivacy: A +generative adversarial network for face anonymization. In Advances in +threshold +count +0 +500 +1000 +1500 +2000 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +Both +Face +Head +None +Fig. 10. +YOLO5Face result on different head label threshold. +threshold +count +0 +500 +1000 +1500 +2000 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +Both +Face +Head +None +Fig. 11. +OpenPifPaf result on different head label threshold. +Visual Computing, pages 565–578. Springer International Publishing, +2019. +[2] Yanjia Zhu, Hongxiang Cai, Shuhan Zhang, Chenhao Wang, and +Yichao Xiong. Tinaface: Strong but simple baseline for face detection. +ArXiv, abs/2011.13183, 2020. +[3] Delong Qi, Weijun Tan, Qi Yao, and Jingfeng Liu. 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In 2016 IEEE International +Conference on Image Processing (ICIP), pages 3464–3468, 2016. +[26] Hamid Rezatofighi, Nathan Tsoi, JunYoung Gwak, Amir Sadeghian, +Ian Reid, and Silvio Savarese. Generalized intersection over union. +June 2019. +[27] Harold W. Kuhn. The hungarian method for the assignment problem. +Naval Research Logistics Quarterly, 2:83–97, 1955. + diff --git a/HNE2T4oBgHgl3EQf-wn9/content/tmp_files/load_file.txt b/HNE2T4oBgHgl3EQf-wn9/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..10df1541f69138fac5dcd80ef6405548605484d1 --- /dev/null +++ b/HNE2T4oBgHgl3EQf-wn9/content/tmp_files/load_file.txt @@ -0,0 +1,547 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf,len=546 +page_content='Robust Human Identity Anonymization using Pose Estimation Hengyuan Zhang∗, Jing-Yan Liao∗, David Paz, Henrik I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' Christensen Abstract— Many outdoor autonomous mobile platforms re- quire more human identity anonymized data to power their data-driven algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' The human identity anonymization should be robust so that less manual intervention is needed, which remains a challenge for current face detection and anonymization systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' In this paper, we propose to use the skeleton generated from the state-of-the-art human pose estimation model to help localize human heads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' We develop criteria to evaluate the performance and compare it with the face detection approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' We demonstrate that the proposed algorithm can reduce missed faces and thus better protect the identity information for the pedestrians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' We also develop a confidence-based fusion method to further improve the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' INTRODUCTION Outdoor mobile robots have the huge potential to benefit human life.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' Applications such as autonomous driving or delivery are coming to our life.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' A critical step before they can be deployed at scale is the pedestrian motion prediction problem, given that human motion is highly uncertain and multi-modal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' To address this problem, recent algorithms rely on data-driven approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' However, collecting these data poses risks of leaking identity information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' Areas having identity information, such as human faces and license plates are required to be anonymized by recent regulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' Recent released datasets follow this approach to blur faces and license plates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' Face identity anonymization consists of two parts, face detection and applying an anonymization algorithm to the detected region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' Face detection can be achieved by applying a standard object detector trained on a face specific dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' Face anonymization methods can then be applied on facial regions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' examples include Gaussian blur and more recent methods such as DeepPrivacy [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' In this work, we are majorly concerned about the face detection phase, where we localize the identity sensitive region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' Over these years of research and development, face de- tection has evolved from using hand-crafted features, such as Haar, to deep-learning ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' Although there are various techniques to tackle this problem, the recent trend in face detection algorithms is treating it as a sub-problem in the object detection field [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' At the time of writing this paper, ∗These members contributed equally to this publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' Affiliation: Contextual Robotics Institute, University of California, San Diego, 9500 Gilman Dr, La Jolla, CA 92093 ©2022 IEEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' Personal use of this material is permitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' the state-of-the-art face detection method is YOLO5Face [3], which is based on the YOLOv5 [4] object detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' Nevertheless, the robustness associated with these methods is subject to their ability to detect facial features across various orientations and small regions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' this can drastically vary across various image resolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' Instead, we propose to infer the human head from the skeleton generated by pose estimation algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' Given that human keypoints across the entire body will cover a much larger region than the face alone, this allows us to anonymize identity information more robustly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' This paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' We review the prior work in face detection, face anonymization, and keypoint detection in section II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' Our proposed method is introduced in detail in section III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' We evaluate the proposed method and discuss the metrics in section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' Finally, we conclude with a number of key takeaways in section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' Our major contributions are: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' We propose to leverage keypoint detectors to infer head location, which increases the identity sensitive region detection range and robustness, thus better protecting the identity of pedestrians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' We propose a metric to evaluate the proposed method and compare it fairly with the face detectors in the anonymization context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' Furthermore, we show that a confidence-based fusion method can further improve the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' RELATED WORK Face Detection: As the first step of the anonymization system, face detection can directly impact the performance and robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' There are several approaches for face de- tection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' Before deep learning based methods were utilized, handcrafted features such as Haar [5] were used to detect faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' Recently, after the benchmark dataset WiderFace [6] was released, face detection grew rapidly and focused on challenges such as multi-scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' To tackle these challenges, many works leverage the knowledge from general object detectors in the face detection task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' FDNet [7], as an example of two-stage detectors, employed multi-scale training, multi- scale testing with light-head Faster RCNN [8] and redesigned anchors to help multi-scale performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' For single-stage methods, TinaFace [2], which is based on RetinaNet [9], introduced the Inception Module to enhance capability of multi-scale detection, and used DIoU [10] as regression loss for small face detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' The state of the art, YOLO5Face [3], modified the YOLOv5 [4] bottleneck part in the architecture to increase robustness for large and small faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' Based on SSD [11], PyramidBox [12] utilized contextual information arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content='04243v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content='CV] 10 Jan 2023 Original Image Boxes fused by confidence Face boxes Head boxes OpenPifPaf skeleton Zoom in view YOLO5Face OpenPifPaf Infer head Fusion Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' We infer head bounding box from the skeleton predicted by a pose estimation algorithm (OpenPifPaf), and detect face bounding boxes use a face detector (YOLO5Face).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' These head and face boxes are fused to produce the final output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' (Results are presented in a zoom in view of the original image) with the proposed Pyramid Anchors (PA), the Low-level Feature Pyramid Network (LFPN), and the Context-sensitive Prediction Module (CPM) to predict multi-scale faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' Face Anonymization: In identity anonymization systems, once faces are detected, an anonymization algorithm can be applied to the detected region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' Even though this is not the focus of our paper, we briefly discuss the prior work to provide the necessary context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' This can be done easily by blocking the region with random pixels, blurring, or pixelation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' However, recent datasets targeting outdoor tasks not only need to anonymize the identity, but also need to reduce its impact on pedestrian detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' Recent approaches use learning to address this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' Instead of trying to hide the identity by destroying the details, the methods focus on changing the appearance, either adding distortion or gener- ating a completely new face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' For example, DeepPrivacy [1] uses a generative adversarial network to produce faces that matches the background context but also looks completely different from the original face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' CIAGAN [13] further brings in control of features of anonymization as well as temporal consistency in a video sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' Pose Estimation: There are two approaches in pose esti- mation: top-down and bottom-up methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' The former meth- ods could be perceived as a person detector with a single- person pose estimator, while the latter predicts body joints individually and associates them afterwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' By leveraging a large amount of data and methods from human detection, top-down methods can achieve competitive results in human pose estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' For example, Mask R-CNN [14] extends instance segmentation to pose estimation by predicting a one- hot map for each joint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' However, top-down methods are not suitable for our case because partial occlusions can lead to missed detections in the human detection stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' This is the result from its two-stage detecting nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' For instance, the performance of human detection could be unstable if only partial observations are presented in the image frame, making it difficult to detect skeletons properly, thus posing challenges to localize head poses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' In contrast, this can be addressed by bottom-up methods in most cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' Moreover, bottom-up methods are also more suitable for real-time applications as they are faster [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' The pioneer work, DeepCut [16], detects keypoints individually and associates detected joints with an integer linear program, which is computationally expensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' Later works accelerate prediction time with greedy decoders, such as OpenPose [17], PersonLab [18], and our method of choice, OpenPifPaf [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' OpenPifPaf handles pose in various scale by predicting the joint location and size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' Although top- down methods generally handle scale variance better than bottom-up methods, we find that OpenPifPaf satisfies our requirements as the human faces that are too small don’t need to be detected for anonymization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' METHOD To overcome the limitations of the face-detection based strategies, we propose to use human body pose estimation methods to infer head position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' The pipeline is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' In this section, we introduce the keypoint detection and association method we use, OpenPifPaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' We then outline the process of estimating head position from the keypoints generated by OpenPifPaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' Finally, we discuss the tracking and fusion method that jointly leverages face and head detections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' OpenPifPaf Our method is based on the human body pose estimation model, OpenPifPaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' OpenPifPaf consists of three parts, back- bone feature extraction, Composite Intensity Fields (CIF) and Composite Association Fields (CAF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' ■口 口口 口The backbone stage is extract feature from image with ResNet [20] or ShuffleNetV2 [21], and the output feature would be shared by CIF and CAF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' The CIF is a 1x1 convo- lution that predicts semantic keypoints, which is identical to the Part Intensity Field (PIF) introduced in [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' For human subjects, the keypoints correspond to body parts such as joints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' For a specific part, the network outputs the confidence score, its two dimensional vector from each pixel and a scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' Given that the feature maps estimated by the CIF prediction head are coarse, a convolution is applied over the coarse targets using a bivariate Gaussian kernel, which yields a high resolution confidence map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' The CAF part shares the same backbone;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' however, this head predicts a confidence, two vectors that point towards the two parts this association connects and two spreads for spatial precision of the regression and two joint sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' The output of OpenPifPaf is a set of keypoint locations and the confidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' Such keypoints are meaningful parts of a human, such as left shoulder, right eye and nose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' Given the predefined connectivity, we can link the key points and produce the skeleton of a human.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' We note that OpenPifPaf generalizes well on our data without fine tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' The detection range of it is much better than the YOLO5Face detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' It is also more robust on the same range, which raises our interests to examine its use for face detection and identity anonymization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' Head Prediction Given the human body pose predictions generated by OpenPifPaf, which consist of skeletons for the COCO [23] dataset, we focus on inferring head positions for anonymiza- tion purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' The actual procedure differs depending on whether facial keypoints are predicted or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' When there are facial keypoints available from OpenPif- Paf, we can readily estimate the head center from these facial keypoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' Then the bounding box dimensions (height and width) of the head are inferred from torso length, which is given by the distance from the shoulder keypoint to the hip keypoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' We assume a fixed ratio between the head dimension to the torso length in terms of pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' This actually varies from person to person but serves as a good approximation for anonymization purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' When facial keypoints are not available but shoulder and hip keypoints are available, the head position needs to be inferred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' This is done simply by centering a bounding box horizontally between the shoulder keypoints, and vertically on top of the shoulder keypoints plus a neck length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' Again, the bounding box dimensions are inferred from the torso length by assuming a fix ratio between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' We also assume the neck length has a fixed ratio with respect to the dimension of the head and experimentally find it using our data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' Head and Face Fusion Given the head boxes predicted by OpenPifPaf, it is intuitive to consider fusion of these head boxes with the face boxes generated by the YOLO5Face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' Since our goal is to reduce missed faces as much as possible to better protect identity, we design our approach to be more tolerant to false positives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' Thus the simplest fusion strategy involves keeping both results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' However, we can also remove face boxes that are covered by the head boxes since the region will be anonymized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' A more risky strategy is to remove a head bounding box if there exists a face prediction within.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' This can potentially increase the probability of exposing identity information;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' however, this strategy is justified by the fact that usually when a face box is associated with a head box, the subject in question is closer to us and we are confident about the face location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' Finally, we can filter the bounding box by its confidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' We keep the bounding box with a higher confidence score if for a head bounding box there exists a face prediction within.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' The confidence score for face bounding box comes directly from the output of YOLO5Face while the confidence score for a head bounding box is the average of the keypoint confidence from OpenPifPaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' Head and Face Tracking Given the predicted head bounding boxes from OpenPif- Paf, we leverage a tracking pipeline to reduce false positives but also to incorporate temporal consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' A Kalman Filter [24] was applied with association based on center-distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' We modify the SORT [25] tracker for our application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' Initially we associated detections with existing tracks based on IOU and GIoU [26];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' however, this often generated poor associations as faces are often too small and may not overlap between adjacent image frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' Thus, we choose an l2 center distance based association which gives better performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' EXPERIMENTS We evaluate the proposed algorithm and present the results in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' To this end, we label two video sequences, one for parameter tuning, which consist of 257 frames, and another 544 frames in different scene for testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' Then, we define a metric that allows fair comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' Finally, we present the results of our algorithm and its comparison with the face detection algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' It is noteworthy that the data was recorded during COVID and masks can be observed but the neural networks were not fine-tuned for this specific case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' Metric and Labeling A fair comparison of our approach against the face de- tectors is challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' Standard evaluations use intersection over union (IOU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' But considering that we focus on identity anonymization, this can be achieved by blurring most of the face and there will be no consequences of including other parts of the head.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' For example, two examples with an IoU score of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content='5 are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' While both achieve the same score, the example on the right evidently better protects the sensitive region compared to the example on the left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' Even though additional regions are also included, applying anonymization to the region has lower risk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' Green solid line boxes are face labels and blue dashed line boxes are detections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' Left and right detection both have 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content='5 IoU with the face label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' The green solid line box is a face label and the red dotted line box is a head label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' Another challenging point is the nature of the algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' Face detectors capture the faces and not the head as depicted by the green bounding boxes, The bounding boxes may change largely when the head facing angle changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' Since our approach mostly captures the bounding boxes of head, which includes hair and is more invariant to the facing direction, evaluating them using a standard IoU metric would lead to inconsistent comparisons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' Therefore we use a two label strategy, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' We label both faces (green solid line) and heads (red dotted line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' We then denote the bounding box for a detection by D, a face label by F, and a head label as H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' Moreover, two criteria for evaluation are introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' First, the face criterion is defined as D ∩ F F > α, which implies that at least a portion α of the face label should be included in D;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' thus most of the identity information can be anonymized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' Second, the head criterion is defined by D ∩ H D > β, which implies that at least a portion β of the detection should be included in the head label;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' thus the background portion within predicted bounding box is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' We collected two video sequences in dense pedestrian walkways in a university area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' For our labeled video se- quences, both heads and faces of the pedestrians are labeled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' Face labels are considered mostly the front of the head with skin, not including the ears and hair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' Head labels include the hair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' We provide labels up to the distance where human annotator believes that the face is potentially recognizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' The video sequence for parameter tuning consists of 259 frames, with 1018 face labels and 1726 head labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' The video sequence for testing consists of 543 frames, with 1949 face labels and 3819 head labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' Comparison with Face Detection Based on the criteria proposed in the previous section, we evaluate the proposed OpenPifPaf based algorithm, the face detector and the fusion method on our testing video sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' The evaluation process are as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' First, we associate each face label with a head label, which is based on the assumption that each face label should be within a head label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' Then for the pedestrians, we have a group of front facing individuals that have both face labels and head labels, and another group only have head labels (back of the pedestrians).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' Second, we associate the detections with the head labels using the Hungarian algorithm [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' Finally, we evaluate the predicted boxes with the labels using the proposed criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' Unless otherwise specified, we set α and β to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' For the targets that have a face label and a head label, we check if the associated detection box satisfies both of the criteria, face only, head only or none.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' We denote the result as match both, face, head or none.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' For the targets that has only a head label, we only evaluate the head criterion for the associated detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' We denote the result as match head or none.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' If a detected box is not associated with any of the labels or doesn’t satisfy any of the criteria with the associated labels, it is counted as a false positive case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' As shown in the Table I, OpenPifPaf predicted detec- tions produce significantly more face matches compared to YOLO5Face;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' however, they also generate considerably more false positives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' The results suggest that OpenPifPaf predicted boxes are doing better in terms of finding face regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' Furthermore, given that we set a higher cost to missing a face than generating false positives, we consider the performance of the OpenPifPaf based method acceptable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' The fusion method further improves the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' If we keep both the face and the head boxes, the approach produces the lowest number of missed faces while also gen- erating the highest number of false positives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' Applying other fusion methods when a predicted face from YOLO5Face is in the predicted head from OpenPifPaf allows us to greatly reduce false positives while the number of missed faces only drops slightly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' Among these fusion methods, fusion by confidence and the keep head method produce similar results, both are slightly better than the keep face method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' Post processing shows that the similarity of the two fusion methods is because OpenPifPaf in general gives higher confidence scores compared to the scores from YOLO5Face, even when the predicted boxes are less accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' To make the confidence fusion more effective, an extension to our work is to tune the confidence scores from the two neural networks using the same dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' Another advantage is that the OpenPifPaf’s predicted boxes miss fewer facial regions when the faces are large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' We Label Face and Head Head FP count FPS Task Match Both Face Head None Head None Detection YOLO5Face 1637 2 26 247 252 1455 551 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content='1 OpenPifPaf Head 1772 5 28 107 1417 290 1540 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content='4 Detection Fusion keep head 1838 5 18 51 1463 244 2545 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content='04 keep face 1829 4 28 51 1465 242 2543 keep both 1840 3 18 51 1465 242 3784 by confidence 1839 4 18 51 1463 244 2545 Tracking YOLO5Face 1468 5 58 381 197 1510 473 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content='1 OpenPifPaf Head 1654 9 38 211 1300 407 1288 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content='4 TABLE I THE RESULT OF MATCHING PREDICTED OPENPIFPAF HEAD BOXES AND YOLO5FACE FACE BOXES WITH THE GROUNDTRUTH HEAD LABELS AND FACE LABELS (DETAILS IN SECTION IV-B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content='Label ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content='Face and Head ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content='Head ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content='FP count ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content='Task ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content='Match ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content='Both ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content='Face ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content='Head ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content='None ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content='Head ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content='None ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content='Unfiltered ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content='Detection ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content='YOLO5Face ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content='1637 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content='26 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content='247 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content='252 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content='1455 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content='551 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content='OpenPifPaf Head ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content='1772 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content='28 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content='107 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content='1417 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content='290 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content='1540 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content='Detection Fusion ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content='by confidence ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content='1839 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content='18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content='51 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content='1463 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content='244 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content='2545 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content='Filtered ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content='Detection ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content='YOLO5Face ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content='1634 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content='26 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content='234 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content='249 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content='1445 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content='557 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content='OpenPifPaf ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content='1728 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content='26 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content='137 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content='1368 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content='326 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content='644 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content='Detection Fusion ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content='by confidence ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content='1818 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content='18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content='56 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content='1443 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content='251 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content='1634 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content='TABLE II ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content='COMPARISON OF RESULTS WITH BOXES FILTERED BY SIZE FOR BOTH HEAD LABELS AND HEAD PREDICTION.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' Filter Size (pixel) Percent of Face Label Missed 0 5 10 15 15 20 25 30 35 40 YOLO5Face OpenPifPaf Head Confidence Fusion Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' Percent of face missed for different methods when filter size changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' verify this by removing head labels whose maximum dimen- sion is smaller than a certain threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' Then we pick those face labels that has a head label associated with it and check how much are these labels not detected by the predicted boxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' We consider it missed when the face criterion is not satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' 4, The fusion results significantly reduce the rate of missed faces for various bounding box size thresholds and OpenPifPaf predicted boxes does better for smaller boxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' The fusion method achieves zero missing rate for heads whose maximum dimension are larger than 40 pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' In terms of the high false positive count, we would like to argue that this does not reflect the true performance of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' This is supported by our observation in the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' For example, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' OpenPifPaf based method Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' OpenPifPaf based method can begin to find head boxes earlier than it is recognizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' was able to infer head boxes (cyan bounding boxes) even when the pedestrians are very far away.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' These faces are so blurry that we did not find it necessary to provide labels for them, since we only provide labels up to the distance where human annotator believes that the face is potentially recognizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' These predictions are counted as false positive because we don’t provide labels for them and it does no harm to anonymize them any way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' This is not the general case where we consider a false positive in a random place, which might cause loss of information in other region if we anonymize it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' We also show this by the following experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' As shown in Table II, we compare the result with or without filtering the label and predicted head boxes by its maximum dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' We set the threshold to 15, which is considered a very small 口 口head size in an image of dimension 1920x1440.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' But with the threshold, we see that the correct cases dropped slightly while the false positive cases dropped significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' It verifies our observation that OpenPifPaf was able to detect heads beyond the recognition range, where there are no labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' For this reason, the resulting high false positive count of OpenPifPaf based results does not imply that it is not a good algorithm for anonymization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' Another interesting findings from the Table I is that the tracking pipeline decreases performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' And the per- formance decreases more for the YOLO5Face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' From our experience working with the labels, this is mostly caused by the head movement when human are walking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' Our head movement is not linear, but coupled with ups and downs for each step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' This does not align with the linear velocity assumption for the Kalman Filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' Given that the YOLO5Face predicts smaller boxes, it is more vulnerable to the error cause by this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' Another reason is the nonlinear motion from the data collection platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' We collected the data using two cameras mounted on top of a vehicle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' Thus the target motion is also coupled with the motion of ego vehicle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' We compare the OpenPifPaf based method with YOLO5Face in challenging cases where the faces are oc- cluded, truncated or hidden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' We present some results in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' 6(a), the person on the right was detected by both methods while the person of the left was only detected by the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' This suggests that the OpenPifPaf based method can handle face occlusion better than YOLO5Face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' But it also requires a significant part of the body to present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' This is verified by Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' 6(b)(c) where multiple faces are missed by both methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' Similarly, the OpenPifPaf based method is not good at handling truncated faces, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' 6(d)(e), since part of the body is also outside of the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' YOLO5Face fails at Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' 6(d) but succeeds in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' 6(e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' 6(f) presents an example where the face was considered hidden but both methods were able to make the detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' It is also worth noting that in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' 6(b), there is a face with mask only detected by OpenPifPaf based method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' This is one of the benefits of using the proposed method, being more robust to facial coverings and decorations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' Result Change for Different Threshold In the following experiments, we would like to show how different face criterion threshold α and head criterion threshold β would influence the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' During these experiments, we set one threshold to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content='5 and increase the other threshold from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content='1 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content='9 by a step size 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' For all pedestrians that have both face label and label, we classify them into match both, face, head or none and count the total number for each category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' The results are presented in figures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' For face criterion threshold α, the higher it is, the more facial region we require the predicted bounding box to cover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' It can be seen from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' 7 that when α is higher than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content='5, the performance of YOLO5Face drops quickly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' This is not the case for OpenPifPaf predicted boxes as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' The result is very natural as OpenPifPaf predicted larger (a) (b) (c) (d) (e) (f) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' Example results of YOLO5Face and the OpenPifPaf based method in challenging scenarios such as occluded (a,b,c), truncated (d,e) and hidden (f) faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' Bounding boxes represent face labels (green), head labels (red), YOLO5Face detections (cyan) and the OpenPifPaf based method detections (blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' threshold count 0 500 1000 1500 2000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content='9 Both Face Head None Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' YOLO5Face result on different face label threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' boxes, which is likely to cover the face box even when the matching criterion becomes more strict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' This implies that using OpenPifPaf predicted box would cover most of the face information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' Thus better protect the identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' For head criterion, YOLO5Face is less influenced by the change of its threshold β as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' Comparatively, OpenPifPaf predicted boxes drops performance significantly when the criterion becomes stricter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' But the overall accuracy is still high when the threshold is set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' Inference Speed Lastly, we would like to discuss the performance trade- off for our anonymization pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' We report frame rate for various methods testing on single 2080ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' As shown in the Table I, the YOLO5Face model runs at 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content='1 frames per second (FPS), which is considered real-time in autonomous driving settings where many datasets are recorded at 10Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' COVID-19 VACCINATION SITE司threshold count 0 500 1000 1500 2000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content='9 Both Face Head None Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' OpenPifPaf predicted head result on different face label threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' threshold count 0 500 1000 1500 2000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content='9 Both Face Head None Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' Fusion by confidence result on different face label threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' When combined with Openpifpaf, the proposed pipeline runs at 4 FPS, which no longer satisfies the real-time requirement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' While the proposed method performs better at the expense of computational costs, we see its value in offline use cases and potential to reduce its run time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' As an offline method, the proposed algorithm with higher accuracy can reduce the risk of identity leak in large-scale datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' It will also require less post-examination that can save human efforts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' Moreover, our proposed algorithm is a plug-and-play pipeline, which means as the state-of-the-art model in pose estimation evolves, the system could be faster and more accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' CONCLUSION We proposed to use the skeleton from pose estima- tion algorithm to infer head bounding boxes for identity anonymization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' Based on this, we further fuse the output with face detector output to get better results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' We show that the proposed methods can significantly reduce the failure cases compare to the face detectors alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' Thus we believe this can be apply to a face anonymization system to better protect the identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' For future work, we will further improve the head prediction localization accuracy and develop more sophisticated fusion method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' REFERENCES [1] H˚akon Hukkel˚as, Rudolf Mester, and Frank Lindseth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' Deepprivacy: A generative adversarial network for face anonymization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' In Advances in threshold count 0 500 1000 1500 2000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content='9 Both Face Head None Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' YOLO5Face result on different head label threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' threshold count 0 500 1000 1500 2000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content='9 Both Face Head None Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' OpenPifPaf result on different head label threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' Visual Computing, pages 565–578.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' Springer International Publishing, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' [2] Yanjia Zhu, Hongxiang Cai, Shuhan Zhang, Chenhao Wang, and Yichao Xiong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQf-wn9/content/2301.04243v1.pdf'} +page_content=' Tinaface: 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100644 index 0000000000000000000000000000000000000000..0bef9258e32be39a0fcee864d9d1ac56df90f8be --- /dev/null +++ b/HtFAT4oBgHgl3EQfuB5J/content/tmp_files/2301.08667v1.pdf.txt @@ -0,0 +1,1280 @@ +Opaque prior distributions in Bayesian latent variable models +Edgar C. Merkle +University of Missouri, USA +Oludare Ariyo +University of Essex, UK +Sonja D. Winter +University of Missouri, USA +Mauricio Garnier-Villarreal +Vrije Universiteit Amsterdam, Nederland +Abstract +We review common situations in Bayesian latent variable models where the +prior distribution that a researcher specifies does not match the prior distri- +bution that the estimation method uses. These situations can arise from the +positive definite requirement on correlation matrices, from the sign indeter- +minacy of factor loadings, and from order constraints on threshold param- +eters. The issue is especially problematic for reproducibility and for model +checks that involve prior distributions, including prior predictive assessment +and Bayes factors. In these cases, one might be assessing the wrong model, +casting doubt on the relevance of the results. The most straightforward solu- +tion to these issues sometimes involves use of informative prior distributions. +We explore other solutions and make recommendations for practice. +Introduction +Bayesian latent variable models, including structural equation models and time series +models, are increasingly becoming more complex to accommodate the complex datasets that +modern technology permits (e.g., Asparouhov, Hamaker, & Muthén, 2018; Bürkner, 2021; +Depaoli, Lai, & Yang, 2021; dos Santos, Azevedo, & Fox, 2022; Driver, Oud, & Voelkle, +2017; Enders, Keller, & Levy, 2018; Haaf, Merkle, & Rouder, 2020; Haaf & Rouder, 2021; +Hoijtink & van de Schoot, 2018; Kaplan, Chen, Yavuz, & Lyu, 2022; Levy & Enders, 2021; +Miočević, Levy, & MacKinnon, 2021; Paganin et al., 2022; Rast, Martin, Liu, & Williams, +2022; Ulitzsch, Pohl, Khorramdel, Kroehne, & von Davier, 2022; Van De Schoot et al., +2013; Van Erp & Browne, 2021; Wilcox, Jacobucci, Zhang, & Ammerman, 2021; Zyphur +This work was supported by the Institute of Education Sciences, U.S. Department of Education, Grant +R305D210044. Correspondence to Edgar Merkle, University of Missouri. Email: merklee@missouri.edu. +arXiv:2301.08667v1 [stat.ME] 20 Jan 2023 + +OPAQUE PRIORS +2 +et al., 2021). To ensure that these complex models are working correctly, we think it is +important to fully understand the problematic issues that can arise in simpler Bayesian +models with latent variables. For example, researchers often overlook the fact that multiple +WAIC metrics are available for the same model, depending on whether or not one includes +latent variables in the likelihood (Merkle, Furr, & Rabe-Hesketh, 2019). The current paper +is devoted to issues with prior distributions that occur for these models, which are easy to +overlook and which can lead to inaccurate results and model summaries. +The issues that we consider here can generally be called opaque prior distributions. +They involve the fact that, for some Bayesian models, the prior distribution that you specify +is not the prior distribution that you actually use. This happens because prior distributions +are influenced by various details surrounding MCMC implementation, beyond any prior that +an analyst specifies. These issues are already known to many statisticians, but they have +some unique manifestations in Bayesian psychometric models with latent variables. Further, +these issues have not received much attention in the context of Bayesian psychometric +modeling. +In psychometric models with latent variables, opaque prior distributions can arise +from positive definite constraints associated with model covariance matrices, from sign +indeterminacies in factor loadings, and from order constraints associated with threshold +parameters. +One example was considered by J. Ghosh and Dunson (2009), who study +a parameter expansion algorithm for estimating Bayesian factor analysis models. +They +developed a Gibbs sampler for an overparameterized model in which the factor loadings +were not identified, then translated the unidentified parameters to identified parameters +via postprocessing. Importantly, they showed that the priors for the identified loadings +(obtained via postprocessing) differed from the priors for the unidentified loadings. We +consider this example in more detail later. +Opaque prior distributions can cause a variety of problems. One way in which these +problems arise is through recent studies where researchers seek to recommend the “best” +prior distributions for various models. For example, specific prior distributions have been +recommended for factor loadings and correlation parameters in confirmatory factor analysis +(Lüdtke, Ulitzsch, & Robitzsch, 2021; Ulitzsch, Lüdtke, & Robitzsch, 2021), factor loadings +and intercept parameters in item factor analysis with dichotomous indicators (Bainter, +2017), regression parameters in mediation models with latent variables (Miočević et al., +2021), covariance matrices in SEM (Liu, Depaoli, & Marvin, 2022; van Zundert, Somer, & +Miočević, 2022), random effect parameters in multilevel SEMs (Van Erp & Browne, 2021; +Zitzmann, Lüdtke, Robitzsch, & Hecht, 2021), and variance parameters in approximate +measurement invariance modeling (Pokropek, Schmidt, & Davidov, 2020). However, as we +will show, opaque priors imply that such recommendations are sometimes software-specific. +In other words, alternative software packages may implement the same model in a different +way, leading to a different implied prior distribution even though the user always specified +the same prior distribution. +We could also experience problems with metrics that explicitly evaluate the prior +distribution, including Bayes factors (e.g., Heck et al., 2022; Kass & Raftery, 1995) and +prior predictive checks (e.g., Vanpaemel, 2020). In both cases, if a researcher specifies a +model with opaque prior distributions, then it is easy to use the wrong prior distributions to +compute the metrics. Relatedly, some methods for verifying the accuracy of one’s MCMC + +OPAQUE PRIORS +3 +algorithm involve generating data from prior distributions. If researchers use the wrong +priors, then they may conclude that their MCMC algorithm is problematic even though it +runs correctly, or vice versa. +The intent of this paper is to illustrate how opaque priors arise and to provide solutions +for avoiding them. This can allow for reproducible results across software implementations, +and it can lead to improved prior predictive assessments and other metrics. In the pages +below, we show how opaque priors can arise from positive definite constraints, from sign +indeterminacies, and from parameter order constraints. For each of these topics, we provide +examples of the problem, discuss how the problem can lead to compounding problems in +model assessment, and provide recommendations for avoiding the problem. +Finally, we +summarize and make general recommendations for practice. +Positive definite constraints +In Bayesian modeling, prior distributions for the covariance matrix often involve the +inverse-Wishart (IW) distribution due to its conditional conjugacy. However, the IW dis- +tribution can be problematic because it assumes the same amount of prior information for +every variance parameter (i.e. a prior relationship between the variance parameter the vari- +ances and correlations). To overcome this challenge, various strategies have been suggested +for separately specifying priors on the variance and correlation parameters underlying the +covariance matrix (Barnard, McCulloch, & Meng, 2000). These priors have been shown +to perform better than the classical IW (Alvarez, Niemi, & Simpson, 2014; Ariyo, Lesaffre, +Verbeke, & Quintero, 2022; Huang, Wand, et al., 2013). However, things become more com- +plicated with three or more random effects because certain restrictions must be imposed to +ensure positive definiteness of the covariance matrix (Barnard et al., 2000; Daniels & Kass, +1999; Huang et al., 2013; Hurtado Rúa, Mazumdar, & Strawderman, 2015; Wei & Higgins, +2013). +The situation becomes even more complicated when the covariance matrix has model- +imposed constraints, which can arise in SEMs with correlated residuals or with across-group +equality constraints. In these models, we cannot impose an IW (or other prior) on the +full covariance matrix because those priors will not respect the constraints imposed by the +model. We could instead place priors on individual parameters within the covariance matrix, +but, when we build a covariance matrix using these parameters, the resulting matrix will +sometimes be non-positive definite. We elaborate on these issues below. +Illustration +An example comes from the popular “Political Democracy” model originally described +by Bollen (1989), shown below in lavaan syntax (Rosseel, 2012). +Intended to describe +countries’ relationships between their levels of industrialization and democracy, the model +contains four observed variables that are measured once during 1960 and again during +1965. The residuals of the 1960 variables are allowed to correlate with the residuals of the +corresponding 1965 variables. Additionally, there exists a pair of similar variables collected +during 1960 and again during 1965, leading to three additional residual correlations. The +full structure of the residual covariance matrix is shown in Figure 1. + +OPAQUE PRIORS +4 +x1 +x2 +x3 +y1 +y2 +y3 +y4 +y5 +y6 +y7 +y8 +x1 +× +x2 +× +x3 +× +y1 +× +× +y2 +× +× +× +y3 +× +× +y4 +× +× +× +y5 +× +× +y6 +× +× +× +y7 +× +× +y8 +× +× +× +Figure 1. Political democracy model, structure of residual covariance matrix. Free param- +eters are marked by ×. +model <- ' +# measurement model +ind60 =~ x1 + x2 + x3 +dem60 =~ y1 + a*y2 + b*y3 + c*y4 +dem65 =~ y5 + a*y6 + b*y7 + c*y8 +# regressions +dem60 ~ ind60 +dem65 ~ ind60 + dem60 +# residual correlations +y1 ~~ y5 +y2 ~~ y4 + y6 +y3 ~~ y7 +y4 ~~ y8 +y6 ~~ y8 +' +As described earlier, we could elect to put a univariate prior distribution on each +variance (or standard deviation or precision) in this matrix, and also on each correlation +associated with off-diagonal entries. But this is problematic because the univariate priors +on correlations will often yield non-positive definite correlation matrices. And if we consider +only the space of positive definite correlation matrices, the univariate priors on correlations +are more informative than one would expect. That is, the prior distributions are opaque: +the analyst specifies a set of priors that are different from the implied prior distributions, +which must obey the constraint of positive definiteness. +How can we characterize the implied prior distributions? We could simply generate +thousands of correlation matrices from the prior, then discard matrices that are not positive +definite, then visualize what is left. As an example of this, we specified a Uniform(−1, 1) +prior for each free correlation in Figure 1. We then generated 100,000 correlation matrices +with the desired structure, discarding the 57,818 matrices that were not positive definite. +Finally, we examined the distributions of the remaining matrices, with a histogram for a +single correlation parameter appearing in Figure 2. We see that the resulting distribution + +OPAQUE PRIORS +5 +0 +500 +1000 +1500 +2000 +−1.0 +−0.5 +0.0 +0.5 +1.0 +Cor(y2,y4) +Frequency +Figure 2. Bollen model, implied prior distribution of a single correlation parameter after +accounting for positive definite constraints. +is no longer uniform; it is approximately symmetric around 0, with more density near 0 +than near −1 or 1. This is because our uniform priors did not account for the fact that +correlation matrices must be positive definite. +To technically describe the positive definite constraint, we can simultaneously permute +the rows and columns of the correlation matrix to obtain a block diagonal matrix. A block +diagonal matrix is useful here because the determinant of the full matrix is implied by the +determinants of each individual block within the matrix. This allows us to examine whether +or not the full matrix is positive definite, by working with submatrices of smaller dimension. +The Cuthill-McKee algorithm (Cuthill & McKee, 1969) allows us to automatically +find an appropriate, block-diagonal permutation. After carrying out this algorithm (via the +netprioR package; Schmich, 2022) and permuting the rows and columns, we arrive at the +matrix in Figure 3. This shows that, to keep the full matrix positive definite, we only need +to worry about the 4 × 4 block in the lower right that involves y2, y4, y6, and y8. The +priors on the (y3, y7) and (y1, y5) correlations have no influence on the positive definiteness +of the full matrix (because a 2 × 2 matrix is positive definite for any correlation in (−1, 1)), +so that we can safely put a uniform prior on each of those correlations. +After applying traditional rules for computing matrix determinants, we can express +the determinant of the 4 × 4 block of correlations as +det(R(4×4)) = 1 + (r1r4 − r2r3)2 − +4 +� +i=1 +r2 +i . +(1) +This shows analytically how the positive definite constraint of our correlation matrix influ- +ences our univariate beta priors. The univariate priors are collectively constrained by the +requirement that Equation (1) be greater than 0. + +OPAQUE PRIORS +6 +x1 +x2 +x3 +y1 +y5 +y3 +y7 +y2 +y4 +y6 +y8 +x1 +1 +x2 +1 +x3 +1 +y1 +1 +× +y5 +× +1 +y3 +1 +× +y7 +× +1 +y2 +1 +× +× +y4 +× +1 +× +y6 +× +1 +× +y8 +× +× +1 +Figure 3. Political democracy model, structure of permuted correlation matrix. Free pa- +rameters are marked by ×. +Implications for Bayes factors +To show how this issue becomes problematic for applied work, we consider the cal- +culation of Bayes factors using the Bollen model. Imagine that we wish to know whether +the two residual correlations involving y2 are necessary. To address this question, we could +compute a Bayes factor comparing a model that includes those two residual correlations, +to a model without those residual correlations. A computationally-cheap way to do this +is the Savage-Dickey method (Dickey & Lientz, 1970; Wagenmakers, Lodewyckx, Kuriyal, +& Grasman, 2010), which requires us to only fit the model that includes the two residual +correlations. The Bayes factor (for the model with correlations versus the model without +correlations) is then the ratio of two densities: the prior density of the two residual covari- +ances at 0, and the posterior density of the two residual covariances at 0. But if we are to +do the correct calculation here, we need to make sure that we use the correct prior densities. +If we set Uniform priors on each individual correlation and ignore the positive definite +constraint, then the prior density of the two correlations at 0 equals 0.25. This corresponds +to a log-density of −1.39. +The true joint prior density on the two correlations (which +respects the positive definite constraint) does not have a simple form, but we can approxi- +mate the density via the positive definite correlation matrices that we randomly generated +earlier (using a density estimation method from the ks package in R; Duong, 2022). Our +approximate joint density at 0 is now 0.46, corresponding to a log-density of −0.78. +These evaluations lead to Bayes factors for the model with correlations, relative to +the model without correlations. Using our incorrect priors that do not account for posi- +tive definite constraints, we obtain a log-Bayes factor of 4.66 in favor of the model with +correlations. Using our priors that do account for positive definite constraints, we obtain +a log-Bayes factor of 5.27 in favor of the model with correlations. Both of these Bayes +factors provide support for the model with correlations, but possibly at different levels of +evidence. For example, if we subscribe to the rules of thumb offered by Kass and Raftery +(1995), then these two Bayes factors lead us to conclude “strong” evidence using the incor- +rect prior calculation, and “very strong” evidence using the correct calculation. We agree + +OPAQUE PRIORS +7 +with you, the reader, that these cutoffs are arbitrary and that the Bayes factors do not +differ by very much. But the main point is that the Bayes factors will systematically differ +depending on how we compute prior densities. These differences will sometimes lead to +different substantive conclusions in practice, with the easier computation being incorrect. +Solutions +While it was straightforward to visualize implied prior distributions in the Bollen +model, the process becomes inefficient for correlation matrices with more than a few di- +mensions. For such correlation matrices, few of the randomly-generated matrices will be +positive definite, and it will take a long time to obtain a sufficient number of positive definite +matrices to describe the implied prior. Additionally, every unique structure of correlation +matrix will have a unique positive definite constraint, similar to the one from Equation (1). +So we desire more general solutions that do not involve a simulation. We describe three +solutions below that differ in complexity and in the extent to which they fully solve the +problem. +Informative priors. +The simplest, partial solution is to maintain the univariate +priors on individual correlations, but make those priors informative around 0. For example, +instead of placing uniform priors on the correlations, we might use Beta(5,5) priors. The +implied prior distributions will then be closer to what the user specifies, because these +informative priors will more often lead to positive definite correlation matrices. But this is +not a full solution because, even with informative priors, we may still encounter non-positive +definite correlation matrices. And it is not straightforward to predict when or how often +this will happen. Additionally, depending on one’s application, certain informative priors +may be inappropriate. +Priors on Cholesky decomposition. +A more general solution to this problem +comes from putting priors on the Cholesky decomposition of the correlation matrix, which is +related to the R. P. Ghosh, Mallick, and Pourahmadi (2021) approach for time series models. +The Cholesky decomposition is advantageous because, to ensure that the correlation matrix +stays positive definite, we only have to ensure that the diagonal elements of the Cholesky +decomposition are positive. For the Political Democracy model, the 4×4 covariance matrix +from the bottom right corner of Figure 2 has a Cholesky decomposition with structure +� +� +� +� +� +c11 +c21 +c22 +c31 +−c21c31/c22 +c33 +0 +c42 +c43 +c44 +� +� +� +� +� , +(2) +where the entries {c11, c22, c33, c44} are constrained to be positive, while {c21, c31, c42, c43} +are unconstrained. Additionally, the entry in row 3, column 2 is fully determined by other +entries. If we place gamma priors (say) on the diagonal entries and normal priors on the +remaining c variates, we can maintain the desired structure of the covariance matrix while +also maintaining positive definiteness. +A disadvantage of this approach is that the entries of the Cholesky decomposition do +not necessarily have intuitive interpretations, so that it is difficult to set informative priors. +Each diagonal entry is related to the portion of the corresponding variable’s variance that + +OPAQUE PRIORS +8 +cannot be accounted for by variables that occur further to the left of the matrix. Each off- +diagonal entry is related to a partial correlation conditioned on variables further to the left +of the matrix (see Joe, 2006; Lewandowski, Kurowicka, & Joe, 2009; Pourahmadi, Daniels, +& Park, 2007). This implies that the order of the variables matters. A further difficulty +is that there does not appear to be an automatic way to obtain a Cholesky structure (like +that of Equation (2)) for arbitrarily-structured covariance matrices. +Combining LKJ with Informative Priors. +A final solution was recently de- +scribed in a blog post by Martin (2021). This solution involves use of a prior distribution +that is the product of (i) an LKJ prior on the full correlation matrix, and (ii) informative +priors on individual entries of the correlation matrix. The resulting prior inherits the posi- +tive definite constraint from (i) while also inheriting the informativeness from (ii). At the +moment, it is not clear that this approach can be used to fix individual entries of the cor- +relation matrix; instead, Martin (2021) recommends highly-informative priors around the +fixed value that is desired (similar to the notion of “approximate zeros” in a factor loading +matrix). At this time, it is not clear whether the highly-informative priors could generally +replace hard constraints, or whether they could cause problems with model convergence. +Sign Indeterminacies +We now turn to a problem that is more specific to SEM: sign indeterminacies of loading +parameters. It is well known that, if we change the signs of all loadings, the SEM likelihood +(usually) stays the same. To avoid this issue, SEM software typically “prefers” positive +loadings. +This preference for positive loadings is implemented in a variety of manners. +First, for both Bayesian and frequentist models, the loadings’ starting values are often set +to positive numbers. Additionally, if a single loading is fixed for identification, it is almost +always fixed to positive 1. This often leads other loadings towards positive values. +Especially when using software like JAGS or Stan, researchers commonly fix the +latent variance to 1 and place truncated normal priors on the factor loadings, where the +distributions are truncated from below at 0 (e.g., Curtis, 2010). This forces all loadings to +be positive and resolves sign indeterminacies in the model. But this solution is problematic +because it does not allow for indicators with “bad” loadings (whose posterior distributions +overlap with zero), and it does not allow for reversed indicators (whose valence is opposite +that of other indicators). Peeters (2012) shows that, to achieve parameter identification of +the likelihood, only one loading per latent variable must be sign restricted (with the latent +variance being fixed to 1). Thus, fixing the signs of all loadings is overly restrictive from a +parameter identification standpoint. +When a single loading per factor is fixed to 1 for identification, we should not need +to fix the signs of any other loadings. If we instead fix the latent variance to 1 for identifi- +cation, then an improved solution (over fixing all signs to positive) is to employ relabeling +algorithms (e.g., Erosheva & Curtis, 2017). Under this approach, we allow factor loadings +to flip between positive and negative values during MCMC estimation. Then, after model +estimation, we change the signs of loadings dependent upon the signs of some focal loading +parameters (and, if the model includes factor correlations or factor regressions, we also need +to change those signs). This strategy leads to positive loading values, while allowing for the +possibility that some loadings are negative. + +OPAQUE PRIORS +9 +The relabeling algorithms’ preferences for positive loadings can conflict with reseach- +ers’ desires to use noninformative prior distributions for the loadings (say, Normal with a +mean of 0 and a large variance). That is, the software’s preference for positive loadings +conflicts with the noninformative prior distributions, which state that both positive and +negative loadings are equally likely. The specific values of the loadings are influenced by the +identification constraints chosen (e.g., Bollen, Lilly, & Luo, 2022), and the prior distribution +should take that into account. +Illustration +To illustrate the interaction between sign indeterminacy and prior distributions, it is +sufficient to consider the usual confirmatory factor model that is fit to the Holzinger and +Swineford (1939) data. We suspect that most people reading this far know the dataset, +which contains scores on various tests of mental ability. We are focusing on the 3-factor +model that is traditionally fit to the version of the data from lavaan (Rosseel, 2012), where +each factor is associated with 3 observed variables. +The Holzinger-Swineford factor model has five types of model parameters: intercepts, +loadings, factor standard deviations, factor correlations, and residual standard deviations. +We assign true (“population”) values to all these parameters. Intercepts receive true values +of 0, factor standard deviations receive true values of 1, factor correlations receive true values +of 0, and residual standard deviations receive true values of 1. Finally and importantly, +loadings receive true values of −1. +Using these true values, we generated a dataset of 1,000 observations and re-fit the +3-factor model back to the data (where true values were treated as unknown). We used +common, non-informative priors for the parameters of the estimated model, which are cur- +rently the defaults in blavaan (Merkle, Fitzsimmons, Uanhoro, & Goodrich, 2021; Merkle +& Rosseel, 2018): +Intercept ∼ N(0, 1000) +Loading ∼ N(0, 100) +Latent covariance matrix ∼ LKJ(1) +Residual SD ∼ Gamma(1, .5), +where the LKJ prior (Lewandowski et al., 2009) is placed on the entire latent covariance +matrix at once and respects the positive definite constraints described earlier. The remaining +priors are placed separately on individual model parameters, with normal distribution being +parameterized by variances. To identify the model, we fixed each latent variance to 1. In +this case, blavaan uses a relabeling algorithm to handle sign indeterminacy. The estimation +had three chains, 500 warmup samples per chain, and 1,000 posterior samples per chain. +The resulting posterior distributions of the loadings appear in Figure 4. These distri- +butions are centered near +1, with the distributions being fully on the positive side of the +space. So we have a situation where the true loading values were −1, we used noninforma- +tive priors that exhibit little influence on the posterior, and the posterior distributions of +loading values are nowhere near the true values. +At this point, readers might object that this is a bogus example, because it just illus- +trates sign indeterminacy. But the example highlights that, for loadings, a noninformative + +OPAQUE PRIORS +10 +speed=~x7 +speed=~x8 +speed=~x9 +textual=~x4 +textual=~x5 +textual=~x6 +visual=~x1 +visual=~x2 +visual=~x3 +0.7 0.8 0.9 1.0 1.1 1.2 +0.8 0.9 1.0 1.1 1.2 +0.7 0.8 0.9 1.0 1.1 1.2 +0.7 0.8 0.9 1.0 1.1 +0.8 0.9 1.0 1.1 1.2 +0.8 0.9 1.0 1.1 1.2 +1.0 +1.2 +1.4 +0.8 0.9 1.0 1.1 1.2 1.3 +0.8 0.9 1.0 1.1 1.2 +Figure 4. Estimated posterior distributions of loadings, Holzinger-Swineford 3-factor model. +prior centered at 0 can be nonsensical. When using these priors, we are usually attempting +to say that we have no idea about the loadings’ values, or to avoid influencing the results of +the model estimation. But this prior ignores the fact that (i) we typically fix a loading to +be positive for identification, and (ii) observed variables are usually positively correlated, +so that we can expect other loadings to be positive. So our original priors, which were in- +tended to be noninformative, may actually state that a “bad” part of the parameter space +is plausible (also see Gelman, Simpson, & Betancourt, 2017; Seaman, Seaman, & Stamey, +2012). We further discuss this issue below, but we first examine how sign indeterminacy +complicates testing and evaluation of MCMC algorithms. +Implications for MCMC Validation +The issue illustrated above also complicates MCMC algorithm validation, which is +used to ensure that MCMC samplers are working correctly. The MCMC validation process +is difficult even without sign indeterminacy, because randomness is inherent in MCMC +sampling. This means that we cannot simply examine whether the posterior means and +standard deviations match other samplers to many decimal places. While it is possible to +obtain analytic posterior distributions for certain models, analytic results are the exception +instead of the rule for models estimated via MCMC. + +OPAQUE PRIORS +11 +Simulation-based calibration (SBC; Modrák et al., 2022; Talts, Betancourt, Simpson, +Vehtari, & Gelman, 2018) is an MCMC validation method that has received recent attention +and implementation. Given a model of interest (including likelihood and priors), simulation- +based calibration can be described in four steps. +1. Generate many sets of parameter values from the prior distribution. +2. For each set of parameters from Step 1, generate an artificial dataset. +3. Fit the model of interest to each artificial dataset from Step 2. +4. Examine whether the resulting posterior distributions look like the prior distribution. +If the MCMC sampler is working correctly, then the posteriors from Step 3 should look like +the priors from which we started. We can graphically examine this idea by comparing the +posterior means from Step 3 to the parameter values from Step 1; we should see an identity +line when plotting the parameter values against the posterior means. +Using the same model from the previous section, we used the SBC package (Kim, +Moon, Modrák, & Säilynoja, 2022) to conduct simulation-based calibration under two sets of +priors, with 1000 simulated data sets each. Set 1 was exactly the same as the noninformative +priors from the previous section. Set 2 was also similar to the previous section, differing +only in the priors for the loadings: instead of Normal(0, 100), the priors for loadings were +Normal(1, 1/16) (where the second number is a variance). This is an informative prior +reflecting the belief that all loadings should be similar to one another (recall that a single +loading is being fixed to 1 for identification). +Results for the Set 1 and Set 2 priors are shown in Figures 5 and 6, respectively. +In both figures, the parameter values simulated from the prior are on the x-axis, and the +posterior means estimated from the artificial data are on the y-axis. Each panel represents +a factor loading (there are 6 free loadings in the model). Each point represents a replication, +and “correct” MCMC algorithms should lead to points that are crowded around the blue +diagonal line. +Figure 5 shows that, for noninformative priors on factor loadings, the points follow a +V or an X pattern. The V pattern occurs for loadings that are constrained to be positive +during estimation, while the X pattern occurs for the remaining loadings. This means that +the estimation algorithm often recovers the true parameter value, but it also often flips the +sign of the parameter value. While this is not a problem if we realize the sign indeterminacy +issue, it is easy to overlook in the context of simulation-based calibration. For example, +typical SBC summaries involve cumulative distributions, which would lead us to conclude +that there are problems with the MCMC, and which would not provide clear clues about +sign indeterminacy. See Merkle et al. (2021) for some discussion of similar analyses. +Figure 6, on the other hand, is closer to what we would hope to see from a correct +MCMC algorithm. In this figure, the points generally form a cloud around the blue diagonal +line, implying that the posterior means match the parameter values that generated the data. +The informative priors on factor loadings work similarly to use of truncated normal priors, +where the signs of the loadings were generally restricted to be positive. But the informative +priors differ from the truncated normals in that they allow for the possibility of negative +loadings. + +OPAQUE PRIORS +12 +speed=~x7 +speed=~x8 +speed=~x9 +textual=~x4 +textual=~x5 +textual=~x6 +visual=~x1 +visual=~x2 +visual=~x3 +−40 +−20 +0 +20 +40 +−40 +−20 +0 +20 +40 +−40 +−20 +0 +20 +40 +−40 +−20 +0 +20 +40 +−40 +−20 +0 +20 +40 +−40 +−20 +0 +20 +40 +−40 +−20 +0 +20 +40 +−40 +−20 +0 +20 +40 +−40 +−20 +0 +20 +40 +−40 +−20 +0 +20 +40 +−40 +−20 +0 +20 +40 +−40 +−20 +0 +20 +40 +−40 +−20 +0 +20 +40 +−40 +−20 +0 +20 +40 +−40 +−20 +0 +20 +40 +−40 +−20 +0 +20 +40 +−40 +−20 +0 +20 +40 +−40 +−20 +0 +20 +40 +Simulated value +Estimate +Figure 5. Prior parameter values versus posterior means for N(0,100) priors on loadings. + +OPAQUE PRIORS +13 +speed=~x7 +speed=~x8 +speed=~x9 +textual=~x4 +textual=~x5 +textual=~x6 +visual=~x1 +visual=~x2 +visual=~x3 +0.0 +0.5 +1.0 +1.5 +2.0 +0.0 +0.5 +1.0 +1.5 +2.0 +0.0 +0.5 +1.0 +1.5 +2.0 +0.0 +0.5 +1.0 +1.5 +2.0 +0.0 +0.5 +1.0 +1.5 +2.0 +0.0 +0.5 +1.0 +1.5 +2.0 +0.0 +0.5 +1.0 +1.5 +2.0 +0.0 +0.5 +1.0 +1.5 +2.0 +0.0 +0.5 +1.0 +1.5 +2.0 +0.0 +0.5 +1.0 +1.5 +2.0 +0.0 +0.5 +1.0 +1.5 +2.0 +0.0 +0.5 +1.0 +1.5 +2.0 +0.0 +0.5 +1.0 +1.5 +2.0 +0.0 +0.5 +1.0 +1.5 +2.0 +0.0 +0.5 +1.0 +1.5 +2.0 +0.0 +0.5 +1.0 +1.5 +2.0 +0.0 +0.5 +1.0 +1.5 +2.0 +0.0 +0.5 +1.0 +1.5 +2.0 +Simulated value +Estimate +Figure 6. Prior parameter values versus posterior means for N(1, 1 +16) priors on loadings. + +OPAQUE PRIORS +14 +Solutions +To address sign indeterminacy in applied work, we advise researchers to explicitly +consider the loadings’ expected signs when setting prior distributions. In many models, +we expect that all the observed variables corresponding to a factor will have the same +direction of relationship with that factor. Additionally, a single loading is often set to 1 +for identification. In a situation like this, it is often reasonable to place priors on the free +loadings that have a mean of 1 and a standard deviation of, say, .5 or 1. These priors look +very informative at first glance, as compared to a normal prior with a mean of 0 and a +variance of 10,000. But our suggested priors better represent what the researcher knows +about the signs of the loadings, combined with the fact that some loadings are being fixed +to 1. +Instead of fixing a single loading to 1 for identification, researchers may fix the latent +variance to 1. In this case, as described by Peeters (2012), one loading per latent variable +must be constrained to be positive (or negative) in order to achieve identification. If that +loading has a normal prior whose support includes the negative (positive) reals, then we +again face a conflict where our stated prior is not the implied prior. The implied prior +for the sign-constrained loading may become truncated normal or truncated t, depending +on other details of the MCMC implementation (see J. Ghosh & Dunson, 2009), while the +priors on remaining loadings are as stated. +While a truncated distribution arises from +the identification constraint here, we think that truncated priors on all loadings should be +avoided, because they prevent the loadings’ posterior distributions from overlapping with +zero. The relabeling strategies described by Erosheva and Curtis (2017) should be preferred. +We could also consider alternative prior distributions that avoid the issue and/or +that make it easier to specify informative prior distributions, though they are not readily +available in popular software. +Graves and Merkle (2022) studied prior distributions on +ratios of factor loadings, as well as prior specification under effect coding (Little, Slegers, & +Card, 2006). Ratios of loadings avoid the sign-switching issue (e.g., the ratio of two negative +loadings remains positive), and effect coding can make it easier to specify informative prior +distributions. Additionally, in the case of exploratory factor analysis, Heaps (2022) recently +proposed a prior distribution for the common factor covariance matrix. +That is, factor +analysis models typically imply a covariance matrix of the form ΛΛ′ + Ψ, where Λ is the +factor loading matrix with many fewer columns than rows. Heaps (2022) proposes to place +a matrix normal prior on ΛΛ′, which encodes researcher knowledge about shared variation +in the observed variables. The term ΛΛ′ remains invariant across factor loadings’ signs and +rotations, so it avoids the issues described here. The proposed priors can also shrink sets +of loadings towards zero, which is similar to confirmatory factor analysis. It remains to be +seen how this prior could be translated to more general SEMs, where the model-implied +covariance matrix becomes more complex. +Order Constraints +Finally, we describe prior distributions for order-constrained parameters, which are +commonly seen in SEMs for ordinal variables with more than two categories. For these +models, there exist threshold parameters that chop each underlying continuous variable +into observed, ordered categories. The threshold parameters must be ordered so that they + +OPAQUE PRIORS +15 +correspond to the ordering of the observed variables. For example, the lowest threshold +chops off the lowest category, the second threshold chops off the bottom two categories, and +so on. +The prior distributions for threshold parameters are often opaque, because the priors +that researchers specify often have no order constraints. This is commonly done to improve +the software’s ease of use: researchers are accustomed to setting univariate normal priors on +individual parameters, and the priors with order constraints typically do not have simple +forms. But the software always imposes order constraints here, which changes the prior +distribution in various manners. Researchers often do not realize that anything happened, +which may be especially problematic when setting informative priors. +Example +Say that a researcher fits a factor analysis model to a set of 4-category ordinal vari- +ables, and that she specifies a Normal(0,5) prior on all threshold parameters in the model. +Because there are four categories per variable, we require three order-constrained thresholds +per variable. We wish to know what these priors look like, after accounting for the order +constraints. +There are at least two ways that we could translate Normal(0,5) priors to three or- +dered parameters. First, we could imagine drawing three separate variates from the normal +distribution, then ordering the variates to obtain ordered thresholds. +Second, we could +imagine drawing the first (lowest) threshold from a Normal(0,5), then adding a Lognor- +mal(0,5) variate to that threshold in order to obtain the second threshold. Once we have +obtained the second threshold, we could add another Lognormal variate to obtain the third +threshold. Lognormal variables can only take positive values, so we are guaranteed to have +an ordered set of thresholds under this approach. +For both of these translations, the threshold parameters’ prior distributions differ +from the Normal(0,5) distribution that the researcher originally declared. We expand on +this point below, separately for the two methods. +Reordering. +When we draw three values from the normal distribution and then +order them, the act of ordering influences the resulting prior distributions. The specific +distributions can be described via statistical theory on order statistics. For our example, +the Normal(0,5) priors translate into the following probability density functions (pdfs) for +individual thresholds: +p(g1) = 3 × φ(g1/5) × [1 − Φ(g1/5)]2 +p(g2) = 6 × φ(g2/5) × Φ(g2/5) × [1 − Φ(g1/5)] +p(g3) = 3 × φ(g2/5) × Φ(g2/5)2, +where φ() is the standard normal pdf and Φ() is the standard normal cdf. These distributions +are visualized in Figure 7, with the Normal(0,5) distribution overlayed for comparison. The +figure shows that the prior for the lower threshold (g1) is centered below 0, while the prior +for the upper threshold (g3) is centered above 0. None of the three distributions matches +the Normal(0,5), despite the fact that the researcher declared a Normal(0,5) prior for all +three parameters. + +OPAQUE PRIORS +16 +0.000 +0.025 +0.050 +0.075 +0.100 +0.125 +−10 +0 +10 +Threshold value +Density +Threshold +g1 +g2 +g3 +N(0,5) +Figure 7. Priors of threshold parameters whose stated priors are Normal(0,5), as implied +by order constraints. +Lognormals. +The blavaan package uses the lognormal approach mentioned pre- +viously. The Normal(0,5) prior goes on the lowest threshold parameter per item. Then, +Normal(0,5) priors are placed on log-differences between subsequent parameters. For ex- +ample, considering the three thresholds from the example, we would have +g1 ∼ Normal(0, 5) +log(g2 − g1) ∼ Normal(0, 5) +log(g3 − g2) ∼ Normal(0, 5), +where we could alternatively say that the differences between thresholds follow Lognormal +priors. +Just like the previous section, the above priors can be translated to priors on individual +thresholds. It is obvious that the prior for g1 continues to be the stated prior, which is +Normal(0,5). But the priors for g2 and g3 involve the sum of a normal distribution and +Lognormal distribution(s). The resulting distributions can be written as +p(g2) = +� g2 +−∞ +φ((g1 − µ)/σ) × logN(g2 − g1, µ, σ)∂g1 +p(g3) = +� g3 +−∞ +� g2 +−∞ +φ((g1 − µ)/σ) × logN(g2 − g1, µ, σ) × logN(g3 − g2, µ, σ)∂g1∂g2, +where logN(x, µ, σ) is the density function of the Lognormal distribution with mean µ and +standard deviation σ (both on the log scale), evaluated at x. +While the priors for g2 and g3 do not have nice forms, we can numerically approximate +the integrals to visualize them. These distributions are shown in Figure 8, which is arranged + +OPAQUE PRIORS +17 +0.00 +0.02 +0.04 +0.06 +0.08 +−10 +−5 +0 +5 +10 +Threshold value +Density +Threshold +g1 +g2 +g3 +Figure 8. Priors of threshold parameters whose stated priors are Normal(0,5), as implied +by placing priors on log-differences. +similarly to Figure 7. As stated before, the lowest threshold, g1, has the stated Normal(0,5) +distribution. The peaks of the distributions of the remaining thresholds are closer to that +of g1, as compared to the distributions in Figure 7. The distributions of g2 and g3 are +also skewed to the right, reflecting the fact that we are adding a positive variate to the +distribution of g1. +Solution +Unlike the previous issues with positive definite constraints and sign constraints, re- +searchers do not have to consider changing their priors in order to address order constraints. +The main solution is to be aware of the fact that, if one places univariate prior distributions +on a set of order-constrained parameters, then some translation will take place to ensure +that the parameters are ordered correctly. And this translation will influence the implied +prior distribution of each parameter. It is worthwhile to understand how this is handled +by one’s software, especially for prior predictive assessments, Bayes factor calculation, and +simulation-based calibration methods. +General Discussion +In this paper, we considered the idea of opaque prior distributions in Bayesian mod- +els with latent variables, where model estimation and prior specification interact to yield +unexpected results. The imprecision arises from the fact that software implementations +must respect various model constraints, while researchers’ prior distribution specifications +do not always respect the same constraints. This leads researchers to declare one set of +prior distributions, which implies a different set of prior distributions depending on the +software implementation. The issue is particularly problematic for software verification, for + +OPAQUE PRIORS +18 +comparing model results across different pieces of software, and for computing metrics that +explictly involve evaluations of the prior distribution. +The three issues that we considered were (i) positive definite constraints on model +covariance matrices; (ii) sign indeterminacy and constraints used to identify model param- +eters (typically factor loadings); and (iii) order constraints on subsets of model parameters +(typically thresholds/intercepts). These issues occur with different frequencies, with (ii) +and (iii) occurring more often than (i). To expand on this, issue (iii) occurs for models that +have ordinal variables with more than two categories, issue (ii) occurs for most measurement +models (with free loadings), and issue (iii) occurs for models with residual covariances, or +other combinations of fixed and free covariances. We could have a worst-case scenario, such +as a multiple group model with ordinal variables and across-group parameter constraints, +where all three issues occur at once. +To avoid problems associated with opaque priors, we offer the following recommen- +dations for practice: +1. If one’s model involves covariance matrices without parameter constraints, use a single +prior for the full covariance matrix (LKJ, inverse Wishart, etc). +2. If one’s model involves a covariance matrix with parameter constraints, consider +putting a prior on the Cholesky decomposition, or use matrix identities to see whether +the full matrix can be broken into blocks that are easier to handle. If these are un- +available, use informative priors on the correlations that place more density close to +0. +3. For factor loadings, consider the expected direction of the relationship between each +observed variable and the corresponding latent variable(s), along with the loading +identification constraints. Use priors that place most density in this expected direc- +tion. +4. Be aware of how order constraints influence priors for thresholds, especially if one is +doing model assessments that directly involve prior evaluation. +Out of these recommendations, the priors on constrained covariance matrices are most +difficult to handle. Future work could make it easier for researchers to place reasonable +priors on constrained covariance matrices. +Importantly, the issue of opaque priors does not mean that all results associated with +affected models are wrong. In the presence of opaque priors, we can still obtain accurate +estimates of posterior distributions, and we can still obtain accurate model information +criteria (like WAIC and LOO). On the other hand, the prior distributions that researchers +describe in their papers may be incorrect, as will prior predictive checks and other model +summaries that directly rely on prior distribution evaluation, such as Bayes factors. In +particular, researchers should be careful when applying Bayes factor computation strategies +(e.g., Gronau, Singmann, & Wagenmakers, 2020) to latent variable models, to ensure that +they are evaluating the correct priors. +Opaque priors are vaguely similar to applied modeling of ordinal variables (e.g., +Bürkner & Vuorre, 2019; Liddell & Kruschke, 2018), where researchers ignore the fact that +they have ordinal variables, treat them as continuous, and sometimes obtain reasonable + +OPAQUE PRIORS +19 +results. Similarly, researchers can ignore the fact that they have opaque priors, estimate +their model, and sometimes obtain reasonable results. In both cases, it is difficult to predict +exactly when the results will be reasonable and when they will not. And ignoring the issues +do not make them disappear. +We conclude by considering that some non-Bayesian researchers may find this paper +appealing, because they can use it to justify phrases like “Bayesian methods are difficult to +use.” We agree that priors present extra complications that do not exist for other methods, +but we find the extra complications to be worthwhile. In our experience, wrestling with +prior distributions can lead to a deeper, more sober understanding of one’s model and how it +interacts with data. This understanding might be achieved via other, non-Bayesian routes, +but it will require the time and effort that Bayesians devote to prior distributions. +Computational Details +All results were obtained using the R system for statistical computing (R Core Team, +2022), version 4.2.2, with major reliance on packages blavaan (Merkle et al., 2021), rstan +(Stan Development Team, 2022), and SBC (Kim et al., 2022). +References +Alvarez, I., Niemi, J., & Simpson, M. 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Supervised latent Dirich- +let allocation with covariates: A Bayesian structural and measurement model of text and +covariates. +(https://psyarxiv.com/62tc3) +Zitzmann, S., Lüdtke, O., Robitzsch, A., & Hecht, M. (2021). On the performance of Bayesian +approaches in small samples: A comment on Smid, McNeish, Miočević, and van de Schoot +(2020). Structural Equation Modeling: A Multidisciplinary Journal, 28(1), 40–50. doi: 10 +.1080/10705511.2020.1752216 +Zyphur, M. J., Hamaker, E. L., Tay, L., Voelkle, M., Preacher, K. J., Zhang, Z., . . . Diener, E. F. +(2021). From data to causes III: Bayesian priors for general cross-lagged panel models (gclm). +Frontiers in Psychology, 12. Retrieved from https://www.frontiersin.org/articles/10 + +OPAQUE PRIORS +23 +.3389/fpsyg.2021.612251 + diff --git a/HtFAT4oBgHgl3EQfuB5J/content/tmp_files/load_file.txt b/HtFAT4oBgHgl3EQfuB5J/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..73a23587a749a3dd6aee24fb7c312ee5ed67a71d --- /dev/null +++ b/HtFAT4oBgHgl3EQfuB5J/content/tmp_files/load_file.txt @@ -0,0 +1,1248 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf,len=1247 +page_content='Opaque prior distributions in Bayesian latent variable models Edgar C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' Merkle University of Missouri, USA Oludare Ariyo University of Essex, UK Sonja D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' Winter University of Missouri, USA Mauricio Garnier-Villarreal Vrije Universiteit Amsterdam, Nederland Abstract We review common situations in Bayesian latent variable models where the prior distribution that a researcher specifies does not match the prior distri- bution that the estimation method uses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' These situations can arise from the positive definite requirement on correlation matrices, from the sign indeter- minacy of factor loadings, and from order constraints on threshold param- eters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' The issue is especially problematic for reproducibility and for model checks that involve prior distributions, including prior predictive assessment and Bayes factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' In these cases, one might be assessing the wrong model, casting doubt on the relevance of the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' The most straightforward solu- tion to these issues sometimes involves use of informative prior distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' We explore other solutions and make recommendations for practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' Introduction Bayesian latent variable models, including structural equation models and time series models, are increasingly becoming more complex to accommodate the complex datasets that modern technology permits (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=', Asparouhov, Hamaker, & Muthén, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' Bürkner, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' Depaoli, Lai, & Yang, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' dos Santos, Azevedo, & Fox, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' Driver, Oud, & Voelkle, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' Enders, Keller, & Levy, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' Haaf, Merkle, & Rouder, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' Haaf & Rouder, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' Hoijtink & van de Schoot, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' Kaplan, Chen, Yavuz, & Lyu, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' Levy & Enders, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' Miočević, Levy, & MacKinnon, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' Paganin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' Rast, Martin, Liu, & Williams, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' Ulitzsch, Pohl, Khorramdel, Kroehne, & von Davier, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' Van De Schoot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=', 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' Van Erp & Browne, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' Wilcox, Jacobucci, Zhang, & Ammerman, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' Zyphur This work was supported by the Institute of Education Sciences, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' Department of Education, Grant R305D210044.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' Correspondence to Edgar Merkle, University of Missouri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' Email: merklee@missouri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content='08667v1 [stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content='ME] 20 Jan 2023 OPAQUE PRIORS 2 et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' To ensure that these complex models are working correctly, we think it is important to fully understand the problematic issues that can arise in simpler Bayesian models with latent variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' For example, researchers often overlook the fact that multiple WAIC metrics are available for the same model, depending on whether or not one includes latent variables in the likelihood (Merkle, Furr, & Rabe-Hesketh, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' The current paper is devoted to issues with prior distributions that occur for these models, which are easy to overlook and which can lead to inaccurate results and model summaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' The issues that we consider here can generally be called opaque prior distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' They involve the fact that, for some Bayesian models, the prior distribution that you specify is not the prior distribution that you actually use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' This happens because prior distributions are influenced by various details surrounding MCMC implementation, beyond any prior that an analyst specifies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' These issues are already known to many statisticians, but they have some unique manifestations in Bayesian psychometric models with latent variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' Further, these issues have not received much attention in the context of Bayesian psychometric modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' In psychometric models with latent variables, opaque prior distributions can arise from positive definite constraints associated with model covariance matrices, from sign indeterminacies in factor loadings, and from order constraints associated with threshold parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' One example was considered by J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' Ghosh and Dunson (2009), who study a parameter expansion algorithm for estimating Bayesian factor analysis models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' They developed a Gibbs sampler for an overparameterized model in which the factor loadings were not identified, then translated the unidentified parameters to identified parameters via postprocessing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' Importantly, they showed that the priors for the identified loadings (obtained via postprocessing) differed from the priors for the unidentified loadings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' We consider this example in more detail later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' Opaque prior distributions can cause a variety of problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' One way in which these problems arise is through recent studies where researchers seek to recommend the “best” prior distributions for various models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' For example, specific prior distributions have been recommended for factor loadings and correlation parameters in confirmatory factor analysis (Lüdtke, Ulitzsch, & Robitzsch, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' Ulitzsch, Lüdtke, & Robitzsch, 2021), factor loadings and intercept parameters in item factor analysis with dichotomous indicators (Bainter, 2017), regression parameters in mediation models with latent variables (Miočević et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=', 2021), covariance matrices in SEM (Liu, Depaoli, & Marvin, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' van Zundert, Somer, & Miočević, 2022), random effect parameters in multilevel SEMs (Van Erp & Browne, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' Zitzmann, Lüdtke, Robitzsch, & Hecht, 2021), and variance parameters in approximate measurement invariance modeling (Pokropek, Schmidt, & Davidov, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' However, as we will show, opaque priors imply that such recommendations are sometimes software-specific.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' In other words, alternative software packages may implement the same model in a different way, leading to a different implied prior distribution even though the user always specified the same prior distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' We could also experience problems with metrics that explicitly evaluate the prior distribution, including Bayes factors (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=', Heck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' Kass & Raftery, 1995) and prior predictive checks (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=', Vanpaemel, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' In both cases, if a researcher specifies a model with opaque prior distributions, then it is easy to use the wrong prior distributions to compute the metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' Relatedly, some methods for verifying the accuracy of one’s MCMC OPAQUE PRIORS 3 algorithm involve generating data from prior distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' If researchers use the wrong priors, then they may conclude that their MCMC algorithm is problematic even though it runs correctly, or vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' The intent of this paper is to illustrate how opaque priors arise and to provide solutions for avoiding them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' This can allow for reproducible results across software implementations, and it can lead to improved prior predictive assessments and other metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' In the pages below, we show how opaque priors can arise from positive definite constraints, from sign indeterminacies, and from parameter order constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' For each of these topics, we provide examples of the problem, discuss how the problem can lead to compounding problems in model assessment, and provide recommendations for avoiding the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' Finally, we summarize and make general recommendations for practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' Positive definite constraints In Bayesian modeling, prior distributions for the covariance matrix often involve the inverse-Wishart (IW) distribution due to its conditional conjugacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' However, the IW dis- tribution can be problematic because it assumes the same amount of prior information for every variance parameter (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' a prior relationship between the variance parameter the vari- ances and correlations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' To overcome this challenge, various strategies have been suggested for separately specifying priors on the variance and correlation parameters underlying the covariance matrix (Barnard, McCulloch, & Meng, 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' These priors have been shown to perform better than the classical IW (Alvarez, Niemi, & Simpson, 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' Ariyo, Lesaffre, Verbeke, & Quintero, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' Huang, Wand, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=', 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' However, things become more com- plicated with three or more random effects because certain restrictions must be imposed to ensure positive definiteness of the covariance matrix (Barnard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=', 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' Daniels & Kass, 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=', 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' Hurtado Rúa, Mazumdar, & Strawderman, 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' Wei & Higgins, 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' The situation becomes even more complicated when the covariance matrix has model- imposed constraints, which can arise in SEMs with correlated residuals or with across-group equality constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' In these models, we cannot impose an IW (or other prior) on the full covariance matrix because those priors will not respect the constraints imposed by the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' We could instead place priors on individual parameters within the covariance matrix, but, when we build a covariance matrix using these parameters, the resulting matrix will sometimes be non-positive definite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' We elaborate on these issues below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' Illustration An example comes from the popular “Political Democracy” model originally described by Bollen (1989), shown below in lavaan syntax (Rosseel, 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' Intended to describe countries’ relationships between their levels of industrialization and democracy, the model contains four observed variables that are measured once during 1960 and again during 1965.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' The residuals of the 1960 variables are allowed to correlate with the residuals of the corresponding 1965 variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' Additionally, there exists a pair of similar variables collected during 1960 and again during 1965, leading to three additional residual correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' The full structure of the residual covariance matrix is shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' OPAQUE PRIORS 4 x1 x2 x3 y1 y2 y3 y4 y5 y6 y7 y8 x1 × x2 × x3 × y1 × × y2 × × × y3 × × y4 × × × y5 × × y6 × × × y7 × × y8 × × × Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' Political democracy model, structure of residual covariance matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' Free param- eters are marked by ×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=" model <- ' # measurement model ind60 =~ x1 + x2 + x3 dem60 =~ y1 + a*y2 + b*y3 + c*y4 dem65 =~ y5 + a*y6 + b*y7 + c*y8 # regressions dem60 ~ ind60 dem65 ~ ind60 + dem60 # residual correlations y1 ~~ y5 y2 ~~ y4 + y6 y3 ~~ y7 y4 ~~ y8 y6 ~~ y8 ' As described earlier, we could elect to put a univariate prior distribution on each variance (or standard deviation or precision) in this matrix, and also on each correlation associated with off-diagonal entries." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' But this is problematic because the univariate priors on correlations will often yield non-positive definite correlation matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' And if we consider only the space of positive definite correlation matrices, the univariate priors on correlations are more informative than one would expect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' That is, the prior distributions are opaque: the analyst specifies a set of priors that are different from the implied prior distributions, which must obey the constraint of positive definiteness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' How can we characterize the implied prior distributions?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' We could simply generate thousands of correlation matrices from the prior, then discard matrices that are not positive definite, then visualize what is left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' As an example of this, we specified a Uniform(−1, 1) prior for each free correlation in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' We then generated 100,000 correlation matrices with the desired structure, discarding the 57,818 matrices that were not positive definite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' Finally, we examined the distributions of the remaining matrices, with a histogram for a single correlation parameter appearing in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' We see that the resulting distribution OPAQUE PRIORS 5 0 500 1000 1500 2000 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content='0 Cor(y2,y4) Frequency Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' Bollen model, implied prior distribution of a single correlation parameter after accounting for positive definite constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' is no longer uniform;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' it is approximately symmetric around 0, with more density near 0 than near −1 or 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' This is because our uniform priors did not account for the fact that correlation matrices must be positive definite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' To technically describe the positive definite constraint, we can simultaneously permute the rows and columns of the correlation matrix to obtain a block diagonal matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' A block diagonal matrix is useful here because the determinant of the full matrix is implied by the determinants of each individual block within the matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' This allows us to examine whether or not the full matrix is positive definite, by working with submatrices of smaller dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' The Cuthill-McKee algorithm (Cuthill & McKee, 1969) allows us to automatically find an appropriate, block-diagonal permutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' After carrying out this algorithm (via the netprioR package;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' Schmich, 2022) and permuting the rows and columns, we arrive at the matrix in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' This shows that, to keep the full matrix positive definite, we only need to worry about the 4 × 4 block in the lower right that involves y2, y4, y6, and y8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' The priors on the (y3, y7) and (y1, y5) correlations have no influence on the positive definiteness of the full matrix (because a 2 × 2 matrix is positive definite for any correlation in (−1, 1)), so that we can safely put a uniform prior on each of those correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' After applying traditional rules for computing matrix determinants, we can express the determinant of the 4 × 4 block of correlations as det(R(4×4)) = 1 + (r1r4 − r2r3)2 − 4 � i=1 r2 i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' (1) This shows analytically how the positive definite constraint of our correlation matrix influ- ences our univariate beta priors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' The univariate priors are collectively constrained by the requirement that Equation (1) be greater than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' OPAQUE PRIORS 6 x1 x2 x3 y1 y5 y3 y7 y2 y4 y6 y8 x1 1 x2 1 x3 1 y1 1 × y5 × 1 y3 1 × y7 × 1 y2 1 × × y4 × 1 × y6 × 1 × y8 × × 1 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' Political democracy model, structure of permuted correlation matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' Free pa- rameters are marked by ×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' Implications for Bayes factors To show how this issue becomes problematic for applied work, we consider the cal- culation of Bayes factors using the Bollen model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' Imagine that we wish to know whether the two residual correlations involving y2 are necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' To address this question, we could compute a Bayes factor comparing a model that includes those two residual correlations, to a model without those residual correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' A computationally-cheap way to do this is the Savage-Dickey method (Dickey & Lientz, 1970;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' Wagenmakers, Lodewyckx, Kuriyal, & Grasman, 2010), which requires us to only fit the model that includes the two residual correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' The Bayes factor (for the model with correlations versus the model without correlations) is then the ratio of two densities: the prior density of the two residual covari- ances at 0, and the posterior density of the two residual covariances at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' But if we are to do the correct calculation here, we need to make sure that we use the correct prior densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' If we set Uniform priors on each individual correlation and ignore the positive definite constraint, then the prior density of the two correlations at 0 equals 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' This corresponds to a log-density of −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content='39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' The true joint prior density on the two correlations (which respects the positive definite constraint) does not have a simple form, but we can approxi- mate the density via the positive definite correlation matrices that we randomly generated earlier (using a density estimation method from the ks package in R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' Duong, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' Our approximate joint density at 0 is now 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content='46, corresponding to a log-density of −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content='78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' These evaluations lead to Bayes factors for the model with correlations, relative to the model without correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' Using our incorrect priors that do not account for posi- tive definite constraints, we obtain a log-Bayes factor of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content='66 in favor of the model with correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' Using our priors that do account for positive definite constraints, we obtain a log-Bayes factor of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content='27 in favor of the model with correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' Both of these Bayes factors provide support for the model with correlations, but possibly at different levels of evidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' For example, if we subscribe to the rules of thumb offered by Kass and Raftery (1995), then these two Bayes factors lead us to conclude “strong” evidence using the incor- rect prior calculation, and “very strong” evidence using the correct calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' We agree OPAQUE PRIORS 7 with you, the reader, that these cutoffs are arbitrary and that the Bayes factors do not differ by very much.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' But the main point is that the Bayes factors will systematically differ depending on how we compute prior densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' These differences will sometimes lead to different substantive conclusions in practice, with the easier computation being incorrect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' Solutions While it was straightforward to visualize implied prior distributions in the Bollen model, the process becomes inefficient for correlation matrices with more than a few di- mensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' For such correlation matrices, few of the randomly-generated matrices will be positive definite, and it will take a long time to obtain a sufficient number of positive definite matrices to describe the implied prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' Additionally, every unique structure of correlation matrix will have a unique positive definite constraint, similar to the one from Equation (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' So we desire more general solutions that do not involve a simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' We describe three solutions below that differ in complexity and in the extent to which they fully solve the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' Informative priors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' The simplest, partial solution is to maintain the univariate priors on individual correlations, but make those priors informative around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' For example, instead of placing uniform priors on the correlations, we might use Beta(5,5) priors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' The implied prior distributions will then be closer to what the user specifies, because these informative priors will more often lead to positive definite correlation matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' But this is not a full solution because, even with informative priors, we may still encounter non-positive definite correlation matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' And it is not straightforward to predict when or how often this will happen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' Additionally, depending on one’s application, certain informative priors may be inappropriate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' Priors on Cholesky decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' A more general solution to this problem comes from putting priors on the Cholesky decomposition of the correlation matrix, which is related to the R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' Ghosh, Mallick, and Pourahmadi (2021) approach for time series models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' The Cholesky decomposition is advantageous because, to ensure that the correlation matrix stays positive definite, we only have to ensure that the diagonal elements of the Cholesky decomposition are positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' For the Political Democracy model, the 4×4 covariance matrix from the bottom right corner of Figure 2 has a Cholesky decomposition with structure � � � � � c11 c21 c22 c31 −c21c31/c22 c33 0 c42 c43 c44 � � � � � , (2) where the entries {c11, c22, c33, c44} are constrained to be positive, while {c21, c31, c42, c43} are unconstrained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' Additionally, the entry in row 3, column 2 is fully determined by other entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' If we place gamma priors (say) on the diagonal entries and normal priors on the remaining c variates, we can maintain the desired structure of the covariance matrix while also maintaining positive definiteness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' A disadvantage of this approach is that the entries of the Cholesky decomposition do not necessarily have intuitive interpretations, so that it is difficult to set informative priors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' Each diagonal entry is related to the portion of the corresponding variable’s variance that OPAQUE PRIORS 8 cannot be accounted for by variables that occur further to the left of the matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' Each off- diagonal entry is related to a partial correlation conditioned on variables further to the left of the matrix (see Joe, 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' Lewandowski, Kurowicka, & Joe, 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' Pourahmadi, Daniels, & Park, 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' This implies that the order of the variables matters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' A further difficulty is that there does not appear to be an automatic way to obtain a Cholesky structure (like that of Equation (2)) for arbitrarily-structured covariance matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' Combining LKJ with Informative Priors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' A final solution was recently de- scribed in a blog post by Martin (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' This solution involves use of a prior distribution that is the product of (i) an LKJ prior on the full correlation matrix, and (ii) informative priors on individual entries of the correlation matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' The resulting prior inherits the posi- tive definite constraint from (i) while also inheriting the informativeness from (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' At the moment, it is not clear that this approach can be used to fix individual entries of the cor- relation matrix;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' instead, Martin (2021) recommends highly-informative priors around the fixed value that is desired (similar to the notion of “approximate zeros” in a factor loading matrix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' At this time, it is not clear whether the highly-informative priors could generally replace hard constraints, or whether they could cause problems with model convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' Sign Indeterminacies We now turn to a problem that is more specific to SEM: sign indeterminacies of loading parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' It is well known that, if we change the signs of all loadings, the SEM likelihood (usually) stays the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' To avoid this issue, SEM software typically “prefers” positive loadings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' This preference for positive loadings is implemented in a variety of manners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' First, for both Bayesian and frequentist models, the loadings’ starting values are often set to positive numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' Additionally, if a single loading is fixed for identification, it is almost always fixed to positive 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' This often leads other loadings towards positive values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' Especially when using software like JAGS or Stan, researchers commonly fix the latent variance to 1 and place truncated normal priors on the factor loadings, where the distributions are truncated from below at 0 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=', Curtis, 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' This forces all loadings to be positive and resolves sign indeterminacies in the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' But this solution is problematic because it does not allow for indicators with “bad” loadings (whose posterior distributions overlap with zero), and it does not allow for reversed indicators (whose valence is opposite that of other indicators).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' Peeters (2012) shows that, to achieve parameter identification of the likelihood, only one loading per latent variable must be sign restricted (with the latent variance being fixed to 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' Thus, fixing the signs of all loadings is overly restrictive from a parameter identification standpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' When a single loading per factor is fixed to 1 for identification, we should not need to fix the signs of any other loadings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' If we instead fix the latent variance to 1 for identifi- cation, then an improved solution (over fixing all signs to positive) is to employ relabeling algorithms (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=', Erosheva & Curtis, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' Under this approach, we allow factor loadings to flip between positive and negative values during MCMC estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' Then, after model estimation, we change the signs of loadings dependent upon the signs of some focal loading parameters (and, if the model includes factor correlations or factor regressions, we also need to change those signs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' This strategy leads to positive loading values, while allowing for the possibility that some loadings are negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' OPAQUE PRIORS 9 The relabeling algorithms’ preferences for positive loadings can conflict with reseach- ers’ desires to use noninformative prior distributions for the loadings (say, Normal with a mean of 0 and a large variance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' That is, the software’s preference for positive loadings conflicts with the noninformative prior distributions, which state that both positive and negative loadings are equally likely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' The specific values of the loadings are influenced by the identification constraints chosen (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=', Bollen, Lilly, & Luo, 2022), and the prior distribution should take that into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' Illustration To illustrate the interaction between sign indeterminacy and prior distributions, it is sufficient to consider the usual confirmatory factor model that is fit to the Holzinger and Swineford (1939) data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' We suspect that most people reading this far know the dataset, which contains scores on various tests of mental ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' We are focusing on the 3-factor model that is traditionally fit to the version of the data from lavaan (Rosseel, 2012), where each factor is associated with 3 observed variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' The Holzinger-Swineford factor model has five types of model parameters: intercepts, loadings, factor standard deviations, factor correlations, and residual standard deviations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' We assign true (“population”) values to all these parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' Intercepts receive true values of 0, factor standard deviations receive true values of 1, factor correlations receive true values of 0, and residual standard deviations receive true values of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' Finally and importantly, loadings receive true values of −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' Using these true values, we generated a dataset of 1,000 observations and re-fit the 3-factor model back to the data (where true values were treated as unknown).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' We used common, non-informative priors for the parameters of the estimated model, which are cur- rently the defaults in blavaan (Merkle, Fitzsimmons, Uanhoro, & Goodrich, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' Merkle & Rosseel, 2018): Intercept ∼ N(0, 1000) Loading ∼ N(0, 100) Latent covariance matrix ∼ LKJ(1) Residual SD ∼ Gamma(1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content='5), where the LKJ prior (Lewandowski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=', 2009) is placed on the entire latent covariance matrix at once and respects the positive definite constraints described earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' The remaining priors are placed separately on individual model parameters, with normal distribution being parameterized by variances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' To identify the model, we fixed each latent variance to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' In this case, blavaan uses a relabeling algorithm to handle sign indeterminacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' The estimation had three chains, 500 warmup samples per chain, and 1,000 posterior samples per chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' The resulting posterior distributions of the loadings appear in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' These distri- butions are centered near +1, with the distributions being fully on the positive side of the space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' So we have a situation where the true loading values were −1, we used noninforma- tive priors that exhibit little influence on the posterior, and the posterior distributions of loading values are nowhere near the true values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' At this point, readers might object that this is a bogus example, because it just illus- trates sign indeterminacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' But the example highlights that, for loadings, a noninformative OPAQUE PRIORS 10 speed=~x7 speed=~x8 speed=~x9 textual=~x4 textual=~x5 textual=~x6 visual=~x1 visual=~x2 visual=~x3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content='8 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content='2 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' Estimated posterior distributions of loadings, Holzinger-Swineford 3-factor model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' prior centered at 0 can be nonsensical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' When using these priors, we are usually attempting to say that we have no idea about the loadings’ values, or to avoid influencing the results of the model estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' But this prior ignores the fact that (i) we typically fix a loading to be positive for identification, and (ii) observed variables are usually positively correlated, so that we can expect other loadings to be positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' So our original priors, which were in- tended to be noninformative, may actually state that a “bad” part of the parameter space is plausible (also see Gelman, Simpson, & Betancourt, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' Seaman, Seaman, & Stamey, 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' We further discuss this issue below, but we first examine how sign indeterminacy complicates testing and evaluation of MCMC algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' Implications for MCMC Validation The issue illustrated above also complicates MCMC algorithm validation, which is used to ensure that MCMC samplers are working correctly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' The MCMC validation process is difficult even without sign indeterminacy, because randomness is inherent in MCMC sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' This means that we cannot simply examine whether the posterior means and standard deviations match other samplers to many decimal places.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' While it is possible to obtain analytic posterior distributions for certain models, analytic results are the exception instead of the rule for models estimated via MCMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' OPAQUE PRIORS 11 Simulation-based calibration (SBC;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' Modrák et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' Talts, Betancourt, Simpson, Vehtari, & Gelman, 2018) is an MCMC validation method that has received recent attention and implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' Given a model of interest (including likelihood and priors), simulation- based calibration can be described in four steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' Generate many sets of parameter values from the prior distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' For each set of parameters from Step 1, generate an artificial dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' Fit the model of interest to each artificial dataset from Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' Examine whether the resulting posterior distributions look like the prior distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' If the MCMC sampler is working correctly, then the posteriors from Step 3 should look like the priors from which we started.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' We can graphically examine this idea by comparing the posterior means from Step 3 to the parameter values from Step 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' we should see an identity line when plotting the parameter values against the posterior means.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' Using the same model from the previous section, we used the SBC package (Kim, Moon, Modrák, & Säilynoja, 2022) to conduct simulation-based calibration under two sets of priors, with 1000 simulated data sets each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' Set 1 was exactly the same as the noninformative priors from the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' Set 2 was also similar to the previous section, differing only in the priors for the loadings: instead of Normal(0, 100), the priors for loadings were Normal(1, 1/16) (where the second number is a variance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' This is an informative prior reflecting the belief that all loadings should be similar to one another (recall that a single loading is being fixed to 1 for identification).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' Results for the Set 1 and Set 2 priors are shown in Figures 5 and 6, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' In both figures, the parameter values simulated from the prior are on the x-axis, and the posterior means estimated from the artificial data are on the y-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' Each panel represents a factor loading (there are 6 free loadings in the model).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' Each point represents a replication, and “correct” MCMC algorithms should lead to points that are crowded around the blue diagonal line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' Figure 5 shows that, for noninformative priors on factor loadings, the points follow a V or an X pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' The V pattern occurs for loadings that are constrained to be positive during estimation, while the X pattern occurs for the remaining loadings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' This means that the estimation algorithm often recovers the true parameter value, but it also often flips the sign of the parameter value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' While this is not a problem if we realize the sign indeterminacy issue, it is easy to overlook in the context of simulation-based calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' For example, typical SBC summaries involve cumulative distributions, which would lead us to conclude that there are problems with the MCMC, and which would not provide clear clues about sign indeterminacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' See Merkle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' (2021) for some discussion of similar analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' Figure 6, on the other hand, is closer to what we would hope to see from a correct MCMC algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' In this figure, the points generally form a cloud around the blue diagonal line, implying that the posterior means match the parameter values that generated the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' The informative priors on factor loadings work similarly to use of truncated normal priors, where the signs of the loadings were generally restricted to be positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' But the informative priors differ from the truncated normals in that they allow for the possibility of negative loadings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' ' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' Prior parameter values versus posterior means for N(0,100) priors on loadings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' OPAQUE PRIORS 13 speed=~x7 speed=~x8 speed=~x9 textual=~x4 textual=~x5 textual=~x6 visual=~x1 visual=~x2 visual=~x3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content='5 2.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content='0 Simulated value Estimate Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' Prior parameter values versus posterior means for N(1, 1 16) priors on loadings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' OPAQUE PRIORS 14 Solutions To address sign indeterminacy in applied work, we advise researchers to explicitly consider the loadings’ expected signs when setting prior distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' In many models, we expect that all the observed variables corresponding to a factor will have the same direction of relationship with that factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' Additionally, a single loading is often set to 1 for identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' In a situation like this, it is often reasonable to place priors on the free loadings that have a mean of 1 and a standard deviation of, say, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content='5 or 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' These priors look very informative at first glance, as compared to a normal prior with a mean of 0 and a variance of 10,000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' But our suggested priors better represent what the researcher knows about the signs of the loadings, combined with the fact that some loadings are being fixed to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' Instead of fixing a single loading to 1 for identification, researchers may fix the latent variance to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' In this case, as described by Peeters (2012), one loading per latent variable must be constrained to be positive (or negative) in order to achieve identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' If that loading has a normal prior whose support includes the negative (positive) reals, then we again face a conflict where our stated prior is not the implied prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' The implied prior for the sign-constrained loading may become truncated normal or truncated t, depending on other details of the MCMC implementation (see J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' Ghosh & Dunson, 2009), while the priors on remaining loadings are as stated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' While a truncated distribution arises from the identification constraint here, we think that truncated priors on all loadings should be avoided, because they prevent the loadings’ posterior distributions from overlapping with zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' The relabeling strategies described by Erosheva and Curtis (2017) should be preferred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' We could also consider alternative prior distributions that avoid the issue and/or that make it easier to specify informative prior distributions, though they are not readily available in popular software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' Graves and Merkle (2022) studied prior distributions on ratios of factor loadings, as well as prior specification under effect coding (Little, Slegers, & Card, 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' Ratios of loadings avoid the sign-switching issue (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=', the ratio of two negative loadings remains positive), and effect coding can make it easier to specify informative prior distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' Additionally, in the case of exploratory factor analysis, Heaps (2022) recently proposed a prior distribution for the common factor covariance matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' That is, factor analysis models typically imply a covariance matrix of the form ΛΛ′ + Ψ, where Λ is the factor loading matrix with many fewer columns than rows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' Heaps (2022) proposes to place a matrix normal prior on ΛΛ′, which encodes researcher knowledge about shared variation in the observed variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' The term ΛΛ′ remains invariant across factor loadings’ signs and rotations, so it avoids the issues described here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' The proposed priors can also shrink sets of loadings towards zero, which is similar to confirmatory factor analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' It remains to be seen how this prior could be translated to more general SEMs, where the model-implied covariance matrix becomes more complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' Order Constraints Finally, we describe prior distributions for order-constrained parameters, which are commonly seen in SEMs for ordinal variables with more than two categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' For these models, there exist threshold parameters that chop each underlying continuous variable into observed, ordered categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' The threshold parameters must be ordered so that they OPAQUE PRIORS 15 correspond to the ordering of the observed variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' For example, the lowest threshold chops off the lowest category, the second threshold chops off the bottom two categories, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' The prior distributions for threshold parameters are often opaque, because the priors that researchers specify often have no order constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' This is commonly done to improve the software’s ease of use: researchers are accustomed to setting univariate normal priors on individual parameters, and the priors with order constraints typically do not have simple forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' But the software always imposes order constraints here, which changes the prior distribution in various manners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' Researchers often do not realize that anything happened, which may be especially problematic when setting informative priors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' Example Say that a researcher fits a factor analysis model to a set of 4-category ordinal vari- ables, and that she specifies a Normal(0,5) prior on all threshold parameters in the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' Because there are four categories per variable, we require three order-constrained thresholds per variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' We wish to know what these priors look like, after accounting for the order constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' There are at least two ways that we could translate Normal(0,5) priors to three or- dered parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' First, we could imagine drawing three separate variates from the normal distribution, then ordering the variates to obtain ordered thresholds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' Second, we could imagine drawing the first (lowest) threshold from a Normal(0,5), then adding a Lognor- mal(0,5) variate to that threshold in order to obtain the second threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' Once we have obtained the second threshold, we could add another Lognormal variate to obtain the third threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' Lognormal variables can only take positive values, so we are guaranteed to have an ordered set of thresholds under this approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' For both of these translations, the threshold parameters’ prior distributions differ from the Normal(0,5) distribution that the researcher originally declared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' We expand on this point below, separately for the two methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' Reordering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' When we draw three values from the normal distribution and then order them, the act of ordering influences the resulting prior distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' The specific distributions can be described via statistical theory on order statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' For our example, the Normal(0,5) priors translate into the following probability density functions (pdfs) for individual thresholds: p(g1) = 3 × φ(g1/5) × [1 − Φ(g1/5)]2 p(g2) = 6 × φ(g2/5) × Φ(g2/5) × [1 − Φ(g1/5)] p(g3) = 3 × φ(g2/5) × Φ(g2/5)2, where φ() is the standard normal pdf and Φ() is the standard normal cdf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' These distributions are visualized in Figure 7, with the Normal(0,5) distribution overlayed for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' The figure shows that the prior for the lower threshold (g1) is centered below 0, while the prior for the upper threshold (g3) is centered above 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' None of the three distributions matches the Normal(0,5), despite the fact that the researcher declared a Normal(0,5) prior for all three parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' OPAQUE PRIORS 16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content='075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content='125 −10 0 10 Threshold value Density Threshold g1 g2 g3 N(0,5) Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' Priors of threshold parameters whose stated priors are Normal(0,5), as implied by order constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' Lognormals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' The blavaan package uses the lognormal approach mentioned pre- viously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' The Normal(0,5) prior goes on the lowest threshold parameter per item.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' Then, Normal(0,5) priors are placed on log-differences between subsequent parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' For ex- ample, considering the three thresholds from the example, we would have g1 ∼ Normal(0, 5) log(g2 − g1) ∼ Normal(0, 5) log(g3 − g2) ∼ Normal(0, 5), where we could alternatively say that the differences between thresholds follow Lognormal priors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' Just like the previous section, the above priors can be translated to priors on individual thresholds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' It is obvious that the prior for g1 continues to be the stated prior, which is Normal(0,5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' But the priors for g2 and g3 involve the sum of a normal distribution and Lognormal distribution(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' The resulting distributions can be written as p(g2) = � g2 −∞ φ((g1 − µ)/σ) × logN(g2 − g1, µ, σ)∂g1 p(g3) = � g3 −∞ � g2 −∞ φ((g1 − µ)/σ) × logN(g2 − g1, µ, σ) × logN(g3 − g2, µ, σ)∂g1∂g2, where logN(x, µ, σ) is the density function of the Lognormal distribution with mean µ and standard deviation σ (both on the log scale), evaluated at x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' While the priors for g2 and g3 do not have nice forms, we can numerically approximate the integrals to visualize them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' These distributions are shown in Figure 8, which is arranged OPAQUE PRIORS 17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content='08 −10 −5 0 5 10 Threshold value Density Threshold g1 g2 g3 Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' Priors of threshold parameters whose stated priors are Normal(0,5), as implied by placing priors on log-differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' similarly to Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' As stated before, the lowest threshold, g1, has the stated Normal(0,5) distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' The peaks of the distributions of the remaining thresholds are closer to that of g1, as compared to the distributions in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' The distributions of g2 and g3 are also skewed to the right, reflecting the fact that we are adding a positive variate to the distribution of g1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' Solution Unlike the previous issues with positive definite constraints and sign constraints, re- searchers do not have to consider changing their priors in order to address order constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' The main solution is to be aware of the fact that, if one places univariate prior distributions on a set of order-constrained parameters, then some translation will take place to ensure that the parameters are ordered correctly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' And this translation will influence the implied prior distribution of each parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' It is worthwhile to understand how this is handled by one’s software, especially for prior predictive assessments, Bayes factor calculation, and simulation-based calibration methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' General Discussion In this paper, we considered the idea of opaque prior distributions in Bayesian mod- els with latent variables, where model estimation and prior specification interact to yield unexpected results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' The imprecision arises from the fact that software implementations must respect various model constraints, while researchers’ prior distribution specifications do not always respect the same constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' This leads researchers to declare one set of prior distributions, which implies a different set of prior distributions depending on the software implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' The issue is particularly problematic for software verification, for OPAQUE PRIORS 18 comparing model results across different pieces of software, and for computing metrics that explictly involve evaluations of the prior distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' The three issues that we considered were (i) positive definite constraints on model covariance matrices;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' (ii) sign indeterminacy and constraints used to identify model param- eters (typically factor loadings);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' and (iii) order constraints on subsets of model parameters (typically thresholds/intercepts).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' These issues occur with different frequencies, with (ii) and (iii) occurring more often than (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' To expand on this, issue (iii) occurs for models that have ordinal variables with more than two categories, issue (ii) occurs for most measurement models (with free loadings), and issue (iii) occurs for models with residual covariances, or other combinations of fixed and free covariances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' We could have a worst-case scenario, such as a multiple group model with ordinal variables and across-group parameter constraints, where all three issues occur at once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' To avoid problems associated with opaque priors, we offer the following recommen- dations for practice: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' If one’s model involves covariance matrices without parameter constraints, use a single prior for the full covariance matrix (LKJ, inverse Wishart, etc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' If one’s model involves a covariance matrix with parameter constraints, consider putting a prior on the Cholesky decomposition, or use matrix identities to see whether the full matrix can be broken into blocks that are easier to handle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' If these are un- available, use informative priors on the correlations that place more density close to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' For factor loadings, consider the expected direction of the relationship between each observed variable and the corresponding latent variable(s), along with the loading identification constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' Use priors that place most density in this expected direc- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' Be aware of how order constraints influence priors for thresholds, especially if one is doing model assessments that directly involve prior evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' Out of these recommendations, the priors on constrained covariance matrices are most difficult to handle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' Future work could make it easier for researchers to place reasonable priors on constrained covariance matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' Importantly, the issue of opaque priors does not mean that all results associated with affected models are wrong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' In the presence of opaque priors, we can still obtain accurate estimates of posterior distributions, and we can still obtain accurate model information criteria (like WAIC and LOO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' On the other hand, the prior distributions that researchers describe in their papers may be incorrect, as will prior predictive checks and other model summaries that directly rely on prior distribution evaluation, such as Bayes factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' In particular, researchers should be careful when applying Bayes factor computation strategies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=', Gronau, Singmann, & Wagenmakers, 2020) to latent variable models, to ensure that they are evaluating the correct priors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' Opaque priors are vaguely similar to applied modeling of ordinal variables (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=', Bürkner & Vuorre, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' Liddell & Kruschke, 2018), where researchers ignore the fact that they have ordinal variables, treat them as continuous, and sometimes obtain reasonable OPAQUE PRIORS 19 results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' Similarly, researchers can ignore the fact that they have opaque priors, estimate their model, and sometimes obtain reasonable results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' In both cases, it is difficult to predict exactly when the results will be reasonable and when they will not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' And ignoring the issues do not make them disappear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' We conclude by considering that some non-Bayesian researchers may find this paper appealing, because they can use it to justify phrases like “Bayesian methods are difficult to use.” We agree that priors present extra complications that do not exist for other methods, but we find the extra complications to be worthwhile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' In our experience, wrestling with prior distributions can lead to a deeper, more sober understanding of one’s model and how it interacts with data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' This understanding might be achieved via other, non-Bayesian routes, but it will require the time and effort that Bayesians devote to prior distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' Computational Details All results were obtained using the R system for statistical computing (R Core Team, 2022), version 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content='2, with major reliance on packages blavaan (Merkle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=', 2021), rstan (Stan Development Team, 2022), and SBC (Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' References Alvarez, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=', Niemi, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=', & Simpson, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content=' (2014).' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content='3389/fpsyg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content='2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} +page_content='612251' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFAT4oBgHgl3EQfuB5J/content/2301.08667v1.pdf'} diff --git a/ItE4T4oBgHgl3EQfhQ0t/vector_store/index.faiss b/ItE4T4oBgHgl3EQfhQ0t/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..9b8549759709ff150bb0256a629a28e65c3c60c4 --- /dev/null +++ b/ItE4T4oBgHgl3EQfhQ0t/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:211f74dc7a8e13f1baba666894119b7d94551a821b4fb19775b61e7b40c3438e +size 1310765 diff --git 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We apply the algebraic consideration of the holonomy system to study the Hermitian man- +ifolds whose Chern connection is Amrose-Singer and prove more refined structure theorems for such +manifolds. The main result asserts that the universal cover of such a Hermitian manifold must be the +product of a complex Lie group and Hermitian symmetric spaces. +Contents +1. +Introduction and main results +1 +2. +Constructions of irreducible subbundles +2 +3. +Symmetric holonomy systems +3 +4. +Hermitian manifolds with a CAS structure +5 +5. +The Bismut Ambrose-Singer manifolds +6 +References +7 +1. Introduction and main results +This is a continuation of the previous paper [12] on Hermitian manifolds whose Chern connection ∇ is +Ambrose-Singer. A connection is called Ambrose-Singer (abbreviated as an AS connection) if its torsion +and its curvature are parallel with respect to this connection. The conditions can be expressed in terms of +more convoluted PDEs in terms of the curvature tensor of the Levi-Civita connection D and the torsion +of the AS connection [2]. This arises naturally in Ambrose-Singer’s characterization of simply-connected +Riemannian homogeneous spaces (cf.[2]), namely the complete Riemannian manifolds whose isometry +group acts transitively, as a generalization of Cartan’s theorem on the symmetric (or locally symmetric) +spaces. We refer the readers to [9, 14, 15, 16, 17] for other related studies on homogeneous complex +manifolds, namely complex manifolds whose biholomorphism group act transitively, and homogeneous +Hermitian manifolds, namely Hermitian manifolds whose biholomorphic isometries act transitively, and +other homogeneous spaces. It is well known that on a Hermitian manifold there exists a unique canonical +connection, namely the Chern connection [6] (in fact the existence extends to any holomorphic vector +bundles). When the Chern connection of (M, g) is Ambrose-Singer we say that the complex manifold M +admits a CAS structure. (Below we also abbreviate them as Hermitian manifolds with a CAS structure). +The examples of manifolds with CAS structure include complex Lie groups endowed with a left invariant +metric and Hermitian symmetric spaces, their products and quotients of products. In [12] conditions are +given when the universal cover of such a manifold admits some K¨ahler (which is Hermitian symmetric +due to ∇R = 0) de Rham factors. Structure theorems were obtained in complex dimension three and +four. For high dimensions it was proved that +Theorem 1.1. Let (M n, g) be a Hermitian manifold with a CAS structure. Assume further that its +universal cover admits no K¨ahler de Rham factor. Then the universal cover of M admits a parallel +holomorphic (n, 0) form, and (M, g) has zero Chern Ricci curvature. +Since the connection is NOT Levi-Civita, the de Rham decomposition theorem no longer holds to its +full generality. We refer to [12] for some detailed discussions when it remains true for the case considered. +The first main result is the following structure theorem for high dimensional case which classifies all +manifolds with a CAS structure whose universal cover does not contain a K¨ahler de Rham factor. +2010 Mathematics Subject Classification. +53C55 (Primary), 53C05 (Secondary). +Key words and phrases. Ambrose-Singer connection, Chern connection, holonomy algebra/group/system, Chern flat, +locally homogeneous Hermitian manifolds. +The research is partially supported by NSFC grants # 12071050 and 12141101, Chongqing grant cstc2021ycjh- +bgzxm0139, and is supported by the 111 Project D21024. +1 + +Theorem 1.2. Let (M n, g) be a Hermitian manifold with a CAS structure. Assume that the universal +cover � +M does not admit any K¨ahler de Rham factor, then (M, g) must be Chern flat. In particular, M +is covered by a complex Lie group. +Alekseevski˘i and Kimel´fel´d proved [1] that any Ricci flat Riemannian homogeneous manifold M n +must be flat, and in fact it is isometric to T k×Rn−k. The authors proved the result by the consideration of +a volume entropy. It can also be derived from Cheeger-Gromoll’s splitting theorem and the consideration +of Clifford translation [19]. The above result can be viewed as a Hermitian analogue. In fact we have +the following corollary. +Corollary 1.3. Let (M n, g) be a Hermitian manifold with a CAS structure. Assume that the Chern +curvature is Ricci flat. Then it is Chern flat. In particular, M is covered by a complex Lie group. +A more general theorem is stated in Theorem 4.2. As in [12], a similar question can be asked for +the Bismut connection, as well as for the Gauduchon connections which is a family of connections +interpolating between the Chern connection and the Bismut connection. The Bismut connection is the +unique metric connection with the property ∇bJ = 0 (any metric connection with this property is called +a Hermitina connection) and whose torsion is totally skew-symmetric. If D denotes the Levi-Civita +connection, one can check that ∇1 = 1 +2(D − JDJ) is a Hermitian connection. Moreover, the Bismut +connection can be written as 2∇1 − ∇ with ∇ being the Chern connection. +We say that the Bismut connection is Ambrose-Singer (which we abbreviate as BAS) if it has parallel +torsion and parallel curvature tensor. Assume that (M, g) is a compact (or complete) Hermitian manifold, +whose Bismut connection has parallel torsion T b and curvature Rb. If the (first) Ricci curvature of Rb +is zero, weather or not Rb = 0, namely, if (M, g) is Bismut flat? Recall that a Hermitian manifold is +Bismut K¨ahler-like if the curvature of Bismut connection enjoys the symmetries of the curvature of a +K¨ahler metric. We refer interested readers to [21] for historic aspects and more details on the Bismut +connection and some initial studies of Bismut K¨ahler-like manifolds. +In [21], a weaker conjecture was proposed: If a compact Bismut K¨ahler-like manifold has zero (first +Bismut) Ricci, then it is Bismut flat. +Hermitian manifolds with vanishing Bismut first Ricci are called CYT manifolds in the literature, +which stands for Calabi-Yau with torsion. So the above conjecture says that, for a Bismut K¨ahler-like +manifold, if it is CYT then it is Bismut flat. The conjecture was confirmed in [21] for n ≤ 3. Clearly +the above question for BAS manifolds and this conjecture of [21] are closely related. In fact the Bismut +K¨ahler-like condition implies that ∇bT b = 0 and the metric is pluriclosed, while BAS means T b and Rb +are both parallel under ∇b. +As another application of the algebraic techniques, we confirm this conjecture of [21] for BAS mani- +folds, namely, we show that if a BAS manifold is Bismut K¨ahler-like and CYT, then it must be Bismut +flat. We refer the readers to Section 5 for additional definitions and the precise statement. +2. Constructions of irreducible subbundles +In [12] (cf. also [20]) the following result was proved. +Proposition 2.1. Suppose that (M n, g) is a Hermitian manifold with a CAS structure. Then the Chern +curvature is K¨ahler-like. In particular, the Chern curvature satisfies the first Bianchi identity. +Let T denote the torsion and R denote the curvature tensor of the Chern connection. Recall first +some constructions in [12] which are needed here. Let Wp be the subspace of T 1,0 +p +spanned by the image +of T : +W := ⟨{T (X, Y ) | X, Y ∈ T 1,0M}⟩. +Let N := {Z ∈ T 1,0M | ⟨T (X, Y ), Z⟩ = 0 ∀ X, Y ∈ T 1,0M}. The following was proved in [12]. +Proposition 2.2. The subbundles W and N of T 1,0M (also denoted as T ′M) are invariant under the +parallel transport with respect to the Chern connection ∇, and T ′M = W⊕N orthogonally. The curvature +restriction R |W = 0. In particular, when M is not a locally Hermitian symmetric space, the action of +the holonomy group G on T ′M is reducible. +The next result extends the construction, mainly Theorem 1.4 and its proof in Section 6 of [12]. +Theorem 2.3. Let (M, g) be a simply-connected Hermitian manifold with a CAS structure. Assume +that it does not admit any K¨ahler factor. Then N decomposes into N1 ⊕ N2 with R |W+N2 = 0 and N1 +decomposes further into ⊕k +j=1Kj, with each of Kj being invariant and irreducible under the action of the +2 + +holonomy group G. On each Kj there exists a parallel (2, 0)-symplectic form which can be identified with +the standard holomorphic symplectic (2, 0) form on C2kj with 2kj = dimC(Kj). +Proof. We first recap the constructions in the proof of Theorem 1.4 in [12]. By the simply-connectedness +assumption and the Ambrose-Singer holonomy theorem, namely Theorem 3.1 stated in the next section, +W is a trivial bundle which admits parallel (holomorphic) sections Z1, · · · , Zℓ1 where ℓ1 = dim(Wp). Let +ξi = ⟨·, Zi⟩ be the dual 1-forms correspondingly. Then let τi(·, ·) = dξi = ∂ξi. It was proved in Lemma +6.2 of [12] that τi(·, ·) = ⟨T (·, ·), Zi⟩, which also equals to −ιZi∂ω. It was proved there, also can be seen +by direct calculation, that τi(·, ·) = ∂ξi and ¯∂ξi = 0, that {τi} are parallel, holomorphic (2, 0)-forms. +The next key step in the proof of Theorem 1.4 of [12] is to pick one τ in span{τi} such that τ|N +is of maximum rank. Without the loss of generality we denote this one by τ1. It then derived from +the assumption that M has no K¨ahler factor and τ1 is of maximum rank, there exists a orthogonal +decomposition of N into N1 ⊕ N2 such that N2 is generated by parallel global sections, hence R |N2 = 0. +Moreover N1 is of even dimension and τ1|N1 is non-degenerate. Let ℓ2 = dimC(N2). Then dimC(N1) = 2k +and 2k+ℓ1+ℓ2 = n. Since W and N2 are generated by parallel (holomorphic) sections/vector fields, both +K0 := W⊕N2 and N1 are invariant under the parallel transport with respect to the Chern connection. Fix +a point p ∈ M the holonomy action of any element h ∈ G splits as ˜h⊕id : (N1)p⊕(K0)p → (N1)p⊕(K0)p. +Now consider τ|N1, a parallel (2, 0) form on this sub-bundle of T ′M, which we shall still denote by τ. +With respect to a chosen unitary frame of N1 at the point p, τ has the following matrix form: +A = + + +a1E +... +akE + + , +E = +� 0 +1 +−1 +0 +� +, +a1 ≥ · · · ≥ ak > 0. +We may collect the ai into distinct numbers of bj. Namely we write +b1E = + + +a1E +... +a1E + + , +A = + + +b1E +... +bℓ3E + + , +b1 > · · · > bℓ3 > 0. +Namely −b2 +j are distinct eigenvalues of AA if A is the matrix representation of τ at p. Let 2nj denote +the dimension of the corresponding eigenspaces Vj. Clearly �ℓ3 +j=1 nj = k. Since τ is parallel, ˜h∗τ = τ +for any h ∈ G. In terms of matrix we have that ˜htrA˜h = A. Here ˜htr denotes the traspose of ˜h. Note +that ˜h ∈ U(2k), A˜h = ¯˜hA. Since A is real we have ˜hA = A¯˜h. From this we have that ˜hA2 = A2˜h, which +implies that ˜h also keep each eigenspace Vj invariant. This then implies that N1 further decomposes +into orthogonal holonomy-invariant subbundles. +The next statement follows from Weyl’s completely +reducibility theorem. +Thus τ is a constant multiple of the standard holomorphic symplectic (2, 0)- +form. +□ +Corollary 2.4. For any x, y ∈ TpM, (Rx,y)|Kj has vanishing trace. In particular, the Ricci curvature +of R |Kj vanishes. +Proof. Since τ|Kj is a holomorphic parallel (2, 0)-form, the restriction of the holonomy action on Kj +satisfies h ∈ SU(2kj) with 2kj = dimC(Kj). +On the other hand, Rx,y |Kj can be obtained as the +derivative of one parameter family h(t) (cf. Lemma 2.2 in [11] and its proof), thus with zero trace. The +final statement is due to that RicR |Kj (v, ¯v) = � +ℓ R(v, ¯v, Ej +ℓ, ¯Ej +ℓ) for v ∈ Γ(Kj) where {Ej +ℓ} is a unitary +base of Kj. +□ +3. Symmetric holonomy systems +Here we work with real numbers and vector bundles over real numbers. To apply the discussion here +to the previous section one can consider the realification of complex bundles. Note that the real part +of a Hermitian metric on a complex bundle is a Riemannian metric on the realification of the complex +bundle. +The concept of holonomy systems was introduced by J. Simons [13] in his intrinsic proof of Berger’s +holonomy theorem for the Riemannian holonomy groups with respect to the Levi-Civita connections. +The Riemannian holonomy system is a triple S = {V, R, G}, which consists of, a Euclidean space V of +dimension ℓ (we call it the degree of S) endowed with an inner product ⟨·, ·⟩ (also denoted by a bilinear +form H), a connected compact subgroup G of SO(ℓ), and an algebraic curvature operator R (defined on +V ) satisfying the 1st Bianchi identity and that Rx,y ∈ g, ∀x, y ∈ V with g ⊂ so(ℓ) being the Lie algebra +3 + +of G. To see the relevance we first recall the Ambrose-Singer’s holonomy theorem [8] (see also [11] for a +simple proof of half of the result). +Theorem 3.1 (Ambrose-Singer). For any connection ∇ on a Riemannian vector bundle (E, h), let R +be its curvature. Given any path γ from q to p, let γ also denote the parallel transport along it. Then +{γ(Rq)} generates the holonomy algebra, where γ(Rq) ∈ so(Ep) is defined γ · +� +Rx,y |Eq +� +· γ−1. Here +x, y ∈ TqM. +In the case that (M, g) is a Hermitian manifold with a CAS structure, with respect to the Chern +connection, which preserves the inner product of the underlying real tangent bundle, for any parallel +invariant subbundle K ⊂ T ′M, one can apply the above to the bundle V = K ⊕ K. Since ∇ R = 0, it is +easy to see that γ(Rq) = R |Vp. Consequently we have +Proposition 3.2. Let (M, g) be a Hermitian manifold whose Chern connection is Ambrose-Singer. Let +V be as above. Let G be the restricted holonomy group. Then S = (Vp, R, G) is a holonomy system. In +fact it is symmetric. +Proof. The first part is a direct consequence of Theorem 3.1 and the observation that the curvature R +satisfying the first Bianchi identity. The second statement follows from the fact that ∇ R = 0. Recall +that a holonomy system is called symmetric if γ(R) = R for any γ ∈ G. +□ +The following result holds the key of the algebraic aspect of the proof. +Proposition 3.3. Assume that S = {V, R, G} is an irreducible symmetric holonomy system. Then the +Ricci flatness of R implies that R is flat. +Proof. Here we follow the argument in Theorem 3.1 of [11]. First we set up the notations and conventions. +Identify so(n) with ∧2V . Here S2(∧2V ) denotes the symmetric transformations of ∧2V . The S2 +B(·) +denotes the subspace satisfying the 1st Bianchi identity. +Define the metric on gl(V ) by +⟨A, B⟩ ≑ 1 +2 +� +i +⟨A(ei), B(ei)⟩ = 1 +2 trace(BtrA) +In particular for A, B ∈ g the above inner product applies. Let P be the projection from ∧2(V ) onto g, +and let T : g → g be the symmetric isomorphism corresponding to the negative definite bilinear form on +g +(3.1) +B(A, A′) ≑ K(A, A′) − 2⟨A, A′⟩ +with K being the Killing form of g (defined as K(A, A′) = trace(adA · adA′)). Namely T is defined by +B(A, A′) = ⟨T (A), A′⟩. Let J = g ⊕ V (orthogonal sum with the inner product of V and ⟨A, B⟩ on g as +elements in so(n)) and define a Lie algebra structure on J by letting +[A, A′] ≑ [A, A′]; +[x, y] ≑ Rx,y; +[A, x] ≑ A(x), ∀A, A′ ∈ g, x, y ∈ V. +Since A(R) = 0, ∀A ∈ g, it is easy to check that the bracket so defined satisfies the Jacobi identity, +namely J is a Lie algebra. Let B′ be the Killing form of J. It is a basic result of Lie algebra that B′ is +adJ-invariant. +Claim 1: B′|g is given by B defined by (3.1), hence is negative definite. By the definition B′(A, B) = +trace(adA · adB) = �n +i=1⟨adA · adB(ei), ei⟩ + � +α⟨adA · adB(Aα), Aα⟩ where {ei} ({Aα}) is an orthonor- +mal frame of V (g respectively). The second summand is K(A, B). By the definition of the Lie bracket +the first summand is −⟨B(ei), A(ei)⟩ = −2⟨A, B⟩. We first need the following computational results. +Claim 2: B′(A, x) = 0 for A ∈ g and x ∈ V . Similarly +B′(A, x) = trace(adA · adx) = +n +� +i=1 +⟨adA · adx(ei), ei⟩ + +� +α +⟨adA · adx(Aα), Aα⟩ +where {ei} ({Aα}) is an orthonormal frame of V (g respectively). The first term vanishes since +⟨adA · adx(ei), ei⟩ = ⟨[A, Rx,ei]g, ei⟩ = 0. +For the second term, ⟨adA · adx(Aα), Aα⟩ = ⟨−A(Aα(x)), Aα⟩ = 0. +Claim 3: B′|V = λH, and λ ̸= 0 if R ̸= 0. Since B′|V is adg-invariant, hence G-invariant. By the +irreducibility of G-action on V , it implies that B′(x, y) = λ⟨x, y⟩ for some λ. If λ = 0, B′([x, y], [x, y]) = +B′(x, [y, [x, y]]) = 0 since [y, [x, y]] ∈ V . Now by Claim 1, which asserts that B′|g is negative definite, we +have that [x, y] = Rx,y = 0, ∀x, y ∈ V . Hence R = 0. +4 + +Now we assume that λ ̸= 0, otherwise we have proved the result. +A linear algebraic calculation +shows that ⟨[[x, y], z], w⟩ = −⟨[x, y], (z ∧w)⟩. Thus ⟨[[x, y], z], w⟩ = 1 +λB′([[x, y], z], w) = 1 +λB′([x, y], [z, w]). +Putting them together we have the equation for x, y, z, w ∈ V +(3.2) +⟨Rx,y z, w⟩ = ⟨[[x, y], z], w⟩ = 1 +λ⟨[x, y], T ([z, w])⟩ = 1 +λB([x, y], [z, w]⟩. +Now the Ricci curvature is given by +RicR(x, x) = 1 +λ +� +B([x, ei], [x, ei]). +Hence if RicR = 0 it implies that [x, ei] = Rx,ei = 0 for any ei, x, namely R = 0. +□ +The above proposition is the algebraic analogue of Theorem 8.6 of Vol II of [8]. +4. Hermitian manifolds with a CAS structure +The subbundle of N0, which is defined as +N0 := {X ∈ N | T (X, Y ) = 0, +∀ Y ∈ T 1,0M}, +was introduced in [12] to capture any K¨ahler de Rham factor in the universal cover of a Hermitian +manifold M with a CAS structure. It is easy to see that if locally there is a de Rham K¨ahler factor for +M, then the corresponding holomorphic tangent subbundle must be N0. Applying it to the universal +cover of M we have the following result. +Proposition 4.1. Let (M, g) be a Hermitian manifold with a CAS structure. Let � +M be its universal +cover. Let N0 be the subset of T ′ � +M defined above. Then � +M splits as M1 × M2 with M2 being K¨ahler and +T ′M2 = N0. Moreover M1 does not admit any K¨ahler factor. +Proof. This is essentially Theorem 3.6 of [12]. +□ +By combining Corollary 2.4, Propositions 3.2, 3.3, we have the following theorem. +Theorem 4.2. Let (M, g) be a Hermitian manifold with a CAS structure. Then its universal cover � +M +splits into M1 × M2 with M1 being a Chern flat Hermitian manifold with a complex Lie group structure, +and M2 is a product of irreducible Hermitian symmetric spaces. +Proof. First split the manifold � +M into two factors as in the statement. By the proof of Lemma 2.2 in [11], +the splitting of the holonomy action implies that splitting of Rx,y. Applying Corollary 2.4, Propositions +3.2, 3.3 to the irreducible G-invariant subbundle of T ′M1 we conclude that the Chern curvature is flat on +each irreducible summand, hence is totally flat. Now since M1 has zero curvature and parallel torsion, +we may find global parallel vector fields X1, · · · , Xk with k = dimC(M1) (which are holomorphic) such +that +[Xi, Xj] = T (Xi, Xj) = T k +ijXk. +The fact that the torsion is parallel implies that {T k +ij} are constants. By Lemma 3.1 of [12], they also +satisfy the Jacobi identity in general. Now we can appeal the result Cartan [5] (pp. 188-192) to assert +that there is a complex Lie group structure on M1. +□ +Proof of Theorem 1.2. The first statement follows from the above result. +The second statement +follows from Boothby’s theorem of [4] when M is compact. When M is not compact, we can apply the +above Theorem 4.2. For a modern treatment of the existence of a complex Lie group structure one can +see [7] pages 20-25, Section 3.1 of Ch1 for the existence of Lie group structure and [10], Theorem on page +212 of Section 11 of Ch IV for details on the existence of complex Lie group structure. This implies that +one can endow a complex Lie group structure on � +M. +□ +Proof of Corollary 1.3. By Theorem 4.2, the universal cover of M is the product of a complex Lie +group M1 with M2 which is products of irreducible simply-connected Hermitian symmetric spaces, namely +M2 = Cn1 × N2 × · · · × Nℓ with Ni being a nonflat irreducible simply-connected Hermitian symmetric +space. However, the nonflat factor must be of semi-simple type hence have its Ricci curvature being +a nonzero factor of the K¨ahler metric. Then the assumption implies that ˜ +M = M1 × Cn1, which is a +complex Lie group itself. +□ +The corollary shows that T ′M1 may miss some complex Lie group factors. To capture it let Fp = +{Z ∈ T 1,0 +p +M | h(Z) = Z, ∀ h ∈ Gp}, where G is the holonomy group of the Chern connection at p. Recall +the following result from [12]. +5 + +Proposition 4.3. Let (M, g) be a Hermitian manifold with a CAS structure. Let F = ∪xFx be the +sub-bundle of T 1,0M. Then F is a holomorphic integrable foliation. Moreover, if Z1, · · · , Zr is a parallel +frame of F, then +[Zi, Zj] = ck +ijZk +for some constant ck +ij. In particular, when M is simply-connected, there exists a complex Lie group F +acting almost freely, holomorphically on M such that T 1,0 +x (F · x) = Fx. +Theorem 4.2 implies that on � +M, F = T ′M1 ⊕ Cn1. +5. The Bismut Ambrose-Singer manifolds +Another application of Proposition 3.3 is to give a partial confirmation to a conjectured raised in [21, +Conjecture 2], which states that: +Conjecture 5.1 ( [21]). Let (M n, g) be a compact Bismut K¨ahler-like manifold without any K¨ahler de +Rham factor of dimension > 1. If g is CYT, then g is Bismut flat. +Recall that a Hermitian manifold is said to be Bismut K¨ahler-like (see [3] and [20]), if the curvature +tensor Rb of its Bismut connection ∇b obeys all the K¨ahler symmetries: Rb +xyz ¯ +w = 0 and Rb +x¯yz ¯ +w = Rb +z¯yx ¯ +w +for any type (1, 0) tangent vectors x, y, z, w. When n ≥ 2, there are examples of non-K¨ahler manifolds +which are Bismut K¨ahler-like, and such metrics were classified in complex dimensions n = 2 and 3 ([22], +[21]). +The manifold is said to be CYT, which stands for Calabi-Yau with torsion, if the first Ricci curvature +of ∇b vanishes identically. This means that the restricted holonomy group of ∇b is contained in SU(n). +The conjecture is known to be true for n ≤ 3 by [21], but open when n ≥ 4. +By the main result of [22], we know that a Hermitian metric g is Bismut K¨ahler-like if and only if ∇b +has parallel torsion and g is pluriclosed (namely, ∂∂ω = 0, where ω is the K¨ahler form of g). +Bismut flat manifolds were fully classified [18]. They are quotients of Samelson spaces, namely, Lie +groups with bi-invariant metrics and compatible left-invariant complex structures. +Now suppose that (M n, g) is a complete Hermitian manifold which is Bismut Ambrose-Singer, meaning +that ∇b is Ambrose-Singer. As an immediate application of Proposition 3.3, we have the following partial +confirmation to the aforementioned conjecture: +Theorem 5.2. Let (M n, g) be a complete Hermitian manifold whose Bismut connection is Ambrose- +Singer. If g is Bismut K¨ahler-like and CYT, then g is Bismut flat. +Proof. Fix any p ∈ M and let V be the tangent space of M at p. Note that the Bismut K¨ahler-like +assumption guarantees that Rb obeys the first Bianchi identity, so the Bismut connection ∇b gives a +holonomy system, which is clearly a symmetric one. In order to apply Proposition 3.3, we need to verify +that that the Ricci curvature vanishes for each irreducible component. Decompose the tangent bundle +into subbundles where the Bismut holonomy group acts irreducibly. Writing in complex frames, say +T 1,0M = ⊕r +j=1Ej, then the Bismut curvature form Θb is block-diagonal: +Θb = + + +Θb +1 +... +Θb +r + + , +Θb +j = (Θb +ik), +ei, ek ∈ Ej. +Since ∇b is assumed to be K¨ahler-like, the entries of Θb +j are all combinations of ϕiϕk for e1, ek ∈ Ej only. +In particular, tr(Θb) = 0 if and only if tr(Θb +j) = 0 for each j. So if we assume that the Bismut curvature +has vanishing Ricci, then each irreducible component will also have vanishing Ricci, thus one can apply +Proposition 3.3 to conclude that the curvature vanishes. This completes the proof. Note that since the +curvature is assumed to be parallel, so any K¨ahler de Rham factor will be locally Hermitian symmetric, +hence it will be flat if it is Ricci flat. +□ +We remark that, when n ≥ 3, there are plenty of examples ([23]) of compact Hermitian manifolds +(M n, g) whose Bismut connection is Ambrose-Singer, and with zero first (and second) Bismut Ricci +curvature, yet it is not Bismut flat. So unlike in the Chern connection case, for Bismut connection +under the Ambrose-Singer condition, the vanishing of (first) Ricci does not guarantee the vanishing of +full Bismut curvature tensor. Nonetheless, we would like to speculate and propose the following: +Conjecture 5.3. Let (M n, g) be a complete Hermitian manifold whose Bismut connection is Ambrose- +Singer. If the first, second, and third Ricci of the Bismut curvature Rb all vanish, then Rb = 0. +6 + +Acknowledgments. We would like to thank Christoph B¨ohm for his interests and bringing our attention +to a result of Alekseevski˘i and B. N. Kimel´fel´d, James Stanfield and Steve Gindi for their interests to +extending the result to the BAS manifolds. The first author would like to thank T. Colding and MIT +mathematics for the hospitality during his visit when the first draft of the paper was completed. +References +[1] D. V. Alekseevski˘i and B. N. Kimel´fel´d, Structure of homogeneous Riemannian spaces with zero Ricci curvature. +(Russian) Funkcional. Anal. i Priloen. 9 (1975), no. 2, 5-11. (cited on page 2) +[2] W. Ambrose and I.M. Singer, On homogeneous Riemannian manifolds. Duke Math. J., 25 (1958), 647-669. (cited on +page 1) +[3] D. Angella, A. Otal, L. Ugarte, R. Villacampa, +On Gauduchon connections with K¨ahler-like curvature. ArXiv: +1809.02632. (cited on page 6) +[4] W. Boothby, Hermitian manifolds with zero curvature. Michigan Math. J., 5 (1958), no. 2, 229-233. (cited on page 5) +[5] E. Cartan, La th´eorie des groupes finis et continus et la g´eom´etrie diff´erentielle trait´ees par m´ethode du rep`ere mobile. +Cahiers Scientifiques, no. 18, Gauthier-Villars, Paris, 1937. (cited on page 5) +[6] S.-S. Chern, Complex manifolds without potential theory. With an appendix on the geometry of characteristic classes. +Second edition. Universitext Springer-Verlag, New York-Heidelberg, 1979. iii+152 pp. (cited on page 1) +[7] B. A. Dubrovin, A. T. Fomenko and S. P. Novikov, Modern geometry-methods and applications. Part II. The geometry +and topology of maniflds. Translated from Russian by Robert G. Burns. Graduate Texts in Mathematics, 104, Sprng- +Verlag, New York, 1985. xv+430 pp. ISBN: 0-387-96162-3-53-01 (cited on page 5) +[8] S. Kobayashi and K. Nomizu, +Foundations of differential geometry. Vol. I, II. Reprint of the 1963 original. Wiley +Classics Library. A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1996. xii+329 pp. (cited on +pages 4, 5) +[9] B. Kostant, Holonomy and the Lie algebra of infinitesimal motions of a Riemannian manifold. Trans. Amer. Math. +Soc. 80 (1955), 528-542. (cited on page 1) +[10] Y. Matsushima, Differentiable manifolds. Translated from the Japanese by E. T. Kobayashi. Pure and Applied Math- +ematics, 9. Marcel Dekker, Inc., New York, 1972. vii+303 pp. (cited on page 5) +[11] L. Ni, An alternatie induction argument in Simons’ proof of holonomy theorem. Analysis and Partial Differential +Euqations on Manifolds, Fractals and Graphs (Nankai, 2019, A. Grigoryan, Y Sun, Eds.) Advances in Analysis and +Geometry, 3 (2021), 443-458. (cited on pages 3, 4, 5) +[12] L. Ni and F. Zheng, On Hermitian manifolds whose Chern connection is Ambrose-Singer. ArXiv: 2208.10040 (cited +on pages 1, 2, 3, 5) +[13] J. Simons, On the transitivity of holonomy systems. Ann. of Math. 76 (1962), 213-234. (cited on page 3) +[14] I.M. Singer, Infinitesimally homogeneous spaces. Comm. Pure Appl. Math., 13 (1960), 685-697. (cited on page 1) +[15] F. Tricerri and L. Vanhecke, Homogeneous Structures on Riemannian Manifolds. London Math. Soc. Lect. Notes, vol. +83. Cambridge Univ. Press, Cambridge (1983). (cited on page 1) +[16] H.-C. Wang, Closed manifolds with homogeneous complex structure. Amer. J. Math. 76 (1954), 1-32. (cited on page +1) +[17] H.-C. Wang, Complex parallisiable manifolds. Proc. Amer. Math. Soc. 5(1954), 771-776. (cited on page 1) +[18] Q. Wang, B. Yang, F. Zheng, On Bismut flat manifolds. Trans. Amer. Math. Soc. 373 (2020), 5747-5772. (cited on +page 6) +[19] J.-A. Wolf, Spaces of constant curvature. Fifth edition. Publish or Perish, Inc., Houston, TX, 1984. xviii+412 pp. +ISBN: 0-914098-07-1 53-02. (cited on page 2) +[20] B. Yang and F. Zheng, On curvature tensors of Hermitian manifolds. Comm. Anal. Geom. 26 (2018), no.5, 1195-1222. +(cited on pages 2, 6) +[21] S.-T. Yau, +Q. Zhao and F. Zheng, +On Strominger K¨ahler-like manifolds with degenerate torsion. ArXiv +preprint:1908.05322v3. (cited on pages 2, 6) +[22] Q. Zhao, F. Zheng, Srominger connection and pluriclosed metrics. ArXiv: 1904.06604. (cited on page 6) +[23] Q. Zhao, F. Zheng, On Hermitian manifolds with Bismut-Srominger parallel torsion. ArXiv;2208.03071. +(cited on +page 6) +Lei Ni. Department of Mathematics, University of California, San Diego, La Jolla, CA 92093, USA +Email address: leni@ucsd.edu +Fangyang Zheng. +School of Mathematical Sciences, Chongqing Normal University, Chongqing 401331, +China +Email address: 20190045@cqnu.edu.cn; +franciszheng@yahoo.com +7 + diff --git a/K9AyT4oBgHgl3EQfsfmx/content/tmp_files/load_file.txt b/K9AyT4oBgHgl3EQfsfmx/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..64a8409861e43197565385f65576cea1e045e19d --- /dev/null +++ b/K9AyT4oBgHgl3EQfsfmx/content/tmp_files/load_file.txt @@ -0,0 +1,512 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf,len=511 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content='00579v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content='DG] 2 Jan 2023 HERMITIAN MANIFOLDS WHOSE CHERN CONNECTION IS AMBROSE-SINGER-II LEI NI AND FANGYANG ZHENG Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' We apply the algebraic consideration of the holonomy system to study the Hermitian man- ifolds whose Chern connection is Amrose-Singer and prove more refined structure theorems for such manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' The main result asserts that the universal cover of such a Hermitian manifold must be the product of a complex Lie group and Hermitian symmetric spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Contents 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Introduction and main results 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Constructions of irreducible subbundles 2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Symmetric holonomy systems 3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Hermitian manifolds with a CAS structure 5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' The Bismut Ambrose-Singer manifolds 6 References 7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Introduction and main results This is a continuation of the previous paper [12] on Hermitian manifolds whose Chern connection ∇ is Ambrose-Singer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' A connection is called Ambrose-Singer (abbreviated as an AS connection) if its torsion and its curvature are parallel with respect to this connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' The conditions can be expressed in terms of more convoluted PDEs in terms of the curvature tensor of the Levi-Civita connection D and the torsion of the AS connection [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' This arises naturally in Ambrose-Singer’s characterization of simply-connected Riemannian homogeneous spaces (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' [2]), namely the complete Riemannian manifolds whose isometry group acts transitively, as a generalization of Cartan’s theorem on the symmetric (or locally symmetric) spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' We refer the readers to [9, 14, 15, 16, 17] for other related studies on homogeneous complex manifolds, namely complex manifolds whose biholomorphism group act transitively, and homogeneous Hermitian manifolds, namely Hermitian manifolds whose biholomorphic isometries act transitively, and other homogeneous spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' It is well known that on a Hermitian manifold there exists a unique canonical connection, namely the Chern connection [6] (in fact the existence extends to any holomorphic vector bundles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' When the Chern connection of (M, g) is Ambrose-Singer we say that the complex manifold M admits a CAS structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' (Below we also abbreviate them as Hermitian manifolds with a CAS structure).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' The examples of manifolds with CAS structure include complex Lie groups endowed with a left invariant metric and Hermitian symmetric spaces, their products and quotients of products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' In [12] conditions are given when the universal cover of such a manifold admits some K¨ahler (which is Hermitian symmetric due to ∇R = 0) de Rham factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Structure theorems were obtained in complex dimension three and four.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' For high dimensions it was proved that Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Let (M n, g) be a Hermitian manifold with a CAS structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Assume further that its universal cover admits no K¨ahler de Rham factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Then the universal cover of M admits a parallel holomorphic (n, 0) form, and (M, g) has zero Chern Ricci curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Since the connection is NOT Levi-Civita, the de Rham decomposition theorem no longer holds to its full generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' We refer to [12] for some detailed discussions when it remains true for the case considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' The first main result is the following structure theorem for high dimensional case which classifies all manifolds with a CAS structure whose universal cover does not contain a K¨ahler de Rham factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' 2010 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' 53C55 (Primary), 53C05 (Secondary).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Ambrose-Singer connection, Chern connection, holonomy algebra/group/system, Chern flat, locally homogeneous Hermitian manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' The research is partially supported by NSFC grants # 12071050 and 12141101, Chongqing grant cstc2021ycjh- bgzxm0139, and is supported by the 111 Project D21024.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' 1 Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Let (M n, g) be a Hermitian manifold with a CAS structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Assume that the universal cover � M does not admit any K¨ahler de Rham factor, then (M, g) must be Chern flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' In particular, M is covered by a complex Lie group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Alekseevski˘i and Kimel´fel´d proved [1] that any Ricci flat Riemannian homogeneous manifold M n must be flat, and in fact it is isometric to T k×Rn−k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' The authors proved the result by the consideration of a volume entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' It can also be derived from Cheeger-Gromoll’s splitting theorem and the consideration of Clifford translation [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' The above result can be viewed as a Hermitian analogue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' In fact we have the following corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Let (M n, g) be a Hermitian manifold with a CAS structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Assume that the Chern curvature is Ricci flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Then it is Chern flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' In particular, M is covered by a complex Lie group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' A more general theorem is stated in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' As in [12], a similar question can be asked for the Bismut connection, as well as for the Gauduchon connections which is a family of connections interpolating between the Chern connection and the Bismut connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' The Bismut connection is the unique metric connection with the property ∇bJ = 0 (any metric connection with this property is called a Hermitina connection) and whose torsion is totally skew-symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' If D denotes the Levi-Civita connection, one can check that ∇1 = 1 2(D − JDJ) is a Hermitian connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Moreover, the Bismut connection can be written as 2∇1 − ∇ with ∇ being the Chern connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' We say that the Bismut connection is Ambrose-Singer (which we abbreviate as BAS) if it has parallel torsion and parallel curvature tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Assume that (M, g) is a compact (or complete) Hermitian manifold, whose Bismut connection has parallel torsion T b and curvature Rb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' If the (first) Ricci curvature of Rb is zero, weather or not Rb = 0, namely, if (M, g) is Bismut flat?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Recall that a Hermitian manifold is Bismut K¨ahler-like if the curvature of Bismut connection enjoys the symmetries of the curvature of a K¨ahler metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' We refer interested readers to [21] for historic aspects and more details on the Bismut connection and some initial studies of Bismut K¨ahler-like manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' In [21], a weaker conjecture was proposed: If a compact Bismut K¨ahler-like manifold has zero (first Bismut) Ricci, then it is Bismut flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Hermitian manifolds with vanishing Bismut first Ricci are called CYT manifolds in the literature, which stands for Calabi-Yau with torsion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' So the above conjecture says that, for a Bismut K¨ahler-like manifold, if it is CYT then it is Bismut flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' The conjecture was confirmed in [21] for n ≤ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Clearly the above question for BAS manifolds and this conjecture of [21] are closely related.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' In fact the Bismut K¨ahler-like condition implies that ∇bT b = 0 and the metric is pluriclosed, while BAS means T b and Rb are both parallel under ∇b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' As another application of the algebraic techniques, we confirm this conjecture of [21] for BAS mani- folds, namely, we show that if a BAS manifold is Bismut K¨ahler-like and CYT, then it must be Bismut flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' We refer the readers to Section 5 for additional definitions and the precise statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Constructions of irreducible subbundles In [12] (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' also [20]) the following result was proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Suppose that (M n, g) is a Hermitian manifold with a CAS structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Then the Chern curvature is K¨ahler-like.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' In particular, the Chern curvature satisfies the first Bianchi identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Let T denote the torsion and R denote the curvature tensor of the Chern connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Recall first some constructions in [12] which are needed here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Let Wp be the subspace of T 1,0 p spanned by the image of T : W := ⟨{T (X, Y ) | X, Y ∈ T 1,0M}⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Let N := {Z ∈ T 1,0M | ⟨T (X, Y ), Z⟩ = 0 ∀ X, Y ∈ T 1,0M}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' The following was proved in [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' The subbundles W and N of T 1,0M (also denoted as T ′M) are invariant under the parallel transport with respect to the Chern connection ∇, and T ′M = W⊕N orthogonally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' The curvature restriction R |W = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' In particular, when M is not a locally Hermitian symmetric space, the action of the holonomy group G on T ′M is reducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' The next result extends the construction, mainly Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content='4 and its proof in Section 6 of [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Let (M, g) be a simply-connected Hermitian manifold with a CAS structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Assume that it does not admit any K¨ahler factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Then N decomposes into N1 ⊕ N2 with R |W+N2 = 0 and N1 decomposes further into ⊕k j=1Kj, with each of Kj being invariant and irreducible under the action of the 2 holonomy group G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' On each Kj there exists a parallel (2, 0)-symplectic form which can be identified with the standard holomorphic symplectic (2, 0) form on C2kj with 2kj = dimC(Kj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' We first recap the constructions in the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content='4 in [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' By the simply-connectedness assumption and the Ambrose-Singer holonomy theorem, namely Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content='1 stated in the next section, W is a trivial bundle which admits parallel (holomorphic) sections Z1, · · · , Zℓ1 where ℓ1 = dim(Wp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Let ξi = ⟨·, Zi⟩ be the dual 1-forms correspondingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Then let τi(·, ·) = dξi = ∂ξi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' It was proved in Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content='2 of [12] that τi(·, ·) = ⟨T (·, ·), Zi⟩, which also equals to −ιZi∂ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' It was proved there, also can be seen by direct calculation, that τi(·, ·) = ∂ξi and ¯∂ξi = 0, that {τi} are parallel, holomorphic (2, 0)-forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' The next key step in the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content='4 of [12] is to pick one τ in span{τi} such that τ|N is of maximum rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Without the loss of generality we denote this one by τ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' It then derived from the assumption that M has no K¨ahler factor and τ1 is of maximum rank, there exists a orthogonal decomposition of N into N1 ⊕ N2 such that N2 is generated by parallel global sections, hence R |N2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Moreover N1 is of even dimension and τ1|N1 is non-degenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Let ℓ2 = dimC(N2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Then dimC(N1) = 2k and 2k+ℓ1+ℓ2 = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Since W and N2 are generated by parallel (holomorphic) sections/vector fields, both K0 := W⊕N2 and N1 are invariant under the parallel transport with respect to the Chern connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Fix a point p ∈ M the holonomy action of any element h ∈ G splits as ˜h⊕id : (N1)p⊕(K0)p → (N1)p⊕(K0)p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Now consider τ|N1, a parallel (2, 0) form on this sub-bundle of T ′M, which we shall still denote by τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' With respect to a chosen unitary frame of N1 at the point p, τ has the following matrix form: A = \uf8ee \uf8ef\uf8f0 a1E .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' akE \uf8f9 \uf8fa\uf8fb , E = � 0 1 −1 0 � , a1 ≥ · · · ≥ ak > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' We may collect the ai into distinct numbers of bj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Namely we write b1E = \uf8ee \uf8ef\uf8f0 a1E .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' a1E \uf8f9 \uf8fa\uf8fb , A = \uf8ee \uf8ef\uf8f0 b1E .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' bℓ3E \uf8f9 \uf8fa\uf8fb , b1 > · · · > bℓ3 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Namely −b2 j are distinct eigenvalues of AA if A is the matrix representation of τ at p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Let 2nj denote the dimension of the corresponding eigenspaces Vj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Clearly �ℓ3 j=1 nj = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Since τ is parallel, ˜h∗τ = τ for any h ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' In terms of matrix we have that ˜htrA˜h = A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Here ˜htr denotes the traspose of ˜h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Note that ˜h ∈ U(2k), A˜h = ¯˜hA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Since A is real we have ˜hA = A¯˜h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' From this we have that ˜hA2 = A2˜h, which implies that ˜h also keep each eigenspace Vj invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' This then implies that N1 further decomposes into orthogonal holonomy-invariant subbundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' The next statement follows from Weyl’s completely reducibility theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Thus τ is a constant multiple of the standard holomorphic symplectic (2, 0)- form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' □ Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' For any x, y ∈ TpM, (Rx,y)|Kj has vanishing trace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' In particular, the Ricci curvature of R |Kj vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Since τ|Kj is a holomorphic parallel (2, 0)-form, the restriction of the holonomy action on Kj satisfies h ∈ SU(2kj) with 2kj = dimC(Kj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' On the other hand, Rx,y |Kj can be obtained as the derivative of one parameter family h(t) (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content='2 in [11] and its proof), thus with zero trace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' The final statement is due to that RicR |Kj (v, ¯v) = � ℓ R(v, ¯v, Ej ℓ, ¯Ej ℓ) for v ∈ Γ(Kj) where {Ej ℓ} is a unitary base of Kj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Symmetric holonomy systems Here we work with real numbers and vector bundles over real numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' To apply the discussion here to the previous section one can consider the realification of complex bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Note that the real part of a Hermitian metric on a complex bundle is a Riemannian metric on the realification of the complex bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' The concept of holonomy systems was introduced by J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Simons [13] in his intrinsic proof of Berger’s holonomy theorem for the Riemannian holonomy groups with respect to the Levi-Civita connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' The Riemannian holonomy system is a triple S = {V, R, G}, which consists of, a Euclidean space V of dimension ℓ (we call it the degree of S) endowed with an inner product ⟨·, ·⟩ (also denoted by a bilinear form H), a connected compact subgroup G of SO(ℓ), and an algebraic curvature operator R (defined on V ) satisfying the 1st Bianchi identity and that Rx,y ∈ g, ∀x, y ∈ V with g ⊂ so(ℓ) being the Lie algebra 3 of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' To see the relevance we first recall the Ambrose-Singer’s holonomy theorem [8] (see also [11] for a simple proof of half of the result).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content='1 (Ambrose-Singer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' For any connection ∇ on a Riemannian vector bundle (E, h), let R be its curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Given any path γ from q to p, let γ also denote the parallel transport along it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Then {γ(Rq)} generates the holonomy algebra, where γ(Rq) ∈ so(Ep) is defined γ · � Rx,y |Eq � γ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Here x, y ∈ TqM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' In the case that (M, g) is a Hermitian manifold with a CAS structure, with respect to the Chern connection, which preserves the inner product of the underlying real tangent bundle, for any parallel invariant subbundle K ⊂ T ′M, one can apply the above to the bundle V = K ⊕ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Since ∇ R = 0, it is easy to see that γ(Rq) = R |Vp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Consequently we have Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Let (M, g) be a Hermitian manifold whose Chern connection is Ambrose-Singer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Let V be as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Let G be the restricted holonomy group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Then S = (Vp, R, G) is a holonomy system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' In fact it is symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' The first part is a direct consequence of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content='1 and the observation that the curvature R satisfying the first Bianchi identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' The second statement follows from the fact that ∇ R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Recall that a holonomy system is called symmetric if γ(R) = R for any γ ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' □ The following result holds the key of the algebraic aspect of the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Assume that S = {V, R, G} is an irreducible symmetric holonomy system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Then the Ricci flatness of R implies that R is flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Here we follow the argument in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content='1 of [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' First we set up the notations and conventions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Identify so(n) with ∧2V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Here S2(∧2V ) denotes the symmetric transformations of ∧2V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' The S2 B(·) denotes the subspace satisfying the 1st Bianchi identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Define the metric on gl(V ) by ⟨A, B⟩ ≑ 1 2 � i ⟨A(ei), B(ei)⟩ = 1 2 trace(BtrA) In particular for A, B ∈ g the above inner product applies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Let P be the projection from ∧2(V ) onto g, and let T : g → g be the symmetric isomorphism corresponding to the negative definite bilinear form on g (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content='1) B(A, A′) ≑ K(A, A′) − 2⟨A, A′⟩ with K being the Killing form of g (defined as K(A, A′) = trace(adA · adA′)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Namely T is defined by B(A, A′) = ⟨T (A), A′⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Let J = g ⊕ V (orthogonal sum with the inner product of V and ⟨A, B⟩ on g as elements in so(n)) and define a Lie algebra structure on J by letting [A, A′] ≑ [A, A′];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' [x, y] ≑ Rx,y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' [A, x] ≑ A(x), ∀A, A′ ∈ g, x, y ∈ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Since A(R) = 0, ∀A ∈ g, it is easy to check that the bracket so defined satisfies the Jacobi identity, namely J is a Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Let B′ be the Killing form of J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' It is a basic result of Lie algebra that B′ is adJ-invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Claim 1: B′|g is given by B defined by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content='1), hence is negative definite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' By the definition B′(A, B) = trace(adA · adB) = �n i=1⟨adA · adB(ei), ei⟩ + � α⟨adA · adB(Aα), Aα⟩ where {ei} ({Aα}) is an orthonor- mal frame of V (g respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' The second summand is K(A, B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' By the definition of the Lie bracket the first summand is −⟨B(ei), A(ei)⟩ = −2⟨A, B⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' We first need the following computational results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Claim 2: B′(A, x) = 0 for A ∈ g and x ∈ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Similarly B′(A, x) = trace(adA · adx) = n � i=1 ⟨adA · adx(ei), ei⟩ + � α ⟨adA · adx(Aα), Aα⟩ where {ei} ({Aα}) is an orthonormal frame of V (g respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' The first term vanishes since ⟨adA · adx(ei), ei⟩ = ⟨[A, Rx,ei]g, ei⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' For the second term, ⟨adA · adx(Aα), Aα⟩ = ⟨−A(Aα(x)), Aα⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Claim 3: B′|V = λH, and λ ̸= 0 if R ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Since B′|V is adg-invariant, hence G-invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' By the irreducibility of G-action on V , it implies that B′(x, y) = λ⟨x, y⟩ for some λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' If λ = 0, B′([x, y], [x, y]) = B′(x, [y, [x, y]]) = 0 since [y, [x, y]] ∈ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Now by Claim 1, which asserts that B′|g is negative definite, we have that [x, y] = Rx,y = 0, ∀x, y ∈ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Hence R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' 4 Now we assume that λ ̸= 0, otherwise we have proved the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' A linear algebraic calculation shows that ⟨[[x, y], z], w⟩ = −⟨[x, y], (z ∧w)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Thus ⟨[[x, y], z], w⟩ = 1 λB′([[x, y], z], w) = 1 λB′([x, y], [z, w]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Putting them together we have the equation for x, y, z, w ∈ V (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content='2) ⟨Rx,y z, w⟩ = ⟨[[x, y], z], w⟩ = 1 λ⟨[x, y], T ([z, w])⟩ = 1 λB([x, y], [z, w]⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Now the Ricci curvature is given by RicR(x, x) = 1 λ � B([x, ei], [x, ei]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Hence if RicR = 0 it implies that [x, ei] = Rx,ei = 0 for any ei, x, namely R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' □ The above proposition is the algebraic analogue of Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content='6 of Vol II of [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Hermitian manifolds with a CAS structure The subbundle of N0, which is defined as N0 := {X ∈ N | T (X, Y ) = 0, ∀ Y ∈ T 1,0M}, was introduced in [12] to capture any K¨ahler de Rham factor in the universal cover of a Hermitian manifold M with a CAS structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' It is easy to see that if locally there is a de Rham K¨ahler factor for M, then the corresponding holomorphic tangent subbundle must be N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Applying it to the universal cover of M we have the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Let (M, g) be a Hermitian manifold with a CAS structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Let � M be its universal cover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Let N0 be the subset of T ′ � M defined above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Then � M splits as M1 × M2 with M2 being K¨ahler and T ′M2 = N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Moreover M1 does not admit any K¨ahler factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' This is essentially Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content='6 of [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' □ By combining Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content='4, Propositions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content='2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content='3, we have the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Let (M, g) be a Hermitian manifold with a CAS structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Then its universal cover � M splits into M1 × M2 with M1 being a Chern flat Hermitian manifold with a complex Lie group structure, and M2 is a product of irreducible Hermitian symmetric spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' First split the manifold � M into two factors as in the statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' By the proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content='2 in [11], the splitting of the holonomy action implies that splitting of Rx,y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Applying Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content='4, Propositions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content='2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content='3 to the irreducible G-invariant subbundle of T ′M1 we conclude that the Chern curvature is flat on each irreducible summand, hence is totally flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Now since M1 has zero curvature and parallel torsion, we may find global parallel vector fields X1, · · · , Xk with k = dimC(M1) (which are holomorphic) such that [Xi, Xj] = T (Xi, Xj) = T k ijXk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' The fact that the torsion is parallel implies that {T k ij} are constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content='1 of [12], they also satisfy the Jacobi identity in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Now we can appeal the result Cartan [5] (pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' 188-192) to assert that there is a complex Lie group structure on M1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' □ Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' The first statement follows from the above result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' The second statement follows from Boothby’s theorem of [4] when M is compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' When M is not compact, we can apply the above Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' For a modern treatment of the existence of a complex Lie group structure one can see [7] pages 20-25, Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content='1 of Ch1 for the existence of Lie group structure and [10], Theorem on page 212 of Section 11 of Ch IV for details on the existence of complex Lie group structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' This implies that one can endow a complex Lie group structure on � M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' □ Proof of Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' By Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content='2, the universal cover of M is the product of a complex Lie group M1 with M2 which is products of irreducible simply-connected Hermitian symmetric spaces, namely M2 = Cn1 × N2 × · · · × Nℓ with Ni being a nonflat irreducible simply-connected Hermitian symmetric space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' However, the nonflat factor must be of semi-simple type hence have its Ricci curvature being a nonzero factor of the K¨ahler metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Then the assumption implies that ˜ M = M1 × Cn1, which is a complex Lie group itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' □ The corollary shows that T ′M1 may miss some complex Lie group factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' To capture it let Fp = {Z ∈ T 1,0 p M | h(Z) = Z, ∀ h ∈ Gp}, where G is the holonomy group of the Chern connection at p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Recall the following result from [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' 5 Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Let (M, g) be a Hermitian manifold with a CAS structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Let F = ∪xFx be the sub-bundle of T 1,0M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Then F is a holomorphic integrable foliation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Moreover, if Z1, · · · , Zr is a parallel frame of F, then [Zi, Zj] = ck ijZk for some constant ck ij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' In particular, when M is simply-connected, there exists a complex Lie group F acting almost freely, holomorphically on M such that T 1,0 x (F · x) = Fx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content='2 implies that on � M, F = T ′M1 ⊕ Cn1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' The Bismut Ambrose-Singer manifolds Another application of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content='3 is to give a partial confirmation to a conjectured raised in [21, Conjecture 2], which states that: Conjecture 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content='1 ( [21]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Let (M n, g) be a compact Bismut K¨ahler-like manifold without any K¨ahler de Rham factor of dimension > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' If g is CYT, then g is Bismut flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Recall that a Hermitian manifold is said to be Bismut K¨ahler-like (see [3] and [20]), if the curvature tensor Rb of its Bismut connection ∇b obeys all the K¨ahler symmetries: Rb xyz ¯ w = 0 and Rb x¯yz ¯ w = Rb z¯yx ¯ w for any type (1, 0) tangent vectors x, y, z, w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' When n ≥ 2, there are examples of non-K¨ahler manifolds which are Bismut K¨ahler-like, and such metrics were classified in complex dimensions n = 2 and 3 ([22], [21]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' The manifold is said to be CYT, which stands for Calabi-Yau with torsion, if the first Ricci curvature of ∇b vanishes identically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' This means that the restricted holonomy group of ∇b is contained in SU(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' The conjecture is known to be true for n ≤ 3 by [21], but open when n ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' By the main result of [22], we know that a Hermitian metric g is Bismut K¨ahler-like if and only if ∇b has parallel torsion and g is pluriclosed (namely, ∂∂ω = 0, where ω is the K¨ahler form of g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Bismut flat manifolds were fully classified [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' They are quotients of Samelson spaces, namely, Lie groups with bi-invariant metrics and compatible left-invariant complex structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Now suppose that (M n, g) is a complete Hermitian manifold which is Bismut Ambrose-Singer, meaning that ∇b is Ambrose-Singer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' As an immediate application of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content='3, we have the following partial confirmation to the aforementioned conjecture: Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Let (M n, g) be a complete Hermitian manifold whose Bismut connection is Ambrose- Singer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' If g is Bismut K¨ahler-like and CYT, then g is Bismut flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Fix any p ∈ M and let V be the tangent space of M at p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Note that the Bismut K¨ahler-like assumption guarantees that Rb obeys the first Bianchi identity, so the Bismut connection ∇b gives a holonomy system, which is clearly a symmetric one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' In order to apply Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content='3, we need to verify that that the Ricci curvature vanishes for each irreducible component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Decompose the tangent bundle into subbundles where the Bismut holonomy group acts irreducibly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Writing in complex frames, say T 1,0M = ⊕r j=1Ej, then the Bismut curvature form Θb is block-diagonal: Θb = \uf8ee \uf8ef\uf8f0 Θb 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Θb r \uf8f9 \uf8fa\uf8fb , Θb j = (Θb ik), ei, ek ∈ Ej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Since ∇b is assumed to be K¨ahler-like, the entries of Θb j are all combinations of ϕiϕk for e1, ek ∈ Ej only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' In particular, tr(Θb) = 0 if and only if tr(Θb j) = 0 for each j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' So if we assume that the Bismut curvature has vanishing Ricci, then each irreducible component will also have vanishing Ricci, thus one can apply Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content='3 to conclude that the curvature vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Note that since the curvature is assumed to be parallel, so any K¨ahler de Rham factor will be locally Hermitian symmetric, hence it will be flat if it is Ricci flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' □ We remark that, when n ≥ 3, there are plenty of examples ([23]) of compact Hermitian manifolds (M n, g) whose Bismut connection is Ambrose-Singer, and with zero first (and second) Bismut Ricci curvature, yet it is not Bismut flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' So unlike in the Chern connection case, for Bismut connection under the Ambrose-Singer condition, the vanishing of (first) Ricci does not guarantee the vanishing of full Bismut curvature tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Nonetheless, we would like to speculate and propose the following: Conjecture 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Let (M n, g) be a complete Hermitian manifold whose Bismut connection is Ambrose- Singer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' If the first, second, and third Ricci of the Bismut curvature Rb all vanish, then Rb = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' 6 Acknowledgments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' We would like to thank Christoph B¨ohm for his interests and bringing our attention to a result of Alekseevski˘i and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Kimel´fel´d, James Stanfield and Steve Gindi for their interests to extending the result to the BAS manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' The first author would like to thank T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Colding and MIT mathematics for the hospitality during his visit when the first draft of the paper was completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' References [1] D.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Second edition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Universitext Springer-Verlag, New York-Heidelberg, 1979.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' iii+152 pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' (cited on page 1) [7] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Dubrovin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Fomenko and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Novikov, Modern geometry-methods and applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Part II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' The geometry and topology of maniflds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Translated from Russian by Robert G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Burns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Graduate Texts in Mathematics, 104, Sprng- Verlag, New York, 1985.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' xv+430 pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' ISBN: 0-387-96162-3-53-01 (cited on page 5) [8] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Kobayashi and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Nomizu, Foundations of differential geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' I, II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Reprint of the 1963 original.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Wiley Classics Library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' A Wiley-Interscience Publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' John Wiley & Sons, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=', New York, 1996.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' xii+329 pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' (cited on pages 4, 5) [9] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Kostant, Holonomy and the Lie algebra of infinitesimal motions of a Riemannian manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Trans.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Zheng, On Hermitian manifolds with Bismut-Srominger parallel torsion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' ArXiv;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content='2208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content='03071.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' (cited on page 6) Lei Ni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' Department of Mathematics, University of California, San Diego, La Jolla, CA 92093, USA Email address: leni@ucsd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content='edu Fangyang Zheng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' School of Mathematical Sciences, Chongqing Normal University, Chongqing 401331, China Email address: 20190045@cqnu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content='cn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content=' franciszheng@yahoo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfsfmx/content/2301.00579v1.pdf'} +page_content='com 7' 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a/L9FJT4oBgHgl3EQfyi3U/content/tmp_files/2301.11639v1.pdf.txt b/L9FJT4oBgHgl3EQfyi3U/content/tmp_files/2301.11639v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..ad112102d5a348e5d180140469cd09ab7ca65bf4 --- /dev/null +++ b/L9FJT4oBgHgl3EQfyi3U/content/tmp_files/2301.11639v1.pdf.txt @@ -0,0 +1,1224 @@ +A general-purpose machine learning Pt interatomic potential for an accurate +description of bulk, surfaces and nanoparticles +Jan Kloppenburg,1, 2, ∗ Livia B. Pártay,3 Hannes Jónsson,4, 5 and Miguel A. Caro1, 2, † +1Department of Electrical Engineering and Automation, Aalto University, 02150 Espoo, Finland +2Department of Chemistry and Materials Science, Aalto University, 02150 Espoo, Finland +3Department of Chemistry, University of Warwick, Coventry CV4 7AL, UK +4Faculty of Physical Sciences, University of Iceland, VR-III, 107 Reykjavík, Iceland +5Department of Applied Physics, Aalto University, FI-00076 Espoo, Finland +(Dated: January 30, 2023) +A Gaussian approximation machine learning interatomic potential for platinum is presented. It +has been trained on DFT data computed for bulk, surfaces and nanostructured platinum, in partic- +ular nanoparticles. Across the range of tested properties, which include bulk elasticity, surface en- +ergetics and nanoparticle stability, this potential shows excellent transferability and agreement with +DFT, providing state-of-the-art accuracy at low computational cost. We showcase the possibilities +for modeling of Pt systems enabled by this potential with two examples: the pressure-temperature +phase diagram of Pt calculated using nested sampling and a study of the spontaneous crystallization +of a large Pt nanoparticle based on classical dynamics simulations over several nanoseconds. +Keywords: machine learning, platinum potential, GAP, SOAP +I. +INTRODUCTION +Platinum belongs to the noble metal family and is of- +ten used in expensive jewelry. But the wider importance +of Pt for the global economy stems from its countless +industrial uses, even in elemental crystalline form. Plat- +inum is commonly used as a catalyst for many chemical +reactions. For instance, Pt is the best known catalyst +for the hydrogen evolution reaction (HER), where it +shows an extremely small overpotential [1, 2]. Pt is also +one of the few catalysts that can withstand the highly +oxidizing environments of the oxygen reduction reaction +(ORR) and oxygen evolution reaction (OER) [3, 4]. At +the same time, Pt is scarce in the Earth’s crust, and +its supply for industrial applications is severely limited +by cost and availability. Still, for some applications, the +use of Pt can be so advantageous compared to the next- +best option, that it remains in wide use. To reduce the +amount of raw Pt that is needed for a given application, +Pt thin films or nanoparticles (NPs) can be used instead +of the bulk material. The catalytic properties of Pt films +are strongly influenced by crystallographic surface ori- +entation [2]; for NPs, size and shape are the parameters +that determine these properties [5–7]. Understanding +the atomic-scale structure of such systems is, therefore, +critical for understanding the catalytic properties. +In this article, we introduce and validate a general- +purpose machine learning (ML)-based Gaussian ap- +proximation potential (GAP) [8, 9] for elemental +Pt. This potential offers similar accuracy as density- +∗ Correspondence email address: jan.kloppenburg@aalto.fi +† mcaroba@gmail.com +functional theory (DFT) for a small fraction of the com- +putational cost. Our potential shows extremely good +transferability, accurately predicting the interatomic in- +teractions in Pt from bulk to surface through NPs. The +paper is organized as follows. We first discuss the GAP +theoretical framework and the generation of training +data. +We then benchmark our potential against the +prediction of the basic material properties of bulk, sur- +face and NP platinum. +Finally, we use the GAP to +compute the pressure-temperature phase diagram of Pt +using the nested sampling (NS) method and to study +the nucleation of face-centered cubic (FCC) Pt during +the solidification of a large Pt NP. +II. +METHODS +A. +Gaussian approximation potentials +Gaussian approximation potentials use kernel-based +ML techniques to regress the potential energy surface +(PES) of an atomistic system. Provided atomic data is +available (typically energies, forces and virials), usually +computed at the DFT level of theory, a GAP can be +trained on that data, from which it learns. A GAP pre- +diction is made by comparing the atomic structure for +which we seek the prediction to a set of structures in the +database. Each of these comparisons yields a kernel, or +measure of similarity, which is bounded between 0 (the +two structures are nothing alike) and 1 (the structures +are identical). Different descriptors, and combinations +thereof, of the atomic structure can be used to describe +the atomic environments. +For instance, in this work +we use a combination of two-body (2b) and many-body +(mb) soap_turbo descriptors [10, 11]. The actual pre- +arXiv:2301.11639v1 [cond-mat.mtrl-sci] 27 Jan 2023 + +2 +diction for the local atomic energy of atom i is expressed +as: +¯ϵi =e0 + (δ(2b))2 � +s +α(2b) +s +k(2b)(i, s) ++ (δ(mb))2 � +s +α(mb) +s +k(mb)(i, s), +(1) +where k(i, s) is the kernel between the atomic environ- +ment of i and the different atomic environments s in +the sparse set (a subset of structures in the training +database), the αs are fitting coefficients obtained dur- +ing training, e0 is a constant energy per atom and δ +gives the energy scale of the model. Forces can be ob- +tained analytically from the derivatives of Eq. (1). More +details about the GAP framework and many-body de- +scriptors are given in Refs. [8–11]. +We generate training data at the DFT level using +the PBE functional approximation [12] and will denote +it as PBE-DFT from now on. +We use a highly con- +verged plane-wave basis set with a 520 eV cutoff and +an adaptive reciprocal-space integration mesh such that +the number of k points is given by Nk = 1000/Natoms. +The VASP software [13–15] is used with input options +given in the appendix. +The composition and gener- +ation of the database are discussed in Sec. II B. The +training and validation of the potential are done with +the QUIP/GAP codes [16]. +Structure manipulation +and database sorting are done with ASE [17]. Molec- +ular dynamics (MD) simulations are carried out using +LAMMPS [18, 19] and TurboGAP [20]. +B. +Database generation and accuracy tests +We want to create a robust Pt GAP that can be used +safely in exploratory work, e.g., to assess the stabil- +ity of previously unknown Pt NPs. +To this end, the +GAP needs to be simultaneously accurate and transfer- +able. Within a data-driven approach, it is important to +note that prior knowledge of physics and chemistry is +not embedded in the form of the potential. That is, a +GAP does not “know” about the Schrödinger equation +– it only knows about data it has seen during training. +Therefore, the training set must be carefully crafted to +contain all the relevant configurations. +This includes +(meta)stable structures, but also, perhaps counterintu- +itively, high-energy structures. High-energy structures +must be present in the database so that the GAP learns +that they are, in fact, of high energy, otherwise the +GAP could spuriously predict previously unseen unsta- +ble structures to be low in energy. +It is also useful to realize that high-energy observables +can be learned with less accuracy than low-energy ones, +because low-energy structures contribute much more to +the partition function of the system at the temperatures +of interest, and thus to the derived thermodynamic +properties. This leads to an efficient database construc- +tion strategy where a few disordered structures, such +as high-temperature liquid or dimers at close range, +are added to sample configuration space sparsely but +comprehensively. +Further to these, many configura- +tions close to the known stable structures, like close- +packed crystal lattices and surfaces thereof, are added +by “rattling” the atoms around equilibrium and apply- +ing small amounts of strain to the periodic cells. This, +in turn, begs the question: what about the unknown +stable structures? +To improve a GAP in a yet unknown region of con- +figuration space, a successful strategy is iterative train- +ing [21]. In iterative training one trains several versions +of the GAP, and each time uses the newest GAP to +predict stable structures. The energy values and atomic +forces for those structures are then computed with PBE- +DFT and fed to the next version of the GAP, which +will learn from its predecessor’s successes and, espe- +cially, failures. This iterative procedure progressively +refines the GAP’s accuracy in the region of configura- +tion space where the target structures (e.g., NPs) reside. +The advantage is that the computationally demanding +procedure, the structure generation, which might re- +quire thousands or millions of energy and force evalua- +tions, is performed with the GAP, inexpensively. The +PBE-DFT calculations are only carried out for the final +structures or, in some cases, a small subset of the struc- +tures selected along the path followed in configuration +space to generate the final ones. +Figure 1 shows the most commonly used numerical +benchmark for machine learning potentials (MLPs), i.e., +a scatterplot of predicted versus reference energies for a +20%/80% test/training sets split. That is, out of the +entire database of structures, 80% are used to train +the GAP and the 20% unseen structures are used to +test the potential outside the training set. +The root +mean-square error (RMSE) is computed to give a sin- +gle numerical score for the overall performance of the +potential. Our Pt GAP shows remarkably low errors in +this simple test, with an RMSE of only 2.2 meV/atom. +Application-specific tests of the GAP are presented in +the following section, more indicative of how this poten- +tial performs for large-scale and high-throughput simu- +lations. +In Fig. 1 we also show the predictions of an +embedded-atom method (EAM) potential [22, 23] on +our test set for comparison. While EAM can satisfacto- +rily reproduce the energetics of bulk FCC near equilib- +rium, all other structure types are modeled significantly +less accurately. +We will show in Sec. III C a further +comparison for NPs, where we will also include predic- +tions by the Gupta potential [24]. Clearly, the improved +accuracy of GAP comes at the expense of additional +CPU time. For instance, to perform a single-point cal- + +3 +Figure 1. Validation of the Pt GAP performed on atomic +configurations unseen during training. The different config- +uration types (FCC, HCP, etc.) are indicated with different +colors. The results of testing an EAM potential on the same +set of structures are given for reference. The GAP errors are: +maximum energy error: 0.073 eV/atom; energy RMSE: 2.2 +meV/atom; and force RMSE: 0.08 eV/Å. The EAM errors +are: maximum energy error: 3.092 eV/atom; energy RMSE: +439.4 meV/atom; and force RMSE: 1.196 eV/Å. +culation for a NP with 147 atoms, our GAP requires +approximately 109 ms of CPU time whereas an EAM +calculation only needs 1.2 ms. The GAP is still signif- +icantly faster than PBE-DFT (using VASP), for which +this calculation requires of the order of 102 CPU hours +(i.e., ∼ 3.5 × 106 times more expensive than the GAP). +C. +Nested sampling +We use the nested sampling (NS) technique [25, 26] +to evaluate the bulk macroscopic thermodynamic prop- +erties of the new Pt GAP model. NS samples the entire +potential energy surface, starting from high-energy ran- +dom configurations (representing the gas phase) down +to the ground-state structure through a series of nested +energy levels, without requiring any advance knowledge +of the stable phases [27, 28]. +A unique advantage of +NS is that it allows the calculation of the partition +function as a simple post-processing step. This gives +access to thermodynamic properties, such as the heat +capacity–which is the second derivative of the parti- +tion function with respect to temperature–and hence +enables us to identify all the phase transitions of the +system. In the current work we perform the NS calcu- +lations at constant pressure, to compute the pressure- +temperature phase diagram [29–32]. Simulations were +carried out using the pymatnest program package [33], +using LAMMPS to perform the dynamics. +−10 +−5 +0 +5 +10 +15 +20 +25 +30 +35 +5 +10 +15 +20 +25 +30 +−6.2 +−6 +−5.8 +−5.6 +−5.4 +−5.2 +−5 +12 +14 +16 +18 +20 +Potential energy (eV/atom) +Volume (Å3/atom) +FCC +HCP +BCC +SC +−6.2 +−6 +−5.8 +−5.6 +−5.4 +−5.2 +−5 +12 +14 +16 +18 +20 +Figure 2. Equation of state for three cubic and one hexago- +nal Pt crystal phases: face-centered cubic (FCC), hexagonal +closed-packed (HCP), body-centered cubic (BCC) and sim- +ple cubic (SC). The inset shows a closeup in the region where +the global minimum is located. +III. +BENCHMARKS +Our Pt GAP has been designed with the goal of gen- +eral applicability in mind. In this section we prove its +transferability across a selection of different applications +representative of common use cases. We test the GAP +for basic bulk properties (equation of state, elasticity +and phonons), surface energetics and NP formation en- +ergies. +While avoiding a too detailed examination of +each application, which could merit on their own more +focused studies, these examples showcase the predictive +power of the new GAP. In Sec. IV we describe two, +more detailed, applied studies: the phase diagram and +a spontaneous crystal nucleation in nanostructured Pt. +A. +Equation of state, elastic properties and +phonons +The equation-of-state calculation shows the expected +minimum for the FCC phase from zero up to very high +pressure, with HCP about 60 meV/atom above FCC +and body-centered cubic (BCC) slightly above HCP. +The simple cubic (SC) phase is significantly higher in +energy than FCC, HCP and BCC, except at large tensile +strain, i.e., at large (and unrealistic) negative pressures, +where it becomes the stable phase. All phases evolve +smoothly as a function of unit cell volume, as shown in +Fig. 2. +Our tests for Pt show that phonons and elastic con- +stants can be learned accurately when the training data +only contains structures created for this specific pur- +pose. However, the trained potential is then only able + +GAP +0 +-2 +4 +6 +EAM +0 +-2 +4 +.6 +-2 +9 +0 +DFT Energy (eV /atom)4 +Figure 3. +Phonon dispersion as computed by GAP and +PBE-DFT with phonopy [34]. +The PBE-DFT trends are +well reproduced in comparison to PBE-DFT except for a +systematic deviation at the W point. +Table I. Comparison of GAP-predicted elastic constants +with PBE-DFT as well as experimental values. +The per- +centage in brackets shows the deviation vs experiment for +PBE-DFT and the deviation vs PBE-DFT for GAP. +Exp. [35] +PBE-DFT +GAP +C11 (GPa) +373 +320 (−14%) +333 (+4%) +C12 (GPa) +242 +218 (−10%) +228 (+5%) +C44 (GPa) +78 +77 (−1%) +80 (+4%) +to describe those properties, and will not have general- +purpose applicability. +When different structures are +added to bring in more general-purpose applicability, +the high accuracy on both phonons and elastic con- +stants is sacrificed. Phonons (Fig. 3) are still described +reasonably well as compared to PBE-DFT results as +far as the main trends are concerned, except for a sys- +tematic deviation at the W point. Table I shows the +elastic constants computed with GAP and compared +to PBE-DFT, as well as to experiment. Overall, the +agreement with PBE-DFT (the GAP’s “ground truth”) +is good, with a systematic deviation of only +4%. This +deviation is smaller than the PBE-DFT error as com- +pared to experiments, highlighting how, in some cases, +the overall accuracy of the GAP is more limited by the +intrinsic accuracy of the reference method (PBE-DFT +in our case) than by the accuracy of the fit. That is, +for the specific purpose of calculating elastic constants, +the GAP is a better representation of PBE-DFT than +PBE-DFT is of reality. +B. +Surfaces +Platinum is a material widely used in interfacial (elec- +tro)catalysis, and thus it is important to ensure that an +interatomic potential for Pt can accurately reproduce +surface formation energies. +The three surfaces most +commonly studied are those defined by the (111), (100) +and (110) crystallographic FCC planes [2]. The (111) +surface is the most stable one and the one most often +used in electrocatalysis, e.g., for hydrogen production, +due to the low overpotential it exhibits for HER [1]. +A comprehensive study of surface energetics for arbi- +trary Miller indices (hkl) becomes prohibitive for DFT, +due to the large number of atoms in the unit cell for +large indices. For example, a 7-atom thick Pt slab with +(10 1 0) indices already contains 280 atoms in the prim- +itive unit cell. With our Pt GAP, studying these sur- +faces with small tilt angles becomes possible. We there- +fore calculated the surface formation energies, with bulk +FCC Pt as reference, for all the symmetry-inequivalent +Miller planes that can be constructed in Pt when letting +each index run up to 10. To ensure that reconstruction +effects beyond the primitive unit cell are considered, we +ran the calculation for the primitive unit cell generated +with ASE [17] as implemented by Buus, Howalt and +Bligaard, its Niggli equivalent cell [36–38], as well as +2×2 supercells built thereof. We included small random +initial displacements of the atoms to avoid biasing the +geometry optimization due to high-symmetry starting +configurations. Altogether, six calculations were carried +out for each set of Miller indices and the obtained sur- +face formation energies per atom were always the same +(except for negligible numerical differences). This in- +dicates that simple relaxation of the atomic positions +takes place as the surfaces are created and that the sur- +faces have the same periodicity as the primitive unit +cell. +Figure 4 (top panel) shows the surface formation +energies for varying Miller indices within the triangle +enclosed by the (111), (110) and (100) planes as end +points. +The values predicted for those planes, 0.091, +0.117 and 0.117 eV/Å2, respectively, are in good agree- +ment with our reference PBE-DFT values (0.096, 0.123 +and 0.120 eV/Å2, respectively) and with recent values +from the literature [39]. +The GAP predicts smooth +transitions as the cleaved (and relaxed) crystal facet +is tilted between the most common facets. The bottom +panel of the figure shows the surprising result that our +Pt GAP is more capable of reproducing the PBE-DFT +surface energies provided by a “high-quality” PBE-DFT +calculation (computed with the same VASP settings as +those reported in Sec. II) than another PBE-DFT cal- +culation with “standard” settings. The GAP tends to +slightly underestimate the surface formation energies +(by about 5%) but the trends, i.e., the relative forma- + +Frequency (THz) +4 +3 +DFT +GAP +K +L +W +X +U +q Point5 +0.09 +0.095 +0.1 +0.105 +0.11 +0.115 +0.12 +Surface energy (eV/Å2) +(1,0,0) +(1,1,0) +(1,1,1) +(2,1,1) +(3,1,1) +(4,1,1) +(3,2,2) +(5,4,4) +(2,2,1) +(3,3,1) +(4,4,1) +(3,3,2) +(5,5,4) +(2,1,0) +(3,1,0) +(6,1,0) +(3,2,0) (5,4,0) +0.06 +0.08 +0.1 +0.12 +0.14 +0.095 +0.1 +0.105 +0.11 +0.115 +0.12 +0.125 +Prediction (eV/Å2) +PBE-DFT surface energy @ 520 eV (eV/Å2) +Pt GAP +PBE-DFT @ 300 eV +Figure 4. (Top) Surface energies computed with the Pt GAP +for a range of crystal orientations resulting from tilting the +faces between the (111), (100) and (110) directions. Each +cross represents an actual data point, with selected Miller +indices indicated, and the color map is drawn by interpolat- +ing between those. (Bottom) Comparison between the Pt +GAP, a standard-quality PBE-DFT calculation (with VASP +defaults and a 300 eV plane-wave cutoff), and our reference +VASP PBE-DFT calculations (which use a larger cutoff of +520 eV). +tion energies, are accurately captured. +C. +GAP accuracy for nanoparticle modeling +We have generated a large database of Pt NPs for +this work. This database is divided into the two follow- +ing subsets, NP-DB01 and NP-DB02. NP-DB01 con- +tains 8000 NPs generated between Natoms = 10 and +Natoms = 349 using an annealing-quenching-relaxation +procedure, starting from a highly disordered precursor, +where the annealing and quenching steps take 20 ps +each and the annealing happens at 1500 K; we call +this a “cooking” protocol. NP-DB02 contains 3400 NPs +between Natoms = 10 and Natoms = 349 (10 for each +size) where the annealing step of the cooking protocol +takes place at the optimal crystallization temperature +of 1150 K (see Sec. IV B) but otherwise generated in the +same way as NP-DB01. This database is freely available +Figure 5. Formation energies for a selection of annealed NPs +computed with different potentials and standard-quality +PBE-DFT versus a benchmark-quality PBE-DFT calcula- +tion. +from the Zenodo repository [40] and will be extended in +subsequent work, in particular with larger NPs beyond +Natoms = 349. +To assess the ability of our GAP to accurately +model Pt NPs and to compare it to previously avail- +able, commonly used, force fields for Pt modeling, we +selected NPs from NP-DB01 up to Natoms = 150. +The energies were computed with our GAP, standard- +quality PBE-DFT (300 eV plane-wave cutoff), the +Gupta potential [24], and the EAM potential [22, +23]. We compare all these numbers to a benchmark- +quality PBE-DFT calculation (520 eV plane-wave cut- +off). The results of this comparison are shown in Fig. 5. +Clearly, the GAP outperforms the other force fields +with errors (∼ 40 meV/atom) one order of magni- +tude smaller than Gupta (∼ 400 meV/atom) and EAM +(∼ 500 meV/atom) around and above 50-atom NPs. +The GAP errors for these NPs are about 5 times larger +than those obtained from standard-quality PBE-DFT. +For very small NPs (< 50 atoms) the GAP results are +still better than for the other force fields but the errors +are significantly higher than for larger NPs. Since the +atomic motifs in small NPs look significantly different +from those of bulk and surfaces, it is not surprising that +the errors are larger. +The accuracy of GAP can be enhanced specifically +for NPs by iteratively training the potential for that +purpose. That is, we can improve the accuracy of the +GAP in the future by adding (some of) these NPs to +the training set and training a new version of the poten- +tial, as exemplified in Fig. 6. In that figure we observe +the performance of two versions of the Pt GAP. The +first one, GAPv1, is initially used to make two sets of +small NPs, with Natoms ≤ 50. One of the sets is used + +GAP - DFT +Gupta - DFT +EAM - DFT +10 +300eV - DFT +102 +08 +101 +25 +50 +75 +100 +125 +150 +Cluster size6 +−5.2 +−5 +−4.8 +−4.6 +−4.4 +−4.2 +−4 +−3.8 +−3.6 +−3.4 +−5.2 +−4.8 +−4.4 +−4 +10 ≤ 푁atoms ≤ 50 +Bigger NPs +−5.2 +−4.8 +−4.4 +−4 +GAP prediction (eV/atom) +DFT prediction (eV/atom) +NPs made with GAPv1 +NPs made with GAPv2 +GAPv1 +GAPv2 +Figure 6. +(Left) Predicted GAPv1 and GAPv2 energies +computed on NPs generated with GAPv1, versus the cor- +responding DFT values, for NPs in the size range from 10 +to 50 atoms. +High energy-per-atom values correspond to +smaller NPs whereas low values correspond to larger NPs. +(Right) Same test as on the left panel but performed on +NPs generated with GAPv2. The followed NP generation +method in both cases is the “cooking” protocol reported in +the text. +to retrain the GAP, giving GAPv2, and the second set +is used to test the predictions of both versions versus +PBE-DFT. The results are shown on Fig. 6 (left) were +GAPv1 is shown to predict too low (i.e., too stable) en- +ergies for the smallest NPs in the test set (Natoms ≲ 40) +whereas GAPv2 correctly orders all of the NPs gener- +ated with GAPv1. On the right-hand side of the figure +we show the energy predictions of GAPv1 and GAPv2 +for NPs that were generated with GAPv2. There are +two features of the GAP accuracy refinement provided +by iterative training which are apparent from this right- +hand panel. +On the one hand, as expected, GAPv2 +produces “better” NPs than GAPv1, in the sense that +they are lower in energy when looking at the PBE-DFT +energy prediction (i.e., the datapoints are shifted hori- +zontally to the left, compared to the left-hand panel), +and there is less data scatter. On the other hand, coun- +terintuitively, the GAPv1 predictions for these GAPv2- +generated NPs are in better agreement with PBE-DFT +than the GAPv2 predictions. While unexpected, this +is a typical result for early iterations in GAP itera- +tive training: a given iteration of the potential, used +in an application-specific simulation, will favor struc- +tures which populate artificially low regions of the PES. +As new iterations of the potential add these low-energy +structures to the database, the PES is refined and the +GAP “unlearns” the spurious minima and the scatter- +plot converges towards optimal agreement with DFT. +Generally, as new training configurations are gener- +ated, we can retrain and refine the accuracy of our GAP. +For reference, we provide in the repository [41] two ver- +sions of the GAP: the one used for most of the simu- +lations presented in this article (v1) and the one that +contains a small amount of NP-specific iterative train- +ing (v2). Any future version of the GAP will be added +to this repository together with a note on any further +additions to the database, compared to the configura- +tions reported here, with all published versions remain- +ing publicly available. This will ensure that the user +base of the potential has easy access to the most accu- +rate (and most recent) Pt GAP while enabling repro- +ducibility of the results produced with all earlier ver- +sions. Upcoming work from our group will focus on a +detailed study of small Pt NP formation and stability, +and we expect to update this repository with a NP- +optimized version of the GAP in the near future. +IV. +APPLICATIONS +A. +Pressure-temperature phase diagram +The NS calculations were performed as presented in +Ref [30]. The simulations were run at constant pressure +in the range of p = 0.07 − 50 GPa, using a simulation +cell of variable shape and size, containing 24 atoms. We +used 1000 walkers and performed 440 steps (8:1:2:2 ra- +tio of total-energy Hamiltonian Monte Carlo, volume, +cell shear and cell stretch steps) to generate the new +configurations during the NS iterations. These parame- +ters ensure convergence of the melting transition within +±40 K. The use of small systems will inevitably cause +some finite-size effects, for example an underestimation +of the boiling curve and an overestimation of the melt- +ing line as compared to the macroscopic value [29]. In +order to estimate this error, we repeated the simula- +tions with 48 atoms at p = 1 GPa, and obtained 2.8% +lower melting temperature as compared to the 24-atom +calculation. +Figure 7 shows the pressure-temperature phase di- +agram. +At low pressure, we observe a heat capacity +peak at high temperature corresponding to the boiling +curve and its extension to the supercritical region, the +Widom line, marked by a shallower and broader peak +(shown by dashed red line in Fig. 7). +To locate the +critical point in the NS calculations, we drew on the +results of Bruce and Wilding [42] and calculated the +density distribution in the temperature region of the +peak. With this, we estimate the critical parameters +to be pc = 0.1 − 0.2 GPa and Tc = 9500 − 10600 K. +The low-pressure melting transition is estimated to be +≈ 1650 K, hence underestimating the experimentally +determined transition. This inaccuracy could be either +due to our GAP or to an inherent error of the PBE +functional used to train it. Since NS calculations at the +PBE level are simply intractable, there is no straight- +forward way to pinpoint the origin of this disagreement +with experiment. The NS calculations found the solid + +7 + 1000 + 2000 + 3000 + 4000 + 5000 + 10 + 20 + 30 + 40 + 50 + 60 +Pressure (GPa) +Nested sampling +Errandonea (2013) +Kavner (1998) + 0 + 1000 + 2000 + 0 + 0.2 + 1 +Temperature (K) + 9000 + 10000 + 11000 + 0.1 + 0.3 +face-centered-cubic +liquid +face-centered-cubic +liquid +gas +Figure 7. +Pressure-temperature phase diagram calculated +by nested sampling (red lines and symbols). Error bars rep- +resent the full widths at half maximum of the heat capacity +curves. Green and blue symbols show experimental melting +temperatures taken from Refs. [43] and [44], respectively. +structure to be FCC as expected, and explored other +close-packed stacking variants only in thermodynami- +cally insignificant proportions. +B. +Spontaneous FCC nucleation and +crystallization +We also used the Pt GAP to study the spontaneous +nucleation of the stable FCC structure and the sponta- +neous formation of facets in a large NP (16384 atoms) +with MD. Figure 8 shows the sequence from the initial +cube carved out of an FCC lattice. This is melted at +3000 K for 40 ps and then the quenching process takes +place by cooling the NP from 3000 K down to 300 K +over 1 ns using a linear temperature profile, controlled +by a Berendsen thermostat with time constant 0.1 ps. +The figure also shows a slice through the middle of the +NP and, for reference, a periodic solid with the same +number of atoms and undergoing the same temperature +profile. For the solid, the pressure is controlled with a +Berendsen barostat with time constant 1 ps and inverse +compressibility equal to 100 times that of water. +To get further insight into the atomistic processes +taking place during crystallization, in Fig. 8 we map +the similarity of the local atomic structures to reference +atomic motifs: bulk FCC and the stable (100), (110) +and (111) FCC surface reconstructions. This is done +by computing the SOAP descriptors of each atom in +the system and calculating the similarity kernel with +the SOAP descriptors of the reference motifs. +These +similarities are indicated by color coding the resulting +structures. As expected, towards the end of the quench +the interior of the NP (as well as the solid) is FCC- +like, and the NP facets are (111)-like. +Interestingly, +the simulation shows that the formation of the FCC +interior is nucleated from the surfaces inwards. There- +Starting configurations +NP shell +NP core (slice) +fcc (slice) +T = 3000 K +T = 800 K +T = 300 K +Time +fcc +(100) +(110) +(111) +Figure 8. Snapshots throughout the process of spontaneous +crystallization from a melted Pt droplet as it cools down +to room temperature, as modeled with our GAP. The left +column shows the resulting NP from the outside, whereas +the central column shows a slice through the middle. The +same process for bulk Pt is shown on the right column. The +color coding indicates the degree of similarity, computed +from SOAP kernels, of each local atomic environment (cen- +tered on the atoms) to the stable bulk FCC motif, as well +as the three most common surface motifs: (100), (110) and +(111), where (111) is the most stable facet. The dark bands +between FCC (red) regions in the final structures correspond +to grain boundaries. +fore, there is grain formation with the (111) direction +pointing approximately from the surface towards the +center of the NP. For this reason, the resulting NP is +polycrystalline, with the grain boundaries indicated by +dark-colored atoms. It is clear from the figure that the +formation of the FCC interior in the NP happens at a +higher temperature than in the solid due to the nucle- +ation effect at the (111) facets. A video animation of +this process is available [45]. +To elucidate the role of quench rate on the results, we +monitored the evolution of the NP’s structure as it was + +8 +−5.84 +−5.8 +−5.76 +−5.72 +−5.68 +−5.64 +−5.6 +800 +900 +1000 +1100 +1200 +1300 +1400 +Potential energy (eV/atom) +Temperature (K) +Ann. +1 ns +2 ns +3 ns +4 ns +5 ns +6 ns +7 ns +8 ns +9 ns +10 ns +Figure 9. Potential energy profile as a function of temper- +ature in a series of melt-quench simulations, for different +cooling rates (1 ns to 10 ns cooling period). +The overall +process starts at 3000 K and ends at 300 K; the shown data +focuses on the region where crystallization takes place, cor- +responding to the formation of stable FCC motifs. The thin +gray line shows the profile of a simulation where the sam- +ple is quenched extremely fast from 3000 K to 1150 K and +annealed at that temperature before being brought down to +room temperature. See text for details. +cooled down from 3000 K to 300 K for additional quench +rates corresponding to 2 ns to 10 ns simulations, with +the same MD settings as before. Figure 9 shows the +evolution of the potential energy as a function of tem- +perature in the 1400 K to 800 K temperature window, +where most of the FCC nucleation takes place in these +simulations (outside of this range the potential energy +evolves linearly with temperature, as expected from the +virial theorem). According to our MD results, the on- +set of significant structure rearrangement favorable to- +wards FCC nucleation takes place at around 1200 K +and continues down to a temperature which depends +on the quench rate (the slower the rate the higher the +final temperature). From these values we infer an op- +timal crystallization temperature around 1150 K. This +is analogous to the graphitization temperature in car- +bon materials [46, 47]. We therefore repeated the MD +simulation starting from the 3000 K melted NP but fix- +ing the thermostat’s target temperature at 1150 K and +annealed for 1 ns (indicated as “Ann.” in the figure). +There is a rapid quench from 3000 K to 1150 K and then +the system equilibrates for a few ps, corresponding to +the loop seen at high potential energy, before it starts +to go down in energy as it crystallizes (the vertical drop +in potential energy at 1150 K). Most of the annealing +process was completed after 250 ps, with no noticeable +further drop in potential energy after 500 ps of MD. +After the 1 ns annealing simulation had ended, we fur- +ther quenched the structure to 300 K over 100 ps using +a linear temperature profile. The results showed good +agreement with the more computationally demanding +slow quenches. This annealing process at 1150 K thus +allows us to minimize the number of MD steps that are +required to generate a reasonably stable NP, generated +from a process mimicking spontaneous solidification. +V. +CONCLUSIONS AND OUTLOOK +We have developed a GAP for Pt with state-of-the-art +force-field accuracy for the description of bulk, surface +and nanostructured systems. We have benchmarked our +GAP against PBE-DFT for general accuracy, elasticity, +phonons, surface energetics and NP formation energies. +Except for small NPs (Natoms ≲ 40), our GAP shows re- +markable agreement with the reference PBE-DFT data. +We have then proceeded to use the GAP in situations +beyond the reach of PBE-DFT calculations. Namely, +we have computed the temperature-pressure phase di- +agram and studied the spontaneous solidification and +FCC-motif nucleation in a large NP. The new GAP and +several other resources have been made freely available. +In the near future, we will further develop our reference +database and the potential itself for improved descrip- +tion of NPs and surface dynamics, with the objective +to get detailed insight into the atomic-scale phenomena +taking place in Pt-based systems of interest in (elec- +tro)catalysis. +ACKNOWLEDGMENTS +J. K. and M. A. C. gratefully acknowledge funding +from the Academy of Finland under the C1 Value Pro- +gramme, project No. 329483. M. A. C. also acknowl- +edges personal funding from the Academy of Finland, +project No. 330488. H. J. acknowledges funding from +the Icelandic Research Fund, project No. 207283-053. +L. B. P. acknowledges support from the EPSRC through +an Early Career Fellowship (EP/T000163/1). Compu- +tational resources for this project were obtained from +CSC - IT Center for Science and Aalto University’s +Science-IT project. +Appendix A: VASP input file +The VASP INCAR input file used for the PBE-DFT +calculations is given below: +PREC = Accurate +ENCUT = 520 +EDIFF = 1.0e-05 +ISMEAR = 0; SIGMA = 0.1 +ALGO = Normal + +9 +LWAVE = .FALSE. +LCHARG = .FALSE. +The k-space sampling is not explicitly set in the INCAR +file. +Instead, k points are chosen by homogeneously +sampling the first Brillouin zone with the total number +of points determined by the relation natoms×nk = 1000. +To enable high-throughput calculations, the Fireworks +framework [48] was used for task automation and single- +point workflows, similar to the implementation in Ato- +mate [49], which rely on Custodian [50] as VASP han- +dler. +[1] J. Solla-Gullón, P. Rodríguez, E. Herrero, A. Aldaz, +and J. M. Feliu, “Surface characterization of platinum +electrodes,” Phys. Chem. Chem. Phys. 10, 1359 (2008). +[2] N. 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' † 1Department of Electrical Engineering and Automation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' Aalto University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' 02150 Espoo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' Finland 2Department of Chemistry and Materials Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' Aalto University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' 02150 Espoo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' Finland 3Department of Chemistry,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' University of Warwick,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' Coventry CV4 7AL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' UK 4Faculty of Physical Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' University of Iceland,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' VR-III,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' 107 Reykjavík,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' Iceland 5Department of Applied Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' Aalto University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' FI-00076 Espoo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' Finland (Dated: January 30,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' 2023) A Gaussian approximation machine learning interatomic potential for platinum is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' It has been trained on DFT data computed for bulk, surfaces and nanostructured platinum, in partic- ular nanoparticles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' Across the range of tested properties, which include bulk elasticity, surface en- ergetics and nanoparticle stability, this potential shows excellent transferability and agreement with DFT, providing state-of-the-art accuracy at low computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' We showcase the possibilities for modeling of Pt systems enabled by this potential with two examples: the pressure-temperature phase diagram of Pt calculated using nested sampling and a study of the spontaneous crystallization of a large Pt nanoparticle based on classical dynamics simulations over several nanoseconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' Keywords: machine learning, platinum potential, GAP, SOAP I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' INTRODUCTION Platinum belongs to the noble metal family and is of- ten used in expensive jewelry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' But the wider importance of Pt for the global economy stems from its countless industrial uses, even in elemental crystalline form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' Plat- inum is commonly used as a catalyst for many chemical reactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' For instance, Pt is the best known catalyst for the hydrogen evolution reaction (HER), where it shows an extremely small overpotential [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' Pt is also one of the few catalysts that can withstand the highly oxidizing environments of the oxygen reduction reaction (ORR) and oxygen evolution reaction (OER) [3, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' At the same time, Pt is scarce in the Earth’s crust, and its supply for industrial applications is severely limited by cost and availability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' Still, for some applications, the use of Pt can be so advantageous compared to the next- best option, that it remains in wide use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' To reduce the amount of raw Pt that is needed for a given application, Pt thin films or nanoparticles (NPs) can be used instead of the bulk material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' The catalytic properties of Pt films are strongly influenced by crystallographic surface ori- entation [2];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' for NPs, size and shape are the parameters that determine these properties [5–7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' Understanding the atomic-scale structure of such systems is, therefore, critical for understanding the catalytic properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' In this article, we introduce and validate a general- purpose machine learning (ML)-based Gaussian ap- proximation potential (GAP) [8, 9] for elemental Pt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' This potential offers similar accuracy as density- ∗ Correspondence email address: jan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content='kloppenburg@aalto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content='fi † mcaroba@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content='com functional theory (DFT) for a small fraction of the com- putational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' Our potential shows extremely good transferability, accurately predicting the interatomic in- teractions in Pt from bulk to surface through NPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' We first discuss the GAP theoretical framework and the generation of training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' We then benchmark our potential against the prediction of the basic material properties of bulk, sur- face and NP platinum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' Finally, we use the GAP to compute the pressure-temperature phase diagram of Pt using the nested sampling (NS) method and to study the nucleation of face-centered cubic (FCC) Pt during the solidification of a large Pt NP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' METHODS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' Gaussian approximation potentials Gaussian approximation potentials use kernel-based ML techniques to regress the potential energy surface (PES) of an atomistic system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' Provided atomic data is available (typically energies, forces and virials), usually computed at the DFT level of theory, a GAP can be trained on that data, from which it learns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' A GAP pre- diction is made by comparing the atomic structure for which we seek the prediction to a set of structures in the database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' Each of these comparisons yields a kernel, or measure of similarity, which is bounded between 0 (the two structures are nothing alike) and 1 (the structures are identical).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' Different descriptors, and combinations thereof, of the atomic structure can be used to describe the atomic environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' For instance, in this work we use a combination of two-body (2b) and many-body (mb) soap_turbo descriptors [10, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' The actual pre- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content='11639v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content='mtrl-sci] 27 Jan 2023 2 diction for the local atomic energy of atom i is expressed as: ¯ϵi =e0 + (δ(2b))2 � s α(2b) s k(2b)(i, s) + (δ(mb))2 � s α(mb) s k(mb)(i, s), (1) where k(i, s) is the kernel between the atomic environ- ment of i and the different atomic environments s in the sparse set (a subset of structures in the training database), the αs are fitting coefficients obtained dur- ing training, e0 is a constant energy per atom and δ gives the energy scale of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' Forces can be ob- tained analytically from the derivatives of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' More details about the GAP framework and many-body de- scriptors are given in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' [8–11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' We generate training data at the DFT level using the PBE functional approximation [12] and will denote it as PBE-DFT from now on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' We use a highly con- verged plane-wave basis set with a 520 eV cutoff and an adaptive reciprocal-space integration mesh such that the number of k points is given by Nk = 1000/Natoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' The VASP software [13–15] is used with input options given in the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' The composition and gener- ation of the database are discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' II B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' The training and validation of the potential are done with the QUIP/GAP codes [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' Structure manipulation and database sorting are done with ASE [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' Molec- ular dynamics (MD) simulations are carried out using LAMMPS [18, 19] and TurboGAP [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' Database generation and accuracy tests We want to create a robust Pt GAP that can be used safely in exploratory work, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=', to assess the stabil- ity of previously unknown Pt NPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' To this end, the GAP needs to be simultaneously accurate and transfer- able.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' Within a data-driven approach, it is important to note that prior knowledge of physics and chemistry is not embedded in the form of the potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' That is, a GAP does not “know” about the Schrödinger equation – it only knows about data it has seen during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' Therefore, the training set must be carefully crafted to contain all the relevant configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' This includes (meta)stable structures, but also, perhaps counterintu- itively, high-energy structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' High-energy structures must be present in the database so that the GAP learns that they are, in fact, of high energy, otherwise the GAP could spuriously predict previously unseen unsta- ble structures to be low in energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' It is also useful to realize that high-energy observables can be learned with less accuracy than low-energy ones, because low-energy structures contribute much more to the partition function of the system at the temperatures of interest, and thus to the derived thermodynamic properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' This leads to an efficient database construc- tion strategy where a few disordered structures, such as high-temperature liquid or dimers at close range, are added to sample configuration space sparsely but comprehensively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' Further to these, many configura- tions close to the known stable structures, like close- packed crystal lattices and surfaces thereof, are added by “rattling” the atoms around equilibrium and apply- ing small amounts of strain to the periodic cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' This, in turn, begs the question: what about the unknown stable structures?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' To improve a GAP in a yet unknown region of con- figuration space, a successful strategy is iterative train- ing [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' In iterative training one trains several versions of the GAP, and each time uses the newest GAP to predict stable structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' The energy values and atomic forces for those structures are then computed with PBE- DFT and fed to the next version of the GAP, which will learn from its predecessor’s successes and, espe- cially, failures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' This iterative procedure progressively refines the GAP’s accuracy in the region of configura- tion space where the target structures (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=', NPs) reside.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' The advantage is that the computationally demanding procedure, the structure generation, which might re- quire thousands or millions of energy and force evalua- tions, is performed with the GAP, inexpensively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' The PBE-DFT calculations are only carried out for the final structures or, in some cases, a small subset of the struc- tures selected along the path followed in configuration space to generate the final ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' Figure 1 shows the most commonly used numerical benchmark for machine learning potentials (MLPs), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=', a scatterplot of predicted versus reference energies for a 20%/80% test/training sets split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' That is, out of the entire database of structures, 80% are used to train the GAP and the 20% unseen structures are used to test the potential outside the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' The root mean-square error (RMSE) is computed to give a sin- gle numerical score for the overall performance of the potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' Our Pt GAP shows remarkably low errors in this simple test, with an RMSE of only 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content='2 meV/atom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' Application-specific tests of the GAP are presented in the following section, more indicative of how this poten- tial performs for large-scale and high-throughput simu- lations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' 1 we also show the predictions of an embedded-atom method (EAM) potential [22, 23] on our test set for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' While EAM can satisfacto- rily reproduce the energetics of bulk FCC near equilib- rium, all other structure types are modeled significantly less accurately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' We will show in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' III C a further comparison for NPs, where we will also include predic- tions by the Gupta potential [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' Clearly, the improved accuracy of GAP comes at the expense of additional CPU time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' For instance, to perform a single-point cal- 3 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' Validation of the Pt GAP performed on atomic configurations unseen during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' The different config- uration types (FCC, HCP, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=') are indicated with different colors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' The results of testing an EAM potential on the same set of structures are given for reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' The GAP errors are: maximum energy error: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content='073 eV/atom;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' energy RMSE: 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content='2 meV/atom;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' and force RMSE: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content='08 eV/Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' The EAM errors are: maximum energy error: 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content='092 eV/atom;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' energy RMSE: 439.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content='4 meV/atom;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' and force RMSE: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content='196 eV/Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' culation for a NP with 147 atoms, our GAP requires approximately 109 ms of CPU time whereas an EAM calculation only needs 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content='2 ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' The GAP is still signif- icantly faster than PBE-DFT (using VASP), for which this calculation requires of the order of 102 CPU hours (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=', ∼ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content='5 × 106 times more expensive than the GAP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' Nested sampling We use the nested sampling (NS) technique [25, 26] to evaluate the bulk macroscopic thermodynamic prop- erties of the new Pt GAP model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' NS samples the entire potential energy surface, starting from high-energy ran- dom configurations (representing the gas phase) down to the ground-state structure through a series of nested energy levels, without requiring any advance knowledge of the stable phases [27, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' A unique advantage of NS is that it allows the calculation of the partition function as a simple post-processing step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' This gives access to thermodynamic properties, such as the heat capacity–which is the second derivative of the parti- tion function with respect to temperature–and hence enables us to identify all the phase transitions of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' In the current work we perform the NS calcu- lations at constant pressure, to compute the pressure- temperature phase diagram [29–32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' Simulations were carried out using the pymatnest program package [33], using LAMMPS to perform the dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' −10 −5 0 5 10 15 20 25 30 35 5 10 15 20 25 30 −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content='2 −6 −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content='8 −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content='6 −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content='4 −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content='2 −5 12 14 16 18 20 Potential energy (eV/atom) Volume (Å3/atom) FCC HCP BCC SC −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content='2 −6 −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content='8 −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content='6 −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content='4 −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content='2 −5 12 14 16 18 20 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' Equation of state for three cubic and one hexago- nal Pt crystal phases: face-centered cubic (FCC), hexagonal closed-packed (HCP), body-centered cubic (BCC) and sim- ple cubic (SC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' The inset shows a closeup in the region where the global minimum is located.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' BENCHMARKS Our Pt GAP has been designed with the goal of gen- eral applicability in mind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' In this section we prove its transferability across a selection of different applications representative of common use cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' We test the GAP for basic bulk properties (equation of state, elasticity and phonons), surface energetics and NP formation en- ergies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' While avoiding a too detailed examination of each application, which could merit on their own more focused studies, these examples showcase the predictive power of the new GAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' IV we describe two, more detailed, applied studies: the phase diagram and a spontaneous crystal nucleation in nanostructured Pt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' Equation of state, elastic properties and phonons The equation-of-state calculation shows the expected minimum for the FCC phase from zero up to very high pressure, with HCP about 60 meV/atom above FCC and body-centered cubic (BCC) slightly above HCP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' The simple cubic (SC) phase is significantly higher in energy than FCC, HCP and BCC, except at large tensile strain, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=', at large (and unrealistic) negative pressures, where it becomes the stable phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' All phases evolve smoothly as a function of unit cell volume, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' Our tests for Pt show that phonons and elastic con- stants can be learned accurately when the training data only contains structures created for this specific pur- pose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' However, the trained potential is then only able GAP 0 2 4 6 EAM 0 2 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content='6 2 9 0 DFT Energy (eV /atom)4 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' Phonon dispersion as computed by GAP and PBE-DFT with phonopy [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' The PBE-DFT trends are well reproduced in comparison to PBE-DFT except for a systematic deviation at the W point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' Comparison of GAP-predicted elastic constants with PBE-DFT as well as experimental values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' The per- centage in brackets shows the deviation vs experiment for PBE-DFT and the deviation vs PBE-DFT for GAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' [35] PBE-DFT GAP C11 (GPa) 373 320 (−14%) 333 (+4%) C12 (GPa) 242 218 (−10%) 228 (+5%) C44 (GPa) 78 77 (−1%) 80 (+4%) to describe those properties, and will not have general- purpose applicability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' When different structures are added to bring in more general-purpose applicability, the high accuracy on both phonons and elastic con- stants is sacrificed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' Phonons (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' 3) are still described reasonably well as compared to PBE-DFT results as far as the main trends are concerned, except for a sys- tematic deviation at the W point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' Table I shows the elastic constants computed with GAP and compared to PBE-DFT, as well as to experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' Overall, the agreement with PBE-DFT (the GAP’s “ground truth”) is good, with a systematic deviation of only +4%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' This deviation is smaller than the PBE-DFT error as com- pared to experiments, highlighting how, in some cases, the overall accuracy of the GAP is more limited by the intrinsic accuracy of the reference method (PBE-DFT in our case) than by the accuracy of the fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' That is, for the specific purpose of calculating elastic constants, the GAP is a better representation of PBE-DFT than PBE-DFT is of reality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' Surfaces Platinum is a material widely used in interfacial (elec- tro)catalysis, and thus it is important to ensure that an interatomic potential for Pt can accurately reproduce surface formation energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' The three surfaces most commonly studied are those defined by the (111), (100) and (110) crystallographic FCC planes [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' The (111) surface is the most stable one and the one most often used in electrocatalysis, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=', for hydrogen production, due to the low overpotential it exhibits for HER [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' A comprehensive study of surface energetics for arbi- trary Miller indices (hkl) becomes prohibitive for DFT, due to the large number of atoms in the unit cell for large indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' For example, a 7-atom thick Pt slab with (10 1 0) indices already contains 280 atoms in the prim- itive unit cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' With our Pt GAP, studying these sur- faces with small tilt angles becomes possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' We there- fore calculated the surface formation energies, with bulk FCC Pt as reference, for all the symmetry-inequivalent Miller planes that can be constructed in Pt when letting each index run up to 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' To ensure that reconstruction effects beyond the primitive unit cell are considered, we ran the calculation for the primitive unit cell generated with ASE [17] as implemented by Buus, Howalt and Bligaard, its Niggli equivalent cell [36–38], as well as 2×2 supercells built thereof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' We included small random initial displacements of the atoms to avoid biasing the geometry optimization due to high-symmetry starting configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' Altogether, six calculations were carried out for each set of Miller indices and the obtained sur- face formation energies per atom were always the same (except for negligible numerical differences).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' This in- dicates that simple relaxation of the atomic positions takes place as the surfaces are created and that the sur- faces have the same periodicity as the primitive unit cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' Figure 4 (top panel) shows the surface formation energies for varying Miller indices within the triangle enclosed by the (111), (110) and (100) planes as end points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' The values predicted for those planes, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content='091, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content='117 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content='117 eV/Å2, respectively, are in good agree- ment with our reference PBE-DFT values (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content='096, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content='123 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content='120 eV/Å2, respectively) and with recent values from the literature [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' The GAP predicts smooth transitions as the cleaved (and relaxed) crystal facet is tilted between the most common facets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' The bottom panel of the figure shows the surprising result that our Pt GAP is more capable of reproducing the PBE-DFT surface energies provided by a “high-quality” PBE-DFT calculation (computed with the same VASP settings as those reported in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' II) than another PBE-DFT cal- culation with “standard” settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' The GAP tends to slightly underestimate the surface formation energies (by about 5%) but the trends, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=', the relative forma- Frequency (THz) 4 3 DFT GAP K L W X U q Point5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content='095 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content='105 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content='115 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content='12 Surface energy (eV/Å2) (1,0,0) (1,1,0) (1,1,1) (2,1,1) (3,1,1) (4,1,1) (3,2,2) (5,4,4) (2,2,1) (3,3,1) (4,4,1) (3,3,2) (5,5,4) (2,1,0) (3,1,0) (6,1,0) (3,2,0) (5,4,0) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content='095 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content='105 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content='115 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content='125 Prediction (eV/Å2) PBE-DFT surface energy @ 520 eV (eV/Å2) Pt GAP PBE-DFT @ 300 eV Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' (Top) Surface energies computed with the Pt GAP for a range of crystal orientations resulting from tilting the faces between the (111), (100) and (110) directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' Each cross represents an actual data point, with selected Miller indices indicated, and the color map is drawn by interpolat- ing between those.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' (Bottom) Comparison between the Pt GAP, a standard-quality PBE-DFT calculation (with VASP defaults and a 300 eV plane-wave cutoff), and our reference VASP PBE-DFT calculations (which use a larger cutoff of 520 eV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' tion energies, are accurately captured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' GAP accuracy for nanoparticle modeling We have generated a large database of Pt NPs for this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' This database is divided into the two follow- ing subsets, NP-DB01 and NP-DB02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' NP-DB01 con- tains 8000 NPs generated between Natoms = 10 and Natoms = 349 using an annealing-quenching-relaxation procedure, starting from a highly disordered precursor, where the annealing and quenching steps take 20 ps each and the annealing happens at 1500 K;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' we call this a “cooking” protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' NP-DB02 contains 3400 NPs between Natoms = 10 and Natoms = 349 (10 for each size) where the annealing step of the cooking protocol takes place at the optimal crystallization temperature of 1150 K (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' IV B) but otherwise generated in the same way as NP-DB01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' This database is freely available Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' Formation energies for a selection of annealed NPs computed with different potentials and standard-quality PBE-DFT versus a benchmark-quality PBE-DFT calcula- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' from the Zenodo repository [40] and will be extended in subsequent work, in particular with larger NPs beyond Natoms = 349.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' To assess the ability of our GAP to accurately model Pt NPs and to compare it to previously avail- able, commonly used, force fields for Pt modeling, we selected NPs from NP-DB01 up to Natoms = 150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' The energies were computed with our GAP, standard- quality PBE-DFT (300 eV plane-wave cutoff), the Gupta potential [24], and the EAM potential [22, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' We compare all these numbers to a benchmark- quality PBE-DFT calculation (520 eV plane-wave cut- off).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' The results of this comparison are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' Clearly, the GAP outperforms the other force fields with errors (∼ 40 meV/atom) one order of magni- tude smaller than Gupta (∼ 400 meV/atom) and EAM (∼ 500 meV/atom) around and above 50-atom NPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' The GAP errors for these NPs are about 5 times larger than those obtained from standard-quality PBE-DFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' For very small NPs (< 50 atoms) the GAP results are still better than for the other force fields but the errors are significantly higher than for larger NPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' Since the atomic motifs in small NPs look significantly different from those of bulk and surfaces, it is not surprising that the errors are larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' The accuracy of GAP can be enhanced specifically for NPs by iteratively training the potential for that purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' That is, we can improve the accuracy of the GAP in the future by adding (some of) these NPs to the training set and training a new version of the poten- tial, as exemplified in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' In that figure we observe the performance of two versions of the Pt GAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' The first one, GAPv1, is initially used to make two sets of small NPs, with Natoms ≤ 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' One of the sets is used GAP - DFT Gupta - DFT EAM - DFT 10 300eV - DFT 102 08 101 25 50 75 100 125 150 Cluster size6 −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content='2 −5 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content='8 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content='6 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content='4 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content='2 −4 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content='8 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content='6 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content='4 −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content='2 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content='8 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content='4 −4 10 ≤ 푁atoms ≤ 50 Bigger NPs −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content='2 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content='8 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content='4 −4 GAP prediction (eV/atom) DFT prediction (eV/atom) NPs made with GAPv1 NPs made with GAPv2 GAPv1 GAPv2 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' (Left) Predicted GAPv1 and GAPv2 energies computed on NPs generated with GAPv1, versus the cor- responding DFT values, for NPs in the size range from 10 to 50 atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' High energy-per-atom values correspond to smaller NPs whereas low values correspond to larger NPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' (Right) Same test as on the left panel but performed on NPs generated with GAPv2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' The followed NP generation method in both cases is the “cooking” protocol reported in the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' to retrain the GAP, giving GAPv2, and the second set is used to test the predictions of both versions versus PBE-DFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' The results are shown on Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' 6 (left) were GAPv1 is shown to predict too low (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=', too stable) en- ergies for the smallest NPs in the test set (Natoms ≲ 40) whereas GAPv2 correctly orders all of the NPs gener- ated with GAPv1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' On the right-hand side of the figure we show the energy predictions of GAPv1 and GAPv2 for NPs that were generated with GAPv2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' There are two features of the GAP accuracy refinement provided by iterative training which are apparent from this right- hand panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' On the one hand, as expected, GAPv2 produces “better” NPs than GAPv1, in the sense that they are lower in energy when looking at the PBE-DFT energy prediction (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=', the datapoints are shifted hori- zontally to the left, compared to the left-hand panel), and there is less data scatter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' On the other hand, coun- terintuitively, the GAPv1 predictions for these GAPv2- generated NPs are in better agreement with PBE-DFT than the GAPv2 predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' While unexpected, this is a typical result for early iterations in GAP itera- tive training: a given iteration of the potential, used in an application-specific simulation, will favor struc- tures which populate artificially low regions of the PES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' As new iterations of the potential add these low-energy structures to the database, the PES is refined and the GAP “unlearns” the spurious minima and the scatter- plot converges towards optimal agreement with DFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' Generally, as new training configurations are gener- ated, we can retrain and refine the accuracy of our GAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' For reference, we provide in the repository [41] two ver- sions of the GAP: the one used for most of the simu- lations presented in this article (v1) and the one that contains a small amount of NP-specific iterative train- ing (v2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' Any future version of the GAP will be added to this repository together with a note on any further additions to the database, compared to the configura- tions reported here, with all published versions remain- ing publicly available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' This will ensure that the user base of the potential has easy access to the most accu- rate (and most recent) Pt GAP while enabling repro- ducibility of the results produced with all earlier ver- sions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' Upcoming work from our group will focus on a detailed study of small Pt NP formation and stability, and we expect to update this repository with a NP- optimized version of the GAP in the near future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' APPLICATIONS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' Pressure-temperature phase diagram The NS calculations were performed as presented in Ref [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' The simulations were run at constant pressure in the range of p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content='07 − 50 GPa, using a simulation cell of variable shape and size, containing 24 atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' We used 1000 walkers and performed 440 steps (8:1:2:2 ra- tio of total-energy Hamiltonian Monte Carlo, volume, cell shear and cell stretch steps) to generate the new configurations during the NS iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' These parame- ters ensure convergence of the melting transition within ±40 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' The use of small systems will inevitably cause some finite-size effects, for example an underestimation of the boiling curve and an overestimation of the melt- ing line as compared to the macroscopic value [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' In order to estimate this error, we repeated the simula- tions with 48 atoms at p = 1 GPa, and obtained 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content='8% lower melting temperature as compared to the 24-atom calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' Figure 7 shows the pressure-temperature phase di- agram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' At low pressure, we observe a heat capacity peak at high temperature corresponding to the boiling curve and its extension to the supercritical region, the Widom line, marked by a shallower and broader peak (shown by dashed red line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' To locate the critical point in the NS calculations, we drew on the results of Bruce and Wilding [42] and calculated the density distribution in the temperature region of the peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' With this, we estimate the critical parameters to be pc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content='1 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content='2 GPa and Tc = 9500 − 10600 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' The low-pressure melting transition is estimated to be ≈ 1650 K, hence underestimating the experimentally determined transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' This inaccuracy could be either due to our GAP or to an inherent error of the PBE functional used to train it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' Since NS calculations at the PBE level are simply intractable, there is no straight- forward way to pinpoint the origin of this disagreement with experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' The NS calculations found the solid 7 1000 2000 3000 4000 5000 10 20 30 40 50 60 Pressure (GPa) Nested sampling Errandonea (2013) Kavner (1998) 0 1000 2000 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content='2 1 Temperature (K) 9000 10000 11000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content='3 face-centered-cubic liquid face-centered-cubic liquid gas Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' Pressure-temperature phase diagram calculated by nested sampling (red lines and symbols).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' Error bars rep- resent the full widths at half maximum of the heat capacity curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' Green and blue symbols show experimental melting temperatures taken from Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' [43] and [44], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' structure to be FCC as expected, and explored other close-packed stacking variants only in thermodynami- cally insignificant proportions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' Spontaneous FCC nucleation and crystallization We also used the Pt GAP to study the spontaneous nucleation of the stable FCC structure and the sponta- neous formation of facets in a large NP (16384 atoms) with MD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' Figure 8 shows the sequence from the initial cube carved out of an FCC lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' This is melted at 3000 K for 40 ps and then the quenching process takes place by cooling the NP from 3000 K down to 300 K over 1 ns using a linear temperature profile, controlled by a Berendsen thermostat with time constant 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content='1 ps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' The figure also shows a slice through the middle of the NP and, for reference, a periodic solid with the same number of atoms and undergoing the same temperature profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' For the solid, the pressure is controlled with a Berendsen barostat with time constant 1 ps and inverse compressibility equal to 100 times that of water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' To get further insight into the atomistic processes taking place during crystallization, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' 8 we map the similarity of the local atomic structures to reference atomic motifs: bulk FCC and the stable (100), (110) and (111) FCC surface reconstructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' This is done by computing the SOAP descriptors of each atom in the system and calculating the similarity kernel with the SOAP descriptors of the reference motifs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' These similarities are indicated by color coding the resulting structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' As expected, towards the end of the quench the interior of the NP (as well as the solid) is FCC- like, and the NP facets are (111)-like.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' Interestingly, the simulation shows that the formation of the FCC interior is nucleated from the surfaces inwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' There- Starting configurations NP shell NP core (slice) fcc (slice) T = 3000 K T = 800 K T = 300 K Time fcc (100) (110) (111) Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' Snapshots throughout the process of spontaneous crystallization from a melted Pt droplet as it cools down to room temperature, as modeled with our GAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' The left column shows the resulting NP from the outside, whereas the central column shows a slice through the middle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' The same process for bulk Pt is shown on the right column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' The color coding indicates the degree of similarity, computed from SOAP kernels, of each local atomic environment (cen- tered on the atoms) to the stable bulk FCC motif, as well as the three most common surface motifs: (100), (110) and (111), where (111) is the most stable facet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' The dark bands between FCC (red) regions in the final structures correspond to grain boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' fore, there is grain formation with the (111) direction pointing approximately from the surface towards the center of the NP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' For this reason, the resulting NP is polycrystalline, with the grain boundaries indicated by dark-colored atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' It is clear from the figure that the formation of the FCC interior in the NP happens at a higher temperature than in the solid due to the nucle- ation effect at the (111) facets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' A video animation of this process is available [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' To elucidate the role of quench rate on the results, we monitored the evolution of the NP’s structure as it was 8 −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content='84 −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content='8 −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content='76 −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content='72 −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content='68 −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content='64 −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content='6 800 900 1000 1100 1200 1300 1400 Potential energy (eV/atom) Temperature (K) Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' 1 ns 2 ns 3 ns 4 ns 5 ns 6 ns 7 ns 8 ns 9 ns 10 ns Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' Potential energy profile as a function of temper- ature in a series of melt-quench simulations, for different cooling rates (1 ns to 10 ns cooling period).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' The overall process starts at 3000 K and ends at 300 K;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' the shown data focuses on the region where crystallization takes place, cor- responding to the formation of stable FCC motifs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' The thin gray line shows the profile of a simulation where the sam- ple is quenched extremely fast from 3000 K to 1150 K and annealed at that temperature before being brought down to room temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' See text for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' cooled down from 3000 K to 300 K for additional quench rates corresponding to 2 ns to 10 ns simulations, with the same MD settings as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' Figure 9 shows the evolution of the potential energy as a function of tem- perature in the 1400 K to 800 K temperature window, where most of the FCC nucleation takes place in these simulations (outside of this range the potential energy evolves linearly with temperature, as expected from the virial theorem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' According to our MD results, the on- set of significant structure rearrangement favorable to- wards FCC nucleation takes place at around 1200 K and continues down to a temperature which depends on the quench rate (the slower the rate the higher the final temperature).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' From these values we infer an op- timal crystallization temperature around 1150 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' This is analogous to the graphitization temperature in car- bon materials [46, 47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' We therefore repeated the MD simulation starting from the 3000 K melted NP but fix- ing the thermostat’s target temperature at 1150 K and annealed for 1 ns (indicated as “Ann.” in the figure).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' There is a rapid quench from 3000 K to 1150 K and then the system equilibrates for a few ps, corresponding to the loop seen at high potential energy, before it starts to go down in energy as it crystallizes (the vertical drop in potential energy at 1150 K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' Most of the annealing process was completed after 250 ps, with no noticeable further drop in potential energy after 500 ps of MD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' After the 1 ns annealing simulation had ended, we fur- ther quenched the structure to 300 K over 100 ps using a linear temperature profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' The results showed good agreement with the more computationally demanding slow quenches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' This annealing process at 1150 K thus allows us to minimize the number of MD steps that are required to generate a reasonably stable NP, generated from a process mimicking spontaneous solidification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' CONCLUSIONS AND OUTLOOK We have developed a GAP for Pt with state-of-the-art force-field accuracy for the description of bulk, surface and nanostructured systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' We have benchmarked our GAP against PBE-DFT for general accuracy, elasticity, phonons, surface energetics and NP formation energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' Except for small NPs (Natoms ≲ 40), our GAP shows re- markable agreement with the reference PBE-DFT data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' We have then proceeded to use the GAP in situations beyond the reach of PBE-DFT calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' Namely, we have computed the temperature-pressure phase di- agram and studied the spontaneous solidification and FCC-motif nucleation in a large NP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' The new GAP and several other resources have been made freely available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' In the near future, we will further develop our reference database and the potential itself for improved descrip- tion of NPs and surface dynamics, with the objective to get detailed insight into the atomic-scale phenomena taking place in Pt-based systems of interest in (elec- tro)catalysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' ACKNOWLEDGMENTS J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' gratefully acknowledge funding from the Academy of Finland under the C1 Value Pro- gramme, project No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' 329483.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' also acknowl- edges personal funding from the Academy of Finland, project No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' 330488.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' acknowledges funding from the Icelandic Research Fund, project No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' 207283-053.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' acknowledges support from the EPSRC through an Early Career Fellowship (EP/T000163/1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' Compu- tational resources for this project were obtained from CSC - IT Center for Science and Aalto University’s Science-IT project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' Appendix A: VASP input file The VASP INCAR input file used for the PBE-DFT calculations is given below: PREC = Accurate ENCUT = 520 EDIFF = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content='0e-05 ISMEAR = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' SIGMA = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content='1 ALGO = Normal 9 LWAVE = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content='FALSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' LCHARG = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content='FALSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' The k-space sampling is not explicitly set in the INCAR file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' Instead, k points are chosen by homogeneously sampling the first Brillouin zone with the total number of points determined by the relation natoms×nk = 1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FJT4oBgHgl3EQfyi3U/content/2301.11639v1.pdf'} +page_content=' To enable high-throughput calculations, the Fireworks framework [48] was 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a/M9E0T4oBgHgl3EQfjQFP/content/tmp_files/2301.02456v1.pdf.txt b/M9E0T4oBgHgl3EQfjQFP/content/tmp_files/2301.02456v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..615414dcd455e3beb677be783f736a097e1f6725 --- /dev/null +++ b/M9E0T4oBgHgl3EQfjQFP/content/tmp_files/2301.02456v1.pdf.txt @@ -0,0 +1,1773 @@ +Relative asymptotic oscillations of the out-of-time-ordered correlator as a quantum +chaos indicator +Jakub Novotn´y1, ∗ and Pavel Str´ansk´y1, † +1Institute of Particle and Nuclear Physics, Faculty of Mathematics and Physics, +Charles University, V Holeˇsoviˇck´ach 2, 18000 Prague, Czech Republic +(Dated: January 9, 2023) +A detailed numerical study reveals that the asymptotic values of the standard deviation-to-mean +ratio of the out-of-time-ordered correlator can be successfully used as a measure of the quantum +chaoticity of the system. We employ a finite-size fully connected quantum system with two degrees +of freedom, namely the algebraic u(3) model, and demonstrate a clear correspondence between the +relative oscillations of the correlators and the ratio of the chaotic part of the volume of phase space +in the classical limit of the system. We also show how the relative oscillations scale with the system +size and conjecture that the scaling exponent can also serve as a robust chaos indicator. +I. +INTRODUCTION +The correspondence between the theory of classical +and quantum chaos has been extensively studied since +the formulation of the Bohigas-Giannoni-Schmit conjec- +ture [1]. While classical chaos is rigorously constructed +mathematically and routinely studied, most often by the +rate of exponential divergence of neighboring trajectories, +the study of quantum chaos is more intriguing. Quan- +tum chaoticity is usually defined indirectly by a compar- +ison of suitable properties of a quantum system—most +often correlations in energy spectra—with the chaotic- +ity of its classical counterpart. In particular scenarios, +the theory of classical-quantum correspondence is well- +established [2, 3]. +In this paper, we move from the static description of +the quantum chaos based on spectral properties to the +dynamical manifestations. We will employ the Out-of- +Time-Ordered Correlators (OTOCs)—four-point corre- +lation functions of two quantum operators taken at dif- +ferent times. +The OTOCs have already gained popularity as an in- +dicator of quantum chaos, especially their short-time +evolution due to their connection with classical insta- +bility. They were introduced long ago as a semiclassi- +cal tool to study superconductivity [4] and later dusted +off by showing their relevance in black-hole physics and +chaos [5, 6]. In the case of a quantum system with the +classical limit, the short-time behavior of the OTOCs +mimics the exponential spreading of neighboring clas- +sical trajectories up to the Ehrenfest time, leading to +the notion of quantum Lyapunov exponent [7–19] and +quantum butterfly effect [5, 18, 20–23]. In quantum sys- +tems with local interactions and local OTOC operators, +the initial time evolution of the OTOCs describes the +spreading (or scrambling) of quantum information [24– +30]. Based on the rate of initial growth of OTOCs, inde- +pendently of the existence of the classical limit, quantum +∗ novotny.jakub@ipnp.mff.cuni.cz +† stransky@ipnp.troja.mff.cuni.cz +systems can be placed into a category of slow scramblers +(maximally polynomial initial growth even in the pres- +ence of chaos) [10, 31–33], or fast scramblers (exponen- +tial scrambling) [8, 10, 12, 20]. The black holes have been +conjectured as the fastest scramblers [5], employing the +AdS/CFT duality [34]. OTOC analysis is being applied +for studying many-body quantum systems, spin systems, +quantum circuits [30]. In recent years, this vast area for +application gave rise to many proposals and measuring +protocols [24, 35–41] followed by experimental measure- +ments [30, 42, 43]. +OTOCs have also been applied to +study many-body quantum scars [17, 44] and quantum +thermalization [26, 45], and as a probe of the excited- +state quantum phase transitions [46]. +Whereas there is a vast amount of literature about +the study of the short-time OTOC evolution, a possible +connection of its long-time properties to chaos has been +hinted at in a few recent works. In the case of finite- +size many-body quantum systems, OTOCs saturate in +the long-time regime and exhibit oscillations around the +mean value. +In Ref. [8], it was suggested for the first +time that the appearance of saturation oscillations could +be linked to quantum chaos. Later, a numerical study +of a quantum Harper map demonstrated that the sup- +pression of the OTOC oscillations in infinite tempera- +ture limit quantitatively corresponds to the ratio of the +chaotic volume of the whole phase space [10]. The same +authors have shown [47] that the suppression of relative +oscillations of infinite-temperature OTOC can reveal the +transition to chaos even in a small chain of four spins +with OTOC studied on time scales attainable by ex- +periment [48]. The relative oscillations of OTOCs as a +qualitative chaos indicator have also been mentioned in +other studies [49, 50]. Some studies propose that even +the asymptotic OTOC mean value can serve as a chaos +indicator [17]. +The short-time exponential behavior of OTOCs is not +a bulletproof sign of quantum chaoticity. Unstable po- +tential stationary points can cause exponential growth +of OTOCs even in integrable systems [15, 45, 51–53], +hence misleadingly indicating chaos. +And vice versa, +the already mentioned slow scramblers show polynomial +arXiv:2301.02456v1 [quant-ph] 6 Jan 2023 + +2 +growth of OTOC even in chaotic regimes [10]. In this +context, one can ask to what extent the relative oscilla- +tions overcome the drawbacks of the short-time OTOCs. +Another point can be whether they allow not only for in- +dicating chaos but also for measuring it, and how well +they compare to other known chaos measures. +Some +clues are already present in the literature. Authors in +Ref. [45] noticed that the size of oscillations of OTOCs in +the long-time regime in the simple Bose-Hubbard dimer +could qualitatively distinguish nonchaotic, mixed, and +fully chaotic regions, while the OTOCs show exponen- +tial growth even in the fully regular regime of the model. +And in the already mentioned work [47], authors show +that chaos suppresses relative oscillations of OTOCs even +in the slowly scrambling spin chains. +In this paper, we will be interested mainly in the long- +time behavior of OTOCs, although we will also briefly +discuss their short-time evolution and compare it with +classical stability. +We will work with the eigenenergy +expectation values of the OTOCs, which will allow us +to analyze the energy dependence of relative oscillations +and link them to other well-established energy-dependent +measures of chaoticity. Specifically, we will show that the +ratio of the asymptotic standard deviation to the asymp- +totic mean value (called wiggliness for brevity) compares +to the volume of the regular part of the phase space in +the classical limit. +We will provide a detailed numerical study in a bosonic +algebraic quantum model based on the u(3) algebra. This +model has been originally introduced to describe bending +modes of linear polyatomic molecules [54–57]. Later it +has been successfully used to investigate purely theoreti- +cal concepts, such as quantum monodromy and quantum +critical effects [58–60]. Recently it has also been applied +to study the properties of spinor Bose-Einstein conde- +sates [39, 61–63], which are experimentally achievable, for +example, as a condensate of cold rubidium atoms [64, 65]. +Some aspects of the OTOCs and their relation to chaos +have already been studied in this model [39]. +We have chosen the u(3) model because its Hilbert +space is finite. The model has just 2 degrees of freedom +and a known classical limit. On top of that, it is nonin- +tegrable with enough parameters to tune the chaoticity. +Due to the algebraic origin, all observables can be easily +constructed from the operators of the u(3) algebra. And +finally, one can vary the number of boson excitations in +the system and study how the asymptotic OTOC behav- +ior changes with the system size. +This paper is organized as follows. In Sec. II we recall +some necessary concepts of the theory of classical chaos; +we introduce the OTOCs and the wiggliness, which is the +primary tool of the present work, and briefly discuss the +correspondence between the classical and quantum chaos +indicators. +Sec. III introduces the u(3) model and its +classical limit. The numerical results and their discussion +is given in Sec. IV. We finally conclude in Sec. V. +II. +THEORY +In this section, we introduce the classical Lyapunov +exponent and the fraction of the regular part of the en- +ergy hypersurface in the phase space. Then we present +energy-dependent microcanonical OTOCs. +Finally, we +identify short-time and long-time evolution in the OTOC +dynamics and specify to which classical notion they can +be compared. +A. +Classical dynamics +We consider a classical system with f degrees of free- +dom described by time-independent Hamiltonian Hcl(x), +which is a function of coordinates and conjugated mo- +menta x ≡ (x1, . . . , x2f) = (p1, . . . , pf, q1, . . . , qf) on a +phase space. The time-independence of the Hamiltonian +implies energy conservation. Therefore, classical trajec- +tories are constrained to (2f − 1)-dimensional hypersur- +faces ΣE determined by constant energy Hcl = E. +One of the prominent signs of classical chaos is the ex- +ponential separation of asymptotically close trajectories +characterized by the positivity of the Lyapunov expo- +nent [66, 67] +λcl = lim +t→∞ lim +|δx|→0 +1 +t log |(x + δx)(t) − x(t)| +|δx| +, +(1) +where x(t) and (x + δx)(t) are trajectories with initial +conditions x0 and x0+δx, respectively, δx is an infinites- +imally small deviation vector in the neighbourhood of the +point x0 on the given energy hypersurface, and |•| is a +norm in the phase space. In the case of regular (stable) +trajectories λcl = 0. +The Lyapunov exponent explains the local properties +of the dynamics. On the other hand, the overall chaotic- +ity of the system with energy E is reflected in the fraction +of regularity +freg(E) = Γreg(E) +Γ(E) +∈ [0, 1], +(2) +where Γ(E) is the entire volume of the energy hyper- +surface ΣE and Γreg(E) is the volume of all the regu- +lar regions in ΣE, which, in general, are well-separated +from the regions of chaotic dynamics. The limit values +correspond to fully chaotic (freg = 0) and fully regular +(freg = 1) dynamics at the given energy E. +B. +Quantum dynamics +Quantum chaos is often studied from static properties +of the spectral correlations [3]. Here we focus on the dy- +namical manifestations, namely on the properties of the +OTOC. In a quantum system described by Hamiltonian + +3 +ˆH, the out-of-time-ordered correlator is introduced as the +expectation value in energy eigenstates |En⟩ +Cn(t) = +� +En +���[ ˆV (t), ˆW(0)]†[ ˆV (t), ˆW(0)] +���En +� +, +(3) +where En is the n-th eigenenergy, ˆH |En⟩ = En |En⟩, and +ˆV , ˆW are for the moment arbitrary quantum operators +in the Heisenberg picture, +ˆV (t) = e +i +ℏ ˆ +Ht ˆV e− i +ℏ ˆ +Ht; +(4) +[ ˆV (t), ˆW(0)] is their commutator. If both operators are +Hermitian, then +Cn(t) = − +� +En +���� +� +ˆV (t), ˆW(0) +�2����En +� +. +(5) +The OTOC properties undoubtedly depend on the choice +of the operators. These are usually relevant physical ob- +servables of the system, such as positions, momenta oper- +ators, or local spin operators in models with finite-range +interaction. The variability of the OTOC dynamics for +several pairs of operators will be demonstrated later in +Sec. IV. +Note that the OTOCs are often considered as ther- +mal averages in the canonical ensemble at inverse tem- +perature β [6, 20], sometimes even at infinite tempera- +ture β = 0 only [8, 10]. For the sake of distinguishing +pure state and thermal state averaging, the OTOCs of +type (3) are called the microcanonical OTOCs in the lit- +erature [15, 68]. +C. +Classical-quantum correspondence +The time dependence of a general OTOC can be di- +vided into short-time and long-time regimes, roughly +limited by the Ehrenfest (or scrambling) time τE ∝ +λ−1 +cl ln N [10, 19, 69, 70], where N is the size parame- +ter of the system. In more detailed analyses, there has +also been described a universal power-law growth at very +short times td < tE [14] (td is called the dephasing time) +for OTOC operators commuting at t = 0, and yet an- +other time scale given by the diffusion time tD ≫ tE, +which is the time of complete saturation of the quantum +dynamics [71]. +Whereas the OTOCs in eigenstates corresponding to +regular dynamics typically show strong oscillations at +all times, OTOCs for chaotic eigenstates are character- +ized by initial exponential growth for t < τE and long- +time saturation regime with small aperiodic fluctuations +around the mean value for t > τD. In the chaotic sys- +tems with a classical limit, the exponent λ of the initial +exponential growth +C(t) ∝ e2λt +(6) +is related to the classical Lyapunov exponent λcl. +Es- +pecially, for operators ˆxi that correspond to the canon- +ical coordinates and momenta xi on the classical phase +space the exponent λ coincides with the Lyapunov expo- +nent [4, 12, 68], +[ˆxi(t), ˆxj(0)]2 ↔ {xi(t), xj(0)}2 ∼ e2λclt, +(7) +where {•, •} are the classical Poisson brackets; the indices +i, j = 1, . . . , 2f can be taken arbitrarily. +In general, however, the equality λ = λcl is not valid +for all choices of OTOC operators. It can be shown that +the Poisson brackets {A(t), A(0)} of a general function A +on the phase space have a Lyapunov term that mimics +the exponential separation of infinitesimally close chaotic +trajectories and which is projected onto the gradient of A. +Therefore, this gradient and other terms containing the +second derivative of A may modify the exact value of λ, +and the corresponding OTOC will not grow exactly with +the classical Lyapunov exponent. +The long-time behavior of OTOCs for a given pair of +operators ˆV , ˆW is captured by the mean value and vari- +ance +Cn = lim +T →∞ +1 +T +� T +0 +Cn(t)dt, +(8) +σ2 +n = lim +T →∞ +� T +0 +C2 +n(t)dt − C +2 +n. +(9) +It will be shown later that the crucial role in quantifying +chaoticity is played not by these quantities themselves +but by their ratio (the coefficient of variation) +νn = σn +Cn +, +(10) +reflecting the relative oscillations of the OTOC. In the +following text, we will, for the sake of brevity, call this +ratio wiggliness, and we will always specify for which op- +erators the OTOCs and the wiggliness are computed. +III. +MODEL +The model Hamiltonian belongs to a class of boson- +interacting systems. It is constructed from nine gener- +ators of the spectrum-generating u(3) Lie algebra [54]. +The generators can be represented by the bilinear prod- +ucts of creation and annihilation operators ˆb† +iˆbj, i, j = +0, 1, 2, where ˆb0 ≡ ˆσ is the scalar boson operator and +ˆb1,2 ≡ ˆτ1,2 form the pair of circular boson operators +ˆτ± = (ˆτ1 ± iˆτ2) / +√ +2, all of them satisfying the boson +commutation relations +� +ˆbi,ˆb† +j +� += δij, +� +ˆbi,ˆbj +� += +� +ˆb† +i,ˆb† +j +� += 0. +(11) +The Hilbert space is spanned over all single-particle +states, +|nσ, n+, n−⟩ ≡ 1 +N +� +ˆσ†�nσ � +ˆτ † ++ +�n+ � +ˆτ † +− +�n− +|0⟩ +(12) + +4 +where nσ, n± = 0, 1, 2, . . . are the numbers of the corre- +sponding boson excitations, |0⟩ is the vacuum state and +N a normalizing factor. +Note that any operator con- +structed from the generators conserves the total number +of boson excitations ˆN = ˆσ†ˆσ+ˆτ † ++ˆτ++ˆτ † +−ˆτ−. The number +N = nσ + nτ, where nτ = n+ + n−, specifies the fully- +symmetric irreducible representation of u(3) and dictates +the dimensionality of the corresponding (finite) Hilbert +space H, +dim H = 1 +2(N + 1)(N + 2). +(13) +Therefore, N serves as a tunable size parameter of the +system. +It is convenient to construct another set of generators +as linear combinations of ˆb† +iˆbj, +ˆnτ = ˆτ † ++ˆτ+ + ˆτ † +−ˆτ−, +ˆns = ˆσ†ˆσ, +ˆD± = ± +√ +2 +� +ˆτ † +±ˆσ − ˆσ†ˆτ∓ +� +, +ˆQ± = +√ +2ˆτ † +±ˆτ∓, +(14) +ˆR± = +√ +2 +� +ˆτ † +±ˆσ + ˆσ†ˆτ∓ +� +, +ˆl = ˆτ † ++ˆτ+ − ˆτ † +−ˆτ−, +where the following subsets are generators of subalgebras +o(3) = span +� +ˆD±, ˆl +� +, +(15a) +o(3) = span +� +ˆR±, ˆl +� +, +(15b) +u(2) = span +� +ˆnτ, ˆl, ˆQ± +� +, +(15c) +o(2) = span +� +ˆl +� +(15d) +that admit three subalgebra chains, +I :u(3) ⊃ u(2) ⊃ o(2), +(16a) +II :u(3) ⊃ o(3) ⊃ o(2), +(16b) +II :u(3) ⊃ o(3) ⊃ o(2). +(16c) +The o(2) is the symmetry algebra reflecting the invari- +ance of the system with respect to the rotation around +the z-axis. The last two chains are equivalent and in- +terchangeable, connected via a unitary transformation. +Depending on the choice of algebra o(3) or o(3), it is +talked about momentum or coordinate realization of the +same Hamiltonian [56]. +A common way to construct the o(3) Hamiltonian is +via the Casimir operators of all the possible subalgebras. +Here we consider only the linear Casimir ˆC1[u(2)] of the +u(2) algebra and the quadratic Casimir ˆC2[o(3)] of the +o(3) algebra, and build the Hamiltonian in the following +form +ˆH0 = 1 − ξ +N +ˆC1[u(2)] +� �� � +ˆnτ +− +ξ +N(N − 1) +ˆC2[o(3)] +� �� � +ˆ +D2 +, +(17) +where +ˆD2 ≡ 1 +2 +� +ˆD+ ˆD− + ˆD− ˆD+ +� ++ ˆl2 +(18) +and ξ ∈ [0, 1] is an adjustable parameter. The factor N +is added so that the Hamiltonian scales correctly in the +infinite-size limit N → ∞. +The system given by ˆH0 has f = 2 degrees of freedom +and is integrable with Hamiltonian and angular momen- +tum operator ˆl2 being the two independent integrals of +motion. This Hamiltonian is frequently used to model +a quantum phase transition between the o(2) (usually +called symmetric) phase and the o(3) (called deformed +or displaced) phase, occurring at ξ = 1/5 in the infinite- +size limit [58]. +The integrability of ˆH0 will be broken by violating the +o(2) symmetry, +ˆH = ˆH0 − ϵ +N +ˆDx, +(19) +where ˆDx = ( ˆD+ + ˆD−)/2 is the dipole operator [ ˆDy +would be defined as ˆDy = ( ˆD+ + ˆD−)/2i]. Note that +ˆH still has a discrete symmetry (parity), so its spectrum +forms two invariant sets of states in the Hilbert space. +A suitable basis for distinguishing these two invariant +subspaces is given by vectors +|N, nτ, l±⟩ ≡ +1 +√ +2 (|N, nτ, l⟩ ± |N, nτ, −l⟩) , +(20) +where l is taken as non-negative. Two subspaces H(1,2) +invariant under the action of the Hamiltonian (19) have +the form +H(1) = span{|N, nτ, l+⟩l-even, |N, nτ, l−⟩l-odd}, +(21a) +H(2) = span{|N, nτ, l+⟩l-odd, |N, nτ, l−⟩l-even}, +(21b) +where nτ = {0, 1, . . . N} and l = {0, 1, . . . , nτ}. The full +Hilbert space is H = H(1) ⊕ H(2). +The perturbed model is integrable for particular val- +ues of ξ = 0 and 1, which correspond to excluding one +of the Casimir operators in (17). Instead of ˆl, other op- +erators play the role of the second integral of motion. +For ξ = 0, the Hamiltonian commutes with the operator +ˆn + ˆQ where ˆQ = +� +ˆQ+ + ˆQ− +� +/ +√ +2, and for ξ = 1, the +Hamiltonian is formed only by operators from algebra +o(3) and commutes with both ˆD2 and ˆDx. +The classical limit of the Hamiltonian can be obtained +by transforming the circular boson operators into the +coordinate-momentum form +ˆbj = +� +N +2 (ˆqj + iˆpj) , +j = 1, 2, +(22) +using the Holstein-Primakoff transformation [72, 73] to +eliminate the degree of freedom connected with the σ +boson due to the conservation of N, and performing the +limit N → ∞, after which the operators with commuta- +tion relations +[ˆqj, ˆpk] = i +N δjk +(23) + +5 +serve as canonically conjugated coordinates and mo- +menta. +The corresponding classical Hamiltonian func- +tion reads as +Hcl ≡ lim +N→∞ +ˆH = (1 − ξ) s2 +2 +− ξ +�� +p2 +1 + p2 +2 +� � +2 − s2� ++ (p1q2 − q1p2)2� +− ϵp2 +� +2 − s2. +(24) +The four-dimensional phase space (p1, p2, q1, q2) is com- +pact, bounded by the condition s2 ≤ 2, where s2 = +p2 +1 + p2 +2 + q2 +1 + q2 +2. +Note that the commutation relations (23) imply that +the size parameter N also serves as an effective Planck +constant ℏeff = 1/N. +Therefore, the infinite-size limit +N → ∞ coincides with the classical limit ℏeff → 0 in this +model. +In the classical limit, the operators ˆDx,y and ˆRx,y be- +have approximately as a momentum and a position co- +ordinate in the phase space [55, 56] as they are mapped +to +1 +N +ˆDx,y +N→∞ +−−−−→ p2,1 +� +2 − s2, +(25a) +1 +N +ˆRx,y +N→∞ +−−−−→ q1,2 +� +2 − s2. +(25b) +Due to the above-mentioned connection of the o(3) and +o(3) operators via a unitary transformation, the spec- +trum of the Hamiltonian is the same no matter whether +constructed of the ˆDx,y or ˆRx,y operators. +IV. +RESULTS +In this section, we begin by illustrating the manifes- +tation of classical chaos in the classical limit of the u(3) +model, and we briefly explain numerical techniques used +to calculate the Lyapunov exponent λcl and the fraction +of regularity freg. Then we fit the short-time evolution of +the OTOC with the exponential function (6) and confirm +that its exponential rate λ corresponds with the classical +Lyapunov exponent for various OTOC operators. +Fi- +nally, we focus on the main topic of this paper, which is +the OTOC fluctuations at long times. +A. +Classical chaos +Except for some particular values of the parameters +described in Sec. III, the model described by Hamilto- +nian (24) is nonintegrable, thus exhibiting chaotic dy- +namics. In Fig. 1, we show the classical fraction of reg- +ularity freg for a fixed value of external perturbation +ϵ = 0.4 and for ξ ∈ [0, 1]. The chaotic regions (freg < 1) +start to arise at ξ ≈ 0.1 and lie approximately on and +above the energy E = 0. +The chaoticity is demon- +strated by Poincar´e sections plotted in panels (a) – (i). +Each section point corresponds to one trajectory cross- +ing through the chosen section plane q1 = 0; the color of +the points indicates the size of the Lyapunov exponent. +Stable trajectories form regular structures in the shape +of deformed lines or circles, whereas chaotic areas are +filled with seemingly randomly located points. In gen- +eral, these 2D Poincar´e sections visualize the 3D energy +manifolds of constant energy embedded in the 4D phase +space. The white areas in panels (b) and (c) are proper +topological holes in the energy hypersurface and depict +the intriguing shape of the sections. +FIG. 1. +Classical fraction of regularity freg(ξ, E) (top panel) +and selected examples of Poincar´e sections by q1 = 0 plane +[panels (a)—(i)] for the classical limit of the u(3) model (24) +with nonintegrable perturbation ϵ = 0.4. The colors of the +section points indicate the values of the Lyapunov exponent +λcl for the corresponding trajectories. Values freg = 0 and +freg = 1 correspond to fully chaotic (all trajectories on the +energy hypersurface ΣE have a positive Lyapunov exponent) +and fully regular (all trajectories have λcl = 0) dynamics, +respectively. + +1.0 +C +0.5 - +(b) +(e) +E 0.0 +(h) +-0.5 +(i) +-1.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +freg +a +b +c +0 +(e) +d +(f) +q2 +0 +1 +(g) +(h) +(i) +1 +0 +0 +-1 +1 +-1 +0 +1 +-1 +0 +1 +p2 +0.0 +0.04 +0.08 +入 +0.12 +0.16 +0.26 +0 +100 +200 +0.0 +0.5 +1.0 +1.5 +2.0 +C(t) +¯C +0.4 × 109 +0.8 × 109 t +0.0 +0.5 +1.0 +1.5 +2.0 +E = −0.54 +E = 0.21 +FIG. 2. +An example of the short-time and long-time OTOC +dynamics for operators [ ˆDx(t), ˆDx(0)]2 in a state from a fully +regular (n = 1, thin blue curve) and chaotic (n = 650, thick +orange curve) part of the quantum energy spectrum of the +u(3) model, respectively, as a function of time. +Model pa- +rameters are ξ = 0.4, ϵ = 0.4, system size is N = 50. OTOC +values are divided by the corresponding mean value C for bet- +ter comparison; C(E1 = −0.54) ≈ 101 and C(E650 = 0.21) ≈ +105. The approximate Ehrenfest time tE for the chaotic state +is indicated by the vertical black dashed line. +The values of the Lyapunov exponents were obtained +by evolving the deviation δx of two neighboring trajec- +tories for a sufficiently long time using the equation of +tangent dynamics [15, 66, 67]. +As the freg(E), we ef- +fectively take the ratio of the regular area (area in the +Poincar´e section crossed by trajectories with λcl = 0) to +the total section area intersected by any of the orbits. We +choose the initial conditions so that the given Poincar´e +section divided into a sufficiently dense mesh of cells is +entirely covered with evolved trajectories. Our detailed +numerical analysis shows that the values of freg obtained +from the Poincar´e sections differ from the freg computed +directly from (2) (i.e. as a ratio of the volumes of the +3D hypersurfaces) by no more than ≈ 5%, which is a fair +price to pay for saving plenty of the computation time. +Note that the same procedure of calculating the freg was +used and discussed in another model [74]. +B. +Short-time OTOCs +As explained in Sec. II C, the short-time behavior of +the OTOCs generally reflects the classical divergence of +neighboring trajectories and can serve to define a quan- +tum analog of the classical Lyapunov exponent. In Fig. 2 +we show the typical time series of the OTOCs for quan- +tum states exhibiting both regular and chaotic dynam- +ics. Regularity is characterized by wide oscillations, with +the OTOC values frequently returning close to the initial +value C(0). In contrast, the chaotic ones initially show +a fast (exponential) increase followed by saturation and +low oscillations around the mean value, usually far from +the initial C(0). +The tight connection between the short-time behavior +of OTOCs and classical chaos is demonstrated in Fig. 3. +We plot there the exponents λ obtained from the fit of +the initial exponential growth of OTOCs (6) for various +−0.6 +−0.4 +−0.2 +0.0 +0.2 +0.4 +0.6 +E +0.05 +0.10 +0.15 +0.20 +0.25 +λ +[ˆl(t), ˆl(0)]2 +[ˆnτ(t), ˆD2(0)]2 +[ ˆDx(t), ˆDx(0)]2 +λcl +FIG. 3. +Instability parameters λ from the exponential fit +of the short-time growth of OTOCs for three choices of the +OTOC operators indicated in the legend, compared with the +classical Lyapunov exponent λcl. +Parameters of the u(3) +model are ξ = 0.4, ϵ = 0.4. System size is N = 50. +choices of the OTOC operators. We can observe that the +values of λ depend on the choice of operators. However, +the overall energy dependence for all OTOC operators +qualitatively agrees with the classical Lyapunov expo- +nent λcl(E). The best agreement between the classical +and quantum instability parameters is achieved for the +OTOC operators [ ˆDx(t), ˆDx]2. As stated before, opera- +tor ˆDx(t) can be linked, in the classical limit (25), to one +of the canonical coordinates on the phase space, leading +to the correspondence between λ and λcl given by Eq. (7). +C. +Long-time OTOCs +Here we focus on the main analysis of this paper, which +is the properties of OTOC wiggliness. A brief glimpse at +Fig. 2 reveals the difference between the relative OTOC +oscillations in the regular and chaotic state at higher +times. +The OTOC function evaluated for the regular +state with energy E1 forms a wide “strip” covering the +graph from C ≈ 0 to some finite value C = Cmax, re- +sulting in a relatively high value of the wiggliness. +In +contrast, in the chaotic state after the initial exponential +explosion, the strip is narrower with minor oscillations +around the saturation value, and the wiggliness is, there- +fore, smaller. +The numerical analysis shows that the suppression of +the wiggliness in energy regions containing chaotic states +is correlated with a decrease in the fraction of regular- +ity freg. +This correlation is depicted in Fig. 4, which +plots the relative OTOC oscillations computed for three +choices of OTOC operators and three values of param- +eter ξ, and compares the wiggliness to the fraction of +regularity freg . +We distinguish three types of wiggliness behavior based +on the chaoticity of the system: + +7 +0.0 +0.2 +0.4 +0.6 +(a) +(b) +(c) +0.0 +0.2 +0.4 +0.6 +(d) +(e) +(f) +ν +λcl +1 +2freg +ν +−0.25 +0.00 +0.25 +0.50 +0.75 +0.0 +0.2 +0.4 +0.6 +(g) +−0.50 +−0.25 +0.00 +0.25 +0.50 +(h) +−1.0 +−0.5 +0.0 +(i) +E +[ ˆDx(t), ˆDx(0)]2 +[ˆl(t), ˆl(0)]2 +[ ˆD2(t), ˆnτ(0)]2 +ξ = 0.8 +ξ = 0.4 +ξ = 0.2 +FIG. 4. +Wiggliness for three distinct pairs of the OTOC operators calculated for all eigenstates |En⟩ along the spectrum +(black dots), the classical fraction of regularity freg (thick red curve) and the classical Lyapunov exponent (thin blue curve). In +addition, the wiggliness is smoothed using the moving average in energy (thick yellow curve labeled as ν) for comparison with +freg. The OTOC operators are indicated at the vertical axes. Parameters of the u(3) model are ξ = {0.2, 0.4, 0.8}, ϵ = 0.4. +System size is N = 100. Note that the Poincar´e sections in Fig. 1 were computed for the same model parameters and can serve +as an additional illustration of the regular versus chaotic dynamics. +• Regular regions (freg = 1) +In the parts of the spectra that correspond to com- +pletely regular dynamics, the values νn are typi- +cally higher in comparison with other regions. They +may form regular structures as νn reflects the ex- +istence of an additional constant of motion, see +Fig. 4 (c),(h),(i). +The shape of these structures +naturally depends on the choice of OTOC opera- +tors. +• Chaotic regions (freg = 0) +For quantum states that lie in completely chaotic +energy intervals, the values νn are generally smaller +compared to other regions. In addition, the wig- +gliness values corresponding to neighboring eigen- +states are much more similar. This can be observed +in Fig. 4 for ξ = 0.4 (second column) at E ≈ 0.2 and +ξ = 0.8 (third column) at E ≈ 0.1, as all the points +in these chaotic regions are “pressed” together. +• Mixed spectrum (freg > 0) +In the partially chaotic energy intervals of the spec- +tra, we have intertwined sets of states with proper- +ties of both regular and chaotic regions. Therefore, +they are characterized by a relatively wide spread +of wiggliness values but without any characteristic +regular pattern. +Remarkable could be the sensi- +tivity of νn to regular islands. Consider the region +0.25 > E > 0.5 in the second column of Fig. 4. The +system is almost entirely chaotic at this energy in- +terval in the classical limit. In the corresponding +Poincar´e section shown Fig. 1(b), we can hardly +notice tiny regular islands immersed in the chaotic +sea. Yet there is a significant amount of eigenstates +with high OTOC wiggliness, reflecting these flecks +of classical regularity. +The principal observation is that the smoothed values +ν(E) of the wiggliness (calculated as a moving average +over a small number of successive energy levels) follow the +trend of freg. It leads us to conjecture that the smoothed +asymptotic relative oscillations of the OTOCs reflect the +chaoticity of the quantum system. + +8 +FIG. 5. Smoothed wiggliness ν along the spectra of the u(3) +model (19) for ξ ∈ [0, 1], ϵ = 0.4 and two pairs of OTOC +operators. +The smoothing window entails five neighboring +levels. +The system size is N = 60, corresponding to 1891 +energy levels. +Roman numbers I and II label two regions +with values of ν for OTOC operator [ ˆDx(t), ˆDx(0)]2. +The structures present for the wiggliness in the regular +regions of energy can be understood within the theory of +Peres lattices [75, 76]; these lattices are formed from en- +ergy expectation values On = ⟨En| ˆO|En⟩ of an operator +ˆO as a set of points (En, On). +In nonchaotic parts of +the energy spectrum, the observable ˆO can be expressed +as a smooth function of all the independent constants of +motion, which leads to regular patterns in the Peres lat- +tice. Chaos arises with the loss of integrals of motion, +which leads to a collapse of the regular Peres lattice. In +the fully chaotic regions, all the points are “pressed” to- +gether with minor differences in the On expectation val- +ues. The differences further diminish with the increase +in the system size. +This behavior can be understood +via the Eigenvector Thermalization Hypothesis [77–79]. +Values Cn and σn given by Eqs. (8) and (9) are, in fact, +expectation values of rather complicated time-averaged +operators provided by the OTOC commutators, and the +wiggliness is a ratio of these two; hence it is expected +that the wiggliness retains some of the properties of the +Peres lattices. +Fig. 5 displays the smoothed wiggliness for two choices +of OTOC operators [ ˆDx(t), ˆDx(0)]2 and [ˆl(t), ˆl(0)]2. We +can directly compare these results with the freg plot- +ted in Fig. 1 and observe that the chaotic regions with +freg ≈ 0 and regions with reduced ν(E) overlap. In the +case of operator ˆDx, we find regions of small smoothed +wiggliness in the regular area of the spectrum (marked I +and II in the figure), spuriously labeling them as chaotic. +These regions of suppressed wiggliness can also be seen +in Fig. 4(i). Low values of ν(E) in region I are related to +the fact that ξ = 0.8 is very close to the value ξ = 1 for +which ˆDx commutes with the Hamiltonian. Region II is +a finite-size effect that diminishes with increasing N. +So far, we have discussed the wiggliness only, but one +0 +1 +2 +3 +4 +¯C/dim H +Regular +0.00 +0.05 +0.10 +0.15 +0.20 +Chaotic +0 +1 +2 +3 +σ/dim H +0.00 +0.01 +0.02 +0.03 +0.04 +−0.2 +0.0 +0.2 +0.4 +0.6 +E +0.0 +0.2 +0.4 +0.6 +0.8 +ν +Smoothed values +1 +2freg +−0.5 +0.0 +0.5 +E +0.0 +0.2 +0.4 +0.6 +0.8 +FIG. 6. Comparison of mean values Cn (first row), standard +deviations σn (second row) and the wiggliness νn of OTOC +operators [ ˆDx(t), ˆDx(0)]2 (third row) in an integrable case +ϵ = 0.0 (left column) and chaotic case ϵ = 0.4 (right column) +of the u(3) model with ξ = 0.4 and system size N = 100. +Values of Cn and σn are divided by the Hilbert space di- +mension dim H = 5151. Yellow lines correspond to smoothed +values using the moving average. The red line is the fraction +regularity freg, scaled by 1/2 for better visualization. +could ask whether just the mean value or variance (or +the standard deviation as the square root of the variance) +would sufficiently indicate and quantify the presence of +chaos. Based on the extensive numerical evidence, the +answer is no. This is illustrated in Fig. 6 where we com- +pare Cn, σn and νn of [ ˆDx(t), ˆDx(0)]2 OTOCs in an in- +tegrable and chaotic regime of the u(3) model. As men- +tioned above, the sets of points (En, Cn) and (En, σn) +form a kind of Peres lattices. Therefore we can infer the +presence of chaos in energy regions where the lattice is +disordered and where the points are vertically “pressed” +together. +In the chaotic regime, the mean value and +the standard deviation are higher in the chaotic regions +around E ≈ 0.25 than in the regular regions close to the +lower and upper energy limits of the spectrum. However, +their values are more than an order of magnitude lower +than in the integrable case ϵ = 0. +Therefore we can- +not simply determine the chaoticity of a general system +based on the concrete values of Cn and σn, even after +smoothening over several neighboring states. It is their +ratio in the form of relative oscillations that reveals the + +[Da(t), Dα(0)]2 +[(t), i(0)]2 +1.0 +0.5 +0.0 +E +-0.5 +II +-1.0 +0 +0.25 +0.5 +0.75 +10 +0.25 +0.5 +0.75 +1 +CS +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +D9 +−0.5 +0.0 +0.5 +E +0.1 +0.2 +0.3 +0.4 +0.5 +¯ν +(a) +N = 10 +N = 20 +N = 44 +N = 80 +N = 120 +N = 160 +1 +2 +3 +4 +5 +log(N) +−2.5 +−2.0 +−1.5 +−1.0 +−0.5 +log(¯ν) +(b) +E = −0.3 +E = −0.2 +E = −0.1 +E = 0.0 +E = 0.1 +E = 0.2 +−0.5 +0.0 +0.5 +E +−0.4 +−0.2 +0.0 +(c) +α +−0.5 +0.0 +0.5 +E +−0.5 +0.0 +0.5 +1.0 +1.5 +(d) +β +FIG. 7. Dependence of the wiggliness on the size N of the +quantum system. Figures are calculated for the u(3) model +in the chaotic regime with ξ = 0.4 and ϵ = 0.4. +As the +OTOC operators we choose [ ˆDx(t), ˆDx(0)]2. +Panel (a) dis- +plays smoothed wiggliness as a function of energy along the +spectrum for several system sizes from N = 10 to N = 160, +as indicated in the legend. Panel (b) shows the dependence of +the smoothed wiggliness on N for several energy values in the +log-log scale (points) and their fit by function (26) (dashed +lines). +degree of chaoticity and that corresponds to the classical +freg, as demonstrated in Figs 4, 5 and 6(c). +D. +Effect of the system size +In this section, we discuss the effect of the system size +on the relative oscillations νn. We take the total number +of boson excitations N of the u(3) model as a parameter +and study the dependence of the smoothed wiggliness +ν(N) both on the energy E and on the size parameter N. +Fig. 7(a) shows an example of the smoothed wiggliness +for one choice of the pairs of OTOC operators for the +whole accessible energy range and for several values of +N. The observed overall behavior is the following: In the +regions of the spectrum with chaotic or mixed dynamics +(freg < 1), ν decreases with increasing size of the system, +whereas in the regular regions (freg = 1) the smoothed +wiggliness remains approximately constant (usually with +a value ≈ 0.6) or slowly increases with N. +This observation is further confirmed by Fig. 7(b), +which shows the dependence ν(N) for six different values +of energy taken from regions of both regular and chaotic +dynamics. The linear dependence in the log-log plot im- +plies algebraic scaling of ν with N, +ν(E, N) = N α(E)e−β(E), +(26) +where the dependence of ν on the energy manifests only +via the exponents α(E) and β(E). These exponents, ob- +tained by the fit as demonstrated in Fig. 7(b), are plotted +in Figs. 7(c) and (d). Since the spectrum of the system is +discrete, we interpolated the values of ν to get precisely +into the desired energy value for the fit. +In Figs. 7(c) and (d), one can immediately notice the +apparent qualitative resemblance of α(E), β(E), and the +wiggliness itself, see Figs. 4(b) and 7(a). The cause of +the similarity between α and β can be deduced from +Fig .7(b): All the fitted lines intersect in a small region +at log(N0) ≈ 2.5, so there exists a linear transformation +between α and β taken at two arbitrarily chosen energies +E and E′, +log(N0) = β(E) − β(E′) +α(E) − α(E′). +(27) +Since the lines do not intersect precisely at one point +(and there is no known reason that they should), this +transformation is only approximate. Nevertheless, even +the approximate relation leads to similar shapes of curves +α(E) and β(E). +More important is the similarity of ν(E) and the scal- +ing exponent α(E) shown in Fig. 7(a) and (c). +That +implies that the suppression rate of the wiggliness scales +with the system size; the more chaotic the system is, the +faster the suppression of ν. Note that this behavior has +already been observed in non-collective spin systems [49]. +The exponent α can, therefore, also serve as a chaos in- +dicator, and because it does not depend on the size N +of the system, it can be considered more robust than the +wiggliness itself. Once α and β are known, they allow for +calculating the wiggliness for any system size N. +The last observation can be summarized in the fol- +lowing way: In the energy regions of the spectrum with +chaotic or mixed dynamics (freg < 1), the suppression +rate is negative, α < 0, whereas in the regular regions +(freg = 1) there is no suppression of the wiggliness or a +light enhancement, α ≳ 0. +We further demonstrate the validity of the scaling con- +jecture for three totally different models, ranging from in- +tegrable to fully chaotic. Fig. 8 shows the scaling of the +smoothed wiggliness for a u(3) model with mixed dynam- +ics (panel a), integrable, hence fully regular u(3) model +(panel b) and a fully chaotic system whose Hamiltonian +is modeled as a random GOE matrix [3] (panel c). Again, +we observe that the wiggliness is algebraically suppressed +in the chaotic regions of spectra and approximately con- +stant in the regular parts with increasing system size N. +Corresponding exponents α and β are shown in Fig. 9. +Regular parts of spectra have almost zero suppression +rate α, while regions with mixed or purely chaotic dy- +namics have negative α, which also mimics the shape of +freg(E) and reflects the level of chaoticity. As mentioned +above, the exponent β behaves in a similar way and could +also serve as an indicator of chaos. +Note that the scaling behavior described in this section +is also observed for other choices of OTOC operators and + +10 +−1.2 +−1.0 +−0.8 +−0.6 +−0.4 +−0.2 +0.0 +0.2 +E +0.3 +0.4 +0.5 +0.6 +¯ν +(a) +N = 9 +N = 18 +N = 27 +N = 36 +N = 60 +N = 100 +−0.2 +0.0 +0.2 +0.4 +0.6 +E +0.2 +0.3 +0.4 +0.5 +0.6 +¯ν +(b) +N = 10 +N = 20 +N = 30 +N = 40 +N = 60 +N = 80 +−1.0 +−0.5 +0.0 +0.5 +1.0 +E +0.05 +0.10 +0.15 +0.20 +¯ν +(c) +N = 20 +N = 30 +N = 40 +N = 50 +N = 60 +N = 70 +FIG. 8. +Smoothed wiggliness ν plotted for various system +sizes N and in three different models. (a) u(3) model with ξ = +0.8, ϵ = 0.4 and OTOC operators [ˆl(t), ˆl(0)]2. (b) u(3) model +with ξ = 0.4, ϵ = 0.0 (integrable case) and OTOC operators +[ ˆDx(t), ˆDx(0)]2. (c) A fully chaotic system with Hamiltonian +modeled by a random GOE matrix [3] and OTOC operators +taken as the u(3) model [ ˆDx(t), ˆDx(0)]2. +−1.0 +−0.5 +0.0 +0.5 +1.0 +E +−1.0 +−0.8 +−0.6 +−0.4 +−0.2 +0.0 +α +(a) +(b) +(c) +−1.0 +−0.5 +0.0 +0.5 +1.0 +E +−0.6 +−0.4 +−0.2 +0.0 +0.2 +0.4 +0.6 +β +(a) +(b) +(c) +FIG. 9. Exponents α and β obtained for the models displayed +in Fig. 8. +other regimes of the algebraic u(3) model. +V. +CONCLUSIONS +While the short-time behavior of the OTOCs testifies +about the “stability” of quantum states and the speed of +their spreading, leading to the definition of the quantum +Lyapunov exponent, we have demonstrated that the long- +time OTOC dynamics accounts for the overall chaoticity +of quantum system. The quantum Lyapunov exponent +thus reflects local properties of the dynamics, while the +asymptotic relative oscillations—the wiggliness—are re- +lated to the global chaos characteristics. It is analogical +to classical mechanics, where the classical Lyapunov ex- +ponent characterizes a single trajectory, while an asymp- +totic time evolution of a chaotic orbit, due to ergodic and +topological properties, covers the chaotic domain of the +phase space energy hypersurface. From the knowledge +of the entire volume of the energy hypersurface, one can +immediately deduce the fraction of regularity freg. +We have performed an extensive study of various +OTOC operators in different regimes of the u(3) model, +ranging from fully integrable to fully chaotic, and have +shown that the wiggliness of the long-time OTOC dy- +namics and its suppression rate α are robust measures +of chaoticity, giving qualitatively the same results for +an almost arbitrary choice of OTOC operators. Due to +quantum fluctuations, some smoothing over neighboring +states is necessary, leading to a finite resolution. How- +ever, the resolution can be improved by increasing the +system size, which makes the energy spectrum denser, +thus allowing for a narrower smoothing window. In sys- +tems with a relatively small size, the wiggliness can sup- +plement the quantum Lyapunov exponent, which, due to +the shortness of the Ehrenfest time, cannot often reli- +ably capture the exponential rate of the initial OTOC +dynamics. +The short-time evolution requires a reasonable time +resolution. Therefore, the wiggliness can be experimen- +tally relevant in systems where the initial dynamics is so +fast that it cannot be properly measured, but the system +does not decohere quickly, so the long-time oscillations +can be captured and analyzed. +ACKNOWLEDGMENTS +Pavel Str´ansk´y thanks Pavel Cejnar, Lea Santos, Jorge +Hirsch, and Jorge Ch´avez for valuable discussions. 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Olshanii, Thermalization +and its mechanism for generic isolated quantum systems, +Nature 452, 854 (2008). + diff --git a/M9E0T4oBgHgl3EQfjQFP/content/tmp_files/load_file.txt b/M9E0T4oBgHgl3EQfjQFP/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..1e9aa4bfa67c16c5ade9d352696b6108cdfc45c0 --- /dev/null +++ b/M9E0T4oBgHgl3EQfjQFP/content/tmp_files/load_file.txt @@ -0,0 +1,1218 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf,len=1217 +page_content='Relative asymptotic oscillations of the out-of-time-ordered correlator as a quantum chaos indicator Jakub Novotn´y1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' ∗ and Pavel Str´ansk´y1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' † 1Institute of Particle and Nuclear Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' Faculty of Mathematics and Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' Charles University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' V Holeˇsoviˇck´ach 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' 18000 Prague,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' Czech Republic (Dated: January 9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' 2023) A detailed numerical study reveals that the asymptotic values of the standard deviation-to-mean ratio of the out-of-time-ordered correlator can be successfully used as a measure of the quantum chaoticity of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' We employ a finite-size fully connected quantum system with two degrees of freedom, namely the algebraic u(3) model, and demonstrate a clear correspondence between the relative oscillations of the correlators and the ratio of the chaotic part of the volume of phase space in the classical limit of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' We also show how the relative oscillations scale with the system size and conjecture that the scaling exponent can also serve as a robust chaos indicator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' INTRODUCTION The correspondence between the theory of classical and quantum chaos has been extensively studied since the formulation of the Bohigas-Giannoni-Schmit conjec- ture [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' While classical chaos is rigorously constructed mathematically and routinely studied, most often by the rate of exponential divergence of neighboring trajectories, the study of quantum chaos is more intriguing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' Quan- tum chaoticity is usually defined indirectly by a compar- ison of suitable properties of a quantum system—most often correlations in energy spectra—with the chaotic- ity of its classical counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' In particular scenarios, the theory of classical-quantum correspondence is well- established [2, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' In this paper, we move from the static description of the quantum chaos based on spectral properties to the dynamical manifestations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' We will employ the Out-of- Time-Ordered Correlators (OTOCs)—four-point corre- lation functions of two quantum operators taken at dif- ferent times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' The OTOCs have already gained popularity as an in- dicator of quantum chaos, especially their short-time evolution due to their connection with classical insta- bility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' They were introduced long ago as a semiclassi- cal tool to study superconductivity [4] and later dusted off by showing their relevance in black-hole physics and chaos [5, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' In the case of a quantum system with the classical limit, the short-time behavior of the OTOCs mimics the exponential spreading of neighboring clas- sical trajectories up to the Ehrenfest time, leading to the notion of quantum Lyapunov exponent [7–19] and quantum butterfly effect [5, 18, 20–23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' In quantum sys- tems with local interactions and local OTOC operators, the initial time evolution of the OTOCs describes the spreading (or scrambling) of quantum information [24– 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' Based on the rate of initial growth of OTOCs, inde- pendently of the existence of the classical limit, quantum ∗ novotny.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='jakub@ipnp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='mff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='cuni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='cz † stransky@ipnp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='troja.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='mff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='cuni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='cz systems can be placed into a category of slow scramblers (maximally polynomial initial growth even in the pres- ence of chaos) [10, 31–33], or fast scramblers (exponen- tial scrambling) [8, 10, 12, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' The black holes have been conjectured as the fastest scramblers [5], employing the AdS/CFT duality [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' OTOC analysis is being applied for studying many-body quantum systems, spin systems, quantum circuits [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' In recent years, this vast area for application gave rise to many proposals and measuring protocols [24, 35–41] followed by experimental measure- ments [30, 42, 43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' OTOCs have also been applied to study many-body quantum scars [17, 44] and quantum thermalization [26, 45], and as a probe of the excited- state quantum phase transitions [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' Whereas there is a vast amount of literature about the study of the short-time OTOC evolution, a possible connection of its long-time properties to chaos has been hinted at in a few recent works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' In the case of finite- size many-body quantum systems, OTOCs saturate in the long-time regime and exhibit oscillations around the mean value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' [8], it was suggested for the first time that the appearance of saturation oscillations could be linked to quantum chaos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' Later, a numerical study of a quantum Harper map demonstrated that the sup- pression of the OTOC oscillations in infinite tempera- ture limit quantitatively corresponds to the ratio of the chaotic volume of the whole phase space [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' The same authors have shown [47] that the suppression of relative oscillations of infinite-temperature OTOC can reveal the transition to chaos even in a small chain of four spins with OTOC studied on time scales attainable by ex- periment [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' The relative oscillations of OTOCs as a qualitative chaos indicator have also been mentioned in other studies [49, 50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' Some studies propose that even the asymptotic OTOC mean value can serve as a chaos indicator [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' The short-time exponential behavior of OTOCs is not a bulletproof sign of quantum chaoticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' Unstable po- tential stationary points can cause exponential growth of OTOCs even in integrable systems [15, 45, 51–53], hence misleadingly indicating chaos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' And vice versa, the already mentioned slow scramblers show polynomial arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='02456v1 [quant-ph] 6 Jan 2023 2 growth of OTOC even in chaotic regimes [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' In this context, one can ask to what extent the relative oscilla- tions overcome the drawbacks of the short-time OTOCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' Another point can be whether they allow not only for in- dicating chaos but also for measuring it, and how well they compare to other known chaos measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' Some clues are already present in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' Authors in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' [45] noticed that the size of oscillations of OTOCs in the long-time regime in the simple Bose-Hubbard dimer could qualitatively distinguish nonchaotic, mixed, and fully chaotic regions, while the OTOCs show exponen- tial growth even in the fully regular regime of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' And in the already mentioned work [47], authors show that chaos suppresses relative oscillations of OTOCs even in the slowly scrambling spin chains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' In this paper, we will be interested mainly in the long- time behavior of OTOCs, although we will also briefly discuss their short-time evolution and compare it with classical stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' We will work with the eigenenergy expectation values of the OTOCs, which will allow us to analyze the energy dependence of relative oscillations and link them to other well-established energy-dependent measures of chaoticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' Specifically, we will show that the ratio of the asymptotic standard deviation to the asymp- totic mean value (called wiggliness for brevity) compares to the volume of the regular part of the phase space in the classical limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' We will provide a detailed numerical study in a bosonic algebraic quantum model based on the u(3) algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' This model has been originally introduced to describe bending modes of linear polyatomic molecules [54–57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' Later it has been successfully used to investigate purely theoreti- cal concepts, such as quantum monodromy and quantum critical effects [58–60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' Recently it has also been applied to study the properties of spinor Bose-Einstein conde- sates [39, 61–63], which are experimentally achievable, for example, as a condensate of cold rubidium atoms [64, 65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' Some aspects of the OTOCs and their relation to chaos have already been studied in this model [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' We have chosen the u(3) model because its Hilbert space is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' The model has just 2 degrees of freedom and a known classical limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' On top of that, it is nonin- tegrable with enough parameters to tune the chaoticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' Due to the algebraic origin, all observables can be easily constructed from the operators of the u(3) algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' And finally, one can vary the number of boson excitations in the system and study how the asymptotic OTOC behav- ior changes with the system size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' This paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' II we recall some necessary concepts of the theory of classical chaos;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' we introduce the OTOCs and the wiggliness, which is the primary tool of the present work, and briefly discuss the correspondence between the classical and quantum chaos indicators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' III introduces the u(3) model and its classical limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' The numerical results and their discussion is given in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' We finally conclude in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' THEORY In this section, we introduce the classical Lyapunov exponent and the fraction of the regular part of the en- ergy hypersurface in the phase space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' Then we present energy-dependent microcanonical OTOCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' Finally, we identify short-time and long-time evolution in the OTOC dynamics and specify to which classical notion they can be compared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' Classical dynamics We consider a classical system with f degrees of free- dom described by time-independent Hamiltonian Hcl(x), which is a function of coordinates and conjugated mo- menta x ≡ (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' , x2f) = (p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' , pf, q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' , qf) on a phase space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' The time-independence of the Hamiltonian implies energy conservation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' Therefore, classical trajec- tories are constrained to (2f − 1)-dimensional hypersur- faces ΣE determined by constant energy Hcl = E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' One of the prominent signs of classical chaos is the ex- ponential separation of asymptotically close trajectories characterized by the positivity of the Lyapunov expo- nent [66,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' 67] λcl = lim t→∞ lim |δx|→0 1 t log |(x + δx)(t) − x(t)| |δx| ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' (1) where x(t) and (x + δx)(t) are trajectories with initial conditions x0 and x0+δx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' respectively,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' δx is an infinites- imally small deviation vector in the neighbourhood of the point x0 on the given energy hypersurface,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' and |•| is a norm in the phase space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' In the case of regular (stable) trajectories λcl = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' The Lyapunov exponent explains the local properties of the dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' On the other hand, the overall chaotic- ity of the system with energy E is reflected in the fraction of regularity freg(E) = Γreg(E) Γ(E) ∈ [0, 1], (2) where Γ(E) is the entire volume of the energy hyper- surface ΣE and Γreg(E) is the volume of all the regu- lar regions in ΣE, which, in general, are well-separated from the regions of chaotic dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' The limit values correspond to fully chaotic (freg = 0) and fully regular (freg = 1) dynamics at the given energy E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' Quantum dynamics Quantum chaos is often studied from static properties of the spectral correlations [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' Here we focus on the dy- namical manifestations, namely on the properties of the OTOC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' In a quantum system described by Hamiltonian 3 ˆH, the out-of-time-ordered correlator is introduced as the expectation value in energy eigenstates |En⟩ Cn(t) = � En ���[ ˆV (t), ˆW(0)]†[ ˆV (t), ˆW(0)] ���En � , (3) where En is the n-th eigenenergy, ˆH |En⟩ = En |En⟩, and ˆV , ˆW are for the moment arbitrary quantum operators in the Heisenberg picture, ˆV (t) = e i ℏ ˆ Ht ˆV e− i ℏ ˆ Ht;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' (4) [ ˆV (t), ˆW(0)] is their commutator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' If both operators are Hermitian, then Cn(t) = − � En ���� � ˆV (t), ˆW(0) �2����En � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' (5) The OTOC properties undoubtedly depend on the choice of the operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' These are usually relevant physical ob- servables of the system, such as positions, momenta oper- ators, or local spin operators in models with finite-range interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' The variability of the OTOC dynamics for several pairs of operators will be demonstrated later in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' Note that the OTOCs are often considered as ther- mal averages in the canonical ensemble at inverse tem- perature β [6, 20], sometimes even at infinite tempera- ture β = 0 only [8, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' For the sake of distinguishing pure state and thermal state averaging, the OTOCs of type (3) are called the microcanonical OTOCs in the lit- erature [15, 68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' Classical-quantum correspondence The time dependence of a general OTOC can be di- vided into short-time and long-time regimes, roughly limited by the Ehrenfest (or scrambling) time τE ∝ λ−1 cl ln N [10, 19, 69, 70], where N is the size parame- ter of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' In more detailed analyses, there has also been described a universal power-law growth at very short times td < tE [14] (td is called the dephasing time) for OTOC operators commuting at t = 0, and yet an- other time scale given by the diffusion time tD ≫ tE, which is the time of complete saturation of the quantum dynamics [71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' Whereas the OTOCs in eigenstates corresponding to regular dynamics typically show strong oscillations at all times, OTOCs for chaotic eigenstates are character- ized by initial exponential growth for t < τE and long- time saturation regime with small aperiodic fluctuations around the mean value for t > τD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' In the chaotic sys- tems with a classical limit, the exponent λ of the initial exponential growth C(t) ∝ e2λt (6) is related to the classical Lyapunov exponent λcl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' Es- pecially, for operators ˆxi that correspond to the canon- ical coordinates and momenta xi on the classical phase space the exponent λ coincides with the Lyapunov expo- nent [4, 12, 68], [ˆxi(t), ˆxj(0)]2 ↔ {xi(t), xj(0)}2 ∼ e2λclt, (7) where {•, •} are the classical Poisson brackets;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' the indices i, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' , 2f can be taken arbitrarily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' In general, however, the equality λ = λcl is not valid for all choices of OTOC operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' It can be shown that the Poisson brackets {A(t), A(0)} of a general function A on the phase space have a Lyapunov term that mimics the exponential separation of infinitesimally close chaotic trajectories and which is projected onto the gradient of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' Therefore, this gradient and other terms containing the second derivative of A may modify the exact value of λ, and the corresponding OTOC will not grow exactly with the classical Lyapunov exponent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' The long-time behavior of OTOCs for a given pair of operators ˆV , ˆW is captured by the mean value and vari- ance Cn = lim T →∞ 1 T � T 0 Cn(t)dt, (8) σ2 n = lim T →∞ � T 0 C2 n(t)dt − C 2 n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' (9) It will be shown later that the crucial role in quantifying chaoticity is played not by these quantities themselves but by their ratio (the coefficient of variation) νn = σn Cn , (10) reflecting the relative oscillations of the OTOC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' In the following text, we will, for the sake of brevity, call this ratio wiggliness, and we will always specify for which op- erators the OTOCs and the wiggliness are computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' MODEL The model Hamiltonian belongs to a class of boson- interacting systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' It is constructed from nine gener- ators of the spectrum-generating u(3) Lie algebra [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' The generators can be represented by the bilinear prod- ucts of creation and annihilation operators ˆb† iˆbj, i, j = 0, 1, 2, where ˆb0 ≡ ˆσ is the scalar boson operator and ˆb1,2 ≡ ˆτ1,2 form the pair of circular boson operators ˆτ± = (ˆτ1 ± iˆτ2) / √ 2, all of them satisfying the boson commutation relations � ˆbi,ˆb† j � = δij, � ˆbi,ˆbj � = � ˆb† i,ˆb† j � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' (11) The Hilbert space is spanned over all single-particle states, |nσ, n+, n−⟩ ≡ 1 N � ˆσ†�nσ � ˆτ † + �n+ � ˆτ † − �n− |0⟩ (12) 4 where nσ, n± = 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' are the numbers of the corre- sponding boson excitations, |0⟩ is the vacuum state and N a normalizing factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' Note that any operator con- structed from the generators conserves the total number of boson excitations ˆN = ˆσ†ˆσ+ˆτ † +ˆτ++ˆτ † −ˆτ−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' The number N = nσ + nτ, where nτ = n+ + n−, specifies the fully- symmetric irreducible representation of u(3) and dictates the dimensionality of the corresponding (finite) Hilbert space H, dim H = 1 2(N + 1)(N + 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' (13) Therefore, N serves as a tunable size parameter of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' It is convenient to construct another set of generators as linear combinations of ˆb† iˆbj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' ˆnτ = ˆτ † +ˆτ+ + ˆτ † −ˆτ−,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' ˆns = ˆσ†ˆσ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' ˆD± = ± √ 2 � ˆτ † ±ˆσ − ˆσ†ˆτ∓ � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' ˆQ± = √ 2ˆτ † ±ˆτ∓,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' (14) ˆR± = √ 2 � ˆτ † ±ˆσ + ˆσ†ˆτ∓ � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' ˆl = ˆτ † +ˆτ+ − ˆτ † −ˆτ−,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' where the following subsets are generators of subalgebras o(3) = span � ˆD±,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' ˆl � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' (15a) o(3) = span � ˆR±,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' ˆl � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' (15b) u(2) = span � ˆnτ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' ˆl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' ˆQ± � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' (15c) o(2) = span � ˆl � (15d) that admit three subalgebra chains,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' I :u(3) ⊃ u(2) ⊃ o(2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' (16a) II :u(3) ⊃ o(3) ⊃ o(2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' (16b) II :u(3) ⊃ o(3) ⊃ o(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' (16c) The o(2) is the symmetry algebra reflecting the invari- ance of the system with respect to the rotation around the z-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' The last two chains are equivalent and in- terchangeable, connected via a unitary transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' Depending on the choice of algebra o(3) or o(3), it is talked about momentum or coordinate realization of the same Hamiltonian [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' A common way to construct the o(3) Hamiltonian is via the Casimir operators of all the possible subalgebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' Here we consider only the linear Casimir ˆC1[u(2)] of the u(2) algebra and the quadratic Casimir ˆC2[o(3)] of the o(3) algebra, and build the Hamiltonian in the following form ˆH0 = 1 − ξ N ˆC1[u(2)] � �� � ˆnτ − ξ N(N − 1) ˆC2[o(3)] � �� � ˆ D2 , (17) where ˆD2 ≡ 1 2 � ˆD+ ˆD− + ˆD− ˆD+ � + ˆl2 (18) and ξ ∈ [0, 1] is an adjustable parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' The factor N is added so that the Hamiltonian scales correctly in the infinite-size limit N → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' The system given by ˆH0 has f = 2 degrees of freedom and is integrable with Hamiltonian and angular momen- tum operator ˆl2 being the two independent integrals of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' This Hamiltonian is frequently used to model a quantum phase transition between the o(2) (usually called symmetric) phase and the o(3) (called deformed or displaced) phase, occurring at ξ = 1/5 in the infinite- size limit [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' The integrability of ˆH0 will be broken by violating the o(2) symmetry, ˆH = ˆH0 − ϵ N ˆDx, (19) where ˆDx = ( ˆD+ + ˆD−)/2 is the dipole operator [ ˆDy would be defined as ˆDy = ( ˆD+ + ˆD−)/2i].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' Note that ˆH still has a discrete symmetry (parity), so its spectrum forms two invariant sets of states in the Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' A suitable basis for distinguishing these two invariant subspaces is given by vectors |N, nτ, l±⟩ ≡ 1 √ 2 (|N, nτ, l⟩ ± |N, nτ, −l⟩) , (20) where l is taken as non-negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' Two subspaces H(1,2) invariant under the action of the Hamiltonian (19) have the form H(1) = span{|N, nτ, l+⟩l-even, |N, nτ, l−⟩l-odd}, (21a) H(2) = span{|N, nτ, l+⟩l-odd, |N, nτ, l−⟩l-even}, (21b) where nτ = {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' N} and l = {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' , nτ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' The full Hilbert space is H = H(1) ⊕ H(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' The perturbed model is integrable for particular val- ues of ξ = 0 and 1, which correspond to excluding one of the Casimir operators in (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' Instead of ˆl, other op- erators play the role of the second integral of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' For ξ = 0, the Hamiltonian commutes with the operator ˆn + ˆQ where ˆQ = � ˆQ+ + ˆQ− � / √ 2, and for ξ = 1, the Hamiltonian is formed only by operators from algebra o(3) and commutes with both ˆD2 and ˆDx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' The classical limit of the Hamiltonian can be obtained by transforming the circular boson operators into the coordinate-momentum form ˆbj = � N 2 (ˆqj + iˆpj) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' j = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' (22) using the Holstein-Primakoff transformation [72,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' 73] to eliminate the degree of freedom connected with the σ boson due to the conservation of N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' and performing the limit N → ∞,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' after which the operators with commuta- tion relations [ˆqj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' ˆpk] = i N δjk (23) 5 serve as canonically conjugated coordinates and mo- menta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' The corresponding classical Hamiltonian func- tion reads as Hcl ≡ lim N→∞ ˆH = (1 − ξ) s2 2 − ξ �� p2 1 + p2 2 � � 2 − s2� + (p1q2 − q1p2)2� − ϵp2 � 2 − s2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' (24) The four-dimensional phase space (p1, p2, q1, q2) is com- pact, bounded by the condition s2 ≤ 2, where s2 = p2 1 + p2 2 + q2 1 + q2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' Note that the commutation relations (23) imply that the size parameter N also serves as an effective Planck constant ℏeff = 1/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' Therefore, the infinite-size limit N → ∞ coincides with the classical limit ℏeff → 0 in this model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' In the classical limit, the operators ˆDx,y and ˆRx,y be- have approximately as a momentum and a position co- ordinate in the phase space [55, 56] as they are mapped to 1 N ˆDx,y N→∞ −−−−→ p2,1 � 2 − s2, (25a) 1 N ˆRx,y N→∞ −−−−→ q1,2 � 2 − s2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' (25b) Due to the above-mentioned connection of the o(3) and o(3) operators via a unitary transformation, the spec- trum of the Hamiltonian is the same no matter whether constructed of the ˆDx,y or ˆRx,y operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' RESULTS In this section, we begin by illustrating the manifes- tation of classical chaos in the classical limit of the u(3) model, and we briefly explain numerical techniques used to calculate the Lyapunov exponent λcl and the fraction of regularity freg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' Then we fit the short-time evolution of the OTOC with the exponential function (6) and confirm that its exponential rate λ corresponds with the classical Lyapunov exponent for various OTOC operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' Fi- nally, we focus on the main topic of this paper, which is the OTOC fluctuations at long times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' Classical chaos Except for some particular values of the parameters described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' III, the model described by Hamilto- nian (24) is nonintegrable, thus exhibiting chaotic dy- namics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' 1, we show the classical fraction of reg- ularity freg for a fixed value of external perturbation ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='4 and for ξ ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' The chaotic regions (freg < 1) start to arise at ξ ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='1 and lie approximately on and above the energy E = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' The chaoticity is demon- strated by Poincar´e sections plotted in panels (a) – (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' Each section point corresponds to one trajectory cross- ing through the chosen section plane q1 = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' the color of the points indicates the size of the Lyapunov exponent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' Stable trajectories form regular structures in the shape of deformed lines or circles, whereas chaotic areas are filled with seemingly randomly located points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' In gen- eral, these 2D Poincar´e sections visualize the 3D energy manifolds of constant energy embedded in the 4D phase space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' The white areas in panels (b) and (c) are proper topological holes in the energy hypersurface and depict the intriguing shape of the sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' Classical fraction of regularity freg(ξ, E) (top panel) and selected examples of Poincar´e sections by q1 = 0 plane [panels (a)—(i)] for the classical limit of the u(3) model (24) with nonintegrable perturbation ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' The colors of the section points indicate the values of the Lyapunov exponent λcl for the corresponding trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' Values freg = 0 and freg = 1 correspond to fully chaotic (all trajectories on the energy hypersurface ΣE have a positive Lyapunov exponent) and fully regular (all trajectories have λcl = 0) dynamics, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='0 C 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='5 - (b) (e) E 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='0 (h) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='5 (i) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='0 freg a b c 0 (e) d (f) q2 0 1 (g) (h) (i) 1 0 0 1 1 1 0 1 1 0 1 p2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='08 入 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='26 0 100 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='0 C(t) ¯C 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='4 × 109 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='8 × 109 t 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='0 E = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='54 E = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='21 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' An example of the short-time and long-time OTOC dynamics for operators [ ˆDx(t), ˆDx(0)]2 in a state from a fully regular (n = 1, thin blue curve) and chaotic (n = 650, thick orange curve) part of the quantum energy spectrum of the u(3) model, respectively, as a function of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' Model pa- rameters are ξ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='4, ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='4, system size is N = 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' OTOC values are divided by the corresponding mean value C for bet- ter comparison;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' C(E1 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='54) ≈ 101 and C(E650 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='21) ≈ 105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' The approximate Ehrenfest time tE for the chaotic state is indicated by the vertical black dashed line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' The values of the Lyapunov exponents were obtained by evolving the deviation δx of two neighboring trajec- tories for a sufficiently long time using the equation of tangent dynamics [15, 66, 67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' As the freg(E), we ef- fectively take the ratio of the regular area (area in the Poincar´e section crossed by trajectories with λcl = 0) to the total section area intersected by any of the orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' We choose the initial conditions so that the given Poincar´e section divided into a sufficiently dense mesh of cells is entirely covered with evolved trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' Our detailed numerical analysis shows that the values of freg obtained from the Poincar´e sections differ from the freg computed directly from (2) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' as a ratio of the volumes of the 3D hypersurfaces) by no more than ≈ 5%, which is a fair price to pay for saving plenty of the computation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' Note that the same procedure of calculating the freg was used and discussed in another model [74].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' Short-time OTOCs As explained in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' II C, the short-time behavior of the OTOCs generally reflects the classical divergence of neighboring trajectories and can serve to define a quan- tum analog of the classical Lyapunov exponent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' 2 we show the typical time series of the OTOCs for quan- tum states exhibiting both regular and chaotic dynam- ics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' Regularity is characterized by wide oscillations, with the OTOC values frequently returning close to the initial value C(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' In contrast, the chaotic ones initially show a fast (exponential) increase followed by saturation and low oscillations around the mean value, usually far from the initial C(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' The tight connection between the short-time behavior of OTOCs and classical chaos is demonstrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' We plot there the exponents λ obtained from the fit of the initial exponential growth of OTOCs (6) for various −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='6 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='6 E 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='25 λ [ˆl(t), ˆl(0)]2 [ˆnτ(t), ˆD2(0)]2 [ ˆDx(t), ˆDx(0)]2 λcl FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' Instability parameters λ from the exponential fit of the short-time growth of OTOCs for three choices of the OTOC operators indicated in the legend, compared with the classical Lyapunov exponent λcl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' Parameters of the u(3) model are ξ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='4, ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' System size is N = 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' choices of the OTOC operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' We can observe that the values of λ depend on the choice of operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' However, the overall energy dependence for all OTOC operators qualitatively agrees with the classical Lyapunov expo- nent λcl(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' The best agreement between the classical and quantum instability parameters is achieved for the OTOC operators [ ˆDx(t), ˆDx]2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' As stated before, opera- tor ˆDx(t) can be linked, in the classical limit (25), to one of the canonical coordinates on the phase space, leading to the correspondence between λ and λcl given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' Long-time OTOCs Here we focus on the main analysis of this paper, which is the properties of OTOC wiggliness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' A brief glimpse at Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' 2 reveals the difference between the relative OTOC oscillations in the regular and chaotic state at higher times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' The OTOC function evaluated for the regular state with energy E1 forms a wide “strip” covering the graph from C ≈ 0 to some finite value C = Cmax, re- sulting in a relatively high value of the wiggliness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' In contrast, in the chaotic state after the initial exponential explosion, the strip is narrower with minor oscillations around the saturation value, and the wiggliness is, there- fore, smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' The numerical analysis shows that the suppression of the wiggliness in energy regions containing chaotic states is correlated with a decrease in the fraction of regular- ity freg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' This correlation is depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' 4, which plots the relative OTOC oscillations computed for three choices of OTOC operators and three values of param- eter ξ, and compares the wiggliness to the fraction of regularity freg .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' We distinguish three types of wiggliness behavior based on the chaoticity of the system: 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='6 (a) (b) (c) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='6 (d) (e) (f) ν λcl 1 2freg ν −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='6 (g) −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='50 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='50 (h) −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='0 (i) E [ ˆDx(t), ˆDx(0)]2 [ˆl(t), ˆl(0)]2 [ ˆD2(t), ˆnτ(0)]2 ξ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='8 ξ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='4 ξ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' Wiggliness for three distinct pairs of the OTOC operators calculated for all eigenstates |En⟩ along the spectrum (black dots), the classical fraction of regularity freg (thick red curve) and the classical Lyapunov exponent (thin blue curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' In addition, the wiggliness is smoothed using the moving average in energy (thick yellow curve labeled as ν) for comparison with freg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' The OTOC operators are indicated at the vertical axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' Parameters of the u(3) model are ξ = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='4, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='8}, ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' System size is N = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' Note that the Poincar´e sections in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' 1 were computed for the same model parameters and can serve as an additional illustration of the regular versus chaotic dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' Regular regions (freg = 1) In the parts of the spectra that correspond to com- pletely regular dynamics, the values νn are typi- cally higher in comparison with other regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' They may form regular structures as νn reflects the ex- istence of an additional constant of motion, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' 4 (c),(h),(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' The shape of these structures naturally depends on the choice of OTOC opera- tors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' Chaotic regions (freg = 0) For quantum states that lie in completely chaotic energy intervals, the values νn are generally smaller compared to other regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' In addition, the wig- gliness values corresponding to neighboring eigen- states are much more similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' This can be observed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' 4 for ξ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='4 (second column) at E ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='2 and ξ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='8 (third column) at E ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='1, as all the points in these chaotic regions are “pressed” together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' Mixed spectrum (freg > 0) In the partially chaotic energy intervals of the spec- tra, we have intertwined sets of states with proper- ties of both regular and chaotic regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' Therefore, they are characterized by a relatively wide spread of wiggliness values but without any characteristic regular pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' Remarkable could be the sensi- tivity of νn to regular islands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' Consider the region 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='25 > E > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='5 in the second column of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' The system is almost entirely chaotic at this energy in- terval in the classical limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' In the corresponding Poincar´e section shown Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' 1(b), we can hardly notice tiny regular islands immersed in the chaotic sea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' Yet there is a significant amount of eigenstates with high OTOC wiggliness, reflecting these flecks of classical regularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' The principal observation is that the smoothed values ν(E) of the wiggliness (calculated as a moving average over a small number of successive energy levels) follow the trend of freg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' It leads us to conjecture that the smoothed asymptotic relative oscillations of the OTOCs reflect the chaoticity of the quantum system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' 8 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' Smoothed wiggliness ν along the spectra of the u(3) model (19) for ξ ∈ [0, 1], ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='4 and two pairs of OTOC operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' The smoothing window entails five neighboring levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' The system size is N = 60, corresponding to 1891 energy levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' Roman numbers I and II label two regions with values of ν for OTOC operator [ ˆDx(t), ˆDx(0)]2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' The structures present for the wiggliness in the regular regions of energy can be understood within the theory of Peres lattices [75, 76];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' these lattices are formed from en- ergy expectation values On = ⟨En| ˆO|En⟩ of an operator ˆO as a set of points (En, On).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' In nonchaotic parts of the energy spectrum, the observable ˆO can be expressed as a smooth function of all the independent constants of motion, which leads to regular patterns in the Peres lat- tice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' Chaos arises with the loss of integrals of motion, which leads to a collapse of the regular Peres lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' In the fully chaotic regions, all the points are “pressed” to- gether with minor differences in the On expectation val- ues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' The differences further diminish with the increase in the system size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' This behavior can be understood via the Eigenvector Thermalization Hypothesis [77–79].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' Values Cn and σn given by Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' (8) and (9) are, in fact, expectation values of rather complicated time-averaged operators provided by the OTOC commutators, and the wiggliness is a ratio of these two;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' hence it is expected that the wiggliness retains some of the properties of the Peres lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' 5 displays the smoothed wiggliness for two choices of OTOC operators [ ˆDx(t), ˆDx(0)]2 and [ˆl(t), ˆl(0)]2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' We can directly compare these results with the freg plot- ted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' 1 and observe that the chaotic regions with freg ≈ 0 and regions with reduced ν(E) overlap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' In the case of operator ˆDx, we find regions of small smoothed wiggliness in the regular area of the spectrum (marked I and II in the figure), spuriously labeling them as chaotic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' These regions of suppressed wiggliness can also be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' 4(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' Low values of ν(E) in region I are related to the fact that ξ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='8 is very close to the value ξ = 1 for which ˆDx commutes with the Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' Region II is a finite-size effect that diminishes with increasing N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' So far, we have discussed the wiggliness only, but one 0 1 2 3 4 ¯C/dim H Regular 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='20 Chaotic 0 1 2 3 σ/dim H 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='04 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='6 E 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='8 ν Smoothed values 1 2freg −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='5 E 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='8 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' Comparison of mean values Cn (first row), standard deviations σn (second row) and the wiggliness νn of OTOC operators [ ˆDx(t), ˆDx(0)]2 (third row) in an integrable case ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='0 (left column) and chaotic case ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='4 (right column) of the u(3) model with ξ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='4 and system size N = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' Values of Cn and σn are divided by the Hilbert space di- mension dim H = 5151.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' Yellow lines correspond to smoothed values using the moving average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' The red line is the fraction regularity freg, scaled by 1/2 for better visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' could ask whether just the mean value or variance (or the standard deviation as the square root of the variance) would sufficiently indicate and quantify the presence of chaos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' Based on the extensive numerical evidence, the answer is no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' This is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' 6 where we com- pare Cn, σn and νn of [ ˆDx(t), ˆDx(0)]2 OTOCs in an in- tegrable and chaotic regime of the u(3) model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' As men- tioned above, the sets of points (En, Cn) and (En, σn) form a kind of Peres lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' Therefore we can infer the presence of chaos in energy regions where the lattice is disordered and where the points are vertically “pressed” together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' In the chaotic regime, the mean value and the standard deviation are higher in the chaotic regions around E ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='25 than in the regular regions close to the lower and upper energy limits of the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' However, their values are more than an order of magnitude lower than in the integrable case ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' Therefore we can- not simply determine the chaoticity of a general system based on the concrete values of Cn and σn, even after smoothening over several neighboring states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' It is their ratio in the form of relative oscillations that reveals the [Da(t), Dα(0)]2 [(t), i(0)]2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='0 E 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='5 II 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='75 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='75 1 CS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='6 D9 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='5 E 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='5 ¯ν (a) N = 10 N = 20 N = 44 N = 80 N = 120 N = 160 1 2 3 4 5 log(N) −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='5 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='0 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='5 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='5 log(¯ν) (b) E = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='3 E = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='2 E = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='1 E = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='0 E = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='1 E = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='5 E −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='0 (c) α −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='5 E −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='5 (d) β FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' Dependence of the wiggliness on the size N of the quantum system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' Figures are calculated for the u(3) model in the chaotic regime with ξ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='4 and ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' As the OTOC operators we choose [ ˆDx(t), ˆDx(0)]2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' Panel (a) dis- plays smoothed wiggliness as a function of energy along the spectrum for several system sizes from N = 10 to N = 160, as indicated in the legend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' Panel (b) shows the dependence of the smoothed wiggliness on N for several energy values in the log-log scale (points) and their fit by function (26) (dashed lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' degree of chaoticity and that corresponds to the classical freg, as demonstrated in Figs 4, 5 and 6(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' Effect of the system size In this section, we discuss the effect of the system size on the relative oscillations νn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' We take the total number of boson excitations N of the u(3) model as a parameter and study the dependence of the smoothed wiggliness ν(N) both on the energy E and on the size parameter N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' 7(a) shows an example of the smoothed wiggliness for one choice of the pairs of OTOC operators for the whole accessible energy range and for several values of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' The observed overall behavior is the following: In the regions of the spectrum with chaotic or mixed dynamics (freg < 1), ν decreases with increasing size of the system, whereas in the regular regions (freg = 1) the smoothed wiggliness remains approximately constant (usually with a value ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='6) or slowly increases with N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' This observation is further confirmed by Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' 7(b), which shows the dependence ν(N) for six different values of energy taken from regions of both regular and chaotic dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' The linear dependence in the log-log plot im- plies algebraic scaling of ν with N, ν(E, N) = N α(E)e−β(E), (26) where the dependence of ν on the energy manifests only via the exponents α(E) and β(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' These exponents, ob- tained by the fit as demonstrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' 7(b), are plotted in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' 7(c) and (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' Since the spectrum of the system is discrete, we interpolated the values of ν to get precisely into the desired energy value for the fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' In Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' 7(c) and (d), one can immediately notice the apparent qualitative resemblance of α(E), β(E), and the wiggliness itself, see Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' 4(b) and 7(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' The cause of the similarity between α and β can be deduced from Fig .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='7(b): All the fitted lines intersect in a small region at log(N0) ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='5, so there exists a linear transformation between α and β taken at two arbitrarily chosen energies E and E′, log(N0) = β(E) − β(E′) α(E) − α(E′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' (27) Since the lines do not intersect precisely at one point (and there is no known reason that they should), this transformation is only approximate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' Nevertheless, even the approximate relation leads to similar shapes of curves α(E) and β(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' More important is the similarity of ν(E) and the scal- ing exponent α(E) shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' 7(a) and (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' That implies that the suppression rate of the wiggliness scales with the system size;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' the more chaotic the system is, the faster the suppression of ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' Note that this behavior has already been observed in non-collective spin systems [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' The exponent α can, therefore, also serve as a chaos in- dicator, and because it does not depend on the size N of the system, it can be considered more robust than the wiggliness itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' Once α and β are known, they allow for calculating the wiggliness for any system size N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' The last observation can be summarized in the fol- lowing way: In the energy regions of the spectrum with chaotic or mixed dynamics (freg < 1), the suppression rate is negative, α < 0, whereas in the regular regions (freg = 1) there is no suppression of the wiggliness or a light enhancement, α ≳ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' We further demonstrate the validity of the scaling con- jecture for three totally different models, ranging from in- tegrable to fully chaotic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' 8 shows the scaling of the smoothed wiggliness for a u(3) model with mixed dynam- ics (panel a), integrable, hence fully regular u(3) model (panel b) and a fully chaotic system whose Hamiltonian is modeled as a random GOE matrix [3] (panel c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' Again, we observe that the wiggliness is algebraically suppressed in the chaotic regions of spectra and approximately con- stant in the regular parts with increasing system size N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' Corresponding exponents α and β are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' Regular parts of spectra have almost zero suppression rate α, while regions with mixed or purely chaotic dy- namics have negative α, which also mimics the shape of freg(E) and reflects the level of chaoticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' As mentioned above, the exponent β behaves in a similar way and could also serve as an indicator of chaos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' Note that the scaling behavior described in this section is also observed for other choices of OTOC operators and 10 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='2 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='8 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='6 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='2 E 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='6 ¯ν (a) N = 9 N = 18 N = 27 N = 36 N = 60 N = 100 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='6 E 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='6 ¯ν (b) N = 10 N = 20 N = 30 N = 40 N = 60 N = 80 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='0 E 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='20 ¯ν (c) N = 20 N = 30 N = 40 N = 50 N = 60 N = 70 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' Smoothed wiggliness ν plotted for various system sizes N and in three different models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' (a) u(3) model with ξ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='8, ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='4 and OTOC operators [ˆl(t), ˆl(0)]2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' (b) u(3) model with ξ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='4, ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='0 (integrable case) and OTOC operators [ ˆDx(t), ˆDx(0)]2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' (c) A fully chaotic system with Hamiltonian modeled by a random GOE matrix [3] and OTOC operators taken as the u(3) model [ ˆDx(t), ˆDx(0)]2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='0 E −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='8 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='6 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='0 α (a) (b) (c) −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='0 E −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='6 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content='6 β (a) (b) (c) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' Exponents α and β obtained for the models displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' other regimes of the algebraic u(3) model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' CONCLUSIONS While the short-time behavior of the OTOCs testifies about the “stability” of quantum states and the speed of their spreading, leading to the definition of the quantum Lyapunov exponent, we have demonstrated that the long- time OTOC dynamics accounts for the overall chaoticity of quantum system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' The quantum Lyapunov exponent thus reflects local properties of the dynamics, while the asymptotic relative oscillations—the wiggliness—are re- lated to the global chaos characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' It is analogical to classical mechanics, where the classical Lyapunov ex- ponent characterizes a single trajectory, while an asymp- totic time evolution of a chaotic orbit, due to ergodic and topological properties, covers the chaotic domain of the phase space energy hypersurface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' From the knowledge of the entire volume of the energy hypersurface, one can immediately deduce the fraction of regularity freg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' We have performed an extensive study of various OTOC operators in different regimes of the u(3) model, ranging from fully integrable to fully chaotic, and have shown that the wiggliness of the long-time OTOC dy- namics and its suppression rate α are robust measures of chaoticity, giving qualitatively the same results for an almost arbitrary choice of OTOC operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' Due to quantum fluctuations, some smoothing over neighboring states is necessary, leading to a finite resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' How- ever, the resolution can be improved by increasing the system size, which makes the energy spectrum denser, thus allowing for a narrower smoothing window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' In sys- tems with a relatively small size, the wiggliness can sup- plement the quantum Lyapunov exponent, which, due to the shortness of the Ehrenfest time, cannot often reli- ably capture the exponential rate of the initial OTOC dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' The short-time evolution requires a reasonable time resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' Therefore, the wiggliness can be experimen- tally relevant in systems where the initial dynamics is so fast that it cannot be properly measured, but the system does not decohere quickly, so the long-time oscillations can be captured and analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' ACKNOWLEDGMENTS Pavel Str´ansk´y thanks Pavel Cejnar, Lea Santos, Jorge Hirsch, and Jorge Ch´avez for valuable discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfjQFP/content/2301.02456v1.pdf'} +page_content=' Both authors acknowledge the support of the 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Manuvie 1, S. Chatterjee 2 +1University College Groningen, University of Groningen, r.manuvie@rug.nl +2Foundation The London Story, Fluwelen Burgwal 58, 2511CJ, Den Haag, The Netherlands, +saikat@thelondonstory.org +Abstract +In recent years, there has been a heightened consensus – both within academia and in the public discourse +– that Social Media Platforms (SMPs), amplify the spread of hateful and negative sentiment content. +Researchers have identified how hateful content, political propaganda, and targeted messaging +contributed to real-world harms including insurrections against democratically elected governments, +genocide, and breakdown of social cohesion due to heightened negative discourse towards certain +communities in parts of the world. To counter these issues, SMPs have created semi-automated systems +that can help identify toxic speech. In this paper we analyse the statistical distribution of hateful and +negative sentiment contents within a representative Facebook dataset (n= 604,703) scrapped through 648 +public Facebook pages which identify themselves as proponents (and followers) of far-right Hindutva +actors. These pages were identified manually using keyword searches on Facebook and on +CrowdTangleand classified as far-right Hindutva pages based on page names, page descriptions, and +discourses shared on these pages. We employ state-of-the-art, open-source XLM-T multilingual +transformer-based language models to perform sentiment and hate speech analysis of the textual contents +shared on these pages over a period of 5.5 years. The result shows the statistical distributions of the +predicted sentiment and the hate speech labels; top actors, and top page categories. We further discuss the +benchmark performances and limitations of these pre-trained language models. +1. Introduction +The spread of textual content with negative sentiment and community-targeted hate speech over +Social Media Platforms (SMPs) like Facebook, Twitter, Reddit etc. has become a matter of serious +concern in recent years (see Belew and Massanari 2018, Matamoros-Fernández and Farkas 2021, Bennett +and Segerberg 2022). Case studies demonstrate a range of analytical results on the organic growth of +hateful content around certain sensitive topics like a state election, COVID-19, the immigration policy of +a country etc.; as well as amplified growth through coordinated rapid link-sharing behaviour by actor +networks. In this study, our goal is to analyse the contents of a selected group of public Facebook fan +pages (648 pages) which returned a total of 604,703 text messages on Facebook. The pages were +manually identified as belonging to actors who are responsible for spreading anti-minority and +anti-muslim narratives in contemporary India. An automated analysis of the textual contents associated +1 + +with these fan pages was conducted to test the performance of current state-of-the-art NLP models in +predicting and identifying contents with negative sentiment and hate speech respectively. Open-source +multilingual Natural Language Processing (NLP) models for the sentiment analysis (Barbieri et al 2021), +and the hate speech identification task (Röttger et al 2022) were employed in their ‘evaluation’ mode in +order to perform a forward prediction task over the social media dataset (n=604,703). The approach +allows a quantitative prediction of the amount of negative sentiment, and the amount of hateful content +that is present in the dataset. Based on the outcome of these experiments, the statistical significance +behind the abundance of hateful and negative sentiment content on the Facebook platform is discussed. +Section 2 of the paper describes the process of data selection, preparation of the datasets, and our main +research questions. Sections 3 and 4, discuss the results of sentiment and hate-speech analysis +respectively. Section 5 is an extended discussion of our findings, concluding remarks and future scopes of +this work. +2. Methodology and the dataset +The content analysis presented in this paper has been performed on Facebook’s historical dataset +that was ingested using the CrowdTangle platform in the date range of 31-12-2016 and 01-07-2022. 648 +Facebook pages were identified using keyword search as pages actively spreading anti-muslim and +anti-minority hate speech in India (see Manuvie and Chaterjee, 2023 on the arXiv). The final dataset +consists of a total of 604,703 entries, on which state-of-the-art open-source sentiment analysis and +hate-speech detection models for forward predictions of the sentiment and hatefulness labels respectively +were applied (see sections 3 and 4 below). The PyTorch code that exploits the transformers libraries and +the Huggingface NLP models for these two forward inference tasks are publicly available at our GitHub +repository.1 +Based on the prediction scores of the NLP models, we investigate the following three main three +research questions in this paper, +A. What are the distributions of sentiment and hate speech labels within our CrowdTangle dataset as +predicted by the respective models? +B. Who are the top actors within our dataset who share hateful content and content with the negative +sentiment? +C. What are the top categories of pages that are identified with hateful content and content with +negative sentiment? +3. Sentiment analysis results +For sentiment analysis, the XLM-T multilingual sentiment analysis model from Cardiff NLP +(Barbieri et al 2021) is employed.2 The multilingual model is used as the Facebook textual dataset is a +2 See the model card and the full model available at the Huggingface repository: +https://huggingface.co/cardiffnlp/twitter-xlm-roberta-base-sentiment. Also check out their GitHub repository: +https://github.com/cardiffnlp/xlm-t and the paper https://arxiv.org/pdf/2104.12250v2.pdf. +1 See the following GitHub page for the Google Colab notebooks on the prediction tasks: +https://github.com/SaikatPhys/CrowdTangle-NLP. +2 + +mixture +of +Hindi +and +English +language +texts. +As +XLM-T +is +a +fine-tuned +model +of the +XLM-roBERTa-base model, which is trained on ~198M tweets and further fine-tuned for sentiment +analysis. According to the authors’ benchmarking of the XLM-T multilingual model’s performance on the +Hindi language “Unified Multilingual Sentiment Analysis Benchmark” (UMSAB) dataset, the model has +an F1 score of 56.39 (see Table 4 in Barbieri et al 2021). Although the performance of this model in the +Hindi language is comparatively poor with respect to other languages (e.g., German and English have F1 +scores of 77.35 and 70.63 respectively), the XLM-T model outperforms the XLM-R model by 7.9 +absolute points in sentiment analysis task. Moreover, particularly because the model is trained on a corpus +of social media datasets (i.e., tweets), we think this is relatively the best state-of-the-art open-source +language model for our task of analyzing the sentiment of the mixed-language CrowdTangle dataset. +From the results of the forward prediction, we see a distribution of 36.74% negative, 40.07% neutral and +23.19% positive sentiment content respectively within our Facebook dataset which we have scrapped +from CrowdTangle. The histogram in figure 1 demonstrates the percentages of these predicted labels. +Figure 1: Distribution of sentiment analysis labels as predicted by CardiffNLP’s XLM-T multilingual +sentiment analysis model. The model predicts 37% negative, 40% neutral and 23% positive sentiment +content within our CrowdTangle database. +In order to identify the most dominant actors that are responsible for creating the negative sentiment +content, the dataset of predicted 36.74% labels is split into the top 10 accounts (see figure 2). The top five +Facebook pages within the negative sentiment set are found to be “We Support Hindutva”, “Pushpendra +Kulshrestha Fans Club”, “I Am Proud To Be A Hindu”, “We support hindutva” and “Sanatan Press” +respectively. It’s noteworthy that several of these pages have identical names however, this does not +automatically imply identical ownership or identical content sharing behaviour. The top negative +sentiment exhibiting pages are self-declared promoter/supporters of Hindutva (a far-right ideology which +seeks to establish a Hindu nation in India) or supporters of Far-right political and ideological leaders. This +3 + +is also because our database is collected from a subset of 648 far-right pages that were first manually +selected. Within this limitation, we found the page Sanatan Press as one of the top 10 pages exhibiting +negative sentiments. It is noteworthy that Sanatan Press is the media wing of Sanatan Sanstha a far-right +organisation which has been identified as a dangerous organisation by Facebook and therefore banned on +their platforms (TIME 2021). +Figure 3 plots the top 10 page categories which share the negative sentiment content. Amongst the 648 +pages that were put through the scanner in this study, several of the pages self-identified as general +activity pages, meaning that Facebook’s content moderation policy should fully apply to these pages. Few +pages identified as political organisations, politicians, Media News company and News sites. It is +noteworthy that for political actors and news media, Facebook applies content moderation exceptions, +meaning thereby that content shared by these pages is not as rigorously held to the content moderation +policy. This has implications for automated and semi-automated content removal processes that we +separately discuss elsewhere. +Figure 2. Predicted 36.74% of negative sentiment content is split into the top 10 actors in the descending +order of their percentage of counts. +4 + +-----+Figure 3: Top 10 page categories which share 36.74% of negative sentiment content. +4. Hate speech detection results +To perform the hate speech detection task, the multilingual XTC model – a fine-tuned version of +the XLM-T model on a multilingual hate speech dataset, was employed (Röttger et al 2022). The XTC +model has an F1-score of 78.5 for the “hateful” label and an F1-score of 37.7 for the non-hateful label in +the Hindi language benchmark dataset (see Table 2 in Röttger et al 2022). These scores imply that +although the XTC model performs better in predicting hateful labels with fewer false positives, it can +predict a significant amount of false negatives. Alternatively stated, it can predict hateful texts to be +non-hateful. +In the current forward inference task, the XTC model predicts 11.71% of the text contents to be hateful +and 88.29% to be non-hateful. Given the limitations of the F1 scores of the XTC model, we suspect that +“non-hateful” content must be lesser in percentage than 88.29%. Because of the lower F1-score in the +non-hateful label, it’s probable that the XTC model has failed to rightly classify them as a “hateful” class. +However, a higher F1 score in the hateful class allows us to confidently claim that almost all of this +11.71% hateful content is correctly predicted as hateful. +5 + +Figure 4. The distribution of predicted labels from the hate speech analysis model. The multilingual +model predicts ~88% non-hateful and ~12% hateful content within the CrowdTangle dataset respectively +Figure 5. The predicted 11.71% of hateful content is further split into the top 10 actors in descending +order. +6 + +Figure 6. Top 10 page categories that share the 12% hateful content. +Table 1. Percentages of predicted labels in different categories of sentiment and hate (percentages are +calculated w.r.t. the total size of the dataset, N=375,922) +Labels +Negative sentiment +Neutral sentiment +Positive sentiment +Total +Non-hateful +28.12% +38.07% +22.10% +88.29% +Hateful +8.62% +2.00% +1.09% +11.71% +Total +36.74% +40.07% +23.19% +100% +Given that the F1 score for the hateful label is 78.5, we have furthermore inspected the top +accounts within our dataset that have shared the most hateful content (figure 5). The top five actors within +the hateful content set are, “We Support Hindutva”, “I Am Proud To Be A Hindu”, “Kapil Mishra Fans”, +”Akhand Bharat” and “Pushpendra Kulshrestha Fans Club”. Three out of the top five actors (namely, +“We Support Hindutva”, “I Am Proud To Be A Hindu” and “Pushpendra Kulshrestha Fans Club”) are +the same both in the sets of negative sentiment content and hateful datasets (see figures 2 and 5). As we +7 + +further see in Table 1, out of the entire set of 11.71% identified hateful content, a majority of 8.62% +belongs to the category of negative sentiment. However, the majority of negative sentiment content (i.e., +28.12% out of 36.74%) belongs to the non-hateful category. These statistics imply that negative sentiment +content doesn’t necessarily qualify to be hateful content but hateful content most likely qualifies to carry a +negative sentiment. However, a match between the list of top actors in both the categories of hateful +content and negative sentiment implies that most active perpetrators of negative sentiment content are +also spreaders of hate speech. +In order to investigate what types of page categories are mostly associated with hateful texts, we +have further split the hateful content into top page categories in Figure 6. If we compare the top 10 page +categories between hate content and negative sentiment content, we find that including the +ACTIVITY_GENERAL category, we have a match of altogether 9 categories (namely, COMMUNITY, +FAN_PAGE, +PERSONAL_BLOG, +PERSON, +POLITICIAN, +NEWS_SITE, +MEDIA_NEWS_COMPANY and RELIGIOUS_ORGANIZATION). This list shows what type of actors +and what kind of pages act as the main perpetrators of the hateful and negative sentiment content within +our dataset. +Table 2. Sample sizes of human annotations for validating the predicted labels of the hate speech model +with a confidence level of 95% and a 5% margin of error. +Labels +Non-hateful +(Population size = 331,892) +Hateful +(Population size = 44,030) +Total +Sample size +384 +381 +765 +In order to validate the accuracy of the hate speech model by human annotators, we used a set of +randomly sampled 765 messages from the CrowdTangle dataset and annotated ourselves (see Table 2). In +order for the annotations to be representative of the dataset, we used the number of model-predicted labels +as our population sizes and kept a benchmark confidence level of 95% with a 5% margin of error. By +employing these annotated labels as ground truth, we evaluated the XTC hate speech model’s +performance. During the process of annotation a non-binary scheme was adopted while allowed the +annotators to label the text as hateful, non-hateful, insufficient context, abusive and negative discourse. +The annotator had the ability to apply multiple labels to the text. Of the 765 datapoints which were +re-labled the annotators marked 14% of messages as having “insufficient context”; and 6% as “abusive”; +23% of messages were identified as having “negative discourse”. The rest of the dataset (~57%) was +annotated as simply hateful or non-hateful. For the confusion matrix and evaluation report we are only +using the data which marked as either hateful or non-hateful , as the model is unable to identify abusive, +or discursive context only the binary labeling is used to re-train the model for the purpose of this paper. +The confusion matrix from the evaluation is shown in figure 7. The evaluation report of the model +is shown in Table 3. The model shows F1-scores of 0.83 and 0.71 in the non-hateful and hateful labels +respectively, which implies that the model is performing better in the hate speech prediction task after +8 + +fine-tuning it through human annotation. Compared to the benchmark Hindi dataset used in the original +paper - which had F1-scores of 0.78 and 0.37 respectively for these two labels the new F1 score is now +0.83 and 0.71 respectively. This already shows that the model is performing slightly better upon fine +tunning, although inherently it still lacks nuances and contextual knowledge. +Figure 7. Normalised confusion matrix showing the performance of the hate speech model on annotated +evaluation dataset +Table 3. Evaluation report showing the performance of the hate speech model +Labels +Precision +Recall +F1-score +Non-hateful +1.00 +0.71 +0.83 +Hateful +0.55 +1.00 +0.71 +5. Discussion and conclusions +While there is an evident correlation between hateful content and negative sentiment, from Table +1 we see that the majority of the non-hateful content is actually neutral in sentiment (i.e., 38.07% of +88.29%). Within the non-hateful category, the amount of negative sentiment content dominates over the +amount of positive sentiment content (i.e, 28.12% compared to 22.10%) – which seems counter-intuitive. +Any evident correlation between positive sentiment and non-hateful content from such statistical +distributions of predicted labels can not be drawn. Given that the XTC model has an F1-score of 37.7 for +the non-hateful label (as we mentioned earlier in section 4), it is suspected that several hateful texts are +9 + +1.0 +0.8 +0.71 +0.29 +non-hate +0.6 +True label +0.4 +hate - +0.00 +1.00 +0.2 +- +0.0 +- +non-hate +hate +Predictedlabelwrongly classified as non-hateful by the model. However, to what extent these false negatives are +distributed amongst the three sentiment categories, is not quantifiable within this paper. +Factors that may contribute to the underperformance of the NLP models include definitions of +hate speech and the categorization of negative/neutral/positive sentiment labels itself – which are +implicitly encapsulated in the weights of the language models through the labelled datasets that were used +for training. Our findings suggest that the state of art open-source technologies in sentiment analysis and +hate speech analysis have an inherent limitation in the forward identification and detection of hate speech +and negative sentiments. This has implications for the automated removal of content from social media +platforms. While it is not known to the authors if similar predicting models are used by Very Large Social +Media Platforms like Facebook, Twitter, and Youtube, it can be drawn that smaller and newer platforms +which may rely on open-source technologies for sentiment analysis will encounter issues of false negative +detection and may not be able to swiftly remove hateful content and negative sentiments from their +platforms as effectively. This means that fully automated detection of hateful content especially in +non-English and multilingual databases is far from being in a mature and deployable technology state. +As the above findings suggest, there is a need to further fine-tune the current state-of-the-art +XLM-T class transformer models in popular downstream tasks like hate speech detection and sentiment +analysis – particularly in multilingual settings where the data consists of a mixed set of one Latin script +(i.e., English) and one Devanagari script language (i.e., Hindi). +Acknowledgements +The authors acknowledge the help received from the CrowdTangle – a Facebook-owned tool that tracks +interactions on public content from Facebook pages and groups, verified profiles, Instagram accounts, and +subreddits. The authors acknowledge the role of members of Stichting The London Story in constructive +discussion on this paper. +6. References +A. Belew and K. Massanari. 2018. “Social Movement Mobilization and Online Hate: A Case +Study of the Alternative Right”. +A. Matamoros-Fernández and J. Farkas. 2021. “Racism, Hate Speech, and Social Media: A +Systematic Review and Critique”. Television & New Media, 22(2), 205–224. +Billy Pergio. 2019 “Facebook Banned a Hindu Extremist Group—Then Left Most of Its Pages +Online for Months”. TIME June 9, 2021. Available at: +https://time.com/6072272/facebook-sanatan-sanstha-india/ +CrowdTangle Team (2022). CrowdTangle. Facebook, Menlo Park, California, United States. +F. Barbieri, L. E. Anke and J. Camacho-Collados. 2021. “XLM-T: Multilingual Language Models +in Twitter for Sentiment Analysis and Beyond”. arXiv:2104.12250v2. +P. Röttger, H. Seelawi, D. Nozza, Z. Talat and B. Vidgen. 2022. “Multilingual HateCheck: +Functional Tests for Multilingual Hate Speech Detection Models”. In Proceedings of the Sixth Workshop +on Online Abuse and Harms (WOAH), pages 154–169, Seattle, Washington (Hybrid). Association for +Computational Linguistics. +10 + +11 + diff --git a/N9FRT4oBgHgl3EQf4TgM/content/tmp_files/load_file.txt b/N9FRT4oBgHgl3EQf4TgM/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..6a78e35229d44c466351abdcf198974664e859ed --- /dev/null +++ b/N9FRT4oBgHgl3EQf4TgM/content/tmp_files/load_file.txt @@ -0,0 +1,251 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf,len=250 +page_content='Automated Sentiment and Hate Speech Analysis of Facebook Data by Employing Multilingual Transformer Models R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=' Manuvie 1, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=' Chatterjee 2 1University College Groningen, University of Groningen, r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content='manuvie@rug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content='nl 2Foundation The London Story, Fluwelen Burgwal 58, 2511CJ, Den Haag, The Netherlands, saikat@thelondonstory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content='org Abstract In recent years, there has been a heightened consensus – both within academia and in the public discourse – that Social Media Platforms (SMPs), amplify the spread of hateful and negative sentiment content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=' Researchers have identified how hateful content, political propaganda, and targeted messaging contributed to real-world harms including insurrections against democratically elected governments, genocide, and breakdown of social cohesion due to heightened negative discourse towards certain communities in parts of the world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=' To counter these issues, SMPs have created semi-automated systems that can help identify toxic speech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=' In this paper we analyse the statistical distribution of hateful and negative sentiment contents within a representative Facebook dataset (n= 604,703) scrapped through 648 public Facebook pages which identify themselves as proponents (and followers) of far-right Hindutva actors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=' These pages were identified manually using keyword searches on Facebook and on CrowdTangleand classified as far-right Hindutva pages based on page names, page descriptions, and discourses shared on these pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=' We employ state-of-the-art, open-source XLM-T multilingual transformer-based language models to perform sentiment and hate speech analysis of the textual contents shared on these pages over a period of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content='5 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=' The result shows the statistical distributions of the predicted sentiment and the hate speech labels;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=' top actors, and top page categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=' We further discuss the benchmark performances and limitations of these pre-trained language models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=' Introduction The spread of textual content with negative sentiment and community-targeted hate speech over Social Media Platforms (SMPs) like Facebook, Twitter, Reddit etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=' has become a matter of serious concern in recent years (see Belew and Massanari 2018, Matamoros-Fernández and Farkas 2021, Bennett and Segerberg 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=' Case studies demonstrate a range of analytical results on the organic growth of hateful content around certain sensitive topics like a state election, COVID-19, the immigration policy of a country etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=' as well as amplified growth through coordinated rapid link-sharing behaviour by actor networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=' In this study, our goal is to analyse the contents of a selected group of public Facebook fan pages (648 pages) which returned a total of 604,703 text messages on Facebook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=' The pages were manually identified as belonging to actors who are responsible for spreading anti-minority and anti-muslim narratives in contemporary India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=' An automated analysis of the textual contents associated 1 with these fan pages was conducted to test the performance of current state-of-the-art NLP models in predicting and identifying contents with negative sentiment and hate speech respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=' Open-source multilingual Natural Language Processing (NLP) models for the sentiment analysis (Barbieri et al 2021), and the hate speech identification task (Röttger et al 2022) were employed in their ‘evaluation’ mode in order to perform a forward prediction task over the social media dataset (n=604,703).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=' The approach allows a quantitative prediction of the amount of negative sentiment, and the amount of hateful content that is present in the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=' Based on the outcome of these experiments, the statistical significance behind the abundance of hateful and negative sentiment content on the Facebook platform is discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=' Section 2 of the paper describes the process of data selection, preparation of the datasets, and our main research questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=' Sections 3 and 4, discuss the results of sentiment and hate-speech analysis respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=' Section 5 is an extended discussion of our findings, concluding remarks and future scopes of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=' Methodology and the dataset The content analysis presented in this paper has been performed on Facebook’s historical dataset that was ingested using the CrowdTangle platform in the date range of 31-12-2016 and 01-07-2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=' 648 Facebook pages were identified using keyword search as pages actively spreading anti-muslim and anti-minority hate speech in India (see Manuvie and Chaterjee, 2023 on the arXiv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=' The final dataset consists of a total of 604,703 entries, on which state-of-the-art open-source sentiment analysis and hate-speech detection models for forward predictions of the sentiment and hatefulness labels respectively were applied (see sections 3 and 4 below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=' The PyTorch code that exploits the transformers libraries and the Huggingface NLP models for these two forward inference tasks are publicly available at our GitHub repository.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content='1 Based on the prediction scores of the NLP models, we investigate the following three main three research questions in this paper, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=' What are the distributions of sentiment and hate speech labels within our CrowdTangle dataset as predicted by the respective models?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=' Who are the top actors within our dataset who share hateful content and content with the negative sentiment?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=' What are the top categories of pages that are identified with hateful content and content with negative sentiment?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=' Sentiment analysis results For sentiment analysis, the XLM-T multilingual sentiment analysis model from Cardiff NLP (Barbieri et al 2021) is employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content='2 The multilingual model is used as the Facebook textual dataset is a 2 See the model card and the full model available at the Huggingface repository: https://huggingface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content='co/cardiffnlp/twitter-xlm-roberta-base-sentiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=' Also check out their GitHub repository: https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content='com/cardiffnlp/xlm-t and the paper https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content='org/pdf/2104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content='12250v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content='pdf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=' 1 See the following GitHub page for the Google Colab notebooks on the prediction tasks: https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content='com/SaikatPhys/CrowdTangle-NLP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=' 2 mixture of Hindi and English language texts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=' As XLM-T is a fine-tuned model of the XLM-roBERTa-base model, which is trained on ~198M tweets and further fine-tuned for sentiment analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=' According to the authors’ benchmarking of the XLM-T multilingual model’s performance on the Hindi language “Unified Multilingual Sentiment Analysis Benchmark” (UMSAB) dataset, the model has an F1 score of 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content='39 (see Table 4 in Barbieri et al 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=' Although the performance of this model in the Hindi language is comparatively poor with respect to other languages (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=', German and English have F1 scores of 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content='35 and 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content='63 respectively), the XLM-T model outperforms the XLM-R model by 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content='9 absolute points in sentiment analysis task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=' Moreover, particularly because the model is trained on a corpus of social media datasets (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=', tweets), we think this is relatively the best state-of-the-art open-source language model for our task of analyzing the sentiment of the mixed-language CrowdTangle dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=' From the results of the forward prediction, we see a distribution of 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content='74% negative, 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content='07% neutral and 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content='19% positive sentiment content respectively within our Facebook dataset which we have scrapped from CrowdTangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=' The histogram in figure 1 demonstrates the percentages of these predicted labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=' Figure 1: Distribution of sentiment analysis labels as predicted by CardiffNLP’s XLM-T multilingual sentiment analysis model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=' The model predicts 37% negative, 40% neutral and 23% positive sentiment content within our CrowdTangle database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=' In order to identify the most dominant actors that are responsible for creating the negative sentiment content, the dataset of predicted 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content='74% labels is split into the top 10 accounts (see figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=' The top five Facebook pages within the negative sentiment set are found to be “We Support Hindutva”, “Pushpendra Kulshrestha Fans Club”, “I Am Proud To Be A Hindu”, “We support hindutva” and “Sanatan Press” respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=' It’s noteworthy that several of these pages have identical names however, this does not automatically imply identical ownership or identical content sharing behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=' The top negative sentiment exhibiting pages are self-declared promoter/supporters of Hindutva (a far-right ideology which seeks to establish a Hindu nation in India) or supporters of Far-right political and ideological leaders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=' This 3 is also because our database is collected from a subset of 648 far-right pages that were first manually selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=' Within this limitation, we found the page Sanatan Press as one of the top 10 pages exhibiting negative sentiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=' It is noteworthy that Sanatan Press is the media wing of Sanatan Sanstha a far-right organisation which has been identified as a dangerous organisation by Facebook and therefore banned on their platforms (TIME 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=' Figure 3 plots the top 10 page categories which share the negative sentiment content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=' Amongst the 648 pages that were put through the scanner in this study, several of the pages self-identified as general activity pages, meaning that Facebook’s content moderation policy should fully apply to these pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=' Few pages identified as political organisations, politicians, Media News company and News sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=' It is noteworthy that for political actors and news media, Facebook applies content moderation exceptions, meaning thereby that content shared by these pages is not as rigorously held to the content moderation policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=' This has implications for automated and semi-automated content removal processes that we separately discuss elsewhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=' Predicted 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content='74% of negative sentiment content is split into the top 10 actors in the descending order of their percentage of counts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=' 4 -----+Figure 3: Top 10 page categories which share 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content='74% of negative sentiment content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=' Hate speech detection results To perform the hate speech detection task, the multilingual XTC model – a fine-tuned version of the XLM-T model on a multilingual hate speech dataset, was employed (Röttger et al 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=' The XTC model has an F1-score of 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content='5 for the “hateful” label and an F1-score of 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content='7 for the non-hateful label in the Hindi language benchmark dataset (see Table 2 in Röttger et al 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=' These scores imply that although the XTC model performs better in predicting hateful labels with fewer false positives, it can predict a significant amount of false negatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=' Alternatively stated, it can predict hateful texts to be non-hateful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=' In the current forward inference task, the XTC model predicts 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content='71% of the text contents to be hateful and 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content='29% to be non-hateful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=' Given the limitations of the F1 scores of the XTC model, we suspect that “non-hateful” content must be lesser in percentage than 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content='29%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=' Because of the lower F1-score in the non-hateful label, it’s probable that the XTC model has failed to rightly classify them as a “hateful” class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=' However, a higher F1 score in the hateful class allows us to confidently claim that almost all of this 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content='71% hateful content is correctly predicted as hateful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=' 5 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=' The distribution of predicted labels from the hate speech analysis model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=' The multilingual model predicts ~88% non-hateful and ~12% hateful content within the CrowdTangle dataset respectively Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=' The predicted 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content='71% of hateful content is further split into the top 10 actors in descending order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=' 6 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=' Top 10 page categories that share the 12% hateful content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=' Percentages of predicted labels in different categories of sentiment and hate (percentages are calculated w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=' the total size of the dataset, N=375,922) Labels Negative sentiment Neutral sentiment Positive sentiment Total Non-hateful 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content='12% 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content='07% 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content='10% 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content='29% Hateful 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content='62% 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content='00% 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content='09% 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content='71% Total 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content='74% 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content='07% 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content='19% 100% Given that the F1 score for the hateful label is 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content='5, we have furthermore inspected the top accounts within our dataset that have shared the most hateful content (figure 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=' The top five actors within the hateful content set are, “We Support Hindutva”, “I Am Proud To Be A Hindu”, “Kapil Mishra Fans”, ”Akhand Bharat” and “Pushpendra Kulshrestha Fans Club”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=' Three out of the top five actors (namely, “We Support Hindutva”, “I Am Proud To Be A Hindu” and “Pushpendra Kulshrestha Fans Club”) are the same both in the sets of negative sentiment content and hateful datasets (see figures 2 and 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=' As we 7 further see in Table 1, out of the entire set of 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content='71% identified hateful content, a majority of 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content='62% belongs to the category of negative sentiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=' However, the majority of negative sentiment content (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=', 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content='12% out of 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content='74%) belongs to the non-hateful category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=' These statistics imply that negative sentiment content doesn’t necessarily qualify to be hateful content but hateful content most likely qualifies to carry a negative sentiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=' However, a match between the list of top actors in both the categories of hateful content and negative sentiment implies that most active perpetrators of negative sentiment content are also spreaders of hate speech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=' In order to investigate what types of page categories are mostly associated with hateful texts, we have further split the hateful content into top page categories in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=' If we compare the top 10 page categories between hate content and negative sentiment content, we find that including the ACTIVITY_GENERAL category, we have a match of altogether 9 categories (namely, COMMUNITY, FAN_PAGE, PERSONAL_BLOG, PERSON, POLITICIAN, NEWS_SITE, MEDIA_NEWS_COMPANY and RELIGIOUS_ORGANIZATION).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=' This list shows what type of actors and what kind of pages act as the main perpetrators of the hateful and negative sentiment content within our dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=' Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=' Sample sizes of human annotations for validating the predicted labels of the hate speech model with a confidence level of 95% and a 5% margin of error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=' Labels Non-hateful (Population size = 331,892) Hateful (Population size = 44,030) Total Sample size 384 381 765 In order to validate the accuracy of the hate speech model by human annotators, we used a set of randomly sampled 765 messages from the CrowdTangle dataset and annotated ourselves (see Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=' In order for the annotations to be representative of the dataset, we used the number of model-predicted labels as our population sizes and kept a benchmark confidence level of 95% with a 5% margin of error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=' By employing these annotated labels as ground truth, we evaluated the XTC hate speech model’s performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=' During the process of annotation a non-binary scheme was adopted while allowed the annotators to label the text as hateful, non-hateful, insufficient context, abusive and negative discourse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=' The annotator had the ability to apply multiple labels to the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=' Of the 765 datapoints which were re-labled the annotators marked 14% of messages as having “insufficient context”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=' and 6% as “abusive”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=' 23% of messages were identified as having “negative discourse”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=' The rest of the dataset (~57%) was annotated as simply hateful or non-hateful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=' For the confusion matrix and evaluation report we are only using the data which marked as either hateful or non-hateful , as the model is unable to identify abusive, or discursive context only the binary labeling is used to re-train the model for the purpose of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=' The confusion matrix from the evaluation is shown in figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=' The evaluation report of the model is shown in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=' The model shows F1-scores of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content='83 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content='71 in the non-hateful and hateful labels respectively, which implies that the model is performing better in the hate speech prediction task after 8 fine-tuning it through human annotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=' Compared to the benchmark Hindi dataset used in the original paper - which had F1-scores of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content='78 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content='37 respectively for these two labels the new F1 score is now 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content='83 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content='71 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=' This already shows that the model is performing slightly better upon fine tunning, although inherently it still lacks nuances and contextual knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=' Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=' Normalised confusion matrix showing the performance of the hate speech model on annotated evaluation dataset Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=' Evaluation report showing the performance of the hate speech model Labels Precision Recall F1-score Non-hateful 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content='71 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content='83 Hateful 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content='55 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content='71 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=' Discussion and conclusions While there is an evident correlation between hateful content and negative sentiment, from Table 1 we see that the majority of the non-hateful content is actually neutral in sentiment (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=', 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content='07% of 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content='29%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=' Within the non-hateful category, the amount of negative sentiment content dominates over the amount of positive sentiment content (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content='e, 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content='12% compared to 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content='10%) – which seems counter-intuitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=' Any evident correlation between positive sentiment and non-hateful content from such statistical distributions of predicted labels can not be drawn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=' Given that the XTC model has an F1-score of 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content='7 for the non-hateful label (as we mentioned earlier in section 4), it is suspected that several hateful texts are 9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content='71 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content='29 non-hate 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content='6 True label 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content='4 hate - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content='0 non-hate hate Predictedlabelwrongly classified as non-hateful by the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=' However, to what extent these false negatives are distributed amongst the three sentiment categories, is not quantifiable within this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=' Factors that may contribute to the underperformance of the NLP models include definitions of hate speech and the categorization of negative/neutral/positive sentiment labels itself – which are implicitly encapsulated in the weights of the language models through the labelled datasets that were used for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=' Our findings suggest that the state of art open-source technologies in sentiment analysis and hate speech analysis have an inherent limitation in the forward identification and detection of hate speech and negative sentiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=' This has implications for the automated removal of content from social media platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=' While it is not known to the authors if similar predicting models are used by Very Large Social Media Platforms like Facebook, Twitter, and Youtube, it can be drawn that smaller and newer platforms which may rely on open-source technologies for sentiment analysis will encounter issues of false negative detection and may not be able to swiftly remove hateful content and negative sentiments from their platforms as effectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=' This means that fully automated detection of hateful content especially in non-English and multilingual databases is far from being in a mature and deployable technology state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=' As the above findings suggest, there is a need to further fine-tune the current state-of-the-art XLM-T class transformer models in popular downstream tasks like hate speech detection and sentiment analysis – particularly in multilingual settings where the data consists of a mixed set of one Latin script (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=', English) and one Devanagari script language (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=', Hindi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=' Acknowledgements The authors acknowledge the help received from the CrowdTangle – a Facebook-owned tool that tracks interactions on public content from Facebook pages and groups, verified profiles, Instagram accounts, and subreddits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=' The authors acknowledge the role of members of Stichting The London Story in constructive discussion on this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=' References A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=' Belew and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=' Massanari.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=' “Social Movement Mobilization and Online Hate: A Case Study of the Alternative Right”.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=' In Proceedings of the Sixth Workshop on Online Abuse and Harms (WOAH), pages 154–169, Seattle, Washington (Hybrid).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=' Association for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} +page_content=' 10 11' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQf4TgM/content/2301.13668v1.pdf'} diff --git a/NNFIT4oBgHgl3EQfcytL/content/tmp_files/2301.11267v1.pdf.txt b/NNFIT4oBgHgl3EQfcytL/content/tmp_files/2301.11267v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..6c476add7bbfb3f45226576712e168207f6815c6 --- /dev/null +++ b/NNFIT4oBgHgl3EQfcytL/content/tmp_files/2301.11267v1.pdf.txt @@ -0,0 +1,3644 @@ +Online Convex Optimization with Stochastic Constraints: +Zero Constraint Violation and Bandit Feedback +Yeongjong Kim 1 +Dabeen Lee 2 * +January 27, 2023 +Abstract +This paper studies online convex optimization with stochastic constraints. We propose a variant of the +drift-plus-penalty algorithm that guarantees O( +√ +T) expected regret and zero constraint violation, after a fixed +number of iterations, which improves the vanilla drift-plus-penalty method with O( +√ +T) constraint violation. Our +algorithm is oblivious to the length of the time horizon T, in contrast to the vanilla drift-plus-penalty method. +This is based on our novel drift lemma that provides time-varying bounds on the virtual queue drift and, as a +result, leads to time-varying bounds on the expected virtual queue length. Moreover, we extend our framework to +stochastic-constrained online convex optimization under two-point bandit feedback. We show that by adapting +our algorithmic framework to the bandit feedback setting, we may still achieve O( +√ +T) expected regret and zero +constraint violation, improving upon the previous work for the case of identical constraint functions. Numerical +results demonstrate our theoretical results. +1 +Introduction +Online convex optimization (OCO) is a general framework for modeling decision-making problems under uncertainty. +OCO can be viewed as a repeated game between a learner and an adversarial environment as follows. At each +iteration, the learner selects a decision without the knowledge of the convex loss function chosen by the environment, +after which the learner receives the loss associated with the decision. Based on the repeated interactions, the +learner adapts to the environment in real-time to minimize cumulative loss. The OCO framework is well-suited for +optimizing a large-scale complex system, as the problem is often tackled by decomposing it into small optimization +problems and solving each piece with limited information to improve tractability. Therefore, OCO is applied to +portfolio management [2], routing [5], display learning [10], recommendation systems [13], binary classification [8], +etc. For a comprehensive account of OCO, we refer the reader to [12, 19] and references therein. +1Department of Mathematical Sciences, KAIST, Daejeon 34126, Republic of Korea +2Department of Industrial and Systems Engineering, KAIST, Daejeon 34126, Republic of Korea +*Correspondence to +1 +arXiv:2301.11267v1 [math.OC] 26 Jan 2023 + +OCO with long-term constraints is an extension of OCO to deal with complex functional constraints [16] and +long-term budget restrictions [15]. When the set of available decisions is given by some complicated functions, +projection onto the feasible set can be difficult. Under such a scenario, finding a feasible solution at each iteration +can be expensive, although we may hope for achieving feasibility on average in the long run. For a resource planning +scenario, it may be possible to pool the budget for multiple periods, over which the budget allocation is flexible. +Motivated by this, we aggregate the constraint functions over the time periods and require satisfying the long-term +constraint aggregation. More precisely, we consider +min +x1,...,xT ∈X +T +� +t=1 +ft(xt) +s.t. +T +� +t=1 +gt(xt) ≤ 0 +(1) +where {ft}T +t=1 and {gt}T +t=1 are the loss and constraint functions chosen by the environment over T time steps while +{xt}T +t=1 is the sequence of decisions selected from a domain X by the learner. Here, the learner selects xt based on +the history up to time step t before observing ft and gt. Applications of OCO with long-term constraints include +online routing in wireless networks [17], multi-objective classification [6], and multi-armed bandits with knapsack +constraints [3]. +As the standard OCO framework, the performance of the learner can be measured by the notion of regret, defined as +regret(T) = +T +� +t=1 +ft(xt) − +T +� +t=1 +ft(x∗) +against some fixed benchmark decision x∗ and the long-term constraint violation given by +violation(T) = +T +� +t=1 +gt(xt). +Basically, the learner’s goal is to perform as well as the benchmark while satisfying the constraints in the long run by +minimizing the regret measure and trying to keep the constraint violation expression below zero. However, Mannor +et al. [17] found an example where {gt}T +t=1 is adversarially chosen and the benchmark x∗ is set to an optimal fixed +solution of (1), for which it is impossible to simultaneously bound the regret and constraint violation by a sublinear +function in T. Given this impossibility result, several works have studied some structured special cases. +Mahdavi et al. [16] considered the special case where gt = g for some fixed function g for all t ∈ [T], for which they +developed an augmented-Lagrangian-based method that achieves O( +√ +T) regret and O(T 3/4) constraint violation. +Jenatton et al. [14] generalized this result to O(T max{β,1−β}) regret and O(T 1−β/2) constraint violation by an +adaptive variant of the augmented Lagrangian algorithm parameterized by β ∈ (0, 1). Later, Yu and Neely [24] +gave an algorithm with O( +√ +T) regret and O(1) constraint violation, but it is not a first-order method. +Yu et al. [25] studied the case where gt’s are time-varying but are independent and identically distributed (i.i.d.) +with an unknown probability distribution, which subsumes the identical constraint function case discussed above. +They set the benchmark x∗ to be the best fixed solution minimizing the cumulative loss among the ones satisfying +the constraint in expectation, and they showed that their drift-plus-penalty (DPP) algorithm achieves O( +√ +T) +expected regret and O( +√ +T) expected constraint violation under Slater’s condition. Later, Wei et al. [22] proposed a +2 + +mirror-descent-type variant of DPP that achieves the same asymptotic performance under slightly more general +settings. +Besides these results on structured cases, there are more works regarding constrained OCO. Liakopoulos et al. +[15], Neely and Yu [18], Valls et al. [21] also consider problem (1) with adversarially chosen constraint functions +but they work over different benchmarks with more restrictions to guarantee sublinear regret and constraint violation +simultaneously. Guo et al. [11], Yi et al. [23], Yuan and Lamperski [26] studied the notion of cumulative constraint +violation, given by �T +t=1 [gt(xt)]+ where [a]+ = max{a, 0} over a ∈ R, instead of long-term constraints. +Note that algorithms for OCO use the information about loss and constraint functions and their gradients to proceed. +However, for some applications, we observe only the function values of the decisions and do not have access to the +functions and their gradients. This setting is called OCO with bandit feedback [9]. Agarwal et al. [1] introduced the +two-point bandit feedback setting, in which we can observe the function values at (at least) two points per iteration. +For the two-point feedback setting, we can achieve O( +√ +T) regret [1, 20]. Cao and Liu [7] studied long-term +constrained OCO with identical constraint functions under two-point bandit feedback and provided an algorithm +with O(T 1/2 ˜∆(T)1/2) regret and O(T 3/4 ˜∆(T)1/4) constraint violation where ˜∆(T) is some sublinear function in +T. +Our contributions +In this paper, we focus on OCO with stochastic constraints under the full information setting +and the two-point bandit feedback setting. We improve upon the results of [22, 25] in three aspects. +1. We develop a variant of the drift-plus-penalty algorithm achieving O( +√ +T) expected regret and zero expected +constraint violation, after a fixed number of iterations. +2. Our algorithm is completely oblivious to the length of the time horizon T, in contrast to the vanilla drift- +plus-penalty algorithm by Wei et al. [22], Yu et al. [25] which sets the penalty parameter V = +√ +T and the +step size parameter α = T. This development is based on our novel drift lemma that provides time-varying +bounds on the drift and leads to time-varying bounds on the expected virtual queue size. +3. We adapt our algorithm to the two-point bandit feedback setting and show that we may still guarantee O( +√ +T) +expected regret and zero expected constraint violation. +The result on stochastic-constrained OCO under two-point bandit feedback improves upon the work of Cao and Liu +[7] because our algorithm guarantees better bounds on regret and constraint violation for a strictly more general +setting. +2 +OCO with Stochastic Constraints +Let X ⊆ Rd be a known fixed compact convex set. Let f1, . . . , fT : Rd → R be a sequence of arbitrary convex +functions. Let ¯g(x) = Eω [g(x, ω)] : Rd → R be a function where g(x, ω) is convex with respect to x ∈ X and the +expectation is taken with ω ∈ Ω from an unknown distribution. Constraint functions g1, . . . , gT : Rd → R are given +by gt(x) := g(x, ωt) for t ∈ [T] where ω1, . . . , ωT are i.i.d. samples of ω. We assume that ft is independent of +ωs for s ≥ t + 1. +3 + +As in [25], we take the benchmark decision x∗ defined as an optimal solution to +min +x∈X +T +� +t=1 +ft(x) s.t. ¯g(x) = 1 +T E +� T +� +t=1 +gt(x) +� +≤ 0. +(2) +Then the goal is to design an algorithm that guarantees a sublinear regret against the benchmark x∗ and a sublinear +constraint violation at the same time. We focus on the single constraint setting for simplicity, but our framework +easily extends to multiple constraints. +Henceforth, we work over a norm ∥ · ∥ in Rd and its dual norm ∥ · ∥∗. Let Φ : C → R be a mirror map over +an open convex set C. We assume that X ⊆ C and consider the corresponding Bregman divergence defined as +D(x, y) = Φ(x) − Φ(y) − Φ(y)⊤(x − y) for any x, y ∈ X. +Assumption 1 (Basic assumptions). There exist positive constants Df, Dg, G, R satisfying the following. +• ∥∇ft(x)∥∗ ≤ Df for all t ∈ [T] and x ∈ X. +• ∥∇g(x, ω)∥∗ ≤ Dg, and |g(x, ω)| ≤ G for all x ∈ X and ω ∈ Ω. +• D(x, y) ≤ R2 for all x, y ∈ X. +For convenience, we assume that Φ is 2-strongly convex with respect to the norm ∥ · ∥, which implies that +∥x − y∥ ≤ +� +D(x, y) ≤ R +(3) +for all x, y ∈ X. +Assumption 2 (Slater’s condition). There exist a solution ˆx ∈ X and ϵ > 0 with ¯g(ˆx) = Eω[g(ˆx, ω)] ≤ −ϵ. +The framework of OCO with stochastic constraints has various applications in optimization and learning theory. +Here we discuss a few that are direct applications of the framework. As discussed before, a special case is when the +constraints functions are not time-varying, i.e., gt is given by gt = g for some fixed function g for all t. Another +direct application is stochastic constrained stochastic optimization, that is formulated as the following optimization +problem. +min +x∈X +¯f(x) = Eω [f(x, ω)] +s.t. +¯g(x) ≤ 0. +An iterative algorithm would obtain an i.i.d. sample ωt of ω at each iteration t and consider ft = ft(·, ωt) and +gt = gt(·, ωt). Given {xt}T +t=1, we may obtain ¯xT = (1/T) �T +t=1 xt. By Jensen’s inequality, the optimality +gap of ¯xT satisfies E +� ¯f(¯xT ) +� +− ¯f(x∗) ≤ E [regret(T)] /T, and the constraint violation is given by ¯g(¯xT ) = +T · E [violation(T)]. +3 +Conservative Drift-Plus-Penalty +3.1 +Algorithm Descriptions and Intuitions +We deduce our algorithm by combining the drift-plus-penalty algorithm in [22, 25] for OCO with stochastic +constraints and the conservative optimization method in [4] achieving zero constraint violation for stochastic +constrained stochastic optimization. +4 + +Algorithm 1 Conservative Drift-Plus-Penalty +Initialize: Initial iterates x1 ∈ X and Q1 = 0, step size parameters {αt}T +t=1, penalty parameters {Vt}T +t=1, and +conservatism parameters {γt}T +t=1. +for t = 1 to T do +Observe ft and gt. +Primal update: set xt+1 to be a solution to +min +x∈X +� +(Vt∇ft(xt) + Qt∇gt(xt))⊤ x + αtD(x, xt) +� +Dual update: set Qt+1 to +� +Qt + gt(xt) + ∇gt(xt)⊤(xt+1 − xt) + γt +� ++ +end for +Algorithm 1, which we refer to as Conservative Drift-Plus-Penalty (CDPP), follows the basic outline of the vanilla +DPP algorithm, while there are two main distinctions. +1. For the primal update, we allow penalty parameter Vt and step size parameter αt to be time-varying. In +contrast, Yu et al. [25] used fixed parameters given by Vt = +√ +T and αt = T for all t. Instead, we set for +t ≥ 1, +Vt = +√ +t +and +αt = t. +2. For the dual update, we use a new parameter γt and add it to the current virtual queue Qt in each step t. We +will set wt to +γt = C +√ +t +for some constant C. The dual update of the vanilla DPP method is the one with C = 0. +Note that our choice of parameters Vt and αt for primal updates is oblivious to the length of time horizon T, +providing flexibility for deciding when to stop the online learning procedure. To demonstrate that the time-varying +penalty and step size parameters still guarantee the desired performance, we refine and improve the analysis of the +DPP algorithm. +For the dual update, we introduce the new parameter γt to better control the (expected) long-term constraint +violation. The basic idea is to consider function gt(·)+γt instead of gt(·). Then, as the algorithm aims to control the +corresponding constraint violation �T +t=1 gt(xt) + �T +t=1 γt, this would encourage a further reduction in the actual +constraint violation �T +t=1 gt(xt). We call γt the conservatism parameter, indicating that we add extra conservatism +toward satisfying the long-term constraint. This idea of conservative optimization was used in [4]. +We will show that the introduction of parameter γt leads to a reduction of O(C +√ +T) in the (expected) long-term +constraint violation while incurring an additional (expected) regret of O(C +√ +T). As the vanilla DPP bounds the +long-term constraint violation and the regret by O( +√ +T), we may properly choose a value for C to achieve zero +long-term constraint violation while still bounding the regret by O( +√ +T). +5 + +Let us also mention that our algorithm, as well as the original DPP method, has a close connection to the online +primal-dual gradient method for the saddle-point problem. Consider +Lt(x, λ) = ft(x) + λ(gt(x) + γt) +for x ∈ X and λ ≥ 0 and the corresponding online primal-dual gradient update with step size 1/ +√ +2T given by +xt+1 = PX +� +xt − +1 +√ +2T +(∇ft(xt) + λt∇gt(xt)) +� +, +λt+1 = +� +λt + +1 +√ +2T +(gt(xt) + γt) +� ++ +where PX denote the Euclidean projection operator onto X. In fact, setting Qt = λt +√ +2T, the primal update of +online primal-dual gradient is equivalent to that of our algorithm with Vt = +√ +2T, αt = T, and Φ(·) = ∥ · ∥2 +2, in +which case D(x, y) = ∥x − y∥2 +2. On the other hand, the dual update of online primal-dual gradient translates to +Qt+1 = [Qt + gt(xt) + γt]+ . +Notice that our dual update has the additional term ∇gt(xt)⊤(xt+1 −xt), which is the distinctive component of the +drift-plus-penalty algorithm. Next, let us briefly elaborate on the intuition behind adding the term ∇gt(xt)⊤(xt+1− +xt) to the dual update. +Following Yu et al. [25], we may regard Qt as the size of a virtual queue at time t. Then we consider the associated +quadratic Lyapunov term Lt = Q2 +t/2 and study the corresponding drift given by ∆t = Lt+1−Lt = (Q2 +t+1−Q2 +t)/2. +Lemma 3.1. For t ≥ 1, +∆t ≤ Qt +� +gt(xt) + ∇gt(xt)⊤(xt+1 − xt) + γt +� ++ (G + DgR + γt)2. +The upper bound on the drift ∆t provided by Lemma 3.1 has the (only) term Qt∇gt(xt)⊤(xt+1 −xt) that depends +on the next iterate xt+1. Hence, by choosing xt+1 that minimizes Qt∇gt(xt)⊤(xt+1 − xt), we may attempt to +control the drift. In fact, the primal update of CDPP sets xt+1 to be the minimizer of +Qt∇gt(xt)⊤(x − xt) +� +�� +� +drift ++ Vt∇ft(xt)⊤(x − xt) + αtD(x, xt) +� +�� +� +penalty +over X. Consequently, at each iteration, we get to choose a solution that minimizes the drift term ∆t and a penalty +term for controlling the objective simultaneously. +3.2 +Performance of Conservative Drift-Plus-Penalty +In this section, we show that the expected regret and constraint violation under CDPP are bounded by O( +√ +T) and +0, respectively. +The following two lemmas provide upper bounds on the expected regret and constraint violation under CDPP. The +bounds exhibit the dependence on the conservative parameter C and the expected virtual queue length E [Qt]. +6 + +Lemma 3.2. Under CDPP (Algorithm 1), the expected regret E [regret(T)] is at most +�C(G + DgR + C) +ϵ +�2 ++ K1 +√ +T + C +T +� +t=1 +E[Qt] +t +for some positive constant K1 that depends only on G, Dg, Df, R, ϵ. +Lemma 3.3. Under CDPP (Algorithm 1), the expected constraint violation E [violation(T)] is at most +DfDg +√ +T + E[QT +1] + D2 +g +2 +T +� +t=1 +E[Qt] +t +− C +√ +T. +Given Lemmas 3.2 and 3.3, what remains is to bound the expected virtual queue length E [Qt], which is carried out +based on the following lemma. Before stating the lemma, let us define filtration {Ft : t ≥ 0} where F0 = {∅, Ω} +and Ft = σ(ω1, . . . , ωt) being the σ-algebra generated by the set of random samples {ω1, . . . , ωt}. Note that xt +and Qt are Ft−1-measurable for all t ≥ 1. +Lemma 3.4. Let t ≥ 1, and let 1 ≤ τ ≤ min +�√ +T, t + 1 +� +. In addition, let θt(τ) be defined as +2(G + DgR)τ + 8R2αt +ϵτ ++ 8DfRVt +ϵ ++ 4(G + DgR + ϵ)2 +ϵ +. +Then the following statements hold. +(a) For any t ≥ 1, Qt ≤ C(2C/ϵ)2 + (G + ϵ)t. +(b) For t ≥ (2C/ϵ)2, Qt+1 − Qt ≤ G + ϵ. +(c) For t ≥ (2C/ϵ)2, +E [Qt+τ − Qt | Ft−1] ≤ +� +� +� +(G + ϵ)τ, +if Qt ≤ θt(τ) +−(ϵ/4)τ, +if Qt > θt(τ) +. +Here, as αt and Vt increase with t, so does θt(τ). Note that Lemma 3.4 is a refinement of the drift lemma [25, +Lemma 7]. The parameter τ can be any number less than or equal to +√ +T and t + 1, in contrast to the previous work +where τ is fixed at +√ +T. Furthermore, we use the drift radius θt(τ) that varies over time. Based on Lemma 3.4, we +show the following lemma that provides a time-varying bound on the expected virtual queue size. +Lemma 3.5. For any t ≥ 1, +E [Qt] ≤ θt(⌈ +√ +t⌉) + 4(G + ϵ) +√ +t + log 128(G + ϵ)2 +ϵ2 ++ C (2C/ϵ)2 + (G + ϵ) +� +9 + (2C/ϵ)4� +. +Plugging in this bound on E [Qt] to Lemmas 3.2 and 3.3, we deduce the proposed bounds on the regret and constraint +violation under CDPP. +Theorem 3.6. For any constant C greater than or equal to +DfDg + (4D2 +g + 8) +� +2G + DgR + ϵ + 2R2 + 2DfR +ϵ +� ++ 1, +7 + +there exists a constant T1 that depends only on Df, Dg, G, R, ϵ such that for any T ≥ T1, we have +E +� T +� +t=1 +gt(xt) +� +≤ 0 +under CDPP (Algorithm 1). +Theorem 3.7. For any choice of constant C, +E +� T +� +t=1 +ft(xt) − +T +� +t=1 +ft(x∗) +� += O( +√ +T) +under CDPP (Algorithm 1). +4 +Two-Point Bandit Feedback +4.1 +Problem Setting and Bandit Drift-Plus-Penalty +As an extension of stochastic constrained OCO studied in Section 3, we consider the two-point bandit optimization +setting proposed in [1]. The extension still follows the basic setup described in Section 2. In particular, we obtain +an adversarial sequence of functions f1, . . . , fT while the constraint functions g1, . . . , gT are i.i.d. realizations of +g(·, ω). Moreover, we use the same notions of regret and constraint violation. +In the previous setup, referred to as the full information setting, we observe not only the function values ft(xt) +and gt(xt) but also the gradients ∇ft(xt) and ∇gt(xt). In contrast, under the bandit optimization setting, we do +not have access to the gradients. Instead, we may take two points yt, zt ∈ X and receive their function values +ft(yt), gt(yt) and ft(zt), gt(zt) in each time slot t. From these bandit feedback, we estimate the gradients ∇ft(xt) +and ∇gt(xt). +Our algorithm for the bandit setting, which we call Bandit Drift-Plus-Penalty (BDPP, Algorithm 2), is a modification +of CDPP (Algorithm 1) based on the framework of Shamir [20]. For a simpler presentation of our results, we assume +that we may observe gt(xt) for t ∈ [T] as well, but we may replace gt(xt) with gt(xt + δtut) or gt(xt − δtut) +and still obtain the same asymptotic performance guarantees. +In Algorithm 2, ˜∇ft and ˜∇gt are the estimates of ∇ft(xt) and ∇gt(xt) computed using the two-point feedback +on ft and gt, respectively. The basic outline of Algorithm 2 is the same as that of Algorithm 1 with parameters +Vt = +√ +t, αt = t, and γt = C/ +√ +t, although we would need to set a different value for the constant C. In addition, +we will set the radius parameter as +δt = 1 +√ +t. +On top of Assumptions 1 and 2, we assume that functions ft, gt are Lipschitz continuous in the ℓ2-norm and that the +fourth moment of ut is bounded, as in [20]. +Assumption 3. There exist positive constants Lf, Lg, p∗ satisfying the following. +• ∥∇ft(x)∥2 ≤ Lf for all t ∈ [T] and x ∈ X. +8 + +Algorithm 2 Bandit Drift-Plus-Penalty +Initialize: Initial iterates x1 ∈ X, Q1 = 0, step size parameters {αt}T +t=1, penalty parameters {Vt}T +t=1, conser- +vatism parameters {γt}T +t=1, and radius parameters {δt}T +t=1. +for t = 1 to T do +Sample ut from +� +u ∈ Rd : ∥u∥2 = 1 +� +uniformly at random. +Observe ft(x) and gt(x) for x ∈ {xt ± δtut} and gt(xt). +Set ˜∇ft and ˜∇gt as +˜∇ft := d +2δt +(ft(xt + δtut) − ft(xt − δtut)) ut +˜∇gt := d +2δt +(gt(xt + δtut) − gt(xt − δtut)) ut +Primal update: set xt+1 to be a solution to +min +x∈X +�� +Vt ˜∇ft + Qt ˜∇gt +�⊤ x + αtD(x, xt) +� +Dual update: set Qt+1 to +� +Qt + gt(xt) + ˜∇g⊤ +t (xt+1 − xt) + γt +� ++ +end for +• ∥∇g(x, ω)∥2 ≤ Lg for all x ∈ X and ω ∈ Ω. +• Eut +� +∥ut∥4 +∗ +� +≤ p4 +∗ for all t ∈ [T]. +When ∥ · ∥ is the ℓ2-norm, we may set Lf = Df, Lg = Dg, and p∗ = 1. If ∥ · ∥ is the ℓ1-norm, we may set +Lf = Df +√ +d and Lg = Dg +√ +d. Moreover, p∗ can be set p +� +log d/d for some constant p when ∥ · ∥ is the ℓ1-norm +[20, Lemma 4]. +4.2 +Performance of Bandit Drift-Plus-Penalty +The distinction between Algorithm 1 and Algorithm 2 is that the latter uses stochastic estimates ˜∇ft and ˜∇gt instead +of the true gradients ∇ft(xt) and ∇gt(xt). To adapt the analysis of CDPP to the bandit setting, the first step is to +bound ∥ ˜∇ft∥∗ and ∥ ˜∇gt∥∗. Regarding this, the following lemma provides a deterministic bound on ∥ ˜∇gt∥∗. We +derive the lemma based on the assumption that gt is Lg-Lipschitz continuous in the ℓ2-norm. +Lemma 4.1. For any t ≥ 1, ∥ ˜∇gt∥∗ ≤ Lgℓ where ℓ is defined as ℓ = d · sup {∥u∥∗ : ∥u∥2 ≤ 1}. +Note that Lgℓ = Dgd if ∥ · ∥ is the ℓ2-norm and Lgℓ = Dgd +√ +d if ∥ · ∥ is the ℓ1-norm. Lemma 4.1 implies +a deterministic bound on ∥ ˜∇gt∥2 +∗, that is, ∥ ˜∇gt∥2 +∗ ≤ L2 +gℓ2, but this bound has a (super)quadratic dependence +on the dimension d. The next lemma shows that in expectation, ∥ ˜∇gt∥2 +∗ can be bounded by a function that has +a strictly lower dependence on the dimension d than L2 +gℓ2. Before we state the lemma, let us define filtration +{H0 +t : t ≥ 0} where H0 +0 = {∅, Ω} and H0 +t is the σ-algebra generated by the set of random samples {ω1, . . . , ωt} ∪ +9 + +{u1, . . . , ut−1}. +Lemma 4.2. [20, Lemma 9] There is a positive constant q that does not depend on d such that for any t ≥ 1, +E +� +∥ ˜∇ft∥2 +∗ | H0 +t +� +≤ qdp2 +∗L2 +f +and +E +� +∥ ˜∇gt∥2 +∗ | H0 +t +� +≤ qdp2 +∗L2 +g. +In particular, when ∥ · ∥ is the ℓ2 norm, the expected bounds in Lemma 4.2 are O(D2 +fd) and O(D2 +gd). When ∥ · ∥ is +the ℓ1 norm, the expected bounds are O(D2 +fd log d) and O(D2 +gd log d). Hereinafter, we treat the dimension d as a +constant to focus on the dependence on the time length T. +Next, as for CDPP, we prove upper bounds on the expected regret and constraint violation under Bandit Drift-Plus- +Penalty that exhibit the dependence on the parameter C and the expected virtual queue length E [Qt]. +Lemma 4.3. Under BDPP (Algorithm 2), the expected regret E [regret(T)] is at most +�C(G + C) +ϵ +�2 ++ K2 +√ +T + (C + 2Lg) +T +� +t=1 +E[Qt] +t +for some positive constant K2 that depends only on G, Dg, Df, R, ϵ and Lg, Lf, q, d, p∗. +Lemma 4.4. Under BDPP (Algorithm 2), +T +� +t=1 +gt(xt) ≤ QT +1 + +T +� +t=1 +∥ ˜∇gt∥2 +∗Qt +2t ++ +T +� +t=1 +∥ ˜∇gt∥2 +∗ + ∥ ˜∇ft∥2 +∗ +4 +√ +t +− C +√ +T. +Note that the expectation of the right-hand side of the inequality in Lemma 4.4 can be bounded using Lemma 4.2, +providing an upper bound on the expected constraint violation under BDPP. Given Lemmas 4.3 and 4.4, the +remaining task is to bound the expected virtual queue length E [Qt]. +As Lemma 3.4, we prove a drift lemma for the bandit setting, which shows time-varying bounds on the drift. In +addition, we define filtration {Ht : t ≥ 0} where H0 = {∅, Ω} and Ht is the σ-algebra generated by the set of +random samples {ω1, . . . , ωt} ∪ {u1, . . . , ut}. +Lemma 4.5. Let t ≥ 1, and let 1 ≤ τ ≤ min +�√ +T, t + 1 +� +. In addition, let θt(τ) be defined as +(2G + R2 + qdp2 +∗L2 +g)τ + 8R2αt +ϵτ ++ +4(R2 + qdp2 +∗L2 +f)Vt +ϵ ++ 4 +� +(G + ϵ)2 + R2qdp2 +∗L2 +g +� +ϵ +. +Then the following statements hold. +(a) For any t ≥ 1, +Qt ≤ C(2(C + 2Lg)/ϵ)2 + (G + RLgℓ + ϵ)t. +(b) For t ≥ (2(C + 2Lg)/ϵ)2, +Qt+1 − Qt ≤ G + RLgℓ + ϵ. +(c) For t ≥ (2(C + 2Lg)/ϵ)2, +E [Qt+τ − Qt | Ht−1] ≤ +� +� +� +(G + R2/2 + qdp2 +∗L2 +g/2 + ϵ)τ, +if Qt ≤ θt(τ) +−(ϵ/4)τ, +if Qt > θt(τ) +. +10 + +Note that the drift radius and the drift upper bounds have a dependence on the dimension d, which is due to the usage +of stochastic estimates ˜∇ft and ˜∇gt. In particular, the deterministic bounds (parts (a) and (b)) rely on Lemma 4.1, +while the stochastic drift bound (part (c)) uses Lemma 4.2. +Based on the drift lemma (Lemma 4.5), we deduce a time-varying bound on the expected virtual queue length +E [Qt]. +Lemma 4.6. For any t ≥ 1, E [Qt] is bounded above by +θt(⌈ +√ +t⌉) + 4(G + RLgℓ + ϵ) +√ +t + log 128(G + RLgℓ + ϵ)2 +ϵ2 ++ C (2(C + 2Lg)/ϵ)2 + (G + RLgℓ + ϵ) +� +9 + (2(C + 2Lg)/ϵ)4� +. +In comparison to the full information setting (Lemma 3.5), the bound on E [Qt] grows as the dimension d gets large, +and the growth rate is given by ℓ. +Theorem 4.7. For any constant C greater than or equal to +(2qdp2 +∗L2 +g + 4) +� +4G + R2 + qdp2 +∗L2 +g + 2ϵ + 2RLgℓ + +12R2 + 4qdp2 +∗L2 +f +ϵ +� ++ +qdp2 +∗(L2 +f + L2 +g) +2 ++ 1. +Then there exists a constant T2 that depends only on Df, Dg, G, R, ϵ and Lf, Lg, q, p∗, d, ℓ such that for any +T ≥ T2, we have +E +� T +� +t=1 +gt(xt) +� +≤ 0 +under BDPP (Algorithm 2). +Theorem 4.8. For any choice of constant C, +E +� T +� +t=1 +ft(xt) − +T +� +t=1 +ft(x∗) +� += O( +√ +T) +under BDPP (Algorithm 2). +5 +Numerical Experiments +We test the performance of our conservative drift-plus-penalty algorithm (Algorithm 1). We consider an online +scheduling problem with loss and constraint functions are given by +ft(x) = c⊤ +t x +and +gt(x) = nt − +15 +� +i=1 +h(xi) +where x ∈ R15 represents the vector of power assigned to 15 locations, ct ∈ R15 ++ is the vector of per unit power +electricity costs at time t, nt is the i.i.d. random number of jobs given at time slot t, and h is a concave function +representing the number of jobs that can be done by power allocation x. We compare the following methods. +• DPP: the vanilla drift-plus-penalty algorithm due to Yu et al. [25] with Vt = +√ +T, αt = T, and γt = 0, +11 + +Figure 1: Average electricity market price +(a) Regret +(b) Constraint violation +Figure 2: Regret and constraint violation under DPP and CDPP with various values of C for the electricity market +price data +• CDPP: our conservative drift-plus-penalty algorithm (Algorithm 1) with Vt = +√ +t, αt = t, and γt = C/ +√ +t. +We test various values for C ∈ {0, 5, 10, 20, 50}. +For the first set of experiments, we use the electricity cost data for vectors {ct}T +t=1 from New York ISO (http: +//www.nyiso.com/), following the experiment setup in [25]. The data consist of electricity costs at 15 different +zones every 5 minutes during 2022/12/01-2023/01/20. This corresponds to T ≃ 15, 000. The average price across +15 zones during this period is shown in Fig. 1. +For the second set of experiments, we synthetically generate random cost vectors. We obtain T = 100, 000 cost +vectors whose coordinates are sampled from the normal distribution with a mean of 30 and standard deviation of +10, followed by projection onto [0, 60]. Under this setting, as cost vectors ct are i.i.d., the loss functions are not +adversarial but stochastic. +We further assume that the power allocation vector x is contained in the set X = [0, 10]15 due to some hardware +restriction and that h(x) = log(1 + x). In addition, nt is sampled from the Poisson distribution with parameter 20 +followed by projection onto [0, 35]. We use the ℓ2-norm, in which case D(x, y) = ∥x − y∥2 +2. +12 + +Electricity market price +4000 +3000 +Price +1400 +I +0 +2400 +4000 +6+00 +8+00 +1200014000 +Time slts (each 5 min)1e7 +Regret +0.0 +DPP +CDPP (C=0) +CDPP (C=5) +CDPP (C=10) +0.5 +CDPP (C=20) +CDPP (C=50) +-1.0 +-1.5 +2.0 +0 +2000 +4000 +6000 +8000 +10000 +12000 +14000 +Time slots (each5 min)Constaint violatian +DPP +40000 +CDPP (C=0) +CDPP (C=5) +CDPP (C=10) +DODE +CDPP (C=20) +CDPP (C=50) +24000 +0 +0 +2400 +4000 +6400 +8:+00 +12000 14000 +Time slcts (each 5 min)(a) Regret +(b) Constraint violation +Figure 3: Regret and constraint violation under DPP and CDPP with various values of C for synthetic data +Results from the experiments are summarized in Fig. 2 and 3. Fig. 2 show the results from the experiments with the +real data, while Fig. 3 is for the second set of experiments with synthetic data. +Fig. 2a and 3a show that increasing C results in a higher regret from CDPP, conforming to the theory. Comparing +DPP and CDPP, we observe that DPP incurs the smallest regret at the beginning but the regret of CDPP with a small +C value (0,5,10) later becomes smaller than that of DPP. +We observe from Fig. 2b that increasing the conservatism parameter C reduces constraint violation. In fact, CDPP +with C = 0 incurs a smaller amount of constraint violation than DPP, which perhaps indicates that using time-varing +penalty parameter Vt and step size parameter αt is helpful for decreasing constraint violation. As in the real data +setting, Fig. 3b for the synthetic data setting shows that DPP incurs the largest constraint violation while CDPP with +a higher C has a smaller constraint violation. +Looking at Fig. 2a and 3a, it may seem strange at first glance that the regret goes below zero. This is because the +benchmark x∗ is an optimal solution of (2) not (1). On the other hand, the regret and constraint violation under +CDPP with C = 50 stay close to zero. This is perhaps because, when C gets large, the decisions made by CDPP +more likely to satisfy the constraint in (2). +Lastly, we briefly discuss the non-uniform growth pattern in Fig. 2b. This is perhaps due to the irregular changes in +cost vectors, shown in Fig. 1, resulting in abrupt changes in the loss functions. More precisely, when ct soars at +t, the algorithms would set xt+1 small accordingly because ct = ∇ft(xt). Then gt+1(xt+1) would be large in +response. +6 +Conclusion +This paper studies online convex optimization with stochastic constraints and develops a varaint of the drift-plus- +penalty algorithm that achieves zero constraint violation in expectation while still bounding the expected regret +by O( +√ +T). The algorithm is oblivious to the time horizon length T, in contrast to the vanilla drift-plus-penalty +algorithm. The numerical experiments show that using the time-varying algorithm parameters and the conservatism +13 + +1e6 +Regret +0.0 +DPP +CDPP (C=0) +0.5 +CDPP (C=5) +1.0 +CDPP (C=10) +CDPP (C=20) +1.5 +CDPP (C=50) +2.0 +2.5 +3.0 +3.5 +0 +DO +40000 +84000 +Time slctsConstraint violatian +ADOSE +DPP +CDPP (C=0) +30000 +CDPP (C=5) +CDPP (C=10) +25000 +CDPP (C=20) +CDPP (C=50) +20000 +1500D +10000 +5000 +0 - +0 +20000 +4000D +8000 +Time slctsparameter leads to a significant reduction in constraint violation, without much sacrifice in regret. We also consider +the two-point bandit feedback setting, for which we deduce the same asymptotic bounds on the expected regret and +constraint violation. +References +[1] Alekh Agarwal, Ofer Dekel, and Lin Xiao. Optimal algorithms for online convex optimization with multi-point +bandit feedback. In Conference on Learning Theory (COLT), pages 28–40, 12 2010. +[2] Amit Agarwal, Elad Hazan, Satyen Kale, and Robert E. Schapire. 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URL https://proceedings. +neurips.cc/paper/2018/file/9cb9ed4f35cf7c2f295cc2bc6f732a84-Paper.pdf. +16 + +A +Performance Analysis of Conservative Drift-Plus-Penalty (Algorithm 1) +A.1 +Proof of Lemma 3.1: Basic Upper Bound on the Drift +As Qt+1 = max +� +Qt + gt(xt) + ∇gt(xt)⊤(xt+1 − xt) + γt, 0 +� +, +Q2 +t+1 ≤ +� +Qt + gt(xt) + ∇gt(xt)⊤(xt+1 − xt) + γt +�2 . +Expanding the right-hand side, we obtain +∆t = Q2 +t+1 +2 +− Q2 +t +2 ≤ Qt(gt(xt) + ∇gt(xt)⊤(xt+1 − xt) + γt) + 1 +2 +� +gt(xt) + ∇gt(xt)⊤(xt+1 − xt) + γt +�2 . +Here, +� +gt(xt) + ∇gt(xt)⊤(xt+1 − xt) + γt +�2 ≤ (|gt(xt)| + ∥∇gt(xt)∥∗∥xt+1 − xt∥ + γt)2 +≤ (G + DgR + γt)2 +where the second inequality is due to Assumption 1. Therefore, +∆t ≤ Qt(gt(xt) + ∇gt(xt)⊤(xt+1 − xt) + γt) + (G + DgR + γt)2, +as required. +A.2 +Proof of Lemma 3.2: Providing an Upper Bound on the Expected Regret +We will use the following useful lemma to prove Lemma 3.2. +Lemma A.1. [22, Equation (22)] For any x ∈ X and t ≥ 1, +(Vt∇ft(xt) + Qt∇gt(xt))⊤ (xt+1 − xt) + αtD(xt+1, xt) +≤ (Vt∇ft(xt) + Qt∇gt(xt))⊤ (x − xt) + αtD(x, xt) − αtD(x, xt+1). +Using Lemma A.1, we will show that +E +� T +� +t=1 +ft(xt) − +T +� +t=1 +ft(x∗) +� +≤ αT +VT +R2 + +T +� +t=1 +VtD2 +f +4αt ++ +T +� +t=1 +γt +Vt +E[Qt] ++ +�C(G + DgR + C) +ϵ +�2 ++ +T +� +t=1 +(G + DgR + ϵ)2 +Vt +(4) +holds. We know from basic calculus that +√ +T ≤ +T +� +t=1 +1 +√ +t ≤ 2 +√ +T, +and therefore, (4) implies that +E +� T +� +t=1 +ft(xt) − +T +� +t=1 +ft(x∗) +� +≤ +�C(G + DgR + C) +ϵ +�2 ++ K +√ +T + C +T +� +t=1 +E[Qt] +t +17 + +where +K = R2 + +D2 +f +2 + 2(G + DgR + ϵ)2. +Now let us prove that (4) holds. Adding Vtft(xt) + Qtgt(xt) to both sides of the inequality given in Lemma A.1, +we obtain +Vtft(xt) + Qtgt(xt) + (Vt∇ft(xt) + Qt∇gt(xt))⊤ (xt+1 − xt) + αtD(xt+1, xt) +≤ Vtft(xt) + Qtgt(xt) + (Vt∇ft(xt) + Qt∇gt(xt))⊤ (x − xt) + αtD(x, xt) − αtD(x, xt+1). +(5) +Here, as ft and gt are convex, we have ft(xt) + ∇ft(xt)⊤(x − xt) ≤ ft(x) and gt(xt) + ∇gt(xt)⊤(x − xt) ≤ +gt(x). Then the right-hand side of (5) is bounded above by +Vtft(x) + Qtgt(x) + αtD(x, xt) − αtD(x, xt+1). +Moreover, by Lemma 3.1, the left-hand side of (5) is greater than or equal to +Vtft(xt) + Vt∇ft(xt)⊤(xt+1 − xt) + αtD(xt+1, xt) + ∆t − Qtγt − (G + DgR + γt)2. +Therefore, it follows that +Vtft(xt) ≤ Vtft(x) + Qtgt(x) + αtD(x, xt) − αtD(x, xt+1) +− Vt∇ft(xt)⊤(xt+1 − xt) − αtD(xt+1, xt) − ∆t + Qtγt + (G + DgR + γt)2. +(6) +In the right-hand side of (6), the part −Vt∇ft(xt)⊤(xt+1 − xt) − αtD(xt+1, xt) can be bounded as follows. +−Vt∇ft(xt)⊤(xt+1 − xt) − αtD(xt+1, xt) ≤ Vt∥∇ft(xt)∥∗∥xt+1 − xt∥ − αt∥xt+1 − xt∥2 +≤ VtDf∥xt+1 − xt∥ − αt∥xt+1 − xt∥2 += +V 2 +t D2 +f +4αt +− αt +�VtDf +2αt +− ∥xt+1 − xt∥ +�2 +≤ +V 2 +t D2 +f +4αt +(7) +where the first inequality holds due to u⊤v ≤ ∥u∥∗∥v∥ and (3) while the second inequality is from Assumption 1. +Then (6) and (7) imply that +Vtft(xt) ≤ Vtft(x) + Qtgt(x) + αtD(x, xt) − αtD(x, xt+1) + +V 2 +t D2 +f +4αt +− ∆t + Qtγt + (G + DgR + γt)2. +(8) +Dividing both sides of (8) and summing the resulting inequality for t = 1, . . . , T, we obtain the following inequality. +T +� +t=1 +ft(xt) ≤ +T +� +t=1 +ft(x) + +T +� +t=1 +Qt +Vt +gt(x) + +T +� +t=1 +αt +Vt +(D(x, xt) − D(x, xt+1)) ++ +T +� +t=1 +VtD2 +f +4αt +− +T +� +t=1 +∆t +Vt ++ +T +� +t=1 +Qtγt +Vt ++ +T +� +t=1 +(G + DgR + γt)2 +Vt +. +(9) +18 + +What remains is to derive the desired upper bound on the right-hand side of (9). First, +E +�Qt +Vt +gt(x∗) +� += E +� +E +�Qt +Vt +gt(x∗) | Ft−1 +�� += E +�Qt +Vt +E [gt(x∗) | Ft−1] +� += E +�Qt +Vt +¯g(x∗) +� +≤ 0 +(10) +where the first equality comes from the tower rule, the second equality is because Qt is Ft−1-measurable, and the +last equality holds as ωt is independent of {ω1, . . . ωt−1}. Next, +T +� +t=1 +αt +Vt +(D(x, xt) − D(x, xt+1)) = α1 +V1 +D(x, x1) + +T +� +t=2 +D(x, xt) +�αt +Vt +− αt−1 +Vt−1 +� +− αT +VT +D(x, xT +1) +≤ α1 +V1 +R2 + +T +� +t=2 +R2 +�αt +Vt +− αt−1 +Vt−1 +� += αT +VT +R2 +(11) +where the second inequality holds because αt/Vt − αt−1/Vt−1 = +√ +t − √t − 1 > 0 and D(x, xt) ≤ R2. +Furthermore, +T +� +t=1 +∆t +Vt += 1 +2 +T +� +t=1 +1 +Vt +(Q2 +t+1 − Q2 +t) += − 1 +2V1 +Q2 +1 + +1 +2VT +Q2 +T +1 + 1 +2 +T +� +t=2 +Q2 +t +� +1 +Vt−1 +− 1 +Vt +� +≥ − 1 +2V1 +Q2 +1 += 0 +(12) +where the inequality holds because Qt ≥ 0 for all t ≥ 0 and 1/Vt−1 − 1/Vt = 1/√t − 1 − 1/ +√ +t ≥ 0. Lastly, +T +� +t=1 +(G + DgR + γt)2 +Vt += +⌊(C/ϵ)2⌋ +� +t=1 +(G + DgR + γt)2 +Vt ++ +T +� +t=⌈(C/ϵ)2⌉ +(G + DgR + γt)2 +Vt +≤ +⌊(C/ϵ)2⌋ +� +t=1 +(G + DgR + C)2 + +T +� +t=⌈(C/ϵ)2⌉ +(G + DgR + ϵ)2 +Vt +≤ (G + DgR + C)2(C/ϵ)2 + +T +� +t=1 +(G + DgR + ϵ)2 +Vt +(13) +where the first inequality holds because for t ≥ (C/ϵ)2, +γt = C +√ +t ≤ ϵ +and Vt ≥ 1 and γt ≤ C for t ≥ 1. Taking the expectation of both sides of (9) with x = x∗ and using the +bounds (10)–(13), +E +� T +� +t=1 +ft(xt) +� +≤ E +� T +� +t=1 +ft(x∗) +� ++ αT +VT +R2 + +T +� +t=1 +VtD2 +f +4αt ++ +T +� +t=1 +γt +Vt +E[Qt] ++ +�C(G + DgR + C) +ϵ +�2 ++ +T +� +t=1 +(G + DgR + ϵ)2 +Vt +, +which proves (4), as required. +19 + +A.3 +Proof of Lemma 3.3: Giving an Upper Bound on the Expected Constraint Violation +We will show that +T +� +t=1 +gt(xt) ≤ QT +1 + +T +� +t=1 +DfDgVt + D2 +gQt +2αt +− +T +� +t=1 +γt +(14) +holds. This would imply that +E +� T +� +t=1 +gt(xt) +� +≤ E [QT +1] + DfDg +√ +T + D2 +g +2 +T +� +t=1 +E [Qt] +t +− C +√ +T, +as required. Therefore, it suffices to show that (14) holds. Since Qt+1 ≥ Qt +gt(xt)+∇gt(xt)⊤(xt+1 −xt)+γt, +we have +gt(xt) ≤ Qt+1 − Qt − ∇gt(xt)⊤(xt+1 − xt) − γt +≤ Qt+1 − Qt + ∥∇gt(xt)∥∗∥xt+1 − xt∥ − γt +≤ Qt+1 − Qt + Dg∥xt+1 − xt∥ − γt +(15) +where the second inequality is due to the fact that u⊤v ≤ ∥u∥∗∥v∥ for any u, v ∈ Rd and the last inequality is +by Assumption 1. Summing (15) over t = 1, . . . , T, we obtain +T +� +t=1 +gt(xt) ≤ QT +1 − Q1 + Dg +T +� +t=1 +∥xt+1 − xt∥ − +T +� +t=1 +γt. +(16) +To provide an upper bound on the term ∥xt+1 − xt∥, we use the inequality in Lemma A.1 with x = xt, which +implies that +αtD(xt+1, xt) + αtD(xt, xt+1) ≤ (Vt∇ft(xt) + Qt∇gt(xt))⊤ (xt − xt+1). +(17) +The left-hand side of (17) is greater than or equal to 2αt∥xt+1 − xt∥2 because D(xt+1, xt) ≥ ∥xt+1 − xt∥2 and +D(xt, xt+1) ≥ ∥xt − xt+1∥2 by (3). The right-hand side can be bounded by +∥Vt∇ft(xt) + Qt∇gt(xt)∥∗ ∥xt+1 − xt∥ ≤ (DfVt + DgQt)∥xt+1 − xt∥ +where the inequality is due to Assumption 1. Then we obtain from (17) that +2αt∥xt+1 − xt∥2 ≤ (DfVt + DgQt)∥xt+1 − xt∥, +implying in turn that +∥xt+1 − xt∥ ≤ +1 +2αt +(DfVt + DgQt). +(18) +Then (16) and (18) together with Q1 = 0 imply +T +� +t=1 +gt(xt) ≤ QT +1 + DfDg +2 +T +� +t=1 +Vt +αt ++ D2 +g +2 +T +� +t=1 +Qt +αt +− +T +� +t=1 +γt, +so (14) holds, as required. +20 + +A.4 +Proof of Lemma 3.4: Time-Varying Drift Lemma +Recall that ˆx ∈ X is a solution satisfying Slater’s condition (Assumption 2), i.e., Eω[g(ˆx, ω)] ≤ −ϵ. Assume that +T ≥ (2C/ϵ)4. Then for any t ≥ ⌈ +√ +T⌉, +γt = C +√ +t ≤ +C +T 1/4 ≤ ϵ +2. +Lemma A.2. For any t ≥ (2C/ϵ)2 and s ≥ 0, +E [Qt+s (gt+s(ˆx) + γt+s) | Ft−1] ≤ − ϵ +2 · E [Qt+s | Ft−1] +Proof. Since t ≥ (2C/ϵ)2, we have +γt = C +√ +t ≤ ϵ +2. +Note that +E [Qt+s (gt+s(ˆx) + γt+s) | Ft−1] = E [E [Qt+s (gt+s(ˆx) + γt+s) | Ft+s−1] | Ft−1] += E [Qt+s · E [gt+s(ˆx) + γt+s | Ft+s−1] | Ft−1] += E [Qt+s (¯g(ˆx) + γt+s) | Ft−1] += (¯g(ˆx) + γt+s) · E [Qt+s | Ft−1] +where the first equality is from the tower rule, the second equality holds because Qt+s is Ft+s−1-measurable, the +third equality holds because ωt+s is independent of {ω1, . . . , ωt+s−1}, and the last equality is because ¯g(ˆx)+γt+s +is constant. By Slater’s condition (Assumption 2), we have ¯g(ˆx) ≤ −ϵ. As γt ≤ ϵ/2 for any t ≥ (C/2ϵ)2, it +follows that ¯g(ˆx) + γt+s ≤ −ϵ/2. Since Qt+s is always nonnegative, +E [Qt+s (gt+s(ˆx) + γt+s) | Ft−1] = (¯g(ˆx) + γt+s) · E [Qt+s | Ft−1] ≤ − ϵ +2 · E [Qt+s | Ft−1] , +as required. +Lemma A.3. For t ≥ 1, +−G − DgR ≤ Qt+1 − Qt ≤ G + γt +Proof. Let us first show the upper bound on Qt+1−Qt. As Qt+1 = max +� +Qt + gt(xt) + ∇gt(xt)⊤(xt+1 − xt) + γt, 0 +� +and gt(xt) + ∇gt(xt)⊤(xt+1 − xt) ≤ gt(xt+1) ≤ G by convexity, Qt+1 ≤ max {Qt + G + γt, 0} holds. Then +it follows that Q2 +t+1 ≤ (Qt + G + γt)2, implying in turn that Qt+1 ≤ Qt + G + γt. For the lower bound, we +deduce from Qt+1 = max +� +Qt + gt(xt) + ∇gt(xt)⊤(xt+1 − xt) + γt, 0 +� +that +Qt+1 ≥ Qt + gt(xt) + ∇gt(xt)⊤(xt+1 − xt) + γt ≥ Qt − G − DgR +where the second inequality is comes from gt(xt) ≥ −G, ∇gt(xt)⊤(xt+1 − xt) ≥ −∥gt(xt)∥∗∥xt+1 − xt∥ ≥ +−DgR, and γt ≥ 0. Therefore, Qt+1 − Qt ≥ −G − DgR holds as required. +Proof of Lemma 3.4. Part (a). Recall that Q1 = 0. Let t ≥ 2. By Lemma A.3, we know that Qs+1 −Qs ≤ G+γs +21 + +holds for s ≥ 1. Summing this inequality over s = 1, . . . , t − 1 and using Q1 = 0, +Qt = Qt − Q1 = +t−1 +� +s=1 +(Qs+1 − Qs) ≤ +t−1 +� +s=1 +(G + γs). +Recall that γs ≤ ϵ/2 for s ≥ (2C/ϵ)2. Moreover, γs ≤ C for any s ≥ 1. Then it follows that +t−1 +� +s=1 +(G + γs) = +⌊(2C/ϵ)2⌋ +� +s=1 +(G + γs) + +t−1 +� +s=⌊(2C/ϵ)2⌋+1 +(G + γs) += (t − 1)G + +⌊(2C/ϵ)2⌋ +� +s=1 +γs + +t−1 +� +s=⌊(2C/ϵ)2⌋+1 +γs +≤ (t − 1)G + +⌊(2C/ϵ)2⌋ +� +s=1 +C + +t−1 +� +s=⌊(2C/ϵ)2⌋+1 +ϵ +2 +≤ (t − 1)G + C(2C/ϵ)2 + (t − 1)(ϵ/2) +Since t − 1 ≤ t, this inequality implies that +Qt ≤ +t−1 +� +s=1 +(G + γs) ≤ C(2C/ϵ)2 + (G + ϵ/2)t, +as required. +Part (b). Since t ≥ (2C/ϵ)2, we have γt ≤ ϵ/2. Then the upper bound Qt+1 − Qt ≤ G + γt from Lemma A.3 +implies that Qt+1 − Qt ≤ G + ϵ/2. +Part (c). Let t ≥ (2C/ϵ)2. By part (b), it is straightforward that for any t and τ, +Qt+τ − Qt = +t+τ−1 +� +s=t +(Qs+1 − Qs) ≤ (G + ϵ)τ +Now suppose that Qt ≥ θt(τ). Recall that 1 ≤ τ ≤ t + 1. We will show that when Qt ≥ θt(τ), +E +� +Q2 +t+τ | Ft−1 +� +≤ +� +Qt − ϵ +4τ +�2 +. +(19) +If (19) holds, as √· is a concave function over R+, we deduce from Jensen’s inequality that +E [Qt+τ | Ft−1] ≤ +� +E +� +Q2 +t+τ | Ft−1 +� +≤ Qt − ϵ +4τ. +Therefore, it is sufficient to show that (19) holds true. Note that +Q2 +t+τ = Q2 +t + 2 +t+τ−1 +� +s=t +∆s, +so to provide the desired bound on Q2 +t+τ, we analyze the drift terms. Lemma 3.1 and the observation that γt ≤ ϵ/2 +for any t ≥ (2C/ϵ)2 imply that +∆s ≤ Qs +� +gs(xs) + ∇gs(xs)⊤(xs+1 − xs) + γs +� ++ (G + DgR + ϵ)2 +(20) +22 + +for s = t, . . . , t + τ − 1. Moreover, +Qs(gs(xs) + ∇gs(xs)⊤(xs+1 − xs) + γs) +≤ Qs(gs(xs) + ∇gs(xs)⊤(ˆx − xs) + γs) ++ Vs∇fs(xs)⊤(ˆx − xs+1) + αs (D(ˆx, xs) − D(ˆx, xs+1) − D(xs+1, xs)) +≤ Qs(gs(ˆx) + γs) + DfRVs + αs (D(ˆx, xs) − D(ˆx, xs+1)) +(21) +where the first inequality comes from the inequality given in Lemma A.1 with x set to ˆx and the second inequality is +because gs is convex, Vs∇fs(xs)⊤(ˆx − xs+1) ≤ Vs∥∇fs(xs)∥∗∥ˆx − xs+1∥ ≤ VsDfR, and D(xs+1, xs) ≥ 0. +Combining (20) and (21), we may deduce the following. +E [∆s | Ft−1] ≤ E [Qs(gs(ˆx) + γs) | Ft−1] + αsE [D(ˆx, xs) − D(ˆx, xs+1) | Ft−1] ++ DfRVs + (G + DgR + ϵ)2 +≤ − ϵ +2 · E [Qs | Ft−1] + αsE [D(ˆx, xs) − D(ˆx, xs+1) | Ft−1] + DfRVs + (G + DgR + ϵ)2 +(22) +where we used Lemma A.2 to obtain the second inequality. Summing (22) over s = t, . . . , t + τ − 1, we get the +following. +t+τ−1 +� +s=t +E [∆s | Ft−1] +≤ − ϵ +2 +t+τ−1 +� +s=t +E [Qs | Ft−1] + E +� +αtD(ˆx, xt) + +t+τ−1 +� +s=t+1 +D(ˆx, xs) (αs − αs−1) − αt+τ−1D(ˆx, xt+τ) | Ft−1 +� ++ DfR +t+τ−1 +� +s=t +Vs + (G + DgR + ϵ)2τ +≤ − ϵ +2 +t+τ−1 +� +s=t +E [Qs | Ft−1] + R2αt+τ−1 + DfR +t+τ−1 +� +s=t +Vs + (G + DgR + ϵ)2τ +(23) +where the second inequality holds because {αt}T +t=1 is an increasing sequence and 0 ≤ D(x, y) ≤ R2 for x, y ∈ X. +Since τ ≤ t + 1, we have t + τ − 1 ≤ 2t. Moreover, as αt = t and Vt = +√ +t, +αt+τ−1 ≤ α2t = 2αt +and +t+τ−1 +� +s=t +Vs ≤ τVt+τ−1 ≤ τV2t ≤ 2τVt. +(24) +Then we deduce from (23) and (24) the following. +t+τ−1 +� +s=t +E [∆s | Ft−1] ≤ − ϵ +2 +t+τ−1 +� +s=t +E [Qs | Ft−1] + 2R2αt + 2DfRτVt + (G + DgR + ϵ)2τ. +(25) +Here, by Lemma A.3, +t+τ−1 +� +s=t +E [Qs | Ft−1] ≥ +t+τ−1 +� +s=t +E [Qt − (G + DgR)(s − t) | Ft−1] +≥ τE [Qt | Ft−1] − (G + DgR)τ 2 += τQt − (G + DgR)τ 2 +(26) +23 + +where the equality holds because Qt is Ft−1-measurable. Then, by (25) and (26), +E +� +Q2 +t+τ | Ft−1 +� += Q2 +t + 2 +t+τ−1 +� +s=t +E [∆s | Ft−1] +≤ Q2 +t − ϵτQt + ϵ(G + DgR)τ 2 + 4R2αt + 4DfRτVt + 2(G + DgR + ϵ)2τ. +(27) +Recall that +θt(τ) = 2(G + DgR)τ + 8R2αt +ϵτ ++ 8DfRVt +ϵ ++ 4(G + DgR + ϵ)2 +ϵ += 2 +ϵτ +� +ϵ(G + DgR)τ 2 + 4R2αt + 4DfRτVt + 2(G + DgR + ϵ)2τ +� +. +Therefore, if Qt ≥ θt(τ), then it follows from (27) that +E +� +Q2 +t+τ | Ft−1 +� +≤ Q2 +t − ϵτ +2 Qt ≤ +� +Qt − ϵτ +4 +�2 +, +which proves (19), as required. +A.5 +Proof of Lemma 3.5: Time-Varying Bound on the Expected Virtual Queue Length +Let δ = (G + ϵ) and ξ = ϵ/4. Moreover, let r and ρ be two constants defined as +r = +ξ +4⌈ +√ +T⌉δ2 +and +ρ = 1 − ξ2 +8δ2 . +Since ξ ≤ δ, we know that 0 < ρ < 1. +Lemma A.4. Let t ≥ 1. For any 1 ≤ τ ≤ +√ +T satisfying t − τ ≥ (2C/ϵ)2 and t ≥ 2τ − 1, we have +E +� +erQt� +≤ ρE +� +erQt−τ � ++ erδτerθt−τ . +Proof. As t ≥ 2τ − 1, we have t − τ ≥ τ − 1. Henceforth, we use the notation wt = Qt − Qt−τ. Note that +wt = +t−1 +� +s=t−τ +Qs+1 − Qs ≤ τδ +(28) +holds due to Lemma 3.4(b) because t − τ ≥ (2C/ϵ)2. Furthermore, as τ ≤ +√ +T, +rwt ≤ rτδ ≤ ξ +4δ ≤ 1. +(29) +Next we observe that ex ≤ 1 + x + 2x2 for any x ≤ 1. It was already observed in [25, Appendix A] that +ex ≤ 1 + x + 2x2 holds for any |x| ≤ 1. When x ≤ −1, we know that ex ≤ 1 and x + 2x2 ≥ 0, which indicates +that ex ≤ 1 + x + 2x2. Then +erwt ≤ 1 + rwt + 2r2w2 +t ≤ 1 + rwt + 2r2τ 2δ2 ≤ 1 + rwt + 1 +2rτξ +where the first inequality is from (29) and the observation that ex ≤ 1 + x + 2x2 holds for any x ≤ 1 while the +second inequality is from (28). This inequality implies that +erQt = er(Qt−τ +wt) ≤ erQt−τ +� +1 + rwt + 1 +2rτξ +� +. +(30) +24 + +Let us first consider the case Qt−τ > θt−τ(τ). For ease of notation, we use θt−τ to denote θt−τ(τ). As t−τ ≥ τ −1 +and t − τ ≥ (2C/ϵ)2, it follows from Lemma 3.4(c) that E [wt | Ft−τ−1] ≤ −ξτ. In this case, we obtain the +following based on (30). +E +� +erQt | Qt−τ > θt−τ +� +≤ E +� +erQt−τ +� +1 + rwt + 1 +2rξτ +� +| Qt−τ > θt−τ +� += E +� +E +� +erQt−τ +� +1 + rwt + 1 +2rξτ +� +| Ft−τ−1 +� +| Qt−τ > θt−τ +� +≤ E +� +erQt−τ +� +1 − rξτ + 1 +2rξτ +� +| Qt−τ > θt−τ +� += E +� +ρerQt−τ | Qt−τ > θt−τ +� +(31) +where the first inequality is from (30), the first equality is from the tower rule, and the second inequality holds due +to E [wt | Ft−τ−1] ≤ −ξτ. If Qt−τ ≤ θt−τ, we may deduce the following based on (28). +E +� +erQt | Qt−τ ≤ θt−τ +� += E +� +erwterQt−τ | Qt−τ ≤ θt−τ +� +≤ E +� +erδτerQt−τ | Qt−τ ≤ θt−τ +� +. +(32) +Note that +E +� +erQt� += P [Qt−τ > θt−τ] · E +� +erQt | Qt−τ > θt−τ +� ++ P [Qt−τ ≤ θt−τ] · E +� +erQt | Qt−τ ≤ θt−τ +� +≤ ρE +� +erQt−τ | Qt−τ > θt−τ +� +· P [Qt−τ > θt−τ] + erδτE +� +erQt−τ | Qt−τ ≤ θt−τ +� +· P [Qt−τ ≤ θt−τ] += ρE +� +erQt−τ � ++ +� +erδτ − ρ +� +E +� +erQt−τ | Qt−τ ≤ θt−τ +� +· P [Qt−τ ≤ θt−τ] +≤ ρE +� +erQt−τ � ++ erδτerθt−τ +(33) +where the first inequality is deduced by (31) and (32) and the second inequality holds because erδτ − ρ ≤ erδτ and +P [Qt−τ ≤ θt−τ] ≤ 1. +Proof of Lemma 3.5. Let t ≥ 1, and let τ = ⌈ +√ +t⌉. For any s, we use θs to denote θs(τ). We first consider the case +where +t ≥ max +� +9, +�2C +ϵ +�4� +. +Then it follows that t ≥ 2τ and τ ≥ (2C/ϵ)2. Moreover, ⌊t/τ⌋ = k for some k ≥ 2, in which case +t − (k − 1)τ ≥ τ ≥ (2C/ϵ)2. +Then, we may apply Lemma A.4 for s = t, t − τ, . . . , t − (k − 2)τ. In particular, we obtain +E +� +erQt� +≤ ρE +� +erQt−τ � ++ erδτerθt−τ +≤ ρk−1E +� +erQt−(k−1)τ � ++ erδτ +k−1 +� +i=1 +ρi−1erθt−iτ . +(34) +As θt increases as t increases, θt−iτ ≤ θt for any i ≥ 1. In addition, rδτ ≤ 2rδτ and erC(2C/ϵ)2 ≥ 1. Hence, we +deduce from (34) that +E +� +erQt� +≤ ρk−1E +� +erQt−(k−1)τ � ++ erC(2C/ϵ)2e2rδτerθt +k−2 +� +i=0 +ρi. +(35) +25 + +Moreover, note that t − (k − 1)τ < 2τ. As Qt−(k−1)τ ≤ C(2C/ϵ)2 + 2δτ by Lemma 3.4(a), +erQt−(k−1)τ ≤ erC(2C/ϵ)2e2rδτ ≤ erC(2C/ϵ)2e2rδτ +� +1 +1 − ρerθt +� +(36) +where the second inequality holds because 0 < ρ < 1 and erθt ≥ 1. Combining (35) and (36), +E +� +erQt� +≤ erC(2C/ϵ)2e2rδτerθt +� +ρk−1 +1 − ρ + +k−2 +� +i=0 +ρi +� += +1 +1 − ρerC(2C/ϵ)2e2rδτerθt. +(37) +As p(x) = erx is convex over x ∈ R, Jensen’s inequality implies that +erE[Qt] ≤ E +� +erQt� +. +Then, by (37), we obtain +E [Qt] ≤ θt + 2δτ + C +�2C +ϵ +�2 ++ log +1 +1 − ρ += θt + 2(G + ϵ)τ + log 128(G + ϵ)2 +ϵ2 ++ C +�2C +ϵ +�2 +≤ θt + 4(G + ϵ) +√ +t + log 128(G + ϵ)2 +ϵ2 ++ C +�2C +ϵ +�2 +≤ θt + 4(G + ϵ) +√ +t + log 128(G + ϵ)2 +ϵ2 ++ C +�2C +ϵ +�2 ++ (G + ϵ) +� +9 + +�2C +ϵ +�4� +where the second inequality holds because τ ≤ 2 +√ +t. +Now consider the case t < max +� +9, (2C/ϵ)4� +. Then t < 9 + (2C/ϵ)4. Note that +Qt ≤ C +�2C +ϵ +�2 ++ (G + ϵ)t +≤ θt + 4(G + ϵ) +√ +t + log 128(G + ϵ)2 +ϵ2 ++ C +�2C +ϵ +�2 ++ (G + ϵ) +� +9 + +�2C +ϵ +�4� +where the first inequality comes from Lemma 3.4(a) and the second inequality holds because t ≤ 9 + (2C/ϵ)4. +A.6 +Completing the Proofs of Theorem 3.6 (Constraint Violation) and Theorem 3.7 (Re- +gret) +Plugging in the formula of θt to Lemma 3.5, it follows that for any t ≥ 1, +E [Qt] ≤ 4 +� +2G + DgR + ϵ + 2R2 + 2DfR +ϵ +� √ +t + 4(G + DgR + ϵ)2 +ϵ ++ log 128(G + ϵ)2 +ϵ2 ++ C +�2C +ϵ +�2 ++ (G + ϵ) +� +9 + +�2C +ϵ +�4� +. +(38) +It follows from basic calculus that +log(T + 1) ≤ +T +� +t=1 +1 +t ≤ 1 + log T. +26 + +Proof of Theorem 3.6. Note that +√ +T + 1 ≤ 2 +√ +T for any T ≥ 1. Based on Lemma 3.3 and the expected queue +size bound (38), we deduce the following. +E +� T +� +t=1 +gt(xt) +� +≤ E [QT +1] + +T +� +t=1 +DfDgVt + D2 +gE [Qt] +2αt +− +T +� +t=1 +γt +≤ +� +DfDg + (4D2 +g + 8) +� +2G + DgR + ϵ + 2R2 + 2DfR +ϵ +�� √ +T − C +√ +T ++ +� +4(G + DgR + ϵ)2 +ϵ ++ log 128(G + ϵ)2 +ϵ2 ++ C +�2C +ϵ +�2 ++ (G + ϵ) +� +9 + +�2C +ϵ +�4�� � +1 + D2 +g(1 + log T) +2 +� +. +Since we set C as +C = DfDg + (4D2 +g + 8) +� +2G + DgR + ϵ + 2R2 + 2DfR +ϵ +� ++ 1, +we have +E +� T +� +t=1 +gt(xt) +� +≤ − +√ +T + +� +4(G + DgR + ϵ)2 +ϵ ++ log 128(G + ϵ)2 +ϵ2 ++ C +�2C +ϵ +�2� � +1 + D2 +g(1 + log T) +2 +� +. +Therefore, there exists some positive constant T1 such that for any T ≥ T1, +E +� T +� +t=1 +gt(xt) +� +≤ 0, +as required. +Proof of Theorem 3.7. If C = O(1), then E[Qt] ≤ B +√ +t for some constant B that depends only on G, Dg, Df, R, ϵ +by (38). By Lemma 3.2, it follows that +E +� T +� +t=1 +ft(xt) − +T +� +t=1 +ft(x∗) +� +≤ +�C(G + DgR + C) +ϵ +�2 ++ K1 +√ +T + C +T +� +t=1 +E[Qt] +t +≤ +�C(G + DgR + C) +ϵ +�2 ++ K1 +√ +T + CB +T +� +t=1 +1 +√ +t += O( +√ +T), +as required. +27 + +B +Performance Analysis of Bandit Drift-Plus-Penalty (Algorithm 2) +B.1 +Bounds on the Gradient Estimates +Proof of Lemma 4.1. By the definition of ˜∇gt, it follows that +∥ ˜∇gt∥∗ = d +2δt +|gt(xt + δtut) − gt(xt − δtut)| · ∥ut∥∗ +(39) +≤ d +2δt +Lg∥2δtut∥2∥ut∥∗ +(40) += dLg∥ut∥∗ +(41) +where the inequality comes from Assumption 3. Since ut is a point in +� +u ∈ Rd : ∥u∥2 ≤ 1 +� +. +Following Shamir [20], let us define functions ˆft and ˆgt as follows. For t ∈ [T] and x ∈ X, +ˆft(x) := Eut [ft(x + δtut)] +and +ˆgt(x) := Eut [gt(x + δtut)] +where ut is sampled from the Euclidean unit sphere +� +u ∈ Rd : ∥u∥2 ≤ 1 +� +uniformly at random. We will use the +following two lemmas shown in [20]. +Lemma B.1. [20, Lemma 8] For any t ≥ 1, ˆft and ˆgt are convex and satisfy +sup +x∈X +��� ˆft(x) − ft(x) +��� ≤ Lfδt +and +sup +x∈X +|ˆgt(x) − gt(x)| ≤ Lgδt. +Moreover, ˆft and ˆgt are differentiable, and +∇ ˆft(x) = Eut +� d +δt +ft(x + δtut)ut +� +and +∇ˆgt(x) = Eut +� d +δt +gt(x + δtut)ut +� +. +Lemma B.2. [20, Lemma 9] For any t ≥ 1, +E +� ˜∇ft | H0 +t +� += ∇ ˆft(xt) and E +� ˜∇gt | H0 +t +� += ∇ˆgt(xt). +B.2 +Proof of Lemma 4.3: Upper Bound on the Expected Regret +The following lemma is analogous to Lemma 3.1. +Lemma B.3. For t ≥ 1, +∆t ≤ Qt +� +gt(xt) + ˜∇g⊤ +t (xt+1 − xt) + γt +� ++ (G + γt)2 + R2∥ ˜∇gt∥2 +∗. +Proof. As Qt+1 = max +� +Qt + gt(xt) + ˜∇g⊤ +t (xt+1 − xt) + γt, 0 +� +, +Q2 +t+1 ≤ +� +Qt + gt(xt) + ˜∇g⊤ +t (xt+1 − xt) + γt +�2 . +Expanding the right-hand side, we obtain +∆t = Q2 +t+1 +2 +− Q2 +t +2 ≤ Qt(gt(xt) + ˜∇g⊤ +t (xt+1 − xt) + γt) + 1 +2 +� +gt(xt) + ˜∇g⊤ +t (xt+1 − xt) + γt +�2 . +28 + +Here, +� +gt(xt) + ˜∇g⊤ +t (xt+1 − xt) + γt +�2 ≤ +� +|gt(xt)| + ∥ ˜∇gt∥∗∥xt+1 − xt∥ + γt +�2 +≤ 2 (|gt(xt)| + γt)2 + 2∥ ˜∇gt∥2 +∗∥xt+1 − xt∥2 +≤ 2(G + γt)2 + 2R2∥ ˜∇gt∥2 +∗ +where the third inequality is due to Assumption 1. Therefore, +∆t ≤ Qt(gt(xt) + ˜∇g⊤ +t (xt+1 − xt) + γt) + (G + γt)2 + R2∥ ˜∇gt∥2 +∗, +as required. +The following lemma is a modification of Lemma A.1 which can be derived based on the fact that +� +Vt ˜∇ft + Qt ˜∇gt +�⊤ (x − xt) + αtD(x, xt) +is 2αt-strongly convex with respect to the norm ∥ · ∥ and that xt+1 is its minimizer over X. +Lemma B.4. For any x ∈ X and t ≥ 1, +� +Vt ˜∇ft + Qt ˜∇gt +�⊤ (xt+1 − xt) + αtD(xt+1, xt) +≤ +� +Vt ˜∇ft + Qt ˜∇gt +�⊤ (x − xt) + αtD(x, xt) − αtD(x, xt+1) +Proof of Lemma 4.3. We will show that +T +� +t=1 +E [ft(xt) − ft(x)] ≤ +T +� +t=1 +2Lfδt + +T +� +t=1 +2Lgδt + γt +Vt +E [Qt] + αT +VT +R2 ++ +T +� +t=1 +Vt +4αt +qdp2 +∗L2 +f + C2(G + C)2 +ϵ2 ++ +T +� +T =1 +(G + ϵ)2 + R2qdp2 +∗L2 +g +Vt +holds. +First, let us add Vt ˆft(xt) + Qtˆgt(xt) to both sides of the inequality given in Lemma B.4. Then we obtain +Vt ˆft(xt) + Qtˆgt(xt) + +� +Vt ˜∇ft + Qt ˜∇gt +�⊤ (xt+1 − xt) + αtD(xt+1, xt) +≤ Vt ˆft(xt) + Qtˆgt(xt) + +� +Vt ˜∇ft + Qt ˜∇gt +�⊤ (x − xt) + αtD(x, xt) − αtD(x, xt+1). +(42) +Based on Lemma 3.1, we may observe that the left-hand side of (42) is greater than or equal to +Vt ˆft(xt)+Vt ˜∇f ⊤ +t (xt+1−xt)+αtD(xt+1, xt)+∆t−Qtγt−Qt(gt(xt)−ˆgt(xt))−(G+γt)2−R2∥ ˜∇gt∥2 +∗. (43) +Moreover, we can find a lower bound on the term Vt ˜∇f ⊤ +t (xt+1 − xt) + αtD(xt+1, xt) as follows. +Vt ˜∇f ⊤ +t (xt+1 − xt) + αtD(xt+1, xt) ≥ −Vt∥ ˜∇ft∥∗∥xt+1 − xt∥ + αt∥xt+1 − xt∥2 +≥ − V 2 +t +4αt +∥ ˜∇ft∥2 +∗ − αt∥xt+1 − xt∥2 + αt∥xt+1 − xt∥2 += − V 2 +t +4αt +∥ ˜∇ft∥2 +∗ +(44) +29 + +where the first inequality holds due to u⊤v ≤ ∥u∥∗∥v∥ and (3) while the second inequality is from pq ≤ +p2/(4αt) + αtq2. Combining (42), (43), and (44), it follows that +Vt ˆft(xt) ≤ Vt +� +ˆft(xt) + ˜∇f ⊤ +t (x − xt) +� ++ Qt +� +ˆgt(xt) + ˜∇g⊤ +t (x − xt) +� ++ αtD(x, xt) − αtD(x, xt+1) ++ V 2 +t +4αt +∥ ˜∇ft∥2 +∗ − αtD(xt+1, xt) − ∆t + Qtγt + Qt(gt(xt) − ˆgt(xt)) + (G + γt)2 + R2∥ ˜∇gt∥2 +∗. +(45) +Note that +Qt(gt(xt) − ˆgt(xt)) ≤ Qt|gt(xt) − ˆgt(xt)| ≤ QtLgδt +(46) +by Lemma B.1. Next, we consider the terms Vt +� +ˆft(xt) + ˜∇f ⊤ +t (x − xt) +� +and Qt +� +ˆgt(xt) + ˜∇g⊤ +t (x − xt) +� +. Note +that +E +� +Vt +� +ˆft(xt) + ˜∇f ⊤ +t (x − xt) +�� += Vt · E +� +ˆft(xt) + E +� ˜∇f ⊤ +t (x − xt) | H0 +t +�� += Vt · E +� +ˆft(xt) + (x − xt)⊤E +� ˜∇ft | H0 +t +�� += Vt · E +� +ˆft(xt) + (x − xt)⊤∇ ˆft(xt) +� +≤ Vt · E +� +ˆft(x) +� +(47) +where the first equality is from the linearity of expectation and the tower rule, the second equality holds because +x − xt is H0 +t -measurable, the third equality is by Lemma B.2, and the inequality is due to the convexity of ˆft +(Lemma B.1). Moreover, +E +� +Qt +� +ˆgt(xt) + ˜∇g⊤ +t (x − xt) +�� += E +� +Qtˆgt(xt) + E +� +Qt ˜∇g⊤ +t (x − xt) | H0 +t +�� += E +� +Qtˆgt(xt) + Qt(x − xt)⊤E +� ˜∇gt | H0 +t +�� += E +� +Qtˆgt(xt) + Qt(x − xt)⊤∇ˆgt(xt) +� +≤ E [Qtˆgt(x)] +(48) +where the first equality is from the linearity of expectation and the tower rule, the second equality holds because +Qt(x − xt) is H0 +t -measurable, the third equality is by Lemma B.2, and the inequality is due to the convexity of ˆgt +(Lemma B.1). Furthermore, the terms involving ∥ ˜∇ft∥2 +∗ and ∥ ˜∇gt∥2 +∗ on the right-hand side of (45) can be dealt with +based on Lemma 4.2. Taking the expectation of both sides of (45), the resulting inequality implies the following. +Using the bounds (46), (47), and (48), +VtE +� +ˆft(xt) +� +≤ VtE +� +ˆft(x) +� ++ E [Qtˆgt(x)] + E [αtD(x, xt) − αtD(x, xt+1)] ++ V 2 +t +4αt +qdp2 +∗L2 +f − E [∆t] + γtE [Qt] + QtLgδt + (G + γt)2 + R2qdp2 +∗L2 +g. +(49) +Next, by Lemma B.1, we know that | ˆft(x) − ft(x)| ≤ Lfδt and |ˆgt(x) − gt(x)| ≤ Lgδt, applying which to (49), +we obtain +VtE [ft(xt) − ft(x)] ≤ 2LfδtVt + 2LgδtQt + E [Qtgt(x)] + E [αtD(x, xt) − αtD(x, xt+1)] ++ V 2 +t +4αt +qdp2 +∗L2 +f − E [∆t] + γtE [Qt] + (G + γt)2 + R2qdp2 +∗L2 +g. +(50) +30 + +Dividing both sides of (50) and summing the resulting inequality for t = 1, . . . , T, we obtain the following +inequality. +T +� +t=1 +E [ft(xt) − ft(x)] +≤ +T +� +t=1 +2Lfδt + +T +� +t=1 +2Lgδt +Vt +E [Qt] + +T +� +t=1 +1 +Vt +E [Qtgt(x)] + +T +� +t=1 +αt +Vt +E [D(x, xt) − D(x, xt+1)] ++ +T +� +t=1 +Vt +4αt +qdp2 +∗L2 +f − +T +� +t=1 +1 +Vt +E [∆t] + +T +� +t=1 +γt +Vt +E [Qt] + +T +� +T =1 +(G + γt)2 +Vt ++ +T +� +T =1 +R2qdp2 +∗L2 +g +Vt +. +(51) +As in the proof of Lemma 3.2, we can argue the following inequalities bounding some terms on the right-hand side +of (51). +E [Qtgt(x∗)] = E [Qt¯g(x∗)] ≤ 0, +(52) +T +� +t=1 +αt +Vt +E [D(x, xt) − D(x, xt+1)] ≤ E +� +α1 +V1 +R2 + +T +� +t=2 +R2 +�αt +Vt +− αt−1 +Vt−1 +�� += αT +VT +R2, +(53) +T +� +t=1 +1 +Vt +E [∆t] = E +� +− 1 +2V1 +Q2 +1 + +1 +2VT +Q2 +T +1 + 1 +2 +T +� +t=2 +Q2 +t +� +1 +Vt−1 +− 1 +Vt +�� +≥ − 1 +2V1 +Q2 +1 = 0, +(54) +T +� +T =1 +(G + γt)2 +Vt +≤ (G + C)2(C/ϵ)2 + +T +� +t=1 +(G + ϵ)2 +Vt +. +(55) +Applying the bounds (52), (53), (54), and (55) to (51) with x = x∗, it follows that +T +� +t=1 +E [ft(xt) − ft(x)] ≤ +T +� +t=1 +2Lfδt + +T +� +t=1 +2Lgδt + γt +Vt +E [Qt] + αT +VT +R2 ++ +T +� +t=1 +Vt +4αt +qdp2 +∗L2 +f + C2(G + C)2 +ϵ2 ++ +T +� +T =1 +(G + ϵ)2 + R2qdp2 +∗L2 +g +Vt +, +as required. +B.3 +Proof of Lemma 4.4: Upper Bound on the Constraint Violation +We will show that +T +� +t=1 +gt(xt) ≤ QT +1 + +T +� +t=1 +Vt +4αt +� +∥ ˜∇gt∥2 +∗ + ∥ ˜∇ft∥2 +∗ +� ++ +T +� +t=1 +Qt +2αt +∥ ˜∇gt∥2 +∗ − +T +� +t=1 +γt +holds. +Since Qt+1 ≥ Qt + gt(xt) + ˜∇g⊤ +t (xt+1 − xt) + γt, we have +gt(xt) ≤ Qt+1 − Qt − ˜∇g⊤ +t (xt+1 − xt) − γt +≤ Qt+1 − Qt + ∥ ˜∇gt∥∗∥xt+1 − xt∥ − γt +(56) +31 + +where the second inequality is due to the fact that u⊤v ≤ ∥u∥∗∥v∥ for any u, v ∈ Rd. To provide an upper bound +on the term ∥xt+1 − xt∥, we use the inequality in Lemma B.4 with x = xt, which implies that +αtD(xt+1, xt) + αtD(xt, xt+1) ≤ +� +Vt ˜∇ft + Qt ˜∇gt +�⊤ (xt − xt+1). +(57) +The left-hand side of (57) is greater than or equal to 2αt∥xt+1 − xt∥2 because D(xt+1, xt) ≥ ∥xt+1 − xt∥2 and +D(xt, xt+1) ≥ ∥xt − xt+1∥2 by (3). The right-hand side can be bounded by +��Vt ˜∇ft + Qt ˜∇gt +�� +∗ ∥xt+1 − xt∥ ≤ +� +Vt +�� ˜∇ft +�� +∗ + Qt +�� ˜∇gt +�� +∗ +� +∥xt+1 − xt∥. +Then we obtain from (57) that +2αt∥xt+1 − xt∥2 ≤ +� +Vt +�� ˜∇ft +�� +∗ + Qt +�� ˜∇gt +�� +∗ +� +∥xt+1 − xt∥, +implying in turn that +∥xt+1 − xt∥ ≤ +1 +2αt +� +Vt +�� ˜∇ft +�� +∗ + Qt +�� ˜∇gt +�� +∗ +� +. +(58) +Then +gt(xt) ≤ Qt+1 − Qt + +1 +2αt +� +Vt∥ ˜∇ft∥∗∥ ˜∇gt∥∗ + Qt∥ ˜∇gt∥2 +∗ +� +− γt +≤ Qt+1 − Qt + +1 +4αt +� +Vt +� +∥ ˜∇ft∥2 +∗ + ∥ ˜∇gt∥2 +∗ +� ++ 2Qt∥ ˜∇gt∥2 +∗ +� +− γt +(59) +where the first inequality is obtained by (56) and (58) and the second inequality holds because 2pq ≤ p2 + q2 for +any p, q. Summing (59) over t = 1, . . . , T, we obtain +T +� +t=1 +gt(xt) ≤ QT +1 − Q1 + +T +� +t=1 +Vt +4αt +∥ ˜∇gt∥2 +∗ +� +∥ ˜∇ft∥2 +∗ + ∥ ˜∇gt∥2 +∗ +� ++ +T +� +t=1 +Qt +2αt +∥ ˜∇gt∥2 +∗ − +T +� +t=1 +γt. +(60) +Finally, as Q1 = 0, (60) completes the proof of this lemma. +B.4 +Proof of Lemma 4.5: Time-Varying Drift Lemma for the Bandit Setting +Lemma B.5. For t ≥ 1, +−G − RLgℓ ≤ Qt+1 − Qt ≤ G + RLgℓ + γt. +Moreover, for any s ≥ t, +−G − R2 +2 − 1 +2qdp2 +∗L2 +g ≤ E [Qs+1 − Qs | Ht−1] ≤ G + R2 +2 + 1 +2qdp2 +∗L2 +g + γt. +Proof. Let us first argue that +− G − R∥ ˜∇gt∥∗ ≤ Qt+1 − Qt ≤ G + R∥ ˜∇gt∥∗ + γt +(61) +holds. Here, for the upper bound, note that Qt+1 = max +� +Qt + gt(xt) + ˜∇g⊤ +t (xt+1 − xt) + γt, 0 +� +and +gt(xt) + ˜∇g⊤ +t (xt+1 − xt) ≤ gt(xt+1) ≤ |gt(xt)| + ∥ ˜∇gt∥∗∥xt+1 − xt∥ ≤ G + R∥ ˜∇gt∥∗ +32 + +by Assumption 1. Then Qt+1 ≤ max +� +Qt + G + R∥ ˜∇gt∥∗ + γt, 0 +� +holds. Since Qt + G + R∥ ˜∇gt∥∗ + γt is +nonnegative, it follows that Qt+1 ≤ Qt + G + R∥ ˜∇gt∥∗ + γt. For the lower bound, we deduce from Qt+1 = +max +� +Qt + gt(xt) + ˜∇g⊤ +t (xt+1 − xt) + γt, 0 +� +that +Qt+1 ≥ Qt + gt(xt) + ˜∇g⊤ +t (xt+1 − xt) + γt ≥ Qt − G − R∥ ˜∇gt∥∗ +where the second inequality is comes from gt(xt) ≥ −G, ˜∇g⊤ +t (xt+1 − xt) ≥ −∥ ˜∇gt∥∗∥xt+1 − xt∥ ≥ +−R∥ ˜∇gt∥∗, and γt ≥ 0. From (61), we obtain the desired bounds on Qt+1 − Qt (the first part) by applying the +bound on ∥ ˜∇gt∥∗ shown in Lemma 4.1. +For the second part, it is sufficient to consider the following. +E +� +R∥ ˜∇gs∥∗ | Ht−1 +� +≤ E +�R2 +4 + ∥ ˜∇gs∥2 +∗ | Ht−1 +� += E +� +E +�R2 +2 + 1 +2∥ ˜∇gs∥2 +∗ | H0 +s +� +| Ht−1 +� += E +�R2 +2 + 1 +2qdp2 +∗L2 +g | Ht−1 +� += R2 +2 + 1 +2qdp2 +∗L2 +g, +where the first inequality holds because 2pq ≤ p2 + q2. +Proof of Lemma 4.5. Parts (a) and (b) Note that Qt+1−Qt ≤ G+γt for the full information setting (Lemma A.3) +whereas the upper bound on Qt+1−Qt due to Lemma B.5 has an additional term RLgℓ. Following the same proof of +Lemma 3.4, we deduce that Qt ≤ C(2(C +2Lg)/ϵ)2 +(G+RLgℓ+ϵ)t for t ≥ 1 and Qt+1 −Qt ≤ G+RLgℓ+ϵ, +as required. +Part (c). Let t ≥ (2(C + 2Lg)/ϵ)2. The case when Qt ≤ θt(τ) is clear thanks to Lemma B.5 and part (b). Now +suppose that Qt ≥ θt(τ). Recall that 1 ≤ τ ≤ t + 1. We will show that +E +� +Q2 +t+τ | Ht−1 +� +≤ +� +Qt − ϵ +4τ +�2 +− ϵ2 +16τ 2. +(62) +If (62) holds, as √· is a concave function over R+, we deduce from Jensen’s inequality that +E [Qt+τ | Ht−1] ≤ +� +E +� +Q2 +t+τ | Ht−1 +� +≤ Qt − ϵ +4τ. +Therefore, it is sufficient to show that (62) holds true. Note that +Q2 +t+τ = Q2 +t + 2 +t+τ−1 +� +s=t +∆s. +As in the proof of Lemma 3.4(c), we analyze the drift terms to provide the desired bound on Q2 +t+τ. By Lemma B.3 +and the observation that γt ≤ ϵ/2 for any t ≥ (2(C + 2Lg)/ϵ)2, +∆s ≤ Qs +� +gs(xs) + ˜∇g⊤ +s (xs+1 − xs) + γs +� ++ (G + ϵ)2 + R2∥ ˜∇gs∥2 +∗ +(63) +33 + +for s = t, . . . , t + τ − 1. Moreover, +Qs(gs(xs) + ˜∇g⊤ +s (xs+1 − xs) + γs) +≤ Qs(gs(xs) + ˜∇g⊤ +s (ˆx − xs) + γs) ++ Vs ˜∇f ⊤ +s (ˆx − xs+1) + αs (D(ˆx, xs) − D(ˆx, xs+1) − D(xs+1, xs)) +≤ Qs(gs(xs) + ˜∇g⊤ +s (ˆx − xs) + γs) + Vs +2 (R2 + ∥ ˜∇fs∥2 +∗) + αs (D(ˆx, xs) − D(ˆx, xs+1)) +(64) +where the first inequality comes from the inequality given in Lemma A.1 with x set to ˆx and the second inequality +is because +Vs ˜∇f ⊤ +s (ˆx − xs+1) ≤ Vs∥ ˜∇fs∥∗∥ˆx − xs+1∥ ≤ Vs +2 +� +∥ ˜∇fs∥2 +∗ + ∥ˆx − xs+1∥2� +. +Moreover, as in the proof of Lemma 4.3, we may argue the following. +E +� +Qs +� +gs(xs) + ˜∇g⊤ +s (ˆx − xs) + γs +� +| Ht−1 +� += E +� +Qs(gs(xs) + γs) + E +� ˜∇g⊤ +s (ˆx − xs) | H0 +s +� +| Ht−1 +� += E +� +Qs(gs(xs) + γs) + ∇ˆgs(xs)⊤(ˆx − xs) | Ht−1 +� +≤ E [Qs(gs(xs) − ˆgs(xs)) | Ht−1] + E [Qtˆgs(ˆx) | Ht−1] += E [Qs(gs(xs) − ˆgs(xs)) | Ht−1] + E [Qs(ˆgs(ˆx) − gs(ˆx)) | Ht−1] + E [Qsgs(ˆx) | Ht−1] += E [Qs(gs(xs) − ˆgs(xs)) | Ht−1] + E [Qs(ˆgs(ˆx) − gs(ˆx)) | Ht−1] + E [E [Qsgs(ˆx) | Hs−1] | Ht−1] +≤ E [Qs (¯g(ˆx) + 2δsLg + γs) | Ht−1] +(65) +where the equalities are due to the linearity of expectation and the tower rule. Furthermore, we have +2δsLg + γs ≤ 2δtLg + γt = 2Lg + C +√ +t +≤ ϵ +2. +(66) +Lastly, +E +� +∥ ˜∇fs∥2 +∗ | Ht−1 +� += E +� +E +� +∥ ˜∇fs∥2 +∗ | H0 +s +� +| Ht−1 +� +≤ qdp2 +∗L2 +f, +E +� +∥ ˜∇gs∥2 +∗ | Ht−1 +� += E +� +E +� +∥ ˜∇gs∥2 +∗ | H0 +s +� +| Ht−1 +� +≤ qdp2 +∗L2 +g. +(67) +Taking the expectation of both sides of (64), applying the bounds (63) and (65), (66), and (67), and using Slater’s +condition (Assumption 2), we deduce the following. +E [∆s | Ht−1] +≤ − ϵ +2 · E [Qs | Ht−1] + αsE [D(ˆx, xs) − D(ˆx, xs+1) | Ht−1] ++ Vs +2 +� +R2 + qdp2 +∗L2 +f +� ++ (G + ϵ)2 + R2qdp2 +∗L2 +g. +(68) +Then, following the argument of the proof of Lemma 3.4, we obtain +t+τ−1 +� +s=t +E [∆s | Ht−1] ≤ − ϵ +2τQt + ϵ +2 +� +G + R2 +2 + 1 +2qdp2 +∗L2 +g +� +τ 2 ++ 2R2αt + +� +R2 + qdp2 +∗L2 +f +� +τVt + +� +(G + ϵ)2 + R2qdp2 +∗L2 +g +� +τ. +(69) +34 + +Then it follows from (69) that +E +� +Q2 +t+τ | Ht−1 +� += Q2 +t + 2 +t+τ−1 +� +s=t +E [∆s | Ht−1] +≤ Q2 +t − ϵτQt + ϵ +� +G + R2 +2 + 1 +2qdp2 +∗L2 +g +� +τ 2 ++ 4R2αt + 2 +� +R2 + qdp2 +∗L2 +f +� +τVt + 2 +� +(G + ϵ)2 + R2qdp2 +∗L2 +g +� +τ. +(70) +Recall that +θt(τ) = (2G + R2 + qdp2 +∗L2 +g)τ + 8R2αt +ϵτ ++ +4(R2 + qdp2 +∗L2 +f)Vt +ϵ ++ 4 +� +(G + ϵ)2 + R2qdp2 +∗L2 +g +� +ϵ += 2 +ϵτ +� +ϵ +� +G + R2 +2 + 1 +2qdp2 +∗L2 +g +� +τ 2 + 4R2αt + 2 +� +R2 + qdp2 +∗L2 +f +� +τVt + 2 +� +(G + ϵ)2 + R2qdp2 +∗L2 +g +� +τ +� +. +Therefore, if Qt ≥ θt(τ), then it follows from (70) that +E +� +Q2 +t+τ | Ht−1 +� +≤ Q2 +t − ϵτ +2 Qt = +� +Qt − ϵτ +4 +�2 +− ϵ2τ 2 +16 , +which proves (62), as required. +B.5 +Proof of Lemma 4.6: Time-Varying Bound on the Expected Virtual Queue Size for +the Bandit Setting +As for the full information setting, let δ = (G + ϵ + RLgℓ) and ξ = ϵ/4. Moreover, let r and ρ be two constants +defined as +r = +ξ +4⌈ +√ +T⌉δ2 +and +ρ = 1 − ξ2 +8δ2 . +Since ξ ≤ δ, we know that 0 < ρ < 1. +Lemma B.6. Let t ≥ 1. For any 1 ≤ τ ≤ +√ +T satisfying t − τ ≥ (2(C + 2Lg)/ϵ)2 and t ≥ 2τ − 1, we have +E +� +erQt� +≤ ρE +� +erQt−τ � ++ erδτerθt−τ . +Proof. As t ≥ 2τ − 1, we have t − τ ≥ τ − 1. Henceforth, we use the notation wt = Qt − Qt−τ. Note that +wt = +t−1 +� +s=t−τ +Qs+1 − Qs ≤ τδ +holds due to Lemma 4.5(b) because t − τ ≥ (2(C + 2Lg)/ϵ)2. Furthermore, as τ ≤ +√ +T, +rwt ≤ rτδ ≤ ξ +4δ ≤ 1. +The rest of the argument is the same as the proof of Lemma A.4. +Proof of Lemma 4.6. Let t ≥ 1, and let τ = ⌈ +√ +t⌉. For any s, we use θs to denote θs(τ). As in the proof of +Lemma 3.5, we first consider the case where +t ≥ max +� +9, +�2(C + 2Lg) +ϵ +�4� +. +35 + +Then t ≥ 2τ and τ ≥ (2(C + 2Lg)/ϵ)2. Moreover, ⌊t/τ⌋ = k for some k ≥ 2, in which case +t − (k − 1)τ ≥ τ ≥ (2(C + 2Lg)/ϵ)2. +Following the same proof argument of Lemma 3.5, we deduce the following. +E +� +erQt� +≤ erC(2(C+2Lg)/ϵ)2e2rδτerθt +� +ρk−1 +1 − ρ + +k−2 +� +i=0 +ρi +� += +1 +1 − ρerC(2(C+2Lg)/ϵ)2e2rδτerθt. +(71) +As p(x) = erx is convex over x ∈ R, Jensen’s inequality implies that +erE[Qt] ≤ E +� +erQt� +. +Then, by (71), we obtain +E [Qt] ≤ θt + 2δτ + C +�2(C + 2Lg) +ϵ +�2 ++ log +1 +1 − ρ += θt + 2(G + RLgℓ + ϵ)τ + log 128(G + RLgℓ + ϵ)2 +ϵ2 ++ C +�2(C + 2Lg) +ϵ +�2 +≤ θt + 4(G + RLgℓ + ϵ) +√ +t + log 128(G + RLgℓ + ϵ)2 +ϵ2 ++ C +�2(C + 2Lg) +ϵ +�2 +≤ θt + 4(G + RLgℓ + ϵ) +√ +t + log 128(G + RLgℓ + ϵ)2 +ϵ2 ++ C +�2(C + 2Lg) +ϵ +�2 ++ (G + RLgℓ + ϵ) +� +9 + +�2(C + 2Lg) +ϵ +�4� +where the second inequality holds because τ ≤ 2 +√ +t. We can also show that the inequality holds even when +t < max{9, (2(C + 2Lg)/ϵ)4}, as required. +B.6 +Completing the Proofs of Theorem 4.7 (Constraint Violation) and Theorem 4.8 (Re- +gret) +Plugging in the formula of θt to Lemma 4.6, it follows that for any t ≥ 1, +E [Qt] ≤ 2 +� +4G + R2 + qdp2 +∗L2 +g + 2ϵ + 2RLgℓ + +12R2 + 4qdp2 +∗L2 +f +ϵ +� +√ +t ++ 4(G + ϵ)2 + 4R2qdp2 +∗L2 +g +ϵ ++ log 128(G + RLgℓ + ϵ)2 +ϵ2 ++ C +�2(C + 2Lg) +ϵ +�2 ++ (G + RLgℓ + ϵ) +� +9 + +�2(C + 2Lg) +ϵ +�4� +. +(72) +36 + +Proof of Theorem 4.7. By Lemma 4.4 and the expected queue size bound (72), we obtain the following. +E +� T +� +t=1 +gt(xt) +� +≤ E [QT +1] + qdp2 +∗(L2 +f + L2 +g) +T +� +t=1 +Vt +4αt ++ qdp2 +∗L2 +g +T +� +t=1 +E[Qt] +2αt +− +T +� +t=1 +γt +≤ +� +qdp2 +∗(L2 +f + L2 +g) +2 ++ (2qdp2 +∗L2 +g + 4) +� +4G + R2 + qdp2 +∗L2 +g + 2ϵ + 2RLgℓ + +12R2 + 4qdp2 +∗L2 +f +ϵ +�� +√ +T +− C +√ +T ++ +� +4(G + ϵ)2 + 4R2qdp2 +∗L2 +g +ϵ ++ log 128(G + RLgℓ + ϵ)2 +ϵ2 +� � +1 + qdp2 +∗L2 +g(1 + log T) +2 +� ++ +� +C +�2(C + 2Lg) +ϵ +�2 ++ (G + RLgℓ + ϵ) +� +9 + +�2(C + 2Lg) +ϵ +�4�� � +1 + qdp2 +∗L2 +g(1 + log T) +2 +� +. +(73) +Since we set C as +C = +qdp2 +∗(L2 +f + L2 +g) +2 ++ (2qdp2 +∗L2 +g + 4) +� +4G + R2 + qdp2 +∗L2 +g + 2ϵ + 2RLgℓ + +12R2 + 4qdp2 +∗L2 +f +ϵ +� ++ 1, +we have +E +� T +� +t=1 +gt(xt) +� +≤ − +√ +T ++ +� +4(G + ϵ)2 + 4R2qdp2 +∗L2 +g +ϵ ++ log 128(G + RLgℓ + ϵ)2 +ϵ2 +� � +1 + qdp2 +∗L2 +g(1 + log T) +2 +� ++ +� +C +�2(C + 2Lg) +ϵ +�2 ++ (G + RLgℓ + ϵ) +� +9 + +�2(C + 2Lg) +ϵ +�4�� � +1 + qdp2 +∗L2 +g(1 + log T) +2 +� +. +Therefore, there exists some positive constant B such that for any T ≥ B, +E +� T +� +t=1 +gt(xt) +� +≤ 0, +as required. +Proof of Theorem 4.8. If C = O(1), then E[Qt] ≤ B +√ +t for some constant B that depends only on G, Dg, Df, R, ϵ +and Lf, Lg, q, p∗, d, ℓ by (72). By Lemma 4.3, it follows that +E +� T +� +t=1 +ft(xt) − +T +� +t=1 +ft(x∗) +� +≤ +�C(G + C) +ϵ +�2 ++ K2 +√ +T + (C + 2Lg) +T +� +t=1 +E[Qt] +t +≤ +�C(G + C) +ϵ +�2 ++ K2 +√ +T + (C + 2Lg)B +T +� +t=1 +1 +√ +t += O( +√ +T), +37 + +as required. +38 + diff --git a/NNFIT4oBgHgl3EQfcytL/content/tmp_files/load_file.txt b/NNFIT4oBgHgl3EQfcytL/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..5a884ef06fbcd4eff5f08a3247202594f3b90497 --- /dev/null +++ b/NNFIT4oBgHgl3EQfcytL/content/tmp_files/load_file.txt @@ -0,0 +1,1134 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFIT4oBgHgl3EQfcytL/content/2301.11267v1.pdf,len=1133 +page_content='Online Convex Optimization with Stochastic Constraints: Zero Constraint Violation and Bandit Feedback Yeongjong Kim 1 Dabeen Lee 2 * January 27, 2023 Abstract This paper studies online convex optimization with stochastic constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFIT4oBgHgl3EQfcytL/content/2301.11267v1.pdf'} +page_content=' We propose a variant of the drift-plus-penalty algorithm that guarantees O( √ T) expected regret and zero constraint violation, after a fixed number of iterations, which improves the vanilla drift-plus-penalty method with O( √ T) constraint violation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFIT4oBgHgl3EQfcytL/content/2301.11267v1.pdf'} +page_content=' Our algorithm is oblivious to the length of the time horizon T, in contrast to the vanilla drift-plus-penalty method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFIT4oBgHgl3EQfcytL/content/2301.11267v1.pdf'} +page_content=' This is based on our novel drift lemma that provides time-varying bounds on the virtual queue drift and, as a result, leads to time-varying bounds on the expected virtual queue length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFIT4oBgHgl3EQfcytL/content/2301.11267v1.pdf'} +page_content=' Moreover, we extend our framework to stochastic-constrained online convex optimization under two-point bandit feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFIT4oBgHgl3EQfcytL/content/2301.11267v1.pdf'} +page_content=' We show that by adapting our algorithmic framework to the bandit feedback setting, we may still achieve O( √ T) expected regret and zero constraint violation, improving upon the previous work for the case of identical constraint functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFIT4oBgHgl3EQfcytL/content/2301.11267v1.pdf'} +page_content=' Numerical results demonstrate our theoretical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFIT4oBgHgl3EQfcytL/content/2301.11267v1.pdf'} +page_content=' 1 Introduction Online convex optimization (OCO) is a general framework for modeling decision-making problems under uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFIT4oBgHgl3EQfcytL/content/2301.11267v1.pdf'} +page_content=' OCO can be viewed as a repeated game between a learner and an adversarial environment as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFIT4oBgHgl3EQfcytL/content/2301.11267v1.pdf'} +page_content=' At each iteration, the learner selects a decision without the knowledge of the convex loss function chosen by the environment, after which the learner receives the loss associated with the decision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFIT4oBgHgl3EQfcytL/content/2301.11267v1.pdf'} +page_content=' Based on the repeated interactions, the learner adapts to the environment in real-time to minimize cumulative loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFIT4oBgHgl3EQfcytL/content/2301.11267v1.pdf'} +page_content=' The OCO framework is well-suited for optimizing a large-scale complex system, as the problem is often tackled by decomposing it into small optimization problems and solving each piece with limited information to improve tractability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFIT4oBgHgl3EQfcytL/content/2301.11267v1.pdf'} +page_content=' Therefore, OCO is applied to portfolio management [2], routing [5], display learning [10], recommendation systems [13], binary classification [8], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFIT4oBgHgl3EQfcytL/content/2301.11267v1.pdf'} +page_content=' For a comprehensive account of OCO, we refer the reader to [12, 19] and references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFIT4oBgHgl3EQfcytL/content/2301.11267v1.pdf'} +page_content=' 1Department of Mathematical Sciences, KAIST, Daejeon 34126, Republic of Korea 2Department of Industrial and Systems Engineering, KAIST, Daejeon 34126, Republic of Korea Correspondence to 0, i.e., is interior to +FA; conversely, PD;A will concentrate toward 0 if the actual f has d(f;FA)>0, i.e., is interior to ¬FA. + +Repeated-sampling properties are however insufficient to ensure single-sample coherency. The +discrepancy d(y;FA) measures a departure of the single sample y from what the constraints A would lead +us to expect. Here are two single-sample criteria we could require for using discrepancy P-values like pd;A +(or strictly monotone transforms of them) as evidence measures: +1) Suppose the data exhibit no discrepancy whatsoever from the constraints in A, in the sense that +fy ε FA; then we should want d(y;FA) = 0 and hence pd;A = 1. +2) Suppose M entails all constraints imposed by A and possibly more, so that FM  FA; then we +should want d(y;FM) ≥ d(y;FA), with pd;M ≤ pd;A as well. +e.g., see Kempthorne (1976, p. 767). Suppose d(y;FA) is derived from a formal divergence measure +d(f1;f0) on FUFU (Amari, 2016); condition (1) then follows immediately from the nonegativity of +divergences. For condition (2), which may be termed subset coherence, suppose that FA is a subset of an +exponential family of distributions (e.g., Y may be a vector of binomial variates). Denoting mean vectors +by μk = E(Y;fk) and covariance matrices by Σk, k=1,0, suppose d(f1;f0) = (μ1−μ0)’Σ0−1(μ1−μ0), the squared f0- +standardized Euclidean distance between the Y-means of f1 and f0, sometime called the Euclidean +divergence (although that label is sometimes applied to half this function; see Amari, 2016).We then +have the divergence of the data from FA is d(y;FA) = inf{(y−μ0)’Σ0−1(y−μ0): f0εFA}, the Pearson χ2 fit statistic, +while the mean μy;A of its minimizer fy;A is the minimum-χ2 or generalized-least-squares (GLS) estimate of +μ. Subset coherence then follows follows exactly when the covariance matrix Σ is constant over FA, with +no need for approximate normality of Y, and follows in a local asymptotic sense given conventional +regularity conditions. + +Suppose instead we use the divergence of FA from the data, d(FA;y) = inf{d(f;fy): fεFA}. Subset coherence +then follows when taking d(f1;f0) = E(ln(f0(Y))−ln(f1(Y));f0), the Kullback-Leibler divergence (KLD) of f1 from +f0, also known as the relative entropy or the discrimination information. The KLD is defined under the + +Page 8 of 49 + +usual conventions that 0∙ln(0) = 0 derived from lim(x∙ln(x): x↓0) = 0, and that f0(y) = 0 when f1(y) = 0 +(absolute continuity of f0 with respect to f1). Because fy(y) = 1 and is 0 elsewhere, we have E(ln(fy(Y));fy) = +0, E(ln(f1(Y));fY) = ln(f1(y)) and thus d(FA;y) = sup{−ln(f(y)): fεFA}, half the deviance statistic used in global +tests of fit for regular GLMs. When it exists, the maximum-likelihood estimate (MLE) max{f(y): fεFA} of f +under A is then fA;y = arginf{d(FA;y): fεFA}, and fy is the MLE of f when A entails no constraint on f, as +when A =  (so that FA = FU). + +Decision P-values +In the second definition, a P-value is merely a by-product of a decision criterion (“test”) for whether to +reject or accept the family FA for further use. The analyst specifies in advance a maximum acceptable +false-rejection (Type-I error) rate α for FA and an alternative distribution family Falt  FU disjoint from FA. +Where possible one then finds a critical region RA(α)  RU to implement the decision rule “reject FA if y ε +RA(α)”, for which +1) The type-I error rate is controlled at level α (test validity): When fεFA, f(RA(α)) ≤ α; i.e., when A +holds, the probability f(RA(α)) of YεRA(α) does not exceed α. +2) The type-II error rate over Falt is minimized among valid tests: When fεFalt, f(RA(α)) ≥ f(R) for any +other R  RU that has f(R) ≤ α whenever fεFA; i.e., among valid tests (1), when A fails, RA(α) +maximizes the rejection probability (power). +3) The type-II error rate never falls below α (unbiasedness): When fεFalt, f(RA(α)) ≥ α. +If a set function RA(α) of α satisfies 1-3 for all α, the decision rule “reject FA if yεRA(α)” is called a +uniformly most powerful unbiased (UMPU) α-level test procedure. In Neyman-Pearson (NP) hypothesis- +testing theory, if this “optimal” (UMPU) function RA(α) exists, nothing more is needed for the decision +about FA. + +The maximum acceptable Type-I error α may be left unspecified, and the reader may be provided +instead the observed decision P-value pinf(α) = inf{α: yεRA(α)}; their decision can then be based on +inserting their own α in the P-rule “reject FA if pinf(α) ≤ α” (Lehman, 1986, p. 70). This methodology is +however usually implemented by finding a test statistic TA = t(Y;FA) with realization t = t(y;FA) and mass +function hA(t) such that pinf(α) = pt;A = hA({TA≥t}) = u≥t hA(u)du ≤ α if and only if y ε RA(α); in basic examples +TA is a log likelihood ratio. Many sources however define a decision P-value for FA as the corresponding +random variable PT;A = u≥T hA(u)du with realizations pt;A. If fεFA and RA(α) defines a valid test, Pr(PT;A ≤ α) = +f({YεR(α)}) ≤ α and thus PT;A is distributed as or exceeds a unit-uniform random variable when fεFA. + +Page 9 of 49 + + +The latter uniformity condition is sometimes used to define P-value validity, for it ensures that for any α +the false-rejection rate (size) of the P-rule never exceeds α. Both the divergence PD;A and decision PT;A +are valid in this sense; a key difference is that PT;A is further optimized for power, which can bring its +distribution closer to uniform from above. Note however that there are no single-sample conditions +applied to the realizations t or pt;A; this neglect is a source of defects of decision P-values as evidence +measures, and may arguably be seen as a defect in their use as decision guides as well. An analogous +objection to decision P-values arises when, due to discreteness, exact uniformity of P-values and an +exact size α UMPU tests cannot be obtained without randomizing decisions at boundary observations. +This leads to examples in which the single-sample decision comes down to a coin toss – an anomaly +avoided by redefining the P-value (Lancaster, 1961). + +The two types of P-values are not equivalent +Most of the literature seems to ignore that there are two definitions and interpretations of P-values, +and to the extent it recognizes them, it treats them as differing only in that a divergence P-value is an +observed (realized) quantity and a decision P-value is the corresponding random variable. This is not +however true in general: When single-sample descriptive-geometric coherence criteria are imposed on +discrepancy P-values (for example by using a divergence measure as the discrepancy) but only repeated- +sampling criteria are used to derive decision P-values, we can have pd;A > pt;A, as in the example below +where pd;A−pt;A approaches ½. While such examples correspond to greater power for the P-rule, +Schervish (1996) showed that, for FA defined by bounded parameter intervals and UMPU procedures, +the resulting decision P-values could violate subset coherence (condition 2) in that one could have FM  +FA and yet have pt;M > pt;A. + +In sum, realizations t(y;FA) of UMPU test statistics can violate basic axioms for divergence measures +when treated as functions of FA, leading to incoherent decision P-values pt;A. This incoherency can be +attributed to giving power priority over single-sample coherence, and arguably should disqualify pt;A as a +measure of data evidence about A. In contrast, a divergence P-value pd;A starts from a coherent +discrepancy measure and thus will coherently indicate the compatibility of the targeted constraint set A +or family FA with the data y (more precisely, fy), at least in large samples and often exactly, depending on +the divergence measure. Consequently, 1−pd;A or −logc(pd;A) will coherently indicate the degree of +refutation of A provided by the divergence measure – all without reference to any alternative. + +Page 10 of 49 + + +Interestingly, Schervish took his result to imply that decision P-values were incoherent measures of +evidence supporting FA. As will be discussed below, there are purely logical reasons for rejecting any +statistic as an absolute or unconditional measure of support. As likelihoodists often argue, support (and +thus the possibility of acceptance of A) can only be relative to or conditional on specified alternatives to +FA (e.g., Edwards, 1992; Royall, 1997). Similarly, decision procedures also need to specify alternatives to +FA, although this need does not dictate imposition of UMPU (Hansen & Rice, 2023). + +Summary +Divergence P-values summarize probabilistic information about how far the data diverge from an +embedding model A, or how far the data filtered through an embedding model A diverge from the data +filtered through a more restrictive target model M. The information is measured using the geometry of +the model family defined by A or M. The definitions of divergence P-values differ from NP definitions: In +NP statistics, observed P-values are intermediate calculations for decisions derived solely to satisfy +resampling criteria; they are defined as the minimum α-level at which the tested model would be +rejected by an optimized NP test (Lehmann, 1986, p. 70). In this sense, claims that P-values are +incoherent or do not measure evidence are circular, being consequences of adopting definitions that +force P-values to conform to repeated-sampling criteria (such as UMPU) without regard to single-sample +coherence. In contrast, definitions derived instead from geometric comparisons of observed data and its +projections onto model regions will automatically satisfy at least approximate coherence requirements. + +Misconceptions about P-values and evidence +There are now hundreds of articles spanning many decades about how everyday researchers +misinterpret and abuse statistical tests and P-values. There are far fewer about how myths about P- +values have been promulgated and perpetuated. For example, it is not uncommon to see claims that P- +values “overstate evidence against null hypotheses”. Yet observed P-values are just basic probability +statements about the location of a statistic in a reference distribution, mute about their own “strength”. +Thus, the claims do not reflect anything intrinsic to the concept of a P-value, but instead reflect a +synergism between user mistakes and philosophical commitments that argue against use of P-values: +Users mistakenly perceive the limited evidence encoded in observing p=0.05 as strong simply because +0.05 is an entrenched standard for declaring “statistical significance” – and, far too often, a decisive +factor for publication, as illustrated in Fig. 1 of van Zwet & Cator (2021). In tandem, some statistical + +Page 11 of 49 + +commentators take posterior probabilities or likelihood ratios as “gold standards” for evidence +measurement; those standards are however rejected by those who see how the conflict of those +“standards” with frequentist testing can reflect failings of the prior or likelihood models, rather than +deficiencies of the frequentist assessments (Ritov et al., 2014; Greenland, 2019a; Bickel, 2021a, 2021b). + +A more subtle cognitive problem arose however as modern mathematical statistics emerged. Originally, +P-values were defined directly as probabilities of observed events, where the events concerned statistics +that gauged data divergence from expectations computed from a target (test) model or hypothesis (e.g., +Pearson, 1900; Fisher, 1934, p. 66). These original P-values made no reference to alternative models or +formal decision rules (Cox, 1977). That view further recognized how closeness of the model predictions +to the data (as indicated by a large goodness-of-fit P-value) did not imply the target model is correct +(Pearson, 1906). But Neyman-Pearson (NP) hypothesis testing altered the definition of a P-value to that +of a type of random variable P whose realization p was the smallest α level at which the criterion “reject +if p≤α” would reject the hypothesis, given the observed data (Lehmann, 1986, p. 70) – a redefinition +which almost inextricably bound P-values to statistical decisions. + +Many subsequent authors wrote as if the original goodness-of-fit and NP definitions were +mathematically equivalent, even though (as will be discussed below) this not always the case. This +subtle mistake parallels but is distinct from two different traditions. One tradition uses P-values as +continuous evidence measures, and again is often called “neo-Fisherian” although traceable to Karl +Pearson (Hurlbert & Lombardi, 2009). In contrast, the NP tradition uses P-values solely as components of +decision criteria, a usage which requires specification of alternatives along with choice of a cut-off or α- +level justified by error costs (Neyman 1977; Lakens et al. 2018, Mayo 2018) – albeit in practice a +justification is rarely seen and instead a default α (most often 0.05) is used with no regard to actual error +costs or to uncertainty about the model used to justify error-rate claims. + +With the ascendance of NP theory in mathematical statistics, many if not most writers now ignore how +P-values can be defined and treated descriptively, unconnected to formal decision rules; they instead +focus on how P-values can malfunction when (as often the case) they are misused to measure support, +or else focus on how they can be properly connected to Bayesian measures. The present paper shows +how descriptive P-values can be derived to provide coherent measures of refutational evidence, without +invoking notions of support. + +Page 12 of 49 + + +Descriptive P-values measure compatibility without measuring support +The scale of a P-value p typically runs from 0 to 1, where 0 implies impossibility of an observed statistic +under the target model (complete discord between the statistic and the model) and 1 corresponds to no +conflict of the statistic with the model (no discord between the statistic and the model). This scaling +seems to suggest P-values measure support, but again only reflects degrees of compatibility between +the data and the model. The description of P-values as compatibility measures can be seen in Box (1980, +p. 387), Bayarri & Berger (2000, 2004), Robins et al. (2000) and Greenland (2019a), and is anticipated by +this passage in Fisher (1935, p. 207; emphasis added): +“If we consider a series of hypotheses with different values for the diminishing return [on crop yield from +adding more phosphate fertilizer], and determine which of these values are compatible, at any given level +of significance, with the observed yields, some of the values which would appear to be acceptable would +be negative…”; +see also Pearson (1900, p. 170-171) and Fisher (1934, p. 66) for similar use of “compatible” with P- +values. The same concept has also been described as goodness-of-fit (Pearson, 1900), consonance +(Kempthorne & Folks, 1971; Folks, 1981), and consistency (Cox, 1977); unfortunately, “consistency” is +more often used for unrelated convergence properties. Other terms for the compatibility concept +include conformity, concordance, and accord (Rice, personal communication). Its opposite may thus be +termed incompatibility or discordance, which can be measured by 1−p, although there are cognitive and +abstract arguments for instead employing the surprisal measure or S-value s = −logc(p) for +incompatibility (Greenland, 2019a; Rafi & Greenland, 2020; Greenland 2021a; Amrhein & Greenland +2022; Greenland et al., 2022, 2023). + +Understanding why compatibility is a logically weaker concept than support requires recognizing why +conformity of observations to a target model – that is, failure of observations to conflict with or refute a +model – does not by itself imply support for the model: There are always many other models that will fit +the data just as well yet contradict the target model; consequently, one must a priori impose very tight +restrictions on those other models to generate a support measure. As an extreme example, upon +observing no difference between treated groups in a randomized trial, we may say the model of no +treatment effect is perfectly compatible with the data; but unless we restrict alternative models to trials +that are competently and honestly conducted, we must face that the same observation is also perfectly + +Page 13 of 49 + +compatible with models in which the data were mishandled or altered to erase the appearance of an +effect. + +In going beyond the simple realm of binary logic and into a world of uncertain and extensive complexity, +support and conflict are not logical complements of one another because they do not encompass all the +possibilities for evidence. For example, evidence may be wholly indeterminate, supplying neither +conflict with nor support for a targeted hypothesis. The conceptual incompleteness of conflict and +support corresponds to the asymmetry between evidence against and evidence for a hypothesis, which +is emphasized heavily in falsificationist philosophy (e.g., Popper, 1959); there, “corroboration” is used to +indicate failure to refute a hypothesis without implying support or confirmation. Corroboration is meant +to be weaker than support; nonetheless, in ordinary language, compatibility is even weaker than +corroboration, for compatibility suggests nothing at all about the power or severity of the criterion being +used for evaluation – it is simply the opposite or negation of incompatibility. For example, making one +coin toss and observing heads is highly compatible with the hypothesis H: “The probability  of heads is +½”; but to say it corroborates or supports H insinuates that the observation has more than trivial +evidential value about that H. On the other hand, incompatibility and refutation do have a parallel: Most +would say observing heads is highly incompatible with and even refutes H: “0 <  ≤ 10−9” even though +heads is not impossible under H. + +Ignoring the difference between compatibility and support can be seen as a source of claims that +evidence measurement and statistical testing require specification of alternatives, one of the major +points of contention in the Fisher-Neyman split (Fisher, 1955; Pearson, 1955; Neyman, 1956). While +addressing this asymmetry is not needed for the mathematical development, it should be borne in mind +when interpreting results. In particular, a reason to refer to P-values as compatibility measures is that +they are more accurately understood when recognized as reversals of conflict orderings, rather than as +measures of support (which require restriction of alternatives to a well-defined family of distributions). + +What do P-values and their transforms describe? +P-values describe a relation of a model to data, or the relation of a more restrictive model M to a less +restrictive embedding model A in light of the data, as seen in comparing observed to expected data, or +more generally in comparing M-expected to A-expected fits to the data when M is nested in A (i.e., +imposes all the constraints in A and more). The latter comparison measures information about M in the + +Page 14 of 49 + +data, given A. Another way to characterize this process is that of measuring divergences among different +degrees of data smoothing, ranging from no smoothing (the raw data, which is the expected data under +a saturated model) to A-smoothing to more severe M-smoothing. + +The greater the smoothing (the more constraints imposed), more data detail will be discarded as “noise” +under the model, and more certainty will be assigned to any structure (apparent signal) remaining in the +expectations (e.g., that a parameter appears to be large enough to signal treatment superiority, where +the latter is defined context-specifically). Of course, the model may be mistaken, in which case +important information may be discarded or smoothed away by the model, and excessive certainty will +be assigned to the remaining apparent structure (signal). Thus, to avoid catastrophic loss while getting +rid of obvious noise and hopelessly weak signals, one may maximize the embedding model (minimize +the number of constraints in A) based on context-specific information (Greenland, 2006). + +Given a model, inference may be based on a model diagnostic, as when judging whether too much +structure is smoothed away (too much information is discarded) by the model using tools like residual +plots and P-values for model fit. Upon adopting a working model that has passed these tests, inference +becomes instead model conditional, as when model expectations are substituted for the data and then +used to make claims about the data generator. Examples include marginal (population-standardized) +effect estimates with components estimated from model-fitted quantities such as balancing scores. The +inferential claims are often supported by P-values for coefficients or their inversion into interval +estimates using the selected model, without accounting for the model selection that took place. As has +long been documented, such preselection will invalidate decision rules that ignore it (Leamer, 1978). A +pertinent question is then: Do these P-values mean anything despite the selection? One answer is that +Fisherian P-values (those defined from observed divergences) still describe an observed discrepancy +between the data and the model when that discrepancy is adjusted (“standardized”) to allow for the +random component of the model, which is to say the noise distribution assumed by the model. This +interpretation holds even though the naïve comparison of such post-selection P-values to a pre- +specified α would lead to repeated-sampling-and-selection error rates far higher than α, and thus brings +to the fore the conflict between descriptive and decision uses of P-values. + +I have avoided using “population” because it suggests to most users that the P-value is describing some +target population from which the data were sampled. In reality, the data often come from a source that + +Page 15 of 49 + +happens to be available but whose connection to the actual decision target may be quite speculative. +Then too, the data are affected not only by intentional (interventional) design elements such as +matching and randomization, but also by unintended influences such as causes of refusal to participate, +loss to follow-up, and other procedural problems, as well as sample and data alterations made in the +course of analysis (e.g., discarding units declared as outliers, or fixing records that have missing or +clearly wrong data items). Thus, to validly infer back to the data source or make decisions with known +error rates requires more than just rigid adherence to a pre-specified analysis protocol; it also requires a +model which adequately reproduces the behavior of the entire physical process that determined the +data used in the analysis (Greenland, 2005, 2022). + +Measuring compatibility and conflict with reference distributions and S-values +In general form, the coherence criterion requires that a measure of evidence against a target model M +or hypothesis H is never less than the measure against an embedding model A that contains the target +model M as the special case of A in which the hypothesis H holds. A special case of this criterion was +introduced above as subset coherence for P-values, which requires that the observed P-value for the fit +of a target model M never exceeds the observed P-value for the fit of a more general, flexible +embedding model A in which M is nested. An analogous criterion for a support measure requires that +the measure for M never exceeds that for A. Such criteria can be traced at least back to Gabriel (1969) +and have been extensively studied since, e.g., see Schervish (1996), Fossaluza et al. (2017), Bickel & +Patriota (2019), and Hansen & Rice (2023). + +A descriptive P-value provides the location of an observed statistic in a reference distribution derived +from the targeted (“test” or “null”) model M and the data, and is thus tailored to the study as observed. +In this sense, it is anchored in the single-sample viewpoint. The reference distribution need not be based +on the goal, structure, or optimization of a decision over repeated sampling; instead, its purpose is to +provide coherent measures of compatibility and conflict between the data and the model. It may be +highly conditioned on the observed data in ways that depend on the targeted model; hence the +reference distribution may vary considerably across resampling from the distribution entailed by the +actual data-generating mechanism. Again, as per Cox (1977, sec. 2.1), “such a procedure is to be +distinguished sharply from a decision problem in which ‘acceptance’ or ‘rejection’ is required”. + + +Page 16 of 49 + +In contrast, in Neyman-Pearson (NP) statistics, P-values are random variables that are components of +the decision rule “reject if P≤α”, or are by-products of the decision “reject if the test statistic falls in a +critical region of size α” (Lehmann, 1986, p. 70). In the pure frequentist view of Neyman (1977), losses +and hence these random variables are evaluated over their actual resampling distributions according to +criteria such as power and “unbiasedness”, without regard to features of single-sample realizations; +they thus can suffer from incoherencies when extended to interval hypotheses (Schervish, 1996; Hansen +& Rice, 2023). Statistical arguments against P-values as evidence measures have been based on these +decision-theoretic definitions and criteria; they have largely ignored or dismissed treatments that start +from the original divergence definition (Pearson, 1900) to interpret P-values or their transforms as +single-sample information summaries or measures of compatibility or conflict between data and +assumptions. The sections below will illustrate and expand on how the descriptive treatment of P-values +differs from the decision-theoretic treatment, focusing on the special case in which a P-value +summarizes relations among models and data within an information geometry defined by the +observations as well as the models. + +The Appendix provides a technically more detailed review of divergence P-values illustrated with basic +generalized-linear models (GLMs). In doing so its focus is on intuitive visualization of actual single- +sample properties of P-values. It further reinforces arguments that common misperceptions of P-values +can be mitigated by transforming an observed p to an unbounded and reversed scale to provide a +measure of incompatibility, conflict, refutation, or surprise. This rescaling also removes evidence +measurement from the 0-to-1 scale on which posterior probabilities live, thus removing a source of +confusion of P-values with posterior probabilities. + +The most common example of such a transform is the negative base-c logarithmic transform to a +surprisal or S-value sc = logc(1/p) = −logc(p) = −ln(p)/ln(c); sc = 0 then says the observed discrepancy +statistic or divergence measure from which the P-value was computed was completely unsurprising +given the target model, or that it does not conflict with or contains no information refuting the model. +As the divergence increases, sc increases to reflect surprise at the data given the model, or conflict with +the target model (Bayarri & Berger, 1999; Greenland, 2019a; Rafi & Greenland, 2020; Cole et al. 2021; +Gibson, 2021; Amrhein & Greenland, 2022; Greenland et al., 2022). The base of the log, c, can be seen as +a scale factor; c=10 is commonly used but has no theoretical justification. Taking instead c=e produces +the natural S-value se = −ln(p) which is a special case of the “E-value” in Grünwald et al. (2021) and the + +Page 17 of 49 + +“betting score” in Shafer (2021); see Greenland (2021a). As explained in the Appendix and elsewhere, +c=2 provides an intuitive mechanical interpretation: The binary S-value s2 = −log2(p) measures bits of +information against a model supplied by the statistic, which translates easily into outcomes of coin- +tossing experiments (Greenland, 2019a; Rafi & Greenland, 2020; Cole et al. 2021; Greenland 2021a; +Greenland et al., 2022). + +Competing concepts of P-values +Coherence vs. decision P-values +For NP frequentists, statistical decisions calibrated to long-run frequencies are the central goal of +analysis, and the paramount definitions and evaluation criteria refer only to resampling frequencies. +Pursuing this goal, modern NP theory treats a P-value for a hypothesis H as a random variable that +follows (or at least approximates from above) a uniform resampling distribution when H is correct, and +concentrating toward 0 otherwise. This theory makes no reference to data description; in practice +however the ensuing single-sample decisions are usually misrepresented as data descriptions, as when a +paper reports “no association was observed” when in fact this only meant that p>0.05. + +An acceptable measure of incompatibility or conflict between data and hypotheses must obey the +single-sample criterion of subset coherence, in that its value for a hypothesis expressed as a subset of a +function or parameter space cannot exceed its value for any of its own subsets. In parallel, a measure of +compatibility is coherent if its value for such a hypothesis cannot exceed its value for any of its +supersets. Again, compatibility is a weaker condition than support, in that support implies compatibility +but compatibility does not imply support. Compatibility and support do however share an analogous +coherence concept: A measure of support by data for a hypotheses is coherent if its value for a +hypothesis cannot exceed its value for any of its supersets. Likelihood maxima over sets satisfy this +requirement and thus are coherent support measures. + +In contrast, using interval hypotheses in a simple normal location model, Schervish (1996) showed that +an extension of uniformly most powerful unbiased (UMPU) P-values to interval hypotheses (Lehmann, +1986, sec. 4.2) is an incoherent measure of support; hence a scale reversal such as 1−p or −logc(p) would +provide an incoherent measure of conflict if p were defined from an UMPU test of an interval +hypothesis. Similar problems can arise with Bayes factors (Lavine and Schervish, 1999), but see also +Good (2001). The UMPU extension goes beyond ensuring that the simple decision rule “reject if p≤α” + +Page 18 of 49 + +has Type-I error rate (size) no more than α over the resampling distribution under the interval +hypothesis, by imposing conditions to optimize power. Key to the present discussion is that the P-value +in the rule for an interval UMPU test (Hodges and Lehmann, 1954; Lehmann, 1986, sec. 4.2; Schervish, +1996) is defined and evaluated solely over resampling, with no attention to single-sample coherence. + +Divergence P-values +Incoherence does not afflict all extensions of P-values, such as those that aim only to measure +information on a compatibility/incompatibility scale. Consider definitions that lead to maximizing a +point-hypothesis P-value over a hypothesized interval for a real-valued parameter: Such max-p (or +supremum) extensions have been disqualified from being “frequentist” by adherents of the NP +definition (e.g., see Remark 1 of Robins et al., 2000 p. 1145). One reason is that the maximum two-sided +P-value will equal 1 whenever the parameter estimate falls in the hypothesized interval, resulting in a +distribution far above uniform when the parameter is in the interval interior. This property led Schervish +(1996, sec. 4) to described max-p extensions as “simple minded” and “not particularly useful”. But these +dismissals are based on criteria that can be rejected by those who use P-values as discrepancy measures +or information summaries rather than as test criteria. As illustrated below and explained in the +Appendix, summarization can start from a geometric derivation of P-values from divergence measures. +Doing so naturally leads to maximizing P-values over intervals, thus enforcing their coherence; for point +hypotheses such as H: μ=m, these P-values simplify to ordinary two-sided P-values. + +The descriptive goal is to summarize data information about models, particularly comparisons of fitted +models against data and each other. A descriptive P-value is thus the observed quantile at which a +discrepancy statistic sits in a reference distribution derived from the model. In particular, when +divergence statistics are used to measure discrepancies, the reference distribution may be derived from +a geometry on a model space that combines information on both the data-generating distribution and +the observations. This distribution serves only to locate the data relative to the model in a particular +direction in the model space, using a coordinate scaling standardized to the estimated sampling +variation. The resulting P-values are numerically identical to observed decision-theory P-values when +the divergence statistic and reference distribution chosen for summarization are identical to the test +statistic and resampling distribution chosen for decisions. While this identity is most common, it is not +dictated by the goals of the two approaches and is violated in the examples below and in the Appendix; +Bayarri and Berger (2004) discuss conditioning as a method of reconciling the approaches. + +Page 19 of 49 + + +As illustrated in the Appendix, the descriptive goal can be rephrased as that of depicting relations among +different degrees of data smoothing or filtration, where the filtrations are through nested models. This +goal stands in sharp contrast to Bayesian methods, whose goal is to make statements (or “inferences”) +about parameters conditional on data; these methods require inputs of fully specified parameter +distributions and can break down spectacularly in high dimensions (Robins and Ritov 1997; Ritov et al. +2014). In terms of goals, NP decision theory is more akin to Bayesian methods in going beyond +descriptions to make direct statements or take actions about parameters or models “in light of the +data”, although they differ from ordinary Bayesian decisions in allowing statements or actions that are +not based on parameter distributions or full conditioning on the observed data. + +A core requirement of NP decision theory and frequentist inference procedures is that the data +generator involves a randomizer (for selection into the data, if population inference is a goal; or for +treatment assignment, if causal inference is a goal) that is known apart from an estimable parameter +vector (such as the coefficients in a sample-selection or treatment-assignment probability function). In +the descriptive view, however, the randomizer is demoted to being simply another assumption in the +model used to compute the P-value, and thus (as with other assumptions) its violation may be brought +forward as an explanation for the size of an observed p. There is then no inference beyond the noting +that a small p can signal a problem with one of the assumptions in the model used to derive the +reference distribution – it may be small because the treatment has an effect different from that +assumed by the model, or because there were uncontrolled nonrandom elements in the actual +treatment assignment, observation selection, or data missingness (e.g., nonrandom censoring). This +unconditional descriptive view does not logically condition on any assumption, and thus is inferentially +mute: It does not single out violation of an assumed (test) hypothesis to explain why p seemed small; +nor does it single out satisfaction of the hypothesis to explain why p seemed large. A descriptive P-value +simply states a factual relation between what was observed and what was expected based on the target +model, where “target model” refers to all the assumptions or constraints used to compute the P-value, +including H. + +The main point is that the conflicting goals of decision and descriptive frequentist approaches lead to +definitions and interpretations of extended P-values that differ and may come into conflict. The two + +Page 20 of 49 + +views can however be at least partially reconciled through the criteria and preferences they share. +Consider the following passage from p. 1144 of Robins et al. (2000; emphasis added): +…a p value is useful for assessing compatibility of the null model with the data only if its +distribution under the null model is known to the analyst; otherwise, the analyst has no way of +assessing whether or not observing p = .25, say, is surprising, were the null model true. That we +specify that distribution to be uniform is largely a matter of convention. +The view presented here agrees in that, in order to properly interpret a P-value, we must know +sufficient detail about the meaning and distribution of its source statistic under the target model and its +violations. Nonetheless, most of the assumptions that compose the target model cannot be assured to +hold in observational studies or in randomized experiments with much drop-out or nonadherence. +Furthermore, when the target model does not involve dimension reduction (so the model has interior +points in the information topology described in the Appendix), uniformity of a random P-value becomes +a useful property only at boundary points of the distribution subspace defined by the targeted model. + +For evaluating statistical performance, the descriptive goal further replaces power by information +content. The descriptive view thus regards the emphasis on uniformity of P-values under assumed +models (as seen in most theoretical literature) as a product of NP goals applied to highly idealized +experimental models. Because those models are usually unrealistic in health and medical research, we +seek instead to describe the relations that hold between observations and models, regardless of the +model’s accuracy. + +Example: A divergence P-value for a simple interval hypothesis +Consider the model for n observations arrayed in a vector y = (y1,…,yn)’ in which the yi are assumed to be +independent draws from a normal distribution with unknown mean μ and known variance σ2; call this +set of assumptions the embedding model A, leaving μ the only free parameter in the model. This +embedding or background model places severe restrictions on the distribution F(y) for the random data +vector Y = (Y1,…,Yn)’: It says that if the physical data generator were left to run indefinitely, it would +behave exactly like a random-vector generator for an uncorrelated n-variate normal distribution with +equal Yi means and variances. Note that A implies the sample mean of the yi can replace the data +without loss of information about the data generator, i.e., it is sufficient for determining the behavior of +the generator to the extent allowed by the data, since A implies that behavior is normal (Gaussian) with +variance σ2. But A does not imply that the actual observed-y histogram would be visually well + +Page 21 of 49 + +approximated by a normal density with mean equal to the sample mean and variance equal to σ2, which +is often what is meant when saying the fitted model is adequate as a data summary or compression; the +latter claim requires evaluation of A against the full data vector y, including graphical as well as +goodness-of-fit diagnostics. + +Accepting A for the moment, we may wish to evaluate the information that the data vector y supplies +against a submodel M nested within A that imposes additional restrictions H on μ. Specifically, suppose +M adds to A the restriction H: μ=m with m known. Evidence against μ=m is usually gauged by finding the +“standardized” distance |μ�−m|/(σ/n½) from the sample mean μ� to the target value m in a standard- +normal tail-area function to obtain the 2-sided P-value pm. Equivalently, to get pm we could find the +divergence statistic dm = d(m;μ�) = (n/σ2)|μ�−m|2 in a 1 degree-of-freedom (df) χ2 distribution F(d), +whence we can describe the standardization as multiplying the squared distance from the data summary +to the hypothesized mean m by n/σ2, which is the amount of Fisher information in the data about μ +given the model A. Furthermore, given normality, d(m;μ�) is identical to the usual likelihood-ratio (LR) +statistic for evaluating M against A, and (as discussed in the Appendix) is twice the Kullback-Leibler +information divergence of M from A. + +Under M, the distributions of the random analogs Dm of dm and Pm of pm are derived by treating the Yi as +independent normal(m,σ2); Pm will then be uniform, implying the NP decision rule “reject H if pm≤α” will +have size α; i.e., Pr(Pm≤α;μ=m) = α. If however μ≠m but A (i.i.d. normality with known variance) still +holds, the distribution of Pm will be increasingly concentrated toward 0 as the noncentrality parameter +d(m;μ) = (n/σ2)|μ−m|2 grows. + +The random divergence statistic Dm and P-value Pm for H: μ=m are identical to the test (decision) statistic +and P-value in NP theory, and their distributions are fixed and known given μ=m. The divergence view +departs from the testing (decision) view when M instead adds to A an interval restriction H: mL ≤ μ ≤ mU. +The squared distance from μ� to [mL,mU] is +d([mL,mU];μ�) = min{d(m;μ�): m є [mL,mU]} +The resulting observed divergence P-value for the model M or for the hypothesis H given A is +pM = max{pm: m є [mL,mU]}. + +Page 22 of 49 + +When mL= mU = m, we get pM = pm. But when mL< mU, the distributions of d([mL,mU];μ�) and of pM are no +longer known given M; they can only be computed conditional on μ and may vary considerably across μ +in the interval [mL,mU] defined by H. + +We can nonetheless summarize the discrepancy of μ� from [mL,mU] using pM. The divergence +d([mL,mU];μ�) goes to infinity as μ� becomes more distant from [mL,mU], and is at its minimum of zero +when μ� is in [mL,mU]. Thus, pM will range from zero (μ� infinitely far from H) to one (μ� zero distance from +H). If we consider the standardized distance d(m;μ�) of μ� from the interval as indicating the extent of +incompatibility between the observations and the hypothesis, pM can be taken as an index of +compatibility of the data summary μ� with the model M that restricts μ to [mL,mU]. To restore a direct +relation to the divergence and incompatibility, we may transform pM to a surprisal or S-value such as sM += −log2(pM), which ranges from zero: μ� zero distance from the interval to infinity: μ� infinitely far; sM +provides other benefits for interpreting the observed divergence in terms of the information it supplies +against H given A (Greenland, 2019a, 2021a; Rafi & Greenland, 2020; Cole et al. 2021; Greenland et al., +2022, 2023). + +Turning to the corresponding random P-value PM, we see that with correct A, mL< mU, and increasing +sample size + +when H is incorrect, μ is exterior to the interval (μ 0. A more general approach which allows for +random zero counts replaces y by counts μS fitted under a highly parameterized, minimally constraining +model that removes zeros without oversmoothing the data, and then uses F(y;μS) instead of F(y;y) +(Greenland, 2006). + +Let Rμ be the set of logically possible μ under the distributional form for F(y;μ) specified by A. Each point +μ in Rμ then corresponds to a distribution F(y;μ) for Y in a distribution space Fμ = {F(y;μ): μ є Rμ}. I will +assume the dimension dim(Rμ) of Rμ is n; A may however include assumptions that logically constrain Rμ. +As examples, if A assumes that the Y components are independent normal then Rμ = Rn; if A instead +assumes that Y components are independent exponential or Poisson then Rμ is the positive orthant of +Rn; and if A instead assumes Y components are independent Bernoulli then Rμ is the positive open unit +cube in Rn. + +Denote by RA the set of μ such that F(y;μ) satisfies all the constraints in A (including for example linearity +constraints as well as logical constraints); dim(RA) is often called the total degrees of freedom (df) for A +and dim(Rμ)−dim(RA) the residual df for evaluating A. As an example, suppose A says Y is multivariate +normal(βx,Σ) with β an unknown scalar β, Σ a known positive-definite covariance matrix, and x an n- +vector of known constants. Then RA is the line in Rn through the origin traced out by letting β vary, +dim(RA)=1, and the residual df is n−1. In contrast, for saturated A there is no constraint beyond the +distributional ones, which implies RA = Rμ and hence dim(RA) = n, leaving zero residual df for evaluating +A. + + +Page 37 of 49 + +Nested model evaluations +Suppose we wish to compare a more restricted model (larger set of constraints) M against a less +restricted reference model A with fewer constraints, so that in terms of constraints A ⊂ M. In terms of +the means μ however this containment is reversed to RM ⊂ RA, so that M is often said to be nested in or +a submodel of A, and A is sometimes called the embedding or reference model for evaluating M. +Classical goodness-of-fit evaluations of M against A such as Pearson’s χ2 test take A as saturated with +dim(RA) = n > dim(RM) and define dfM|A = dim(RA)−dim(RM) as the df for the evaluation . Nonetheless, as +with interval hypothesis examples, we may have dfM|A = 0, yet evaluation is still possible. + +The set of constraints added by M above A is the set difference H = A–M (usually H0 is used, but that +invites confusion with more specific constraints assigning zero to a parameter). M then equals the set +union H+A, while A = M−H. A is sometimes called the set of auxiliary constraints for evaluating H. I will +further assume that M confines μ to a fixed finite-dimensional subset of Rn as n increases, and unless +stated otherwise (as with saturated models) I will assume the same for A. The total df for H given A is +sometimes defined as dfM|A but again, this can be zero even if H is nonempty, as when H consists of +inequalities. + +As an example, suppose again the embedding model A includes (along with distributional assumptions) +the constraint that μ = βx for an unknown scalar β and a known vector x of constants, so that RA is a line +through the origin and dim(RM) = 1, while M adds above A the further assumption that β=0, +corresponding to H: β=0 (equivalent to the constraint that μ is the zero vector). RM then is a single point +(the origin) and dim(RM) = 0, while the df for H given A is dfM|A = 1. Now instead suppose M adds to A the +constraint H: β≥0 (rather than β=0); then RM has the same dimension as RA and will contain interior +points (i.e., points bounded away from RA−RM) within the ordinary induced topology of RA. More +generally, if the constraints in A beyond the distributional assumptions are defined using kA functionally +independent equations, and H comprises kH more independent equations for a total of kM = kH+kA +constraints in M, then RA and RM are n−kA and n−kM dimensional manifolds traced out in Rn and dfM|A = +kH. But if instead RM adds only independent inequalities, dfM|A = 0 and RM will ordinarily have interior +points relative to RA. + +Divergence measures + +Page 38 of 49 + +While there is no single formal definition of model regularity, under the usual definitions and models RA +and RM will have the properties required for ease of analysis such as being closed and simply connected +with smooth boundaries, as will be assumed here. Nonetheless, even with such regularity, divergence +and NP-decision P-values can come into serious conflict when (as with interval H) some of the +constraints in H are inequalities. The conflicts may be described as arising from differences in goals +which lead to different treatments of RM when H contains inequalities or other problematic constraints. + +NP theory aims to “test” M against A based solely on resampling criteria, making a decision about +whether to reject M for poor fit relative to A (model checking) or equivalently whether to reject H given +A (hypothesis testing). In contrast, divergence statistics aims to describe the relation of M to A in +geometric terms by quantifying how much M diverges from A, according to some divergence measure +between projections (images) of y on RM and RA. These goals can be recast in terms of how much the +information in y given M diverges from the information in y given A; or equivalently, how much +information about F(y;μ) is lost when our estimate of it is constrained to the smaller distribution +subspace FM = {F(y;μ): μ є RM} instead of the larger subspace FA = {F(y;μ): μ є RA}. As will be illustrated, +technical details of information measurement, loss, and preservation via divergence measures can be +represented via long-established concepts of loglikelihood information and its generalizations (Kullback, +1997; Amari, 2016). + +Geometric and information concepts turn out to track each other closely in the setting considered here: +Throughout most of practice if not theory, divergence is measured based on compatibility or +information criteria that compare distributions in Fμ = {F(y;μ): μ є Rμ} using transforms of summary +statistics such as standardized distances in Rμ or likelihood ratios in Fμ. These lead to two major classes +of divergence measures: Euclidean (squared-deviation) and information-theoretic (deviance) based. +These classes approximate each other in common settings such as GLMs, and coincide in classical +Gaussian linear models. Thus in typical large-sample settings they can be treated as different ways of +viewing the same underlying discrepancies: The geometric measure expresses divergence in terms of +differences in fitted data vectors in Rμ, while the information measure expresses divergence in terms of +differences in information content (expected loglikelihoods) in Fμ. + +Let d(λ;θ) represent a divergence measure from F(y;θ) to F(y;λ) on Fμ, with d(λ;θ) smooth and +nonnegative over both λ and θ, and zero if and only if the compared distributions are equal almost + +Page 39 of 49 + +everywhere; it may however be asymmetric in that d(θ;λ) is distinct from d(λ;θ). In the information- +theory literature d(θ;λ) is more often written as D(λ‖θ). The present notation is adopted to parallel the +logical ordering in F(y;μ) where μ is the index of the distribution being used to evaluate y, and also to +avoid confusion with a random divergence which is denoted below using D. Information-divergence +definitions may further specify that F(y;θ) is absolutely continuous with respect to F(y;λ); this restriction +is however often left out of statistical discussions in favor of other devices such as setting 0∙ln(0) to zero +(e.g., see Bishop et al., 1975). + + A geometric example is the F(y;θ)-standardized squared Euclidean distance SSD(λ;θ) = +(λ−θ)’cov(Y;θ)−1(λ−θ), or Fisher-information divergence, which is asymmetric if cov(Y;μ) varies over Rμ. +An information-theoretic example is the deviance dev(θ;λ) = 2KLD(θ;λ) where KLD(θ;λ) is the Kullback- +Leibler discrimination information, defined as the expectation over F(y;λ) of the log-likelihood ratio for +F(y;λ) versus F(y;θ). KLD(θ;λ) is sometimes described as the expected information loss from using F(y;θ) +as the distribution of Y when the true distribution is F(y;λ), or the entropy of F(y;θ) relative to F(y;λ). + +Basic SSD and deviance measures will suffice to illustrate the present points, although there are many +extensions and variations on them derived by marginalizing, conditioning or penalizing the distributions +to improve some measure of statistical performance, ease computations, or introduce external (prior) +information. + +Divergence statistics +Suppose now we have selected a divergence measure d(y;μ) to efficiently capture information in the +discrepancy of the observed data y from μ, scaled (standardized) by the corresponding distribution +F(y;μ). The observed divergence d(y;RA) of y from RA is then the infimum infA[d(y;a)] over RA. If unique, +the minimizer of d(y;m) over RA, μA = arginfA[d(y;a)], is often taken as a point estimate of μ under the +model A; μA = y if y is an interior point of RA and will be a boundary point of RA otherwise. One may also +use d(y;RA) = d(y;μA) as a goodness of fit statistic for evaluating A against a mean-saturated alternative +(for which y is the estimate of μ). + +As an example, suppose d(y;μ) is the μ-standardized sum of squared deviations SSD(y;μ) which is the +squared distance from μ to y under the metric (scaling) defined by cov(Y;μ); d(y;RA) is then the minimum +squared F(y;μ)-standardized distance to y over μ in RA, and the closest point to y in RA is μA = + +Page 40 of 49 + +arginfA[SSD(y;m)]. Furthermore, SSD(y;μA) is a score statistic for the fit of model A and reduces to +Pearson’s χ2 fit statistic. If cov(Y;μ) and hence the distance metric may vary with μ, iteration over μ in RA +may be required to find d(y;RA). The resulting SSD minimizer μA is sometimes called an iteratively +reweighted least-squares (IRLS) estimate of μ under A, and equals the maximum-likelihood estimate +(MLE) when A is a family of ordinary GLMs; however, use of IRLS on a link-transformed deviation z−g(μ) +instead of y−μ requires adjustment of z at each iteration (McCullagh & Nelder, 1989). + +Further simplifications occur when using instead d(μ;y). For example, taking d(μ;y) = SSD(μ;y), the +squared F(y;y)-standardized distance from y to μ, we get d(RA;y) = infA[d(μ;y)] as the squared length of +the orthogonal projection of y onto RA under the cov(Y;y) metric; μA = arginfA[SSD(μ;y)] is then a +generalized least-squares (GLS) estimate of μ under A, leading to the Neyman χ2 statistic d(RA;y) = +SSD(μA;y). With standard fixed-dimensional GLMs for count Y, the Pearson and Neyman χ2 are first-order +locally equivalent, since under A their covariance components converge to the same large-n limit; and +for binomial-logistic and Poisson-loglinear models, they are score statistics using expected and observed +information, respectively. Nonetheless, simulation studies and numerical considerations have suggested +that the χ2 approximation to the fit statistic and the normal approximation to μA is better in small +samples when using SSD(y;μ) and IRLS than when using SSD(μA;y) and GLS, thanks to the greater stability +of the resulting cov(Y;μA) compared to cov(Y;y) (Maldonado & Greenland, 1994). For example, in Poisson +models cov(Y;μ) is undefined if any component of μ is zero, and typically μA will have zeros much less +frequently than y. Taking instead d(μ;y) = dev(μ;y), d(μA;y) becomes the likelihood-ratio (LR) statistic for +the fit of A, while μA = arginfA[d(μ;y)] becomes the MLE of μ under A, which also appears to perform +better than SSD(μA;y). + +Consider now a submodel M of A. One measure of the divergence d(RA;RM) of y from M given A takes +the minimizer μA = arginfA[d(y;a)] and then minimizes over RM, so that +d(RA;RM) = infM[d(μA;m)] = d(μA;μM) +where μM = arginfM[d(μA;m)] is the point estimate of μ under M given A; μM will be a boundary point of +RM if μA is exterior to or on the boundary of RM. We may call d(RA;RM) a fit statistic for M given A, or a fit +statistic for M against alternatives in A, or a test statistic for H = M−A given A. This case is also described +by saying the data are filtered through the model A before being used to evaluate the submodel M. + + +Page 41 of 49 + +Taking d(RA;RM) = SSD(RA;RM) we obtain a score statistic SSD(μA;μM), where μA and μM are the IRLS +estimates of μ, again equal to the MLEs of μ under A and M when both are ordinary GLMs. A reason for +choosing F(d;μM) for the standardization is that it makes μM the closest one can come to μA without +leaving RM and thus violating H. Nonetheless, taking instead take d(RM;RA) = dev(RM;RA), μA = +arginfA[dev(a;y)] and μM = arginfM[dev(m;μA)] automatically become the MLEs of μ under A and under M, +with dev(RM;RA) = dev(μM;μA) the deviance (likelihood-ratio) statistic for the fit of M given A. + +The deviance and Neyman χ2 obey the additive (Pythagorean) relation d(μM;y) = d(μA;y) + d(μM;μA), +which reveals that M may appear to fit well or even perfectly given A and yet be a very poor fit to the +actual data y, since we may have d(μM;μA) ≈ 0 with d(μM;y) ≈ d(μA;y) arbitaritly large. The same caution +applies to the Pearson χ2, although additivity does not hold exactly if cov(Y;μ) varies with μ (as with +subset coherence, it may arise as a local asymptotic approximation which however breaks down when +d(μM;y) is large). Nonetheless, numeric examples suggest that the usual χ2 approximations may produce +less accurate P-values from dev(μM;μA) and SSD(μM;μA) than from SSD(μA;μM), perhaps unsurprising given +that those measures fail to use H when scaling the divergences, whereas SSD(μA;μM) does use H. + +Parametric models +Most often, A is a family of distributions indexed by a finite parameter vector β with parameter space B +of dimension kA << n that remains fixed as n increases. The above concepts are then transformed into +the much smaller, computationally more manageable space B in place of Rμ. Elements of RA can then be +written as μ(β), with a point estimate of β under A of βA = arginfB{d[y;μ(β)]} and under M given A of βM = +arginfB{d[μ(βA);μ(β)]}. The resulting fit statistics d(y;RA) = d[y;μ(βA)] and d(RA;RM) = d[μ(βA);μ(βM)] +measure the divergence of y from RA and of μ(βA) from RM. Parallel notations follow when using instead +d[μ(β);y]. The parameter-estimation choice then maps to the divergence choice, with IRLS +corresponding to SSD[μ(βA);μ(βM)], GLS corresponding to SSD[μ(βM);μ(βA)], and ML corresponding to +dev[μ(βM);μ(βA)]. + +Given the parametric structure of RA, we can also define measures of divergence of observations from M +given A in the parameter space B. When evaluating a hypothesis H expressed as a linear constraint on β, +most software uses a divergence in B in the form of the Wald statistic for H given A, defined as the βA- +standardized squared distance from βA to βM, +SSD(βM;βA) = (βA−βM)’cov[BA;μ(βA)]−1(βA−βM) + +Page 42 of 49 + +where BA = arginfB{d[Y;μ(β)]} is the random analog of βA; using instead SSD(βA;βM) corresponds to +replacing the covariance matrix by cov[BA;μ(βM)]. When β=μ and A is saturated, βA = μA = y and these +statistics become the Neyman and Pearson χ2 statistics for the fit of M. More generally, under regular +models both Wald statistics are first-order locally equivalent to the score SSD[μ(βA);μ(βM)] and deviance +dev[μ(βM);μ(βA)]. Nonetheless, a long-standing literature on higher-order asymptotics (e.g., Efron & +Hinkley, 1978) as well as simulation studies (e.g., Maldonado & Greenland, 1994) suggest that in finite +samples the usual χ2 approximations for the Wald statistics are inferior to those for the other statistics. + +Divergence distributions +Let DM be the random variable obtained by applying a divergence statistic d(RA;RM) or d(RM;RA) to +repeated samples from F(y;μ). If RM contains more than one point, the distribution F(d;μ) of DM is +unknown even if we are given that M is correct (μ є RM); hence it may be unclear what distribution for +DM should be used to evaluate M. One common device is plug-in estimation, which takes μ = μM with +adjustments for using the random model F(d;μM) in place of a fully prespecified model. An example is +Fisher’s reduction of the degrees of freedom for Pearson’s χ2 statistic for M, SSD(Y;μM), from Pearson’s +original incorrect value to the residual dfM|A = n−dim(RM); here, A is saturated and hence dim(A) = n. + +In other cases, deficiencies of the plug-in approach according to resampling criteria may lead to +modifications of A or the model family FA by finding suitable pivots or by marginalizing, conditioning, or +penalizing the F(y;μ), thus altering the divergence comparisons in ways that may depend on the data, as +well as requiring adjustments to degrees of freedom. These modifications can become difficult if H +involves more than simple dimension reduction, as with inequalities or with penalty functions (even if +the latter vary smoothly over the model space). Regardless of these difficulties, in the divergence +approach the reference distribution serves only to provide a scale for evaluating the observed +divergence dM allowing for random error constrained as specified by the embedding model for d(RM;RA) +or the target model for d(RA;RM). + +Compatibility P-values +Divergence measures are difficult to interpret directly because, first, they correspond to squared +distances rather than distances, and second, their size depends on many “nuisance” aspects of the +problem including effective degrees of freedom. For example, one could replace the Pearson χ2 statistic +SSD(y;μM) with its square root to make it a Euclidean distance in a space scaled by standard deviations, + +Page 43 of 49 + +but its tail behavior and the identification of extreme values would still vary with specifics of M and A. +Thus, to facilitate consistent interpretations, we transform each divergence into a P-value. + +In the remainder of this review, I will use F(dM;M) to denote a derived reference distribution for DM. This +will not be limited to a plug-in distribution, and may be a function of the data not only through μM but +also through ancillary statistics or other data features; it may even have a Bayesian derivation, but is +selected to meet frequency criteria (Bayarri & Berger, 1999, 2000, 2004). Let μt denote the unknown +true value of μ. To avoid extensive technical complications, I will focus on large-sample behavior and +assume F(dM;M) converges to F(dM;μt) as n increases when μt is in RM, and does so at a rate sufficient for +the proposed usage. + +We can now define an observed approximate divergence P-value pM for evaluating M as Pr(DM≥dM;M), +the probability under F(dM;M) that DM would be at least as big as the observed value dM of DM; pM = +1−F(dM;M) when DM is continuous at dM. With H: bL≤β≤bU, pM reduces to the interval P-value described +earlier. A key point is that pM is a descriptive statistic showing the ordinal location of the observed +divergence dM in an explicit reference distribution whose choice may be dictated by information- +summarization goals rather than decision goals. Confusion of these goals has fueled controversy over +choice of sample space, as illustrated by the vast literature on whether we should condition on random +margins of a 2x2 table (e.g., contrast Fisher, 1955 p. 70 with Shuster, 1992), with information +summarization arguing for conditioning and decision criteria arguing against. + +For descriptive purposes, pM can be treated as an index of compatibility of the data y with M given A. +From a more precise geometric perspective, pM is a unit-scaled index of compatibility of the data image +μA on RA with RM, where compatibility is measured by a divergence scaled to the reference distribution +F(dM;M), and information is measured either through inverse-covariance (information or precision) +matrices or through log likelihoods. It should be noted that this perspective makes no reference to +“inference”, “significance”, “confidence”, or other subjective judgements about the contextual +implications or importance of the observation. In particular, pM=1 only means that μA is in RM and thus in +this descriptive sense M is perfectly compatible with the data y given A, while ever smaller values of p +correspond to ever larger divergence of μA from RM and thus ever more discrepancy between the data as +fitted under the embedding model A and as fitted under the restricted model M. + + +Page 44 of 49 + +Note that perfect compatibility with data given A does not mean that M is supported by the data, let +alone correct or even plausible: Any other submodel M* of A with μA є RM* will have pM* = 1 and thus will +also be perfectly compatible with y given A by this measure, even if it otherwise conflicts with M. +Suppose for example n=1, A says Y is normal(μ,1), M further says μ ≤ 0, M* further says μ ≥ 0, and y = μA += 0. Then both M and M* are perfectly compatible with y, in that pM = pM* = 1; yet any prior for μ +continuous at zero would assign zero probability to both models being true at once (M∩M*), leaving M +and M* mutually exclusive almost everywhere. + +Finally, like squared distance and the χ2 distribution from which pM is often derived, pM is “multi-tailed” +or omnibus in that it does not distinguish directions of departure of y or μA from RM. + +Properties of random divergence P-values +To establish a reference distribution, we may ask what repeated-sampling properties can aid the +interpretation of a single observation as an information summary rather than as a decision criterion. +These properties include a type of conditional uniformity in the distribution of the random analog PM of +pM when the true value μt is on the boundary of RM, reflecting the fact that data generated from +boundary points will more often appear ambivalent about whether M is correct than data generated +from other points. This uniformity in divergence P-values is a consequence of limiting divergence +behavior at boundary points, and leads to correspondence with NP P-values when M has no interior +point; it is not however taken as a defining or even necessary feature of P-values. + +In contrast, in NP theory uniformity is treated as a desirable or even defining property of P-values, as it +ensures the test size (type-1 error rate over resampling) of the decision rule “reject if p≤α” does not +exceed α, while maximizing power in typical settings. Conflict with divergence P-values thus arises when +the distribution of PM is not approximately uniform under M. In particular, if μt є RM and Pr(μA є RM;μt) = +Pr(DM=0;μt) > 0, the distribution of PM will have a point mass at 1 and appear excessively conservative +(dominating but not converging to uniformity). The mass can however be dealt with in descriptions by +conditioning on whether DM>0 or PM<1. + +To illustrate, consider a generalization of an interval hypothesis to the case in which M is the region +between two parallel (nonintersecting) hyperplanes in RA. If μt is exterior to RM (or equivalently, given RM +is closed, if μt є RA−RM) then, as n increases, the probability of μA falling in RM approaches zero, DM will in + +Page 45 of 49 + +probability increase without bound, and PM will converge to zero. On the other hand, if μt is interior to +RM (i.e., bounded away from RA−RM) the probability of μA falling in RM will approach 1 and so PM will +converge to 1. These properties seem natural and desirable: Given a linear boundary, the standardized +or information distance between μA and RM should increase with n if μt is outside of RM, and should go to +zero with increasing n if μt is inside RM. + +Suppose however μt is on a linear boundary segment of RM and bounded away from other segments, as +in the parallel case. Then Y will typically have limiting nonzero probabilities of falling outside and inside +of RM, and no amount of data can lead to Y consistently falling in or out of RM. When n is large enough so +that the divergence statistic concentrates away from any other segment, the two probabilities will +approach 50% as n further increases, so that the masses Pr(DM=0;μt) and Pr(PM=1;μt) will converge to +0.50. Thus, again assuming our reference distribution F(dM;M) converges to F(dM;μt), the distribution of +PM will develop a spike of height 0.50 at pM=1 as n increases, while the remaining 50% of its mass will +approach uniformity. + +Outside of such examples, a subtle point for proper interpretation of pM is that if RM is convex and μt is +on the boundary of RM but that boundary is not linear within most of the mass of F(y;μt), Pr(DM=0;μt) can +be much less than 0.50; this happens in the interval case when βt = bL and there is high probability of βM +> bU (making y exterior to RM on the side opposite μt). Note also that if dim(RM) 0, and μM is a boundary point, with +smaller p corresponding to greater standardized distance between μA and μM. + + +Page 46 of 49 + +While pM at first appears convenient as a 0-to-1 index of compatibility, it is poorly scaled in that, with +typical distributions, small pM differences near 1 represent trivial divergence differences whereas small +pM differences near 0 represent dramatic divergence differences. This observation leads to rescaling pM +with a negative log transform. As an example, the binary S-value or surprisal at seeing μA given M, sM = +log2(1/pM) = −log2(pM), can be used as a measure of the information against M supplied by the observed +divergence. Specifically, if whenever M holds, PM stochastically dominates a unit-uniform variate, sM is a +lower bound on the Shannon information in pM against M given A, expressed in log base 2 units (bits), or +equivalently a lower bound on the number of bits of statistical information against M supplied by the +event DM=dM, which is within 1 bit of the information when Pr(DM=0;μM) ≤ 0.50, with equality when P is +unit uniform. Rafi & Greenland (2000) illustrate how sM varies with likelihood ratios and deviance +statistics. + +We can also compare the binary S-value to the information in a coin-tossing experiment about the +hypothesis H: “the tossing is not biased toward heads”: With sM rounded to the nearest integer sN, the +information against M supplied by the observed divergence dM is (within a half bit) the same as the +information against H supplied by seeing all heads in sN tosses of a coin (Greenland 2019a; Rafi & +Greenland 2020; Cole et al. 2021; Amrhein & Greenland, 2022; Greenland et al. 2022). Note again +however that for more complex settings, absence of information against H need not correspond to +presence of supporting information for H if the apparent absence could be due to violations of auxiliary +assumptions in A = M−H (Greenland & Rafi, 2021; Bickel 2022). + +Compatibility regions +Suppose now that within A we have a family of models Mβ indexed by a parameter β (which may be a +vector) in a space B, each defining a subset RMβ of Rμ with its own divergence dMβ and an accompanying +reference distribution F(d;μMβ) and P-value pMβ. Generalizing the notion of compatibility intervals +(Amrhein et al., 2019; Rafi & Greenland, 2020; Cole et al., 2021; Amrhein & Greenland, 2022; Greenland +et al. 2022, 2023), we may then define a π-compatibility set for β as the set of all β for which pMβ > π. +Usually this set will be a connected region in B, and so can be called an π-compatibility region for β, or π- +compatibility interval if β is a scalar. The use of π rather than α is to avoid confusion with the usual 1−α +coverage or “confidence” region, the latter being equivalent to the region in B in which an α-level NP +test fails to reject Mβ. + + +Page 47 of 49 + +When A is correct, the rule “reject Mβ if pMβ≤α” provides a valid α-level test for all β, and thus the +corresponding α-compatibility region will be a 1−α coverage region as well. As with divergence P-values, +however, compatibility regions are not optimized according to NP criteria (e.g., for uniformly most +accurate symmetric coverage), and so more generally the two types of regions may fail to coincide, with +compatibility regions being conservative in the NP sense of overcoverage. Paralleling arguments for +divergence P-values, in such cases one may see that a compatibility interval conveys more information +about what was actually observed than do realized NP decision (“confidence”) intervals. The failings of +observed NP decision intervals are unsurprising given that they are outputs of algorithms optimized only +for resampling criteria, without regard to individual sample anomalies such as empty intervals (which +have led some authors to reject frequentist methods entirely). + +Multiple comparisons +The divergence view offers a relatively simple approach to multiple comparisons when the latter can be +expressed in terms of a multidimensional constraint in a model. To illustrate, suppose the individual +observations are row vectors yi = (yi1,…,yiJ) and the embedding model A states the corresponding Yi are +independent identical J-variate normal row vectors with unknown mean vector μ and known covariance +matrix Σ. Then the classic Scheffe’-type simultaneous (multiple-comparison) P-value for H: μ=0 (or more +precisely, for M = H ∪ A given A) reduces to the percentile location (tail area) of a J df χ2 (score) statistic +comparing the observed-mean vector y� to the zero vector; more precisely, this statistic is the squared Σ- +standardized distance in the J-space RA from the origin 0 to y�. Here, subspace RM contains only the +origin. + +The simultaneous P-value should be contrasted to the J separate P-values p1 ,…, pJ for the single- +component hypotheses Hj: μj = 0. Each pj is a percentile location of a 1 df χ2 statistic comparing one +component mean y�j to 0; that statistic is the squared standardized distance to y�j from the J−1 +dimensional subspace RMj defined by Hj. Viewed in this geometric fashion, we may see that, whether or +not one is interested in multiple comparisons and error control over multiple hypotheses, the +simultaneous P-value and the single-component P-values are describing relations among very different +data summaries and subspaces. Only the research question can determine which (if any) of these or +other P-values should be presented – a point that has been made repeatedly in purely logical terms +(e.g., Greenland & Hofman, 2019; Greenland, 2021b; Rubin, 2021). If mapped carefully into the research + +Page 48 of 49 + +context, geometric descriptions clarify the difference among the targets of multiple and single +comparisons. + +Relations to α-level decision P-values +Under the regular GLM set-up assumed here, suppose H is a pair of linear inequalities that make the +boundaries of RM parallel hyperplanes (as when H is an interval). Pr(PM≤α;μt) then depends only on +precision in the direction orthogonal to the two boundary hyperplanes. In large samples with μt on one +of these boundary segments, Pr(DM>0;μM) will approach 0.50; thus, with PM uniform given DM>0, +Pr(PM≤α;μt) will approach α/2. This suggests a bound on the inefficiency or information loss and +consequent test conservatism from using pM to form a decision rule “Reject H if pM≤α”. + +Suppose we have a decision-optimized P-value PT that is uniform under M. Upon observing PT = pT we +may take sT = −log2(pT) as an upper bound on the bits of information against H given A, and compare this +to the lower bound provided by sM = −log2(pM). When μt is interior to RM, both PT and PM approach 1 as n +grows and so the gap between these information bounds approaches zero. When μt is exterior to RM, +the ratio PM/PT approaches 2 as n grows, corresponding to a one-bit difference in Shannon information +against M given A. + +Summary and Conclusion +When M has no interior point relative to the embedding model A, as when H = M−A corresponds to +dimension reduction with RA and RM proper vector subspaces of Rμ, resampling-decision (NP) criteria and +information-summarization (divergence) criteria may lead to an identical observed P-value p (modulo +small-sample differences, as in unconditional vs. conditional tests in 2x2 tables). This numeric identity +arises in most applications and may well explain why few statistics texts distinguish the two types of P- +values. Nonetheless, they will differ in important examples of interval H; in those cases the question of +which (if either) is more relevant will arise. A simple answer is that if the analyst is only trying to +summarize model fits to the data in hand and wishes to focus on coherent measures of fit, the +divergence P-value is the more relevant: It is a quantile transform of a divergence measure, and as such +automatically meets the coherence criterion that the P-value for a nested model cannot exceed the P- +values for its embedding model when, for example, it uses the the deviance as the measure. + + +Page 49 of 49 + +As seen in simple examples where RM has interior points, a divergence P-value will sacrifice repeated- +sampling optimality to maintain single-sample coherency. But those pure repeated-sampling criteria +have no bearing on its interpretation as a geometric information summary: pM=1 tells us the image μA of +y in the embedding model A is also in M, while a very small pM indicates μA is stochastically far from M. +In this manner, it measures how compatible y is with M under an embedding model, where pM=1 means +zero divergence and hence complete compatibility, with more divergence and hence lower compatibility +indicated by smaller pM. This compatibility interpretation is logically weaker than support, and is +consistent with the conclusion of Schervish (1996) and others that P-values are poor measures of +support, regardless of the definition of “P-value”. + diff --git a/SdE0T4oBgHgl3EQfkwHK/content/tmp_files/load_file.txt b/SdE0T4oBgHgl3EQfkwHK/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..cb5430a65a483222bb7f4a4f0aab1aa1700b3326 --- /dev/null +++ b/SdE0T4oBgHgl3EQfkwHK/content/tmp_files/load_file.txt @@ -0,0 +1,1970 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf,len=1969 +page_content='Page 1 of 49 Divergence vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Decision P-values: A Distinction Worth Making in Theory and Keeping in Practice – or, How Divergence P-values Measure Evidence Even When Decision P-values Do Not SANDER GREENLAND Department of Epidemiology and Department of Statistics, University of California, Los Angeles, California, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=', lesdomes@ucla.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content='edu RUNNING HEAD: Divergence vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Decision P-values With corrections up to 06 January 2022 ABSTRACT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' There are two distinct definitions of “P-value” for evaluating a proposed hypothesis or model for the process generating an observed dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' The original definition starts with a measure of the divergence of the dataset from what was expected under the model, such as a sum of squares or a deviance statistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' A P-value is then the ordinal location of the measure in a reference distribution computed from the model and the data, and is treated as a unit-scaled index of compatibility between the data and the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' In the other definition, a P-value is a random variable on the unit interval whose realizations can be compared to a cutoff α to generate a decision rule with known error rates under the model and specific alternatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' It is commonly assumed that realizations of such decision P- values always correspond to divergence P-values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' But this need not be so: Decision P-values can violate intuitive single-sample coherence criteria where divergence P-values do not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' It is thus argued that divergence and decision P-values should be carefully distinguished in teaching, and that divergence P- values are the relevant choice when the analysis goal is to summarize evidence rather than implement a decision rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Key Words: Decision theory;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Divergence measures;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Falsificationism;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Hypothesis testing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Model checking;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' P-value;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Significance test;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Statistical geometry;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Statistical information;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' S-values;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Surprisals Page 2 of 49 Introduction: Two nonequivalent concepts of P-values There are at least two distinct definitions and uses of P-values within frequentist statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' The first historically is the divergence P-value, which in its basic form is a descriptive geometric summary of the observed deviation of a discrepancy statistic from a model prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' The second is the decision P- value, which is either a random variable used for compact expression of discrete-choice rules over an entire sample space, or the observed value of such a random variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' In simple problems divergence P-values equal realizations of decision P-values;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' hence the two types are usually not distinguished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' The present paper explores how they nonetheless do not correspond in general, either mathematically or in meaning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' It also provides some theoretical development for divergence P-values to explain in detail why criticisms of P-values as evidence measures apply to decision P-values but not to divergence P-values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' In particular, claims that P-values are incoherent evidence measures arise from the use of uniformly most powerful unbiasedness (UMPU) as a criterion to select decision rules, and use of the resulting decision P-values to measure support of hypotheses or models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' The claims do not apply when instead P-values are selected to satisfy single-sample coherence properties and used to measure compatibility, consonance, or consistency of data with hypotheses or models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' The distinction is not new – see for example Kempthorne (1976), and also Cox (1977) in this journal over 45 years ago.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' The present paper elaborates the distinction in both mathematical and philosophical terms to respond to attacks on the use of P-values as evidence measures, which are based on a confusion of divergence and decision P-values, and a confusion of compatibility with support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' As an extreme example, one recent article entitled “P-values Don’t Measure Evidence” claimed that all descriptions of P-values as evidence measures are erroneous (Lavine, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' The examples used to argue this claim are however based on decision-theoretic P-values and likelihood-based criteria for evidence, and overlook entirely definitions and criteria aimed instead at summarizing geometric fit of observations to models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' In the latter definitions, P-values simply describe the divergences between projections of data onto models for sampling probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' A P-value is then seen as one measure of one limited dimension of evidence against a statistical hypothesis or model, although it better serves this role if log- transformed to a more equal-interval scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Page 3 of 49 Lavine’s claim should be contrasted to (for example) Casella & Berger (1987a, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' 106), who state “to a frequentist, evidence takes the form of the p value, or the observed level of significance of the result”, and to the writings of Karl Pearson, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Fisher, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Cox, and many others who used P-values as part of descriptions of refutational evidence measures (evidence against statistical hypotheses).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Here are some examples with added emphases;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' note that 20th-century British statisticians often used “level of significance” and “value of P” to refer to what American statisticians came to call P-values, the latter term having appeared by the 1920s (see Shafer, 2020): “…the larger the value of [the observed test statistic] t the stronger the evidence of departure from H0 of the type it is required to test… For given observations y we calculate t = tobs = t(y), say, and the level of significance pobs by pobs = pr(T≥ tobs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content='H0)… We use pobs as a measure of the consistency of the data with H0 with the following hypothetical interpretation: Suppose that we were to accept the available data as evidence against H0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Then we would be bound to accept all data with a larger value of t as even stronger evidence.” – Cox & Hinkley, Theoretical Statistics (1974, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' 66).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' “…if the value of ν [the test statistic] is in the lower or central part of the distribution [of ν under H0], the data are consistent with H0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' or if ν is in the extreme upper tail of the distribution then this is evidence against H0…the p-value is in a sense the unique measure of the extremity of the value of ν in light of H0.” – Cox & Donnelly (2011, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' 146-147).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' In these sources the observed P-value is a strictly monotone transform of the test statistic via the latter’s cumulative distribution function under H0, which makes pobs the quantile at which tobs fell (Perezgonzalez, 2015a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Thus, in the above quotes the random statistic T orders possible samples by the amount of evidence against H0, and pobs translates tobs to the observed sample rank in that evidence ordering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Notably, a large value of pobs indicates the observations provide little evidence against H0, but cannot by itself constitute evidence for H0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' it can only be taken as indicating consistency, compatibility, concordance, or consonance of the data with H0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Discrepancy P-values for evidence summarization have been traditionally discussed under the heading of pure or absolute significance tests, and labeled “observed significance levels” (Cox & Hinkley, 1974, Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Cox, 1977, sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Cox & Hinkley (1974, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' 66) did invite confusion of evidential and decision P- values by saying that “pobs is the probability that we would mistakenly declare there to be evidence against H0, were we to regard the data under analysis as just decisive against H0.” Subsequently however Cox (1977, sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content="1) noted that “such a procedure is to be distinguished sharply from a decision Page 4 of 49 problem in which 'acceptance' or 'rejection' is required”, and in sec." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content='3 emphasized the distinction in saying that “The contrast between significance tests, as an aid in the summarization of evidence, and decision procedures is implicit or explicit in much recent discussion.” Cox (1977, sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content='3) further explained that such evidential P-values were applicable in the absence of explicit alternatives to the tested constraint (hypothesis): “At this stage in the development [of a statistical model], the null hypothesis is the only aspect of the problem explicitly formulated, so that such general considerations as the likelihood principle are inapplicable.” Regarding the last bolded statement, Cox & Hinkley (1974, example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content='42 and exercise 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content='16) give illustrations in which the likelihood function fails to capture evidence in a sensible or complete manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' More recent examples in which approaches obeying the strong likelihood principle (Cox & Hinkley, Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' 2) fail to meet basic repeated-sampling (frequency-calibration) criteria, yet which admit valid P-values, are given by Robins & Ritov (1997), Robins & Wasserman (2000, sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' 5), and Ritov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Among philosophical criticisms of likelihood for capturing all aspects of evidence are Hacking (1980) and Mayo (2018, sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Thus, criticisms of P-values derived from violations of the likelihood principle are themselves disputable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' None of these failings of relative likelihood rule it out as one evidence summary to use among others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Rather, they should remind us that there is no single accepted definition of “statistical evidence”, nor is there a single formalization of evidence concept that is flawless or adequate for all purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' This fact should be no surprise: How could one justify restriction of a complex concept like “evidence” to one formalization when there is no single sufficient definition or measure of simpler concepts?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Consider clothing size: To buy pants one needs at least waist size and leg length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Even some fundamental concepts in our most exact of sciences, physics, require multiple dimensions to capture (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=', kinetic vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' potential energy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' As Casella & Berger (1987b, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' 135) wrote, “Bayesians and frequentists may never agree on the appropriate way to analyze data and interpret results, but there is no reason why they cannot learn from one another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Whether or not measures of evidence can be reconciled is probably a minor consideration;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' understanding what affects a measure of evidence is a major consideration”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' In line with the bolded statement, the present paper uses the P-value controversy to illustrate the split between reference (neoFisherian) frequentism, which focuses on describing relations of sampling Page 5 of 49 distributions to observed data, and decision-theoretic (Neyman-Pearson-Wald) frequentism, which focuses on optimizing decisions over a sample space given a family of distributions on that space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' This split is usually obscured in teaching and research reports, despite attempts to bring to the fore (Goodman 1993, 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Hubbard & Bayarri 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Schneider, 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Perezgonzalez, 2015b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' It is mirrored in the division between reference (“objective”) Bayesianism, which focuses on summarizing parameter information in a data set, and “subjective” or operational Bayesianism, which focuses on optimizing decisions given external (“prior”) information and, again, a family of distributions on the sample space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Formal decisions are however suspect if not misleading when there are important sources of uncertainty about the models on which they are based (Leamer, 1978;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Stark, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' In contrast, divergence P-values can remain useful pieces of information for evaluating models under those conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' The present treatment may in fact be seen as alternative generalization of the original concept of nondirectional P- values for model fit (Pearson, 1900), with the common 2-sided P-value for point hypotheses being a special case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Theoretical Background The present section describes the central arguments and findings in terms of general distribution- function spaces and may be skipped by less mathematically inclined readers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Subsequent sections will repeat its essential elements in less technical form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Suppose the observed data y is idealized as an n-dimensional object that is a realization of random vector Y with unknown probability mass function f(y) on a sample space RU defined by the logically possible range of Y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' usually Y is a real n-vector variable and hence RU \uf0cd Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' This f(y) may be a composite of a density and a discrete mass function on RU, with f(y) denoting the density at continuity points of the distribution and the probability for mass points (jumps);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' f(RA) will also be used to denote the probability \uf0f2R(A) f(y)dy of Yε R(A) = RA for an arbitrary subset RA of RU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Let FU be the unrestricted or “saturated” space of all possible probability mass functions on RU, including single-point mass (degenerate) functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Consider the case of evaluating a set A of constraints on or assumptions about the distribution of data generated by an observation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Ideally, A is derived from a focused research hypothesis about the actual causal processes generating observations (Greenland, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Taken together, the constraints in A may not fully specify f, but will define a set or Page 6 of 49 family FA \uf0cd FU of the functions that satisfy all the constraints entailed by A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' The complement of FA in FU is the set difference FU – FA, which will be denoted ¬FA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Addition of further constraint sets H (traditional “null hypotheses”) beyond A will be considered below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' In model explorations, the constraint set A is sometimes called a working or embedding model for f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Discrepancy and Divergence P-values In its most basic form, a discrepancy P-value is an ordinal description of a real-valued summary measure d(y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content='FA) of the deviation or divergence of y from the set {μf: fεFA} of Y-expectations μf = E(Y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content='f) of the functions in FA, where the expectations are assumed to be finite (the use of a semicolon rather than a comma or vertical bar within function arguments is to emphasize the asymmetry of the inputs without suggesting probabilistic conditioning, as in Cox and Hinkley, 1974 and Cox, 1977).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Examples of such d(y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content='FA) include Pearson’s χ2 statistic and the sum of squared standardized residuals in a regression analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' The constraints in A are then used to derive a “reference" mass function gy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content='A(d) = gA(d;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content='y) for DA = d(Y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content='FA) which may partially depend on y (as in conditional tests).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' To allow for distributions without expectations we may start from a general measure d(f1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content='f0) of deviation of f1 from f0 in FU and define d(y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content='FA) = inf{d(fy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content='f): fεFA} where fy places all mass on y, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=', fy(y) = 1, so that E(y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content='fy) = y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' In all the settings considered here, gy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content='A is derived from fy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content='A = arginf{d(fy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content='f): fεFA} when the latter exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Assuming DA is a real scalar random variable such as a χ2 statistic, the observed upper-tail P-value derived from the realization DA = d is pd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content='A = gy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content='A({DA≥d}) = \uf0f2u≥d gy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content='A(u)du, which is the ordinal location or rank of d in the reference distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' It is interpreted as a unit-scaled index of consonance, consistency or compatibility of y with FA (Kempthorne & Folks, 1971;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Cox & Hinkley, 1974, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' 66;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Cox, 1977;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Box, 1980, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' 347;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Folks, 1981;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Bayarri & Berger, 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Robins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=', 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Greenland, 2019a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Amrhein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Rafi & Greenland, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Amrhein & Greenland 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Greenland et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=', 2022, 2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' In a complementary fashion, 1−pd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content='A is a unit-scaled index of incompatibility or inconsistency which summarizes the divergence of y from FA as measured by d(y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content='FA), although a more equal-interval measurement of incompatibility or refutational evidence is provided by the S-value −logc(pd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content='A) where c is the log base (Kempthorne, 1990;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Greenland, 2019a, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Rafi & Greenland, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Greenland et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=', 2022, 2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' These definitions and interpretations are sometimes called Fisherian or neoFisherian, although they can be seen in writings before Fisher’s, such as Pearson (1900), and could be called evidential or reference frequentist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Page 7 of 49 Note that, contrary to some discussions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=', Cox & Hinkley, 1974, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' 66) there is no requirement or implication that the random variable PD;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content='A = \uf0f2u≥D gy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content='A(u)du must approach uniformity when fεFA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' in particular, PD;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content='A will instead have a mass at 1 when gA has a mass at 0, as it will in the examples below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Nonetheless, under mild conditions PD;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content='A will be distributed as or exceed a unit-uniform random variable when fεFA and will have sampling properties that are intuitively desirable for a compatibility measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' For example, as n increases PD;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content='A will concentrate toward 1 if the actual f has d(f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content='¬FA)>0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=', is interior to FA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' conversely, PD;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content='A will concentrate toward 0 if the actual f has d(f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content='FA)>0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=', is interior to ¬FA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Repeated-sampling properties are however insufficient to ensure single-sample coherency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' The discrepancy d(y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content='FA) measures a departure of the single sample y from what the constraints A would lead us to expect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Here are two single-sample criteria we could require for using discrepancy P-values like pd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content='A (or strictly monotone transforms of them) as evidence measures: 1) Suppose the data exhibit no discrepancy whatsoever from the constraints in A, in the sense that fy ε FA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' then we should want d(y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content='FA) = 0 and hence pd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content='A = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' 2) Suppose M entails all constraints imposed by A and possibly more, so that FM \uf0cd FA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' then we should want d(y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content='FM) ≥ d(y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content='FA), with pd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content='M ≤ pd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content='A as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=', see Kempthorne (1976, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' 767).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Suppose d(y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content='FA) is derived from a formal divergence measure d(f1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content='f0) on FU\uf0b4FU (Amari, 2016);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' condition (1) then follows immediately from the nonegativity of divergences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' For condition (2), which may be termed subset coherence, suppose that FA is a subset of an exponential family of distributions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=', Y may be a vector of binomial variates).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Denoting mean vectors by μk = E(Y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content='fk) and covariance matrices by Σk, k=1,0, suppose d(f1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content='f0) = (μ1−μ0)’Σ0−1(μ1−μ0), the squared f0- standardized Euclidean distance between the Y-means of f1 and f0, sometime called the Euclidean divergence (although that label is sometimes applied to half this function;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' see Amari, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content='We then have the divergence of the data from FA is d(y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content='FA) = inf{(y−μ0)’Σ0−1(y−μ0): f0εFA}, the Pearson χ2 fit statistic, while the mean μy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content='A of its minimizer fy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content='A is the minimum-χ2 or generalized-least-squares (GLS) estimate of μ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Subset coherence then follows follows exactly when the covariance matrix Σ is constant over FA, with no need for approximate normality of Y, and follows in a local asymptotic sense given conventional regularity conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Suppose instead we use the divergence of FA from the data, d(FA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content='y) = inf{d(f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content='fy): fεFA}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Subset coherence then follows when taking d(f1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content='f0) = E(ln(f0(Y))−ln(f1(Y));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content='f0), the Kullback-Leibler divergence (KLD) of f1 from f0, also known as the relative entropy or the discrimination information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' The KLD is defined under the Page 8 of 49 usual conventions that 0∙ln(0) = 0 derived from lim(x∙ln(x): x↓0) = 0, and that f0(y) = 0 when f1(y) = 0 (absolute continuity of f0 with respect to f1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Because fy(y) = 1 and is 0 elsewhere, we have E(ln(fy(Y));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content='fy) = 0, E(ln(f1(Y));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content='fY) = ln(f1(y)) and thus d(FA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content='y) = sup{−ln(f(y)): fεFA}, half the deviance statistic used in global tests of fit for regular GLMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' When it exists, the maximum-likelihood estimate (MLE) max{f(y): fεFA} of f under A is then fA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content='y = arginf{d(FA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content='y): fεFA}, and fy is the MLE of f when A entails no constraint on f, as when A = \uf0c6 (so that FA = FU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Decision P-values In the second definition, a P-value is merely a by-product of a decision criterion (“test”) for whether to reject or accept the family FA for further use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' The analyst specifies in advance a maximum acceptable false-rejection (Type-I error) rate α for FA and an alternative distribution family Falt \uf0cc FU disjoint from FA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Where possible one then finds a critical region RA(α) \uf0cc RU to implement the decision rule “reject FA if y ε RA(α)”, for which 1) The type-I error rate is controlled at level α (test validity): When fεFA, f(RA(α)) ≤ α;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=', when A holds, the probability f(RA(α)) of YεRA(α) does not exceed α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' 2) The type-II error rate over Falt is minimized among valid tests: When fεFalt, f(RA(α)) ≥ f(R) for any other R \uf0cd RU that has f(R) ≤ α whenever fεFA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=', among valid tests (1), when A fails, RA(α) maximizes the rejection probability (power).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' 3) The type-II error rate never falls below α (unbiasedness): When fεFalt, f(RA(α)) ≥ α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' If a set function RA(α) of α satisfies 1-3 for all α, the decision rule “reject FA if yεRA(α)” is called a uniformly most powerful unbiased (UMPU) α-level test procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' In Neyman-Pearson (NP) hypothesis- testing theory, if this “optimal” (UMPU) function RA(α) exists, nothing more is needed for the decision about FA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' The maximum acceptable Type-I error α may be left unspecified, and the reader may be provided instead the observed decision P-value pinf(α) = inf{α: yεRA(α)};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' their decision can then be based on inserting their own α in the P-rule “reject FA if pinf(α) ≤ α” (Lehman, 1986, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' 70).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' This methodology is however usually implemented by finding a test statistic TA = t(Y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content='FA) with realization t = t(y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content='FA) and mass function hA(t) such that pinf(α) = pt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content='A = hA({TA≥t}) = \uf0f2u≥t hA(u)du ≤ α if and only if y ε RA(α);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' in basic examples TA is a log likelihood ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Many sources however define a decision P-value for FA as the corresponding random variable PT;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content='A = \uf0f2u≥T hA(u)du with realizations pt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' If fεFA and RA(α) defines a valid test, Pr(PT;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content='A ≤ α) = f({YεR(α)}) ≤ α and thus PT;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content='A is distributed as or exceeds a unit-uniform random variable when fεFA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Page 9 of 49 The latter uniformity condition is sometimes used to define P-value validity, for it ensures that for any α the false-rejection rate (size) of the P-rule never exceeds α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Both the divergence PD;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content='A and decision PT;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content='A are valid in this sense;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' a key difference is that PT;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content='A is further optimized for power, which can bring its distribution closer to uniform from above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Note however that there are no single-sample conditions applied to the realizations t or pt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content='A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' this neglect is a source of defects of decision P-values as evidence measures, and may arguably be seen as a defect in their use as decision guides as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' An analogous objection to decision P-values arises when, due to discreteness, exact uniformity of P-values and an exact size α UMPU tests cannot be obtained without randomizing decisions at boundary observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' This leads to examples in which the single-sample decision comes down to a coin toss – an anomaly avoided by redefining the P-value (Lancaster, 1961).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' The two types of P-values are not equivalent Most of the literature seems to ignore that there are two definitions and interpretations of P-values, and to the extent it recognizes them, it treats them as differing only in that a divergence P-value is an observed (realized) quantity and a decision P-value is the corresponding random variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' This is not however true in general: When single-sample descriptive-geometric coherence criteria are imposed on discrepancy P-values (for example by using a divergence measure as the discrepancy) but only repeated- sampling criteria are used to derive decision P-values, we can have pd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content='A > pt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content='A, as in the example below where pd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content='A−pt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content='A approaches ½.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' While such examples correspond to greater power for the P-rule, Schervish (1996) showed that, for FA defined by bounded parameter intervals and UMPU procedures, the resulting decision P-values could violate subset coherence (condition 2) in that one could have FM \uf0cc FA and yet have pt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content='M > pt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' In sum, realizations t(y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content='FA) of UMPU test statistics can violate basic axioms for divergence measures when treated as functions of FA, leading to incoherent decision P-values pt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' This incoherency can be attributed to giving power priority over single-sample coherence, and arguably should disqualify pt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content='A as a measure of data evidence about A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' In contrast, a divergence P-value pd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content='A starts from a coherent discrepancy measure and thus will coherently indicate the compatibility of the targeted constraint set A or family FA with the data y (more precisely, fy), at least in large samples and often exactly, depending on the divergence measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Consequently, 1−pd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content='A or −logc(pd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content='A) will coherently indicate the degree of refutation of A provided by the divergence measure – all without reference to any alternative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Page 10 of 49 Interestingly, Schervish took his result to imply that decision P-values were incoherent measures of evidence supporting FA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' As will be discussed below, there are purely logical reasons for rejecting any statistic as an absolute or unconditional measure of support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' As likelihoodists often argue, support (and thus the possibility of acceptance of A) can only be relative to or conditional on specified alternatives to FA (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=', Edwards, 1992;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Royall, 1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Similarly, decision procedures also need to specify alternatives to FA, although this need does not dictate imposition of UMPU (Hansen & Rice, 2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Summary Divergence P-values summarize probabilistic information about how far the data diverge from an embedding model A, or how far the data filtered through an embedding model A diverge from the data filtered through a more restrictive target model M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' The information is measured using the geometry of the model family defined by A or M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' The definitions of divergence P-values differ from NP definitions: In NP statistics, observed P-values are intermediate calculations for decisions derived solely to satisfy resampling criteria;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' they are defined as the minimum α-level at which the tested model would be rejected by an optimized NP test (Lehmann, 1986, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' 70).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' In this sense, claims that P-values are incoherent or do not measure evidence are circular, being consequences of adopting definitions that force P-values to conform to repeated-sampling criteria (such as UMPU) without regard to single-sample coherence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' In contrast, definitions derived instead from geometric comparisons of observed data and its projections onto model regions will automatically satisfy at least approximate coherence requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Misconceptions about P-values and evidence There are now hundreds of articles spanning many decades about how everyday researchers misinterpret and abuse statistical tests and P-values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' There are far fewer about how myths about P- values have been promulgated and perpetuated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' For example, it is not uncommon to see claims that P- values “overstate evidence against null hypotheses”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Yet observed P-values are just basic probability statements about the location of a statistic in a reference distribution, mute about their own “strength”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Thus, the claims do not reflect anything intrinsic to the concept of a P-value, but instead reflect a synergism between user mistakes and philosophical commitments that argue against use of P-values: Users mistakenly perceive the limited evidence encoded in observing p=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content='05 as strong simply because 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content='05 is an entrenched standard for declaring “statistical significance” – and, far too often, a decisive factor for publication, as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' 1 of van Zwet & Cator (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' In tandem, some statistical Page 11 of 49 commentators take posterior probabilities or likelihood ratios as “gold standards” for evidence measurement;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' those standards are however rejected by those who see how the conflict of those “standards” with frequentist testing can reflect failings of the prior or likelihood models, rather than deficiencies of the frequentist assessments (Ritov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=', 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Greenland, 2019a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Bickel, 2021a, 2021b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' A more subtle cognitive problem arose however as modern mathematical statistics emerged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Originally, P-values were defined directly as probabilities of observed events, where the events concerned statistics that gauged data divergence from expectations computed from a target (test) model or hypothesis (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=', Pearson, 1900;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Fisher, 1934, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' 66).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' These original P-values made no reference to alternative models or formal decision rules (Cox, 1977).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' That view further recognized how closeness of the model predictions to the data (as indicated by a large goodness-of-fit P-value) did not imply the target model is correct (Pearson, 1906).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' But Neyman-Pearson (NP) hypothesis testing altered the definition of a P-value to that of a type of random variable P whose realization p was the smallest α level at which the criterion “reject if p≤α” would reject the hypothesis, given the observed data (Lehmann, 1986, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' 70) – a redefinition which almost inextricably bound P-values to statistical decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Many subsequent authors wrote as if the original goodness-of-fit and NP definitions were mathematically equivalent, even though (as will be discussed below) this not always the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' This subtle mistake parallels but is distinct from two different traditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' One tradition uses P-values as continuous evidence measures, and again is often called “neo-Fisherian” although traceable to Karl Pearson (Hurlbert & Lombardi, 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' In contrast, the NP tradition uses P-values solely as components of decision criteria, a usage which requires specification of alternatives along with choice of a cut-off or α- level justified by error costs (Neyman 1977;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Lakens et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' 2018, Mayo 2018) – albeit in practice a justification is rarely seen and instead a default α (most often 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content='05) is used with no regard to actual error costs or to uncertainty about the model used to justify error-rate claims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' With the ascendance of NP theory in mathematical statistics, many if not most writers now ignore how P-values can be defined and treated descriptively, unconnected to formal decision rules;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' they instead focus on how P-values can malfunction when (as often the case) they are misused to measure support, or else focus on how they can be properly connected to Bayesian measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' The present paper shows how descriptive P-values can be derived to provide coherent measures of refutational evidence, without invoking notions of support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Page 12 of 49 Descriptive P-values measure compatibility without measuring support The scale of a P-value p typically runs from 0 to 1, where 0 implies impossibility of an observed statistic under the target model (complete discord between the statistic and the model) and 1 corresponds to no conflict of the statistic with the model (no discord between the statistic and the model).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' This scaling seems to suggest P-values measure support, but again only reflects degrees of compatibility between the data and the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' The description of P-values as compatibility measures can be seen in Box (1980, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' 387), Bayarri & Berger (2000, 2004), Robins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' (2000) and Greenland (2019a), and is anticipated by this passage in Fisher (1935, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' 207;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' emphasis added): “If we consider a series of hypotheses with different values for the diminishing return [on crop yield from adding more phosphate fertilizer], and determine which of these values are compatible, at any given level of significance, with the observed yields, some of the values which would appear to be acceptable would be negative…”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' see also Pearson (1900, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' 170-171) and Fisher (1934, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' 66) for similar use of “compatible” with P- values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' The same concept has also been described as goodness-of-fit (Pearson, 1900), consonance (Kempthorne & Folks, 1971;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Folks, 1981), and consistency (Cox, 1977);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' unfortunately, “consistency” is more often used for unrelated convergence properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Other terms for the compatibility concept include conformity, concordance, and accord (Rice, personal communication).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Its opposite may thus be termed incompatibility or discordance, which can be measured by 1−p, although there are cognitive and abstract arguments for instead employing the surprisal measure or S-value s = −logc(p) for incompatibility (Greenland, 2019a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Rafi & Greenland, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Greenland 2021a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Amrhein & Greenland 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Greenland et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=', 2022, 2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Understanding why compatibility is a logically weaker concept than support requires recognizing why conformity of observations to a target model – that is, failure of observations to conflict with or refute a model – does not by itself imply support for the model: There are always many other models that will fit the data just as well yet contradict the target model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' consequently, one must a priori impose very tight restrictions on those other models to generate a support measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' As an extreme example, upon observing no difference between treated groups in a randomized trial, we may say the model of no treatment effect is perfectly compatible with the data;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' but unless we restrict alternative models to trials that are competently and honestly conducted, we must face that the same observation is also perfectly Page 13 of 49 compatible with models in which the data were mishandled or altered to erase the appearance of an effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' In going beyond the simple realm of binary logic and into a world of uncertain and extensive complexity, support and conflict are not logical complements of one another because they do not encompass all the possibilities for evidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' For example, evidence may be wholly indeterminate, supplying neither conflict with nor support for a targeted hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' The conceptual incompleteness of conflict and support corresponds to the asymmetry between evidence against and evidence for a hypothesis, which is emphasized heavily in falsificationist philosophy (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=', Popper, 1959);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' there, “corroboration” is used to indicate failure to refute a hypothesis without implying support or confirmation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Corroboration is meant to be weaker than support;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' nonetheless, in ordinary language, compatibility is even weaker than corroboration, for compatibility suggests nothing at all about the power or severity of the criterion being used for evaluation – it is simply the opposite or negation of incompatibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' For example, making one coin toss and observing heads is highly compatible with the hypothesis H: “The probability \uf071 of heads is ½”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' but to say it corroborates or supports H insinuates that the observation has more than trivial evidential value about that H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' On the other hand, incompatibility and refutation do have a parallel: Most would say observing heads is highly incompatible with and even refutes H: “0 < \uf071 ≤ 10−9” even though heads is not impossible under H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Ignoring the difference between compatibility and support can be seen as a source of claims that evidence measurement and statistical testing require specification of alternatives, one of the major points of contention in the Fisher-Neyman split (Fisher, 1955;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Pearson, 1955;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Neyman, 1956).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' While addressing this asymmetry is not needed for the mathematical development, it should be borne in mind when interpreting results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' In particular, a reason to refer to P-values as compatibility measures is that they are more accurately understood when recognized as reversals of conflict orderings, rather than as measures of support (which require restriction of alternatives to a well-defined family of distributions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' What do P-values and their transforms describe?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' P-values describe a relation of a model to data, or the relation of a more restrictive model M to a less restrictive embedding model A in light of the data, as seen in comparing observed to expected data, or more generally in comparing M-expected to A-expected fits to the data when M is nested in A (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=', imposes all the constraints in A and more).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' The latter comparison measures information about M in the Page 14 of 49 data, given A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Another way to characterize this process is that of measuring divergences among different degrees of data smoothing, ranging from no smoothing (the raw data, which is the expected data under a saturated model) to A-smoothing to more severe M-smoothing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' The greater the smoothing (the more constraints imposed), more data detail will be discarded as “noise” under the model, and more certainty will be assigned to any structure (apparent signal) remaining in the expectations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=', that a parameter appears to be large enough to signal treatment superiority, where the latter is defined context-specifically).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Of course, the model may be mistaken, in which case important information may be discarded or smoothed away by the model, and excessive certainty will be assigned to the remaining apparent structure (signal).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Thus, to avoid catastrophic loss while getting rid of obvious noise and hopelessly weak signals, one may maximize the embedding model (minimize the number of constraints in A) based on context-specific information (Greenland, 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Given a model, inference may be based on a model diagnostic, as when judging whether too much structure is smoothed away (too much information is discarded) by the model using tools like residual plots and P-values for model fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Upon adopting a working model that has passed these tests, inference becomes instead model conditional, as when model expectations are substituted for the data and then used to make claims about the data generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Examples include marginal (population-standardized) effect estimates with components estimated from model-fitted quantities such as balancing scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' The inferential claims are often supported by P-values for coefficients or their inversion into interval estimates using the selected model, without accounting for the model selection that took place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' As has long been documented, such preselection will invalidate decision rules that ignore it (Leamer, 1978).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' A pertinent question is then: Do these P-values mean anything despite the selection?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' One answer is that Fisherian P-values (those defined from observed divergences) still describe an observed discrepancy between the data and the model when that discrepancy is adjusted (“standardized”) to allow for the random component of the model, which is to say the noise distribution assumed by the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' This interpretation holds even though the naïve comparison of such post-selection P-values to a pre- specified α would lead to repeated-sampling-and-selection error rates far higher than α, and thus brings to the fore the conflict between descriptive and decision uses of P-values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' I have avoided using “population” because it suggests to most users that the P-value is describing some target population from which the data were sampled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' In reality, the data often come from a source that Page 15 of 49 happens to be available but whose connection to the actual decision target may be quite speculative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Then too, the data are affected not only by intentional (interventional) design elements such as matching and randomization, but also by unintended influences such as causes of refusal to participate, loss to follow-up, and other procedural problems, as well as sample and data alterations made in the course of analysis (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=', discarding units declared as outliers, or fixing records that have missing or clearly wrong data items).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Thus, to validly infer back to the data source or make decisions with known error rates requires more than just rigid adherence to a pre-specified analysis protocol;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' it also requires a model which adequately reproduces the behavior of the entire physical process that determined the data used in the analysis (Greenland, 2005, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Measuring compatibility and conflict with reference distributions and S-values In general form, the coherence criterion requires that a measure of evidence against a target model M or hypothesis H is never less than the measure against an embedding model A that contains the target model M as the special case of A in which the hypothesis H holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' A special case of this criterion was introduced above as subset coherence for P-values, which requires that the observed P-value for the fit of a target model M never exceeds the observed P-value for the fit of a more general, flexible embedding model A in which M is nested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' An analogous criterion for a support measure requires that the measure for M never exceeds that for A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Such criteria can be traced at least back to Gabriel (1969) and have been extensively studied since, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=', see Schervish (1996), Fossaluza et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' (2017), Bickel & Patriota (2019), and Hansen & Rice (2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' A descriptive P-value provides the location of an observed statistic in a reference distribution derived from the targeted (“test” or “null”) model M and the data, and is thus tailored to the study as observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' In this sense, it is anchored in the single-sample viewpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' The reference distribution need not be based on the goal, structure, or optimization of a decision over repeated sampling;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' instead, its purpose is to provide coherent measures of compatibility and conflict between the data and the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' It may be highly conditioned on the observed data in ways that depend on the targeted model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' hence the reference distribution may vary considerably across resampling from the distribution entailed by the actual data-generating mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Again, as per Cox (1977, sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content='1), “such a procedure is to be distinguished sharply from a decision problem in which ‘acceptance’ or ‘rejection’ is required”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Page 16 of 49 In contrast, in Neyman-Pearson (NP) statistics, P-values are random variables that are components of the decision rule “reject if P≤α”, or are by-products of the decision “reject if the test statistic falls in a critical region of size α” (Lehmann, 1986, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' 70).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' In the pure frequentist view of Neyman (1977), losses and hence these random variables are evaluated over their actual resampling distributions according to criteria such as power and “unbiasedness”, without regard to features of single-sample realizations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' they thus can suffer from incoherencies when extended to interval hypotheses (Schervish, 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Hansen & Rice, 2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Statistical arguments against P-values as evidence measures have been based on these decision-theoretic definitions and criteria;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' they have largely ignored or dismissed treatments that start from the original divergence definition (Pearson, 1900) to interpret P-values or their transforms as single-sample information summaries or measures of compatibility or conflict between data and assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' The sections below will illustrate and expand on how the descriptive treatment of P-values differs from the decision-theoretic treatment, focusing on the special case in which a P-value summarizes relations among models and data within an information geometry defined by the observations as well as the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' The Appendix provides a technically more detailed review of divergence P-values illustrated with basic generalized-linear models (GLMs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' In doing so its focus is on intuitive visualization of actual single- sample properties of P-values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' It further reinforces arguments that common misperceptions of P-values can be mitigated by transforming an observed p to an unbounded and reversed scale to provide a measure of incompatibility, conflict, refutation, or surprise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' This rescaling also removes evidence measurement from the 0-to-1 scale on which posterior probabilities live, thus removing a source of confusion of P-values with posterior probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' The most common example of such a transform is the negative base-c logarithmic transform to a surprisal or S-value sc = logc(1/p) = −logc(p) = −ln(p)/ln(c);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' sc = 0 then says the observed discrepancy statistic or divergence measure from which the P-value was computed was completely unsurprising given the target model, or that it does not conflict with or contains no information refuting the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' As the divergence increases, sc increases to reflect surprise at the data given the model, or conflict with the target model (Bayarri & Berger, 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Greenland, 2019a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Rafi & Greenland, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Cole et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Gibson, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Amrhein & Greenland, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Greenland et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' The base of the log, c, can be seen as a scale factor;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' c=10 is commonly used but has no theoretical justification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Taking instead c=e produces the natural S-value se = −ln(p) which is a special case of the “E-value” in Grünwald et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' (2021) and the Page 17 of 49 “betting score” in Shafer (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' see Greenland (2021a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' As explained in the Appendix and elsewhere, c=2 provides an intuitive mechanical interpretation: The binary S-value s2 = −log2(p) measures bits of information against a model supplied by the statistic, which translates easily into outcomes of coin- tossing experiments (Greenland, 2019a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Rafi & Greenland, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Cole et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Greenland 2021a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Greenland et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Competing concepts of P-values Coherence vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' decision P-values For NP frequentists, statistical decisions calibrated to long-run frequencies are the central goal of analysis, and the paramount definitions and evaluation criteria refer only to resampling frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Pursuing this goal, modern NP theory treats a P-value for a hypothesis H as a random variable that follows (or at least approximates from above) a uniform resampling distribution when H is correct, and concentrating toward 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' This theory makes no reference to data description;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' in practice however the ensuing single-sample decisions are usually misrepresented as data descriptions, as when a paper reports “no association was observed” when in fact this only meant that p>0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' An acceptable measure of incompatibility or conflict between data and hypotheses must obey the single-sample criterion of subset coherence, in that its value for a hypothesis expressed as a subset of a function or parameter space cannot exceed its value for any of its own subsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' In parallel, a measure of compatibility is coherent if its value for such a hypothesis cannot exceed its value for any of its supersets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Again, compatibility is a weaker condition than support, in that support implies compatibility but compatibility does not imply support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Compatibility and support do however share an analogous coherence concept: A measure of support by data for a hypotheses is coherent if its value for a hypothesis cannot exceed its value for any of its supersets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Likelihood maxima over sets satisfy this requirement and thus are coherent support measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' In contrast, using interval hypotheses in a simple normal location model, Schervish (1996) showed that an extension of uniformly most powerful unbiased (UMPU) P-values to interval hypotheses (Lehmann, 1986, sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content='2) is an incoherent measure of support;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' hence a scale reversal such as 1−p or −logc(p) would provide an incoherent measure of conflict if p were defined from an UMPU test of an interval hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Similar problems can arise with Bayes factors (Lavine and Schervish, 1999), but see also Good (2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' The UMPU extension goes beyond ensuring that the simple decision rule “reject if p≤α” Page 18 of 49 has Type-I error rate (size) no more than α over the resampling distribution under the interval hypothesis, by imposing conditions to optimize power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Key to the present discussion is that the P-value in the rule for an interval UMPU test (Hodges and Lehmann, 1954;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Lehmann, 1986, sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content='2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Schervish, 1996) is defined and evaluated solely over resampling, with no attention to single-sample coherence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Divergence P-values Incoherence does not afflict all extensions of P-values, such as those that aim only to measure information on a compatibility/incompatibility scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Consider definitions that lead to maximizing a point-hypothesis P-value over a hypothesized interval for a real-valued parameter: Such max-p (or supremum) extensions have been disqualified from being “frequentist” by adherents of the NP definition (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=', see Remark 1 of Robins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=', 2000 p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' 1145).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' One reason is that the maximum two-sided P-value will equal 1 whenever the parameter estimate falls in the hypothesized interval, resulting in a distribution far above uniform when the parameter is in the interval interior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' This property led Schervish (1996, sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' 4) to described max-p extensions as “simple minded” and “not particularly useful”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' But these dismissals are based on criteria that can be rejected by those who use P-values as discrepancy measures or information summaries rather than as test criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' As illustrated below and explained in the Appendix, summarization can start from a geometric derivation of P-values from divergence measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Doing so naturally leads to maximizing P-values over intervals, thus enforcing their coherence;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' for point hypotheses such as H: μ=m, these P-values simplify to ordinary two-sided P-values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' The descriptive goal is to summarize data information about models, particularly comparisons of fitted models against data and each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' A descriptive P-value is thus the observed quantile at which a discrepancy statistic sits in a reference distribution derived from the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' In particular, when divergence statistics are used to measure discrepancies, the reference distribution may be derived from a geometry on a model space that combines information on both the data-generating distribution and the observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' This distribution serves only to locate the data relative to the model in a particular direction in the model space, using a coordinate scaling standardized to the estimated sampling variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' The resulting P-values are numerically identical to observed decision-theory P-values when the divergence statistic and reference distribution chosen for summarization are identical to the test statistic and resampling distribution chosen for decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' While this identity is most common, it is not dictated by the goals of the two approaches and is violated in the examples below and in the Appendix;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Bayarri and Berger (2004) discuss conditioning as a method of reconciling the approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Page 19 of 49 As illustrated in the Appendix, the descriptive goal can be rephrased as that of depicting relations among different degrees of data smoothing or filtration, where the filtrations are through nested models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' This goal stands in sharp contrast to Bayesian methods, whose goal is to make statements (or “inferences”) about parameters conditional on data;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' these methods require inputs of fully specified parameter distributions and can break down spectacularly in high dimensions (Robins and Ritov 1997;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Ritov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' In terms of goals, NP decision theory is more akin to Bayesian methods in going beyond descriptions to make direct statements or take actions about parameters or models “in light of the data”, although they differ from ordinary Bayesian decisions in allowing statements or actions that are not based on parameter distributions or full conditioning on the observed data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' A core requirement of NP decision theory and frequentist inference procedures is that the data generator involves a randomizer (for selection into the data, if population inference is a goal;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' or for treatment assignment, if causal inference is a goal) that is known apart from an estimable parameter vector (such as the coefficients in a sample-selection or treatment-assignment probability function).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' In the descriptive view, however, the randomizer is demoted to being simply another assumption in the model used to compute the P-value, and thus (as with other assumptions) its violation may be brought forward as an explanation for the size of an observed p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' There is then no inference beyond the noting that a small p can signal a problem with one of the assumptions in the model used to derive the reference distribution – it may be small because the treatment has an effect different from that assumed by the model, or because there were uncontrolled nonrandom elements in the actual treatment assignment, observation selection, or data missingness (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=', nonrandom censoring).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' This unconditional descriptive view does not logically condition on any assumption, and thus is inferentially mute: It does not single out violation of an assumed (test) hypothesis to explain why p seemed small;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' nor does it single out satisfaction of the hypothesis to explain why p seemed large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' A descriptive P-value simply states a factual relation between what was observed and what was expected based on the target model, where “target model” refers to all the assumptions or constraints used to compute the P-value, including H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' The main point is that the conflicting goals of decision and descriptive frequentist approaches lead to definitions and interpretations of extended P-values that differ and may come into conflict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' The two Page 20 of 49 views can however be at least partially reconciled through the criteria and preferences they share.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Consider the following passage from p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' 1144 of Robins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' (2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' emphasis added): …a p value is useful for assessing compatibility of the null model with the data only if its distribution under the null model is known to the analyst;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' otherwise, the analyst has no way of assessing whether or not observing p = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content='25, say, is surprising, were the null model true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' That we specify that distribution to be uniform is largely a matter of convention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' The view presented here agrees in that, in order to properly interpret a P-value, we must know sufficient detail about the meaning and distribution of its source statistic under the target model and its violations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Nonetheless, most of the assumptions that compose the target model cannot be assured to hold in observational studies or in randomized experiments with much drop-out or nonadherence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Furthermore, when the target model does not involve dimension reduction (so the model has interior points in the information topology described in the Appendix), uniformity of a random P-value becomes a useful property only at boundary points of the distribution subspace defined by the targeted model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' For evaluating statistical performance, the descriptive goal further replaces power by information content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' The descriptive view thus regards the emphasis on uniformity of P-values under assumed models (as seen in most theoretical literature) as a product of NP goals applied to highly idealized experimental models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Because those models are usually unrealistic in health and medical research, we seek instead to describe the relations that hold between observations and models, regardless of the model’s accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Example: A divergence P-value for a simple interval hypothesis Consider the model for n observations arrayed in a vector y = (y1,…,yn)’ in which the yi are assumed to be independent draws from a normal distribution with unknown mean μ and known variance σ2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' call this set of assumptions the embedding model A, leaving μ the only free parameter in the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' This embedding or background model places severe restrictions on the distribution F(y) for the random data vector Y = (Y1,…,Yn)’: It says that if the physical data generator were left to run indefinitely, it would behave exactly like a random-vector generator for an uncorrelated n-variate normal distribution with equal Yi means and variances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Note that A implies the sample mean of the yi can replace the data without loss of information about the data generator, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=', it is sufficient for determining the behavior of the generator to the extent allowed by the data, since A implies that behavior is normal (Gaussian) with variance σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' But A does not imply that the actual observed-y histogram would be visually well Page 21 of 49 approximated by a normal density with mean equal to the sample mean and variance equal to σ2, which is often what is meant when saying the fitted model is adequate as a data summary or compression;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' the latter claim requires evaluation of A against the full data vector y, including graphical as well as goodness-of-fit diagnostics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Accepting A for the moment, we may wish to evaluate the information that the data vector y supplies against a submodel M nested within A that imposes additional restrictions H on μ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Specifically, suppose M adds to A the restriction H: μ=m with m known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Evidence against μ=m is usually gauged by finding the “standardized” distance |μ�−m|/(σ/n½) from the sample mean μ� to the target value m in a standard- normal tail-area function to obtain the 2-sided P-value pm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Equivalently, to get pm we could find the divergence statistic dm = d(m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content='μ�) = (n/σ2)|μ�−m|2 in a 1 degree-of-freedom (df) χ2 distribution F(d), whence we can describe the standardization as multiplying the squared distance from the data summary to the hypothesized mean m by n/σ2, which is the amount of Fisher information in the data about μ given the model A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Furthermore, given normality, d(m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content='μ�) is identical to the usual likelihood-ratio (LR) statistic for evaluating M against A, and (as discussed in the Appendix) is twice the Kullback-Leibler information divergence of M from A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Under M, the distributions of the random analogs Dm of dm and Pm of pm are derived by treating the Yi as independent normal(m,σ2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Pm will then be uniform, implying the NP decision rule “reject H if pm≤α” will have size α;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=', Pr(Pm≤α;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content='μ=m) = α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' If however μ≠m but A (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' normality with known variance) still holds, the distribution of Pm will be increasingly concentrated toward 0 as the noncentrality parameter d(m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content='μ) = (n/σ2)|μ−m|2 grows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' The random divergence statistic Dm and P-value Pm for H: μ=m are identical to the test (decision) statistic and P-value in NP theory, and their distributions are fixed and known given μ=m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' The divergence view departs from the testing (decision) view when M instead adds to A an interval restriction H: mL ≤ μ ≤ mU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' The squared distance from μ� to [mL,mU] is d([mL,mU];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content='μ�) = min{d(m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content='μ�): m є [mL,mU]} The resulting observed divergence P-value for the model M or for the hypothesis H given A is pM = max{pm: m є [mL,mU]}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Page 22 of 49 When mL= mU = m, we get pM = pm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' But when mL< mU, the distributions of d([mL,mU];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content='μ�) and of pM are no longer known given M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' they can only be computed conditional on μ and may vary considerably across μ in the interval [mL,mU] defined by H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' We can nonetheless summarize the discrepancy of μ� from [mL,mU] using pM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' The divergence d([mL,mU];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content='μ�) goes to infinity as μ� becomes more distant from [mL,mU], and is at its minimum of zero when μ� is in [mL,mU].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Thus, pM will range from zero (μ� infinitely far from H) to one (μ� zero distance from H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' If we consider the standardized distance d(m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content='μ�) of μ� from the interval as indicating the extent of incompatibility between the observations and the hypothesis, pM can be taken as an index of compatibility of the data summary μ� with the model M that restricts μ to [mL,mU].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' To restore a direct relation to the divergence and incompatibility, we may transform pM to a surprisal or S-value such as sM = −log2(pM), which ranges from zero: μ� zero distance from the interval to infinity: μ� infinitely far;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' sM provides other benefits for interpreting the observed divergence in terms of the information it supplies against H given A (Greenland, 2019a, 2021a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Rafi & Greenland, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Cole et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Greenland et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=', 2022, 2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQfkwHK/content/2301.02478v1.pdf'} +page_content=' Turning to the corresponding random P-value PM, we see that with correct A, mL< mU, and increasing sample size \uf0b7 when H is incorrect, μ is exterior to the interval (μ 0 is a regularization parameter. The backward mapping loss can be written as +Lh2l(Xh) = ∥yl − f h +θ∗(Xh)(Xl)∥2 , +(11) +3 + +Under review +Figure 1: Illustration of bidirectional learning Chen et al. (2022b) where (Xl, yl) denotes the static +dataset, yh is a large predefined target score and Xh is the high-scoring design we aim to find. +where θ∗(Xh) is given by +θ∗(Xh) = arg min +θ +∥yh − f h +θ (Xh)∥2 + β∥θ∥2 . +(12) +The high-scoring design Xh can be optimized by minimizing the bidirectional learning loss L(Xh) = +Ll2h(Xh) + Lh2l(Xh). +3 +METHOD +In this section, we first illustrate how to leverage deep linearization to compute the bidirectional +learning loss in a closed form. Subsequently, we introduce a hyperparameter γ to control the +balance between the forward mapping and the backward mapping. We then develop a novel bi-level +optimization framework Adaptive-γ, which leverages a weak supervision signal from an auxiliary +model to effectively update γ. Last but not least, we extend this framework to Adaptive-η, which +enables us to adapt the learning rate η for all gradient-based offline model-based algorithms. We +summarize our method in Algorithm 1. +3.1 +DEEP LINEARIZATION FOR BIDIRECTIONAL LEARNING +In bidirectional learning, the backward mapping loss is intractable for a finite neural network, so Chen +et al. (2022b) employ a neural network with infinite width, which yields a closed-form loss via the +NTK. This however makes it impossible to incorporate the rich biophysical information that has been +distilled into a pre-trained LM (Yang & Hu, 2021). Considering this, we construct a proxy model +by combining a finite-width pre-trained LM with an additional layer. We then linearize the resultant +proxy model, inspired by the recent progress in deep linearization which has established that an +overparameterized DNN model is close to its linearization (Achille et al., 2021; Dukler et al., 2022). +Denote by θ0 = (θpt, θlin +init) ∈ RD×1 the proxy model parameters derived by combining the +parameters of the pre-trained LM θpt and a random initialization of the linear layer θlin +init. In this +paper, we adopt the pre-trained DNABERT (Ji et al., 2021) and Prot-BERT (Elnaggar et al., 2021) +models, and compute the average of token embeddings as the extracted feature, which is fed into the +linear layer to build the proxy. Then we can construct a linear approximation for the proxy model: +fθ(X) ≈ fθ0(X) + ▽θfθ0(X) · (θ − θ0) , +(13) +where fθ(X), fθ0(X) ∈ R, ▽θfθ0(X) ∈ R1×D and ▽θfθ0(X) ∈ RD×1. Intuitively, if the +fine-tuning does not significantly change θ0, then this linearization is a good approximation. By +leveraging this linearization, we can obtain a closed-form solution for Eq.(12) as: +θ∗(Xh) = (▽θfθ0(Xh)⊤ ▽θ fθ0(Xh) + βI)−1 ▽θ fθ0(Xh)⊤(yh − fθ0(Xh)) + θ0. +(14) +Building on this result, we can compute the bidirectional learning loss as: +Lbi(Xh) = 1 +2(∥yh − KXhXl(KXlXl + βI)−1(yl − fθ0(Xl))∥2 ++ ∥yl − KXlXh(KXhXh + βI)−1(yh − fθ0(Xh))∥2) , +(15) +4 + +The static dataset: TFBind8(r) +Forward mapping +The high-scoring data +A +A +A +G +G +T +Li2h(Xh) = Ilyh - f*(Xh)I2, +c +c +T +0* = arg min llyi - fä(Xi)2 + βliI2 +T +Xi +Xh +T +T +A +A +A +Unknown +LCh2i(Xh) = Ilyl - fb*(Xh)(Xi)I2, +G +A +A +c +G +G +0*(Xn) = arg min llyh - f(Xn)I2 + βI01I2 +T +T +T +16 +2.1 +4.5 +3.9 +Backward mapping +Yh +10.0Under review +where K(Xi, Xj) = ▽θfθ0(Xi)⊤ ▽θ fθ0(Xj). Following (Dukler et al., 2022), we can also only +linearize the last layer of the network for simplicity, which defines the following kernel, +K(Xi, Xj) = BERT(Xi)⊤BERT(Xj), +(16) +where BERT(X) denotes the feature of the sequence X extracted by BERT. Its kernel nature makes +this approach suitable for small-data tasks (Arora et al., 2020), especially in drug discovery where the +labeling cost of DNA/proteins is high. +Algorithm 1 Bidirectional Learning for Offline Model-based Biological Sequence Design +Input: Static dataset D = (Xl, yl), predefined target score yh = 10, # of iterations T, +pre-trained biological LM parameterized by θ0, auxiliary model faux(·), regularization β. +Output: High-scoring design X∗ +h. +1: Initialize X0 as the sequence with the highest score in D +2: for τ ← 0 to T − 1 do +3: +Leverage Adaptive-γ in Sec 3.2 to update the balance γ by Eq. (21) +4: +if Adapt learning rate then +5: +Leverage Adaptive-η in Sec 3.3 to update the learning rate η by Eq. (23) +6: +Optimize X by minimizing the bidirectional learning loss Lbi(Xτ, γ) in Eq. (17): +7: +Xτ+1 = Xτ − ηOPT(∇XLbi(Xτ, γ)) +8: Return X∗ +h = XT +3.2 +ADAPTIVE-γ +The forward mapping and the backward mapping play different roles in the sequence optimization +process: the forward mapping encourages the high-scoring sequence to search for a higher target +score (exploitation) and the backward mapping serves as a constraint. Since different sequences +require different degrees of constraint, we introduce an extra hyperparameter γ ∈ [0, 1] to control the +balance between the corresponding terms in the loss function: +Lbi(Xh, γ) = γLl2h(Xh) + (1 − γ)Lh2l(Xh) . +(17) +Thus γ = 1.0 corresponds to the forward mapping alone, γ = 0 results in backward mapping, and +γ = 0.5 leads to the bidirectional loss of (Chen et al., 2022b). +It is non-trivial to determine the most suitable value for γ since we do not know the ground-truth +score for a new design. One possible solution is to train an auxiliary faux(·) to serve as a proxy +evaluation. A reasonable auxiliary is a simple regression model fitted to the offline dataset. Although +this auxiliary model cannot yield ground-truth scores, it can provide weak supervision signals to +update γ, since the auxiliary model and the bidirectional learning provide complementary information. +This is similar to co-teaching (Han et al., 2018) where two models leverage each other’s view. +Formally, we introduce the Adaptive-γ framework. Given a good choice of γ, the produced Xh is +expected to have a high score faux(Xh), based on which we can choose γ. To make the search for γ +more efficient, we can formulate this process as a bi-level optimization problem (Chen et al., 2022a; +2021; 2022c): +γ∗ = arg max +γ +faux(X∗ +h(γ)) , +(18) +s.t. +X∗ +h(γ) = arg min +Xh +Lbi(Xh, γ) . +(19) +We can then use the hyper-gradient ∂faux(X∗ +h(γ)) +∂γ +to update γ. Specifically, the inner level solution +can be approximated via a gradient descent step with a learning rate η: +X∗ +h(γ) = Xh − η dLbi(Xh, γ) +dX⊤ +h +. +(20) +For the outer level, we update γ by hyper-gradient ascent: +γ = γ + η +′ dfaux(X∗ +h(γ)) +dγ += γ + η +′ dfaux(Xh) +dxh +dLbi(Xh, γ) +dx⊤ +h +, +(21) +where we unroll the matrix Xh as a vector xh for better illustration. +5 + +Under review +3.3 +ADAPTIVE-η +We now extend the Adaptive-γ framework to Adaptive-η. As the first learning rate adaptation module +for offline model-based optimization, Adaptive-η is compatible with all gradient-based algorithms +and can effectively finetune the learning rate η via the auxiliary model’s weak supervision signal. All +gradient-based methods that maximize Lθ(X) with respect to X have the following general form: +Xt+1 = Xt + ηOPT(∇XLθ(X)|X=Xt) , +for t ∈ [0, T − 1] , +(22) +where η represents the learning rate of the optimizer. For methods such as simple gradient ascent +(Grad), COMs (Trabucco et al., 2021), ROMA (Yu et al., 2021) and NEMO (Fu & Levine, 2021), +Lθ(·) is related to the proxy model fθ(·); for BDI Chen et al. (2022b) and our proposed method, +BIB, Lθ(·) is the negative of the bidirectional learning loss, i.e., Lθ = −Lbi. +Though the learning rate η can be adapted in some optimizers such as Adam (Kingma & Ba, 2015), +these adaptations rely on only the past optimization history and do not consider the weak supervision +signal from the auxiliary model. Our Adaptive-η optimizes η by solving: +η∗ = arg max +η +faux(X∗ +h(η)) , +(23) +where η can be updated via gradient ascent methods. Considering the sequence optimization procedure +is highly sensitive to the learning rate η, we reset η to η0 at each iteration and update η from η0, +η = η0 − η +′ dfaux(X∗ +h(η)) +dη +. +(24) +In general, this serves to stabilize the optimization procedure. +4 +EXPERIMENTS +We conduct extensive experiments on DNA and protein design tasks, and aim to answer three research +questions: (1) How does BIB compare with state-of-the-art algorithms? (2) Is every design component +necessary in BIB? (3) Does the Adaptive-η module improve gradient-based methods? +4.1 +BENCHMARK +We conduct experiments on two DNA tasks: TFBind8(r) and TFBind10(r), following (Chen et al., +2022b) and three protein tasks: avGFP, AAV and E4B, in (Ren et al., 2022) which have the most data +points. See See Appendix A.2 for details. +Following (Trabucco et al., 2021), we select the top N = 128 most promising sequences for each +comparison method. Among these sequences, we report the maximum normalized ground truth score +as the evaluation metric following (Ren et al., 2022). +4.2 +COMPARISON METHODS +We compare BIB with two groups of baselines: the gradient-based methods and the non-gradient- +based methods. For a fair comparison, the pre-trained LM is used for all methods involving a +proxy and we don’t finetune the LM. The gradient-based methods include: 1) Grad: gradient ascent +on existing sequences to obtain new sequences; 2) COMs (Trabucco et al., 2021): lower bounds the +DNN model by the ground-truth values and then applies gradient ascent; 3) ROMA (Yu et al., 2021): +incorporates a smoothness prior into the DNN model before gradient ascent steps; 4) NEMO (Fu +& Levine, 2021): leverages the normalized maximum-likelihood estimator to bound the distance +between the DNN model and the ground-truth values; 5) BDI (Chen et al., 2022b): adopts the +infinitely wide neural network and its NTK to yield a closed-form bidirectional learning loss. +The non-gradient-based methods include: 1) BO-qEI (Wilson et al., 2017): builds an acquisition +function for sequence exploration; 2) CMA-ES (Hansen, 2006): estimates the covariance matrix to +adjust the sequence distribution towards the high-scoring region; 3) AdaLead (Sinai et al., 2020): +performs a hill-climbing search on the proxy and then queries the sequences with high predictions; 4) +CbAS (Brookes et al., 2019): builds a generative model for sequences above a property threshold and +gradually adapts the distribution by increasing the threshold; 5) PEX (Ren et al., 2022): prioritizes +the evolutionary search for protein sequences with low mutation counts; 6) GENH (Chan et al., 2021): +enhances the score through a learned latent space. +6 + +Under review +0 +10 +20 +30 +40 +T +0.4 +0.6 +0.8 +1.0 +Performance +0.4 +0.6 +0.8 +1.0 +Trade-off +Performance +Trade-off +(a) TFBind8(r). +0 +10 +20 +30 +40 +T +0.4 +0.6 +0.8 +1.0 +1.2 +Performance +0.4 +0.6 +0.8 +1.0 +Trade-off +Performance +Trade-off +(b) avGFP. +0 +10 +20 +30 +40 +T +0.5 +1.0 +1.5 +Performance +0.4 +0.6 +0.8 +1.0 +Trade-off +Performance +Trade-off +(c) E4B. +Figure 3: Trend of performance and trade-off γ as a function of T. +4.3 +TRAINING DETAILS +We follow the training setting in (Chen et al., 2022b) if not specified. We choose OPT as the Adam +optimizer (Kingma & Ba, 2015) for all gradient-based methods. We implement the auxiliary model +as a linear layer with the feature from the pre-trained LM. We set the number of iterations T as 25 for +all experiments following (Norn et al., 2021) and η0 as 0.1 following (Chen et al., 2022b). We run +every setting over 16 trials and report the average score. See Appendix A.3 for other details. +4.4 +RESULTS AND ANALYSIS +We report all experimental results in Table 1 and plot the ranking statistics in Figure 2. We make +the following observations. (1) As shown in Table 1, BIB consistently outperforms the Grad +method on all tasks, which demonstrates that our BIB can effectively mitigate the out-of-distribution +issue. (2) Furthermore, BIB outperforms BDI on 4 out of 5 tasks, which demonstrates the ef- +fectiveness of the pre-trained biological LM over NTK. The reason why BDI outperforms BIB +on TFBind10(r) may be that short sequences do not rely much on the rich sequential informa- +tion from the pre-trained LM. (3) As shown in Figure 2, the gradient-based methods generally +perform better than the non-gradient-based methods, as also observed by Trabucco et al. (2021). +1 2 3 4 5 6 7 8 9 101112 +Rank +BO-qEI +CMA-ES +AdaLead +CbAS +PEX +GENH +Grad +COMs +ROMA +NEMO +BDI +BIB(ours) +Figure 2: Rank minima and max- +ima are represented by whiskers; +vertical lines and black triangles de- +note medians and means. +(4) The gradient-based methods are inferior for the AAV task. +One possible reason is that the design space of AAV (2028) is +much smaller than those of avGFP (20239) and E4B (20102), +which makes the generative modeling and evolutionary algo- +rithms more suitable. (5) This conjecture is also supported +by the experimental results on two DNA design tasks. We +compute the average ranking of gradient-based methods and +non-grad-based methods on TFBind10(r) as 3.5 and 9.5, re- +spectively, and the average ranking of gradient-based methods +and non-grad-based methods on TFBind8(r) as 5.8 and 6.8, re- +spectively. The advantage of gradient-based methods are larger +(9.5 − 3.5 = 6.0) in TFBind10(r) than that (6.8 − 5.8 = 1.0) +in TFBind8(r). (6) The generative modeling methods CbAS +and GENH yield poor results on all tasks, probably because +the high-dimensional data distribution is very hard to model. (7) Overall, BIB attains the best +performance in 3 out of 5 tasks and achieves the best ranking results as shown in Table 1 and Figure 2. +We also visualize the trend of performance (the maximum normalized ground truth score) and trade- +off γ as a function of T on TFBind8(r) in Figure 3(a) and avGFP in Figure 3(b). The performance +generally increases with the time step T and then stabilizes, which demonstrates the effectiveness +and robustness of BIB. Furthermore, we find that the γ values of TFBind8(r) and avGFP generally +increase at first. This means that BIB reduces the impact of the constraint to encourage a more +aggressive search for a high target value during the initial phase. Then γ of TFBind8(r) continues to +increase while the γ of avGFP decreases. We conjecture that the difference is caused by the sequence +length. Small mutations of a biological sequence are enough to yield a good candidate (Ren et al., +2022). For the length-239 protein in avGFP, dramatic mutations 1) are not necessary and 2) can easily +lead to out-of-distribution points. The weak supervision signal from the auxiliary model therefore +encourages a tighter constraint towards the static dataset. By contrast, the DNA sequence is relatively +short and a more widespread search of the sequence space can yield better results. To investigate this +7 + +Under review +Table 1: Experimental results (maximum normalized ground truth score) for comparison. +Method +TFBind8(r) +TFBind10(r) +avGFP +AAV +E4B +Rank Mean +Rank Median +D(best) +0.242 +0.248 +0.314 +0.452 +0.224 +BO-qEI +0.940 ± 0.032 +0.595 ± 0.028 +0.888 ± 0.015 +0.591 ± 0.002 +0.436 ± 0.004 +6.0/12 +7.0/12 +CMA-ES +0.930 ± 0.034 +0.617 ± 0.031 +0.909 ± 0.004 +0.470 ± 0.006 +0.748 ± 0.009 +6.6/12 +6.0/12 +AdaLead +0.941 ± 0.032 +0.602 ± 0.028 +0.885 ± 0.016 +0.581 ± 0.002 +0.433 ± 0.003 +6.2/12 +8.0/12 +CbAS +0.878 ± 0.049 +0.610 ± 0.035 +0.785 ± 0.057 +0.543 ± 0.002 +0.349 ± 0.003 +9.0/12 +10.0/12 +PEX +0.924 ± 0.041 +0.612 ± 0.026 +0.874 ± 0.033 +0.588 ± 0.002 +0.397 ± 0.004 +7.2/12 +8.0/12 +GENH +0.323 ± 0.000 +0.448 ± 0.000 +0.793 ± 0.000 +0.452 ± 0.000 +0.228 ± 0.000 +11.4/12 +11.0/12 +Grad +0.941 ± 0.026 +0.630 ± 0.029 +0.913 ± 0.027 +0.463 ± 0.005 +1.219 ± 0.061 +5.0/12 +5.0/12 +COMs +0.921 ± 0.039 +0.637 ± 0.065 +0.938 ± 0.048 +0.511 ± 0.005 +0.829 ± 0.026 +5.0/12 +5.0/12 +ROMA +0.926 ± 0.032 +0.634 ± 0.061 +0.975 ± 0.133 +0.471 ± 0.005 +1.198 ± 0.042 +4.8/12 +4.0/12 +NEMO +0.930 ± 0.038 +0.632 ± 0.024 +0.914 ± 0.026 +0.505 ± 0.005 +1.036 ± 0.046 +4.8/12 +5.0/12 +BDI +0.823 ± 0.000 +0.678 ± 0.000 +0.873 ± 0.000 +0.452 ± 0.000 +0.224 ± 0.000 +9.0/12 +11.0/12 +BIB(ours) +0.952 ± 0.033 +0.639 ± 0.032 +1.060 ± 0.016 +0.501 ± 0.007 +1.255 ± 0.029 +2.4/12 +1.0/12 +Table 2: Ablation studies on BIB components. +Task +γ = 0.0 +γ = 1.0 +γ = 0.5 +γ = 0.5 + Joint +BIB +BIB + Ada-η +TFBind8(r) +0.936 +0.933 +0.947 +0.935 +0.952 +0.956 +TFBind10(r) +0.611 +0.637 +0.616 +0.622 +0.639 +0.639 +avGFP +0.920 +0.966 +1.018 +1.006 +1.060 +1.082 +AAV +0.449 +0.458 +0.480 +0.420 +0.501 +0.525 +E4B +0.778 +0.903 +1.198 +1.176 +1.255 +1.301 +conjecture, we further visualize the trend of E4B in Figure 3(c). E4B also has long sequences (102) +and we can observe its similar first-increase-then-decrease trend, although it is not as pronounced. +4.5 +ABLATION STUDIES +In this subsection, we conduct ablation studies to verify the effectiveness of the forward mapping, the +backward mapping, and the Adaptive-γ module of BIB. We report the experimental results in Table 2. +Forward mapping & Backward mapping. We can observe that bidirectional learning (γ = 0.5) +performs better than both forward mapping (γ = 1.0) and backward mapping (γ = 0.0) alone in +most tasks, which demonstrates the effectiveness of forward mapping and backward mapping. The +advantage of bidirectional mappings over the forward mapping is larger in the long-sequence tasks +like avGFP (238) and E4B (102) compared with the short-sequence tasks. A possible explanation is +that the constraint is more important for long sequence tasks than short sequence design since the +search space is large and many mutations can easily go out of distribution. +Adaptive-γ. BIB learns γ and this leads to improvements over bidirectional mappings (γ = 0.5) for +all tasks, verifying the effectiveness of Adaptive-γ. We also consider the following variant, +X∗ = arg min +Xh Lbi(Xh, 0.5) − faux(Xh) , +(25) +which jointly optimizes the bidirectional learning loss Lbi(Xh, 0.5) and the auxiliary term faux(Xh). +We found this yields similar or even worse results than pure bidirectional learning. The reason may +be that the weak supervision signal from faux(Xh) can serve as a guide to update the scalar γ but +not as a component of the main optimization objective that directly updates the sequence. +In the final column of Table 2, we examine the performance of the Adaptive-η module. Adding this +module leads to improvements on all five tasks, which demonstrates its effectiveness. +4.6 +ADAPTIVE-η +In this subsection, we aim to further demonstrate the effectiveness of the Adaptive-η module on all +six gradient-based methods. We conduct experiments on two tasks: TFBind8(r) and avGFP. Since +the use of the infinitely wide neural network leads to poor performance for BDI, we modify its +implementation via deep linearization so that it can make use of the pre-trained LM. +As shown in Table 3, Adaptive-η provides a consistent gain for all scenarios, which demonstrates +the widespread applicability and effectiveness of the module. Furthermore, Adaptive-η leads to a +maximum improvement of 1.4% in TFBind8(r) and 12.5% in avGFP. ROMA is the algorithm that +8 + +Under review +Table 3: Adaptive-η on all gradient-based methods. +Method +TFBind8(r) +avGFP +Grad +COMs +ROMA +NEMO +BDI +BIB +Grad +COMs +ROMA +NEMO +BDI +BIB +Normal +0.941 +0.921 +0.926 +0.930 +0.947 +0.952 +0.913 +0.938 +0.975 +0.914 +1.018 +1.060 +Joint +0.941 +0.921 +0.931 +0.932 +0.935 +0.925 +0.913 +0.905 +0.923 +0.906 +1.006 +1.009 +Gain +0.000 +0.000 +0.005 +0.002 +−0.008 +−0.027 +0.000 +−0.033 +−0.052 +−0.008 +−0.012 +−0.051 +Ada-η +0.941 +0.928 +0.939 +0.935 +0.951 +0.956 +0.916 +0.952 +0.998 +0.920 +1.024 +1.082 +Gain +0.000 +0.007 +0.013 +0.005 +0.004 +0.004 +0.003 +0.014 +0.023 +0.006 +0.006 +0.022 +benefits the most. One possible explanation is that ROMA incorporates a local smoothness prior that +leads to more stable gradients, with which Adaptive-η can be more effective. Similar to Sec 4.5, we +consider the following variant, +X∗ = arg max +Xh Lθ(Xh) + faux(Xh) , +(26) +which performs joint optimization instead of bi-level optimization on two objectives. As shown +in Table 3, joint optimization generally deteriorates the performance. This again verifies that the +auxiliary model can only serve as a guide instead of contributing to the main objective. +5 +RELATED WORK +Biological sequence design. There has been a wide range of algorithms for biological sequence +design. Evolutionary algorithms (Sinai et al., 2020; Ren et al., 2022) leverage the learned surrogate +model to provide evolution guidance towards the high-scoring region. Angermueller et al. (2019) +propose a flexible reinforcement learning framework where sequence design is a sequential decision- +making problem. Bayesian optimization methods propose candidate solutions via an acquisition +function (Terayama et al., 2021). Deep generative model methods design sequences in the latent +space (Chan et al., 2021) or gradually adapt the distribution towards the high-scoring region (Brookes +et al., 2019). GFlowNets (Jain et al., 2022) amortize the cost of search over learning and encourage +diversity. Gradient-based methods leverage a surrogate model and its gradient information to +maximize the desired property (Chen et al., 2022b; Norn et al., 2021; Tischer et al., 2020; Linder +& Seelig, 2020). Our proposed BIB belongs to the last category and leverages the rich biophysical +information (Ji et al., 2021; Elnaggar et al., 2021) to directly optimize the biological sequence. +Offline model-based optimization. A majority of sequence design algorithms (Angermueller et al., +2019; Sinai et al., 2020; Ren et al., 2022) focus on the online setting where wet-lab experimental +results in the current round are analyzed to propose candidates in the next round. The problem +of this setting is that wet-lab experiments are often very expensive, and thus a pure data-driven, +offline approach is attractive and has received substantial research attention recently (Trabucco et al., +2022; Kolli et al., 2022). Gradient-based methods have proven to be effective (Trabucco et al., +2021; Yu et al., 2021; Fu & Levine, 2021; Chen et al., 2022b). Among these algorithms, Chen +et al. (2022b) propose bidirectional mappings to distill information from the static dataset into a +high-scoring design, which achieves state-of-the-art performances on a variety of tasks. However, +this bidirectional learning is designed for general tasks, like robot and material design, and the rich +biophysical information in millions of biological sequences is ignored. In this paper, we leverage +recent advances in deep linearization to incorporate the rich biophysical information into bidirectional +learning. +6 +CONCLUSION +In this paper, we propose bidirectional learning for offline model-based biological sequence. Our +work is built on the recently proposed bidirectional learning approach (Chen et al., 2022b), which is +designed for general inputs and relies on the NTK of an infinitely wide network to yield a closed-form +loss computation. Though effective, the NTK cannot learn features. We build a proxy model using the +pre-trained LM model with a linear head and apply the deep linearization scheme to the proxy, which +can yield a closed-form loss and incorporate the wealth of biophysical information at the same time. +In addition, we propose Adaptive-γ to maintain a proper balance between the forward mapping and +the backward mapping by leveraging the weak supervision signal from an auxiliary model. Based on +this framework, we further propose Adaptive-η, the first learning rate adaptation strategy compatible +9 + +Under review +with all gradient-based offline model-based algorithms. Experimental results on DNA and protein +sequence design tasks verify the effectiveness of BIB and Adaptive-η. +7 +ETHICS STATEMENT +Protein sequence design aims to find a protein sequence with a particular biological function, which +has a broad application scope. This can lead to improved drugs that are highly beneficial to society. +For instance, designing the antibody protein for SARS-COV-2 can potentially save millions of human +lives (Kumar et al., 2021) and designing novel anti-microbial peptides (short protein sequences) is +central to tackling the growing public health risks caused by anti-microbial resistance (Murray et al., +2022). Unfortunately, it is possible to direct the research results towards harmful purposes such as the +design of biochemical weapons. As researchers, we believe that we must be aware of the potential +harm of any research outcomes, and carefully consider whether the possible benefits outweigh the +risks of harmful consequences. We also must recognize that we cannot control how the research +may be used. In the case of this paper, we are confident that there is a much greater chance that the +research outcomes will have a beneficial effect. We do not consider that there are any immediate +ethical concerns with the research endeavour. +8 +REPRODUCIBILITY STATEMENT +We provide the code implementation of BIB and Adaptive-η here and we also attach the code in +the supplementary material. We describe the DNA/protein benchmarks in Sec. 4.1 and the training +details in Sec. 4.3. 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Machine Learning (ICML), 2021. +Sihyun Yu, Sungsoo Ahn, Le Song, and Jinwoo Shin. Roma: robust model adaptation for offline +model-based optimization. Proc. Adv. Neur. Inf. Proc. Syst (NeurIPS), 2021. +A +APPENDIX +A.1 +DNA EMBEDDING +To incorporate richer contextual information, the DNA LM Ji et al. (2021) adopts the k-mer +sequence representation, which is widely used in DNA sequence analysis. For example, the sequence +ATGGCT has its 3-mer representation as {ATG, TGG, GGC, GCT}. In this paper, we adopt its +3-mer representation and compute the probability of the 3-mer token by multiplying the probabilities +of the three individual bases. The 3-mer representation is then sent to the pre-trained DNA LM. +A.2 +DATASET DETAILS +We conduct experiments on two DNA tasks following (Chen et al., 2022b) and three protein tasks in +(Ren et al., 2022) which have the most data points. We report the dataset details in Table 4. +DNA Task 1 TFBind8(r). The goal is to find a length-8 DNA sequence to maximize the binding +activity score with a particular transcription factor, SIX6REFR1 (Barrera et al., 2016). We sample +5000 data points for the offline algorithms following (Chen et al., 2022b). +DNA Task 2 TFBind10(r). The task TFBind10(r) is the same as TFBind8(r) except that the goal is +to find a length-10 DNA sequence. Both DNA tasks measure the entire search space and we adopt +these measurements as the approximate ground-truth evaluation. +Protein Task 1 avGFP. This task aims to find a protein sequence with approximately 239 amino +acids to maximize the fluorescence level of Green Fluorescent Proteins (Sarkisyan et al., 2016). The +task oracle is constructed by using the full unobserved dataset (around 52,000 points) following (Ren +et al., 2022). The oracle passes the average of the residue embeddings from the pre-trained Prot-T5 +12 + +Under review +Table 4: Dataset details. +Task +Metric +Min of D +Max of D +Min of Dentire +Max of Dentire +TFBind8(r) +binding activity +0.000 +0.242 +0.000 +1.000 +TFBind10(r) +binding activity +−1.859 +−0.869 +−1.859 +2.129 +avGFP +fluorescence level +1.283 +2.175 +1.283 +4.123 +AAV +viruses viability +−11.176 +−1.814 +−11.176 +9.536 +E4B +ubiquitination rate +−3.589 +−0.770 +−3.589 +8.998 +Table 5: Experimental results on different pre-trained LMs for comparison. +Pre-trained LM +avGFP +AAV +E4B +ProtAlbert +0.907 ± 0.004 +0.478 ± 0.004 +0.552 ± 0.023 +ProtBert(adopted) +1.060 ± 0.016 +0.501 ± 0.007 +1.255 ± 0.029 +ProtBert-BFD +1.119 ± 0.116 +0.549 ± 0.009 +1.880 ± 0.054 +(Elnaggar et al., 2021) into a linear layer and then fits the dataset. The following two task oracles +take the same form. The offline algorithms can only access the lowest-scoring 26,000 data points. +Protein Task 2 AAV. The goal is to engineer a 28-amino acid segment (positions 561–588) of the +VP1 protein to remain viable for gene therapy (Bryant et al., 2021). We use the entire 284, 000 data +points to build the oracle and the lowest-scoring 142, 000 points for the offline algorithms. +Protein Task 3 E4B. This task aims to design a protein (around 102 amino acids) to maximize the +ubiquitination rate to the target protein (Starita et al., 2013). The full dataset consisting of around +100, 000 points is used to build the oracle and the bottom half is used for the offline algorithms. +The parameterization of the oracle is different from that of the regression model from two aspects: +1) model architecture; 2) pre-trained information source. First, the oracle adopts the Prot-T5 model +which consists of an encoder and a decoder, while the regression model adopts the Prot-BERT model +which only has an encoder. Second, Prot-T5 is trained on the BFD and UniRef100 datasets and +ProtBert is trained on the UniRef50 dataset. These two points demonstrate that the oracle and the +regression model are different function classes. We choose the Prot-T5 model as the oracle because +this is the state-of-the-art protein LM to extract features and recent work Elnaggar et al. (2021) has +demonstrated its effectiveness. In order to test how related the Prot-T5 (oracle)/Prot-BERT(proxy) +models are, we trained them on a sampled training dataset and compared the test predictions of the +testing set. By evaluating the Pearson correlation coefficient (PCC) between the two prediction errors +PCC(ProtT5 predictions - test labels, ProtBERT predictions -test labels), we obtain 0.0104 on avGFP, +−0.0005 on AAV, and −0.0062 on E4B. These results suggest that the two models are not strongly +related in terms of the predictions they form. +Following (Trabucco et al., 2021), we select the top N = 128 most promising sequences for each +comparison method. Among these sequences, we report the maximum normalized ground truth score +as the evaluation metric following (Ren et al., 2022). +A.3 +TRAINING DETAILS +We use Pytorch (Paszke et al., 2019) to run all experiments on one V100 GPU. Following the setting +in Norn et al. (2021), we introduce a length-L protein sequence as a continuous random matrix +Xh ∈ RL×20 (Xh ∈ RL×4 for DNA), initialized using a normal distribution with the mean 0 and +the standard deviation of 0.01. To make this sequence correspond correctly to the candidate sequence, +we exchange the largest value in X[l, :] with the value in the amino acid index. +A.4 +DIFFERENT PRETRAINED LMS +As shown in Table 5, we have tested the ProtBERT, ProtAlbert, and ProtBert-BFD models and found +that better-quality models generally work better. The publicly available pre-trained DNA models +are limited and thus we only perform experiments on the protein tasks. +Elnaggar et al. (2021) +demonstrate that the language model performances follow the ordering: ProtBert-BFD > ProtBert > +ProtAlbert. We can see that the performance ranks over the three protein tasks avGFP, AAV, and E4B +are the same. +13 + +Under review +Table 6: Experimental results on different size datasets for comparison. +Dataset size +20 +40 +60 +80 +100 +TFBind8(r) +0.849 ± 0.027 +0.883 ± 0.036 +0.890 ± 0.033 +0.911 ± 0.042 +0.923 ± 0.049 +TFBind10(r) +0.248 ± 0.000 +0.596 ± 0.035 +0.602 ± 0.023 +0.616 ± 0.024 +0.632 ± 0.036 +Table 7: Mean squared prediction losses for comparison. +Method +TFBind8(r) +TFBind10(r) +avGFP +AAV +E4B +Finetuned NN +0.101 ± 0.001 +1.130 ± 0.041 +0.323 ± 0.006 +5.148 ± 0.074 +0.683 ± 0.012 +Linearized NN +0.107 ± 0.000 +1.618 ± 0.000 +3.956 ± 0.000 +23.041 ± 0.000 +1.050 ± 0.000 +NTK +0.111 ± 0.000 +1.840 ± 0.000 +4.866 ± 0.000 +24.451 ± 0.000 +1.075 ± 0.000 +A.5 +DIFFERENT DATASET SIZE +As shown in Table 6, we have tested the performance of BDI as a function of dataset size (N= 20, 40, +60, 80, 100) in TFBind8(r) and TFBind10(r) since they have exact oracle evaluations. We see that +performance is already good for N=20 for TFBind8(r) and N=40 for TFBind10(r). +A.6 +RANKING PERFORMANCE +As for prediction performances, the rank should be: a NN > linearized pre-trained LM > NTK. We +have conducted experiments to verify this. We sample half of the data, train a model to predict another +half data, and report the mean squared loss here and in Appendix A.6 Table 7. A small mean squared +loss indicates a good prediction performance; thus, we have verified the above ranking order. +14 + diff --git a/SdE1T4oBgHgl3EQfHwPN/content/tmp_files/load_file.txt b/SdE1T4oBgHgl3EQfHwPN/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..06fb3547cce30eeeed4a93e9786eede229ab1554 --- /dev/null +++ b/SdE1T4oBgHgl3EQfHwPN/content/tmp_files/load_file.txt @@ -0,0 +1,1030 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf,len=1029 +page_content='Under review BIDIRECTIONAL LEARNING FOR OFFLINE MODEL- BASED BIOLOGICAL SEQUENCE DESIGN Can (Sam) Chen1∗, Yingxue Zhang2, Xue Liu1, Mark Coates1 1McGill University, 2 Huawei Noah’s Ark Lab can.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='chen@mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='mcgill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='ca, yingxue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='zhang@huawei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='com, xueliu@cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='mcgill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='ca, mark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='coates@mcgill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='ca ABSTRACT Offline model-based optimization aims to maximize a black-box objective func- tion with a static dataset of designs and their scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' In this paper, we focus on biological sequence design to maximize some sequence score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' A recent approach employs bidirectional learning, combining a forward mapping for exploitation and a backward mapping for constraint, and it relies on the neural tangent kernel (NTK) of an infinitely wide network to build a proxy model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' Though effective, the NTK cannot learn features because of its parametrization, and its use prevents the incorporation of powerful pre-trained Language Models (LMs) that can capture the rich biophysical information in millions of biological sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' We adopt an alternative proxy model, adding a linear head to a pre-trained LM, and propose a linearization scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' This yields a closed-form loss and also takes into account the biophysical information in the pre-trained LM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' In addition, the forward mapping and the backward mapping play different roles and thus deserve different weights during sequence optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' To achieve this, we train an auxiliary model and leverage its weak supervision signal via a bi-level optimization framework to effec- tively learn how to balance the two mappings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' Further, by extending the framework, we develop the first learning rate adaptation module Adaptive-η, which is com- patible with all gradient-based algorithms for offline model-based optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' Experimental results on DNA/protein sequence design tasks verify the effectiveness of our algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' Our code is available here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' 1 INTRODUCTION Offline model-based optimization aims to maximize a black-box objective function with a static dataset of designs and their scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' This offline setting is realistic since in many real-world scenarios we do not have interactive access to the ground-truth evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' The design tasks of interest include material, aircraft, and biological sequence (Trabucco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' In this paper, we focus on biological sequence design, including DNA sequence and protein sequence, with the goal of maximizing some specified property of these sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' A wide variety of methods have been proposed for biological sequence design, including evolutionary algorithms (Sinai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' Ren et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=', 2022), reinforcement learning methods (Angermueller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=', 2019), Bayesian optimization (Terayama et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=', 2021), search/sampling using generative models (Brookes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' Chan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=', 2021), and GFlowNets (Jain et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' Recently, gradient-based techniques have emerged as an effective alternative (Trabucco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' These approaches first train a deep neural network (DNN) on the static dataset as a proxy and then obtain the new designs by directly performing gradient ascent steps on the existing designs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' Such methods have been widely used in biological sequence design (Norn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' Tischer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' Linder & Seelig, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' One obstacle is the out-of-distribution issue, where the trained proxy model is inaccurate for the newly generated sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' ∗Corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='02931v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='CE] 7 Jan 2023 Under review To mitigate the out-of-distribution issue, recent work proposes regularization of the model (Trabucco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' Fu & Levine, 2021) or the design itself (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=', 2022b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' The first category focuses on training a better proxy by introducing inductive biases such as robustness (Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' The second category introduces bidirectional learning (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=', 2022b), which consists of a forward mapping and a backward mapping, to optimize the design directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' Specifically, the backward mapping leverages the high-scoring design to predict the static dataset and vice versa for the forward mapping, which distills the information of the static dataset into the high-scoring design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' This approach achieves state-of-the-art performances on a variety of tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' Though effective, the proposed bidirectional learning relies on the neural tangent kernel (NTK) of an infinite-width model to yield a closed-form loss, which is a key component of its successful operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' The NTK cannot learn features due to its parameterization (Yang & Hu, 2021) and thus the bidirectional learning cannot incorporate the wealth of biophysical information from Language Models (LMs) pre-trained over a vast corpus of unlabelled sequences (Elnaggar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' Ji et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' To solve this issue, we construct a proxy model by combining a finite-width pre-trained LM with an additional layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' We then linearize the resultant proxy model, inspired by the recent progress in deep linearization (Achille et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' Dukler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' This scheme not only yields a closed-form loss but also exploits the rich biophysical information that has been distilled in the pre-trained LM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' In addition, the forward mapping encourages exploitation in the sequence space and the backward mapping serves as a constraint to mitigate the out-of-distribution issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' It is important to maintain an appropriate balance between exploitation and constraint, and this can vary across design tasks as well as during the optimization process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' We introduce a hyperparameter γ to control the balance, and develop a bi-level optimization framework Adaptive-γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' In this framework, we train an auxiliary model and leverage its weak supervision signal to effectively update γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' To sum up, we propose BIdirectional learning for model-based Biological sequence design (BIB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' Last but not least, since the offline nature prohibits standard cross-validation strategies for hyperparameter tuning, all gradient-based offline model-based algorithms preset the learning rate η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' There is a danger of a poor selection, and to address this, we propose to extend Adaptive-γ to Adaptive-η, which effectively adapts the learning rate η via the weak supervision signal from the trained auxiliary model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' To the best of our knowledge, Adaptive-η is the first learning rate adaptation module for gradient-based algorithms on offline model-based optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' Experiments on DNA and protein sequence design tasks verify the effectiveness of BIB and Adaptive-η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' To summarize, our contributions are three-fold: Instead of adopting the NTK, we propose to construct a proxy model by combining a pre-trained biological LM with an additional trainable layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' We then linearize the proxy model, leveraging the recent progress on deep linearization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' This yields a closed-form loss computation in bidirectional learning and allows us to exploit the rich biophysical information distilled into the LM via pre- training over millions of biological sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' We propose a bi-level optimization framework Adaptive-γ where we leverage weak signals from an auxiliary model to achieve a satisfactory trade-off between exploitation and constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' We further extend this bi-level optimization framework to Adaptive-η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' As the first learning rate tuning scheme in offline model-based optimization, Adaptive-η allows learning rate adaptation for any gradient-based algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' 2 PRELIMINARIES 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='1 OFFLINE MODEL-BASED OPTIMIZATION Offline model-based optimization aims to find a design X to maximize some unknown objective f(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' This can be formally written as, X∗ = arg max X f(X) , (1) where we have access to a size-N dataset D = {(X1, y1)}, · · · , {(XN, yN)} with Xi representing a certain design and yi denoting the design score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' In this paper, Xi represents a biological sequence design, including DNA and protein sequences, and yi represents a property of the biological sequence such as the fluorescence level of the green fluorescent protein (Sarkisyan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' 2 Under review 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='2 BIOLOGICAL SEQUENCE REPRESENTATION Following (Norn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' Killoran et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' Linder & Seelig, 2021), we adopt the position- specific scoring matrix to represent a length-L protein sequence as X ∈ RL×20, where 20 represents 20 different kinds of amino acids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' For a real-world protein sequence, X[l, :] (0 ≤ l ≤ L − 1) is a one-hot vector denoting one kind of amino acid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' During optimization, X[l, :] is a continuous vector and softmax(X[l, :]) represents the probability distribution of all 20 amino acids in the position l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' Similarly, for a DNA sequence, we have X ∈ RL×4 where 4 represents 4 different DNA bases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' The protein sequence X is fed into the embedding layer of the LM, which produces the embedding, e = EMB(softmax(X)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' (2) The main block of the LM takes e as input and outputs biophysical features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' The DNA LM, which adopts the k-mer representation, is a little different from protein LMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' See Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='1 for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='3 GRADIENT ASCENT ON SEQUENCE A common approach to the posed offline model-based optimization problem is to train a proxy fθ(X) on the offline dataset, θ∗ = arg min θ 1 N N � i=1 (fθ(Xi) − yi)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' (3) Then we can obtain the high-scoring design Xh by T gradient ascent steps: Xt+1 = Xt + η∇Xfθ∗(X)|X=Xt , for t ∈ [0, T − 1] , (4) where the high-scoring design Xh can be obtained as XT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' Considering the discrete nature of biological sequences, the input of fθ(·) should be discrete one-hot vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' Following (Norn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=', 2021), we can perform the conversion and predict the score via: ˆ Xi = softmax(Xi) , (5) Zi = onehot(argmax( ˆ Xi)) , (6) ˆy = fθ(Zi) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' (7) Then the gradient regarding Xi can be approximated as, dfθ(Zi) dxi ≈ dfθ(Zi) dzi dˆxi dxi , (8) where we unroll the matrices Xi, ˆ Xi and Zi as vectors xi, ˆxi and zi for notational convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' This approximation allows us to use backpropagation directly from the proxy to the sequence design Xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' For brevity, we will still use fθ(Xi) to represent the proxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='4 BIDIRECTIONAL LEARNING As shown in Figure 1, bidirectional learning (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=', 2022b), consists of two mappings: the forward mapping leverages the static dataset (Xl, yl) to predict the score yh of the high-scoring design Xh, and the backward mapping leverages the high-scoring design data (Xh, yh) to predict the static dataset (Xl, yl).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' The forward mapping loss can be written as Ll2h(Xh) = ∥yh − f l θ∗(Xh)∥2 , (9) where θ∗ is given by θ∗ = arg min θ ∥yl − f l θ(Xl)∥2 + β∥θ∥2 , (10) where β > 0 is a regularization parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' The backward mapping loss can be written as Lh2l(Xh) = ∥yl − f h θ∗(Xh)(Xl)∥2 , (11) 3 Under review Figure 1: Illustration of bidirectional learning Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' (2022b) where (Xl, yl) denotes the static dataset, yh is a large predefined target score and Xh is the high-scoring design we aim to find.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' where θ∗(Xh) is given by θ∗(Xh) = arg min θ ∥yh − f h θ (Xh)∥2 + β∥θ∥2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' (12) The high-scoring design Xh can be optimized by minimizing the bidirectional learning loss L(Xh) = Ll2h(Xh) + Lh2l(Xh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' 3 METHOD In this section, we first illustrate how to leverage deep linearization to compute the bidirectional learning loss in a closed form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' Subsequently, we introduce a hyperparameter γ to control the balance between the forward mapping and the backward mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' We then develop a novel bi-level optimization framework Adaptive-γ, which leverages a weak supervision signal from an auxiliary model to effectively update γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' Last but not least, we extend this framework to Adaptive-η, which enables us to adapt the learning rate η for all gradient-based offline model-based algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' We summarize our method in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='1 DEEP LINEARIZATION FOR BIDIRECTIONAL LEARNING In bidirectional learning, the backward mapping loss is intractable for a finite neural network, so Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' (2022b) employ a neural network with infinite width, which yields a closed-form loss via the NTK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' This however makes it impossible to incorporate the rich biophysical information that has been distilled into a pre-trained LM (Yang & Hu, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' Considering this, we construct a proxy model by combining a finite-width pre-trained LM with an additional layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' We then linearize the resultant proxy model, inspired by the recent progress in deep linearization which has established that an overparameterized DNN model is close to its linearization (Achille et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' Dukler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' Denote by θ0 = (θpt, θlin init) ∈ RD×1 the proxy model parameters derived by combining the parameters of the pre-trained LM θpt and a random initialization of the linear layer θlin init.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' In this paper, we adopt the pre-trained DNABERT (Ji et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=', 2021) and Prot-BERT (Elnaggar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=', 2021) models, and compute the average of token embeddings as the extracted feature, which is fed into the linear layer to build the proxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' Then we can construct a linear approximation for the proxy model: fθ(X) ≈ fθ0(X) + ▽θfθ0(X) · (θ − θ0) , (13) where fθ(X), fθ0(X) ∈ R, ▽θfθ0(X) ∈ R1×D and ▽θfθ0(X) ∈ RD×1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' Intuitively, if the fine-tuning does not significantly change θ0, then this linearization is a good approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' By leveraging this linearization, we can obtain a closed-form solution for Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' (12) as: θ∗(Xh) = (▽θfθ0(Xh)⊤ ▽θ fθ0(Xh) + βI)−1 ▽θ fθ0(Xh)⊤(yh − fθ0(Xh)) + θ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' (14) Building on this result,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' we can compute the bidirectional learning loss as: Lbi(Xh) = 1 2(∥yh − KXhXl(KXlXl + βI)−1(yl − fθ0(Xl))∥2 + ∥yl − KXlXh(KXhXh + βI)−1(yh − fθ0(Xh))∥2) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' (15) 4 The static dataset: TFBind8(r) Forward mapping The high-scoring data A A A G G T Li2h(Xh) = Ilyh - f*(Xh)I2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' c c T 0* = arg min llyi - fä(Xi)2 + βliI2 T Xi Xh T T A A A Unknown LCh2i(Xh) = Ilyl - fb*(Xh)(Xi)I2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' G A A c G G 0*(Xn) = arg min llyh - f(Xn)I2 + βI01I2 T T T 16 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='9 Backward mapping Yh 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='0Under review where K(Xi, Xj) = ▽θfθ0(Xi)⊤ ▽θ fθ0(Xj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' Following (Dukler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=', 2022), we can also only linearize the last layer of the network for simplicity, which defines the following kernel, K(Xi, Xj) = BERT(Xi)⊤BERT(Xj), (16) where BERT(X) denotes the feature of the sequence X extracted by BERT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' Its kernel nature makes this approach suitable for small-data tasks (Arora et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=', 2020), especially in drug discovery where the labeling cost of DNA/proteins is high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' Algorithm 1 Bidirectional Learning for Offline Model-based Biological Sequence Design Input: Static dataset D = (Xl, yl), predefined target score yh = 10, # of iterations T, pre-trained biological LM parameterized by θ0, auxiliary model faux(·), regularization β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' Output: High-scoring design X∗ h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' 1: Initialize X0 as the sequence with the highest score in D 2: for τ ← 0 to T − 1 do 3: Leverage Adaptive-γ in Sec 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='2 to update the balance γ by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' (21) 4: if Adapt learning rate then 5: Leverage Adaptive-η in Sec 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='3 to update the learning rate η by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' (23) 6: Optimize X by minimizing the bidirectional learning loss Lbi(Xτ, γ) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' (17): 7: Xτ+1 = Xτ − ηOPT(∇XLbi(Xτ, γ)) 8: Return X∗ h = XT 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='2 ADAPTIVE-γ The forward mapping and the backward mapping play different roles in the sequence optimization process: the forward mapping encourages the high-scoring sequence to search for a higher target score (exploitation) and the backward mapping serves as a constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' Since different sequences require different degrees of constraint, we introduce an extra hyperparameter γ ∈ [0, 1] to control the balance between the corresponding terms in the loss function: Lbi(Xh, γ) = γLl2h(Xh) + (1 − γ)Lh2l(Xh) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' (17) Thus γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='0 corresponds to the forward mapping alone, γ = 0 results in backward mapping, and γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='5 leads to the bidirectional loss of (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=', 2022b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' It is non-trivial to determine the most suitable value for γ since we do not know the ground-truth score for a new design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' One possible solution is to train an auxiliary faux(·) to serve as a proxy evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' A reasonable auxiliary is a simple regression model fitted to the offline dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' Although this auxiliary model cannot yield ground-truth scores, it can provide weak supervision signals to update γ, since the auxiliary model and the bidirectional learning provide complementary information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' This is similar to co-teaching (Han et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=', 2018) where two models leverage each other’s view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' Formally, we introduce the Adaptive-γ framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' Given a good choice of γ, the produced Xh is expected to have a high score faux(Xh), based on which we can choose γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' To make the search for γ more efficient, we can formulate this process as a bi-level optimization problem (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=', 2022a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' 2022c): γ∗ = arg max γ faux(X∗ h(γ)) , (18) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' X∗ h(γ) = arg min Xh Lbi(Xh, γ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' (19) We can then use the hyper-gradient ∂faux(X∗ h(γ)) ∂γ to update γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' Specifically, the inner level solution can be approximated via a gradient descent step with a learning rate η: X∗ h(γ) = Xh − η dLbi(Xh, γ) dX⊤ h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' (20) For the outer level, we update γ by hyper-gradient ascent: γ = γ + η ′ dfaux(X∗ h(γ)) dγ = γ + η ′ dfaux(Xh) dxh dLbi(Xh, γ) dx⊤ h , (21) where we unroll the matrix Xh as a vector xh for better illustration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' 5 Under review 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='3 ADAPTIVE-η We now extend the Adaptive-γ framework to Adaptive-η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' As the first learning rate adaptation module for offline model-based optimization, Adaptive-η is compatible with all gradient-based algorithms and can effectively finetune the learning rate η via the auxiliary model’s weak supervision signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' All gradient-based methods that maximize Lθ(X) with respect to X have the following general form: Xt+1 = Xt + ηOPT(∇XLθ(X)|X=Xt) , for t ∈ [0, T − 1] , (22) where η represents the learning rate of the optimizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' For methods such as simple gradient ascent (Grad), COMs (Trabucco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=', 2021), ROMA (Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=', 2021) and NEMO (Fu & Levine, 2021), Lθ(·) is related to the proxy model fθ(·);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' for BDI Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' (2022b) and our proposed method, BIB, Lθ(·) is the negative of the bidirectional learning loss, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=', Lθ = −Lbi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' Though the learning rate η can be adapted in some optimizers such as Adam (Kingma & Ba, 2015), these adaptations rely on only the past optimization history and do not consider the weak supervision signal from the auxiliary model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' Our Adaptive-η optimizes η by solving: η∗ = arg max η faux(X∗ h(η)) , (23) where η can be updated via gradient ascent methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' Considering the sequence optimization procedure is highly sensitive to the learning rate η, we reset η to η0 at each iteration and update η from η0, η = η0 − η ′ dfaux(X∗ h(η)) dη .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' (24) In general, this serves to stabilize the optimization procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' 4 EXPERIMENTS We conduct extensive experiments on DNA and protein design tasks, and aim to answer three research questions: (1) How does BIB compare with state-of-the-art algorithms?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' (2) Is every design component necessary in BIB?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' (3) Does the Adaptive-η module improve gradient-based methods?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='1 BENCHMARK We conduct experiments on two DNA tasks: TFBind8(r) and TFBind10(r), following (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=', 2022b) and three protein tasks: avGFP, AAV and E4B, in (Ren et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=', 2022) which have the most data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' See See Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='2 for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' Following (Trabucco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=', 2021), we select the top N = 128 most promising sequences for each comparison method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' Among these sequences, we report the maximum normalized ground truth score as the evaluation metric following (Ren et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='2 COMPARISON METHODS We compare BIB with two groups of baselines: the gradient-based methods and the non-gradient- based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' For a fair comparison, the pre-trained LM is used for all methods involving a proxy and we don’t finetune the LM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' The gradient-based methods include: 1) Grad: gradient ascent on existing sequences to obtain new sequences;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' 2) COMs (Trabucco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=', 2021): lower bounds the DNN model by the ground-truth values and then applies gradient ascent;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' 3) ROMA (Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=', 2021): incorporates a smoothness prior into the DNN model before gradient ascent steps;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' 4) NEMO (Fu & Levine, 2021): leverages the normalized maximum-likelihood estimator to bound the distance between the DNN model and the ground-truth values;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' 5) BDI (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=', 2022b): adopts the infinitely wide neural network and its NTK to yield a closed-form bidirectional learning loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' The non-gradient-based methods include: 1) BO-qEI (Wilson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=', 2017): builds an acquisition function for sequence exploration;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' 2) CMA-ES (Hansen, 2006): estimates the covariance matrix to adjust the sequence distribution towards the high-scoring region;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' 3) AdaLead (Sinai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=', 2020): performs a hill-climbing search on the proxy and then queries the sequences with high predictions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' 4) CbAS (Brookes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=', 2019): builds a generative model for sequences above a property threshold and gradually adapts the distribution by increasing the threshold;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' 5) PEX (Ren et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=', 2022): prioritizes the evolutionary search for protein sequences with low mutation counts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' 6) GENH (Chan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=', 2021): enhances the score through a learned latent space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' 6 Under review 0 10 20 30 40 T 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='0 Performance 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='0 Trade-off Performance Trade-off (a) TFBind8(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' 0 10 20 30 40 T 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='2 Performance 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='0 Trade-off Performance Trade-off (b) avGFP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' 0 10 20 30 40 T 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='5 Performance 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='0 Trade-off Performance Trade-off (c) E4B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' Figure 3: Trend of performance and trade-off γ as a function of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='3 TRAINING DETAILS We follow the training setting in (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=', 2022b) if not specified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' We choose OPT as the Adam optimizer (Kingma & Ba, 2015) for all gradient-based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' We implement the auxiliary model as a linear layer with the feature from the pre-trained LM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' We set the number of iterations T as 25 for all experiments following (Norn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=', 2021) and η0 as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='1 following (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=', 2022b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' We run every setting over 16 trials and report the average score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' See Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='3 for other details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='4 RESULTS AND ANALYSIS We report all experimental results in Table 1 and plot the ranking statistics in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' We make the following observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' (1) As shown in Table 1, BIB consistently outperforms the Grad method on all tasks, which demonstrates that our BIB can effectively mitigate the out-of-distribution issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' (2) Furthermore, BIB outperforms BDI on 4 out of 5 tasks, which demonstrates the ef- fectiveness of the pre-trained biological LM over NTK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' The reason why BDI outperforms BIB on TFBind10(r) may be that short sequences do not rely much on the rich sequential informa- tion from the pre-trained LM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' (3) As shown in Figure 2, the gradient-based methods generally perform better than the non-gradient-based methods, as also observed by Trabucco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' 1 2 3 4 5 6 7 8 9 101112 Rank BO-qEI CMA-ES AdaLead CbAS PEX GENH Grad COMs ROMA NEMO BDI BIB(ours) Figure 2: Rank minima and max- ima are represented by whiskers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' vertical lines and black triangles de- note medians and means.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' (4) The gradient-based methods are inferior for the AAV task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' One possible reason is that the design space of AAV (2028) is much smaller than those of avGFP (20239) and E4B (20102), which makes the generative modeling and evolutionary algo- rithms more suitable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' (5) This conjecture is also supported by the experimental results on two DNA design tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' We compute the average ranking of gradient-based methods and non-grad-based methods on TFBind10(r) as 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='5 and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='5, re- spectively, and the average ranking of gradient-based methods and non-grad-based methods on TFBind8(r) as 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='8 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='8, re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' The advantage of gradient-based methods are larger (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='5 − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='5 = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='0) in TFBind10(r) than that (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='8 − 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='8 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='0) in TFBind8(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' (6) The generative modeling methods CbAS and GENH yield poor results on all tasks, probably because the high-dimensional data distribution is very hard to model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' (7) Overall, BIB attains the best performance in 3 out of 5 tasks and achieves the best ranking results as shown in Table 1 and Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' We also visualize the trend of performance (the maximum normalized ground truth score) and trade- off γ as a function of T on TFBind8(r) in Figure 3(a) and avGFP in Figure 3(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' The performance generally increases with the time step T and then stabilizes, which demonstrates the effectiveness and robustness of BIB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' Furthermore, we find that the γ values of TFBind8(r) and avGFP generally increase at first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' This means that BIB reduces the impact of the constraint to encourage a more aggressive search for a high target value during the initial phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' Then γ of TFBind8(r) continues to increase while the γ of avGFP decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' We conjecture that the difference is caused by the sequence length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' Small mutations of a biological sequence are enough to yield a good candidate (Ren et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' For the length-239 protein in avGFP, dramatic mutations 1) are not necessary and 2) can easily lead to out-of-distribution points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' The weak supervision signal from the auxiliary model therefore encourages a tighter constraint towards the static dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' By contrast, the DNA sequence is relatively short and a more widespread search of the sequence space can yield better results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' To investigate this 7 Under review Table 1: Experimental results (maximum normalized ground truth score) for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' Method TFBind8(r) TFBind10(r) avGFP AAV E4B Rank Mean Rank Median D(best) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='242 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='248 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='314 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='452 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='224 BO-qEI 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='940 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='032 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='595 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='028 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='888 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='591 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='436 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='004 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='0/12 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='0/12 CMA-ES 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='930 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='034 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='617 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='031 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='909 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='470 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='748 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='009 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='6/12 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='0/12 AdaLead 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='941 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} 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+page_content='0/12 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='0/12 BIB(ours) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='952 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='033 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='639 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='032 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='060 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='016 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='501 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='007 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='255 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='029 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='4/12 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='0/12 Table 2: Ablation studies on BIB components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' Task γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='0 γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='0 γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='5 γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='5 + Joint BIB BIB + Ada-η TFBind8(r) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='936 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='933 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='947 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='935 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='952 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='956 TFBind10(r) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='611 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='637 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='616 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='622 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='639 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='639 avGFP 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='920 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='966 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='018 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='006 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='060 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='082 AAV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='449 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='458 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='480 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='420 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='501 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='525 E4B 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='778 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='903 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='198 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='176 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='255 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='301 conjecture, we further visualize the trend of E4B in Figure 3(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' E4B also has long sequences (102) and we can observe its similar first-increase-then-decrease trend, although it is not as pronounced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='5 ABLATION STUDIES In this subsection, we conduct ablation studies to verify the effectiveness of the forward mapping, the backward mapping, and the Adaptive-γ module of BIB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' We report the experimental results in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' Forward mapping & Backward mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' We can observe that bidirectional learning (γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='5) performs better than both forward mapping (γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='0) and backward mapping (γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='0) alone in most tasks, which demonstrates the effectiveness of forward mapping and backward mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' The advantage of bidirectional mappings over the forward mapping is larger in the long-sequence tasks like avGFP (238) and E4B (102) compared with the short-sequence tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' A possible explanation is that the constraint is more important for long sequence tasks than short sequence design since the search space is large and many mutations can easily go out of distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' Adaptive-γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' BIB learns γ and this leads to improvements over bidirectional mappings (γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='5) for all tasks, verifying the effectiveness of Adaptive-γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' We also consider the following variant, X∗ = arg min Xh Lbi(Xh, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='5) − faux(Xh) , (25) which jointly optimizes the bidirectional learning loss Lbi(Xh, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='5) and the auxiliary term faux(Xh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' We found this yields similar or even worse results than pure bidirectional learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' The reason may be that the weak supervision signal from faux(Xh) can serve as a guide to update the scalar γ but not as a component of the main optimization objective that directly updates the sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' In the final column of Table 2, we examine the performance of the Adaptive-η module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' Adding this module leads to improvements on all five tasks, which demonstrates its effectiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='6 ADAPTIVE-η In this subsection, we aim to further demonstrate the effectiveness of the Adaptive-η module on all six gradient-based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' We conduct experiments on two tasks: TFBind8(r) and avGFP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' Since the use of the infinitely wide neural network leads to poor performance for BDI, we modify its implementation via deep linearization so that it can make use of the pre-trained LM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' As shown in Table 3, Adaptive-η provides a consistent gain for all scenarios, which demonstrates the widespread applicability and effectiveness of the module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' Furthermore, Adaptive-η leads to a maximum improvement of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='4% in TFBind8(r) and 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='5% in avGFP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' ROMA is the algorithm that 8 Under review Table 3: Adaptive-η on all gradient-based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' Method TFBind8(r) avGFP Grad COMs ROMA NEMO BDI BIB Grad COMs ROMA NEMO BDI BIB 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='051 Ada-η 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='941 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='928 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='939 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='935 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='951 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='956 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='007 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='013 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='014 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='023 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='022 benefits the most.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' One possible explanation is that ROMA incorporates a local smoothness prior that leads to more stable gradients, with which Adaptive-η can be more effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' Similar to Sec 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='5, we consider the following variant, X∗ = arg max Xh Lθ(Xh) + faux(Xh) , (26) which performs joint optimization instead of bi-level optimization on two objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' As shown in Table 3, joint optimization generally deteriorates the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' This again verifies that the auxiliary model can only serve as a guide instead of contributing to the main objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' 5 RELATED WORK Biological sequence design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' There has been a wide range of algorithms for biological sequence design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' Evolutionary algorithms (Sinai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' Ren et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=', 2022) leverage the learned surrogate model to provide evolution guidance towards the high-scoring region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' Angermueller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' (2019) propose a flexible reinforcement learning framework where sequence design is a sequential decision- making problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' Bayesian optimization methods propose candidate solutions via an acquisition function (Terayama et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' Deep generative model methods design sequences in the latent space (Chan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=', 2021) or gradually adapt the distribution towards the high-scoring region (Brookes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' GFlowNets (Jain et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=', 2022) amortize the cost of search over learning and encourage diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' Gradient-based methods leverage a surrogate model and its gradient information to maximize the desired property (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=', 2022b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' Norn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' Tischer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' Linder & Seelig, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' Our proposed BIB belongs to the last category and leverages the rich biophysical information (Ji et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' Elnaggar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=', 2021) to directly optimize the biological sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' Offline model-based optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' A majority of sequence design algorithms (Angermueller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' Sinai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' Ren et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=', 2022) focus on the online setting where wet-lab experimental results in the current round are analyzed to propose candidates in the next round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' The problem of this setting is that wet-lab experiments are often very expensive, and thus a pure data-driven, offline approach is attractive and has received substantial research attention recently (Trabucco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' Kolli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' Gradient-based methods have proven to be effective (Trabucco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' Fu & Levine, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=', 2022b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' Among these algorithms, Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' (2022b) propose bidirectional mappings to distill information from the static dataset into a high-scoring design, which achieves state-of-the-art performances on a variety of tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' However, this bidirectional learning is designed for general tasks, like robot and material design, and the rich biophysical information in millions of biological sequences is ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' In this paper, we leverage recent advances in deep linearization to incorporate the rich biophysical information into bidirectional learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' 6 CONCLUSION In this paper, we propose bidirectional learning for offline model-based biological sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' Our work is built on the recently proposed bidirectional learning approach (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=', 2022b), which is designed for general inputs and relies on the NTK of an infinitely wide network to yield a closed-form loss computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' Though effective, the NTK cannot learn features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' We build a proxy model using the pre-trained LM model with a linear head and apply the deep linearization scheme to the proxy, which can yield a closed-form loss and incorporate the wealth of biophysical information at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' In addition, we propose Adaptive-γ to maintain a proper balance between the forward mapping and the backward mapping by leveraging the weak supervision signal from an auxiliary model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' Based on this framework, we further propose Adaptive-η, the first learning rate adaptation strategy compatible 9 Under review with all gradient-based offline model-based algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' Experimental results on DNA and protein sequence design tasks verify the effectiveness of BIB and Adaptive-η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' 7 ETHICS STATEMENT Protein sequence design aims to find a protein sequence with a particular biological function, which has a broad application scope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' This can lead to improved drugs that are highly beneficial to society.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' For instance, designing the antibody protein for SARS-COV-2 can potentially save millions of human lives (Kumar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=', 2021) and designing novel anti-microbial peptides (short protein sequences) is central to tackling the growing public health risks caused by anti-microbial resistance (Murray et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' Unfortunately, it is possible to direct the research results towards harmful purposes such as the design of biochemical weapons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' As researchers, we believe that we must be aware of the potential harm of any research outcomes, and carefully consider whether the possible benefits outweigh the risks of harmful consequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' We also must recognize that we cannot control how the research may be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' In the case of this paper, we are confident that there is a much greater chance that the research outcomes will have a beneficial effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' We do not consider that there are any immediate ethical concerns with the research endeavour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' 8 REPRODUCIBILITY STATEMENT We provide the code implementation of BIB and Adaptive-η here and we also attach the code in the supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' We describe the DNA/protein benchmarks in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='1 and the training details in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' We also explain how to obtain the sequence embedding from the pre-trained LM and how to perform gradient ascent steps on the sequence in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' 2.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' Can Chen, Xi Chen, Chen Ma, Zixuan Liu, and Xue Liu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' Gradient-based bi-level optimization for deep learning: A survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' arXiv preprint arXiv:2207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='11719, 2022a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' Can Chen, Yingxue Zhang, Jie Fu, Xue Liu, and Mark Coates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' 10 Under review Can Chen, Jingbo Zhou, Fan Wang, Xue Liu, and Dejing Dou.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' Structure-aware protein self-supervised learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' arXiv preprint arXiv:2204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='04213, 2022c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' Yonatan Dukler, Alessandro Achille, Giovanni Paolini, Avinash Ravichandran, Marzia Polito, and Stefano Soatto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' DIVA: Dataset derivative of a learning task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' Learning Rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' (ICLR), 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' Machine Learning (ICML), 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' Sihyun Yu, Sungsoo Ahn, Le Song, and Jinwoo Shin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' Roma: robust model adaptation for offline model-based optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' Neur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' Inf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' Syst (NeurIPS), 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' A APPENDIX A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='1 DNA EMBEDDING To incorporate richer contextual information, the DNA LM Ji et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' (2021) adopts the k-mer sequence representation, which is widely used in DNA sequence analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' For example, the sequence ATGGCT has its 3-mer representation as {ATG, TGG, GGC, GCT}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' In this paper, we adopt its 3-mer representation and compute the probability of the 3-mer token by multiplying the probabilities of the three individual bases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' The 3-mer representation is then sent to the pre-trained DNA LM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='2 DATASET DETAILS We conduct experiments on two DNA tasks following (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=', 2022b) and three protein tasks in (Ren et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=', 2022) which have the most data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' We report the dataset details in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' DNA Task 1 TFBind8(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' The goal is to find a length-8 DNA sequence to maximize the binding activity score with a particular transcription factor, SIX6REFR1 (Barrera et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' We sample 5000 data points for the offline algorithms following (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=', 2022b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' DNA Task 2 TFBind10(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' The task TFBind10(r) is the same as TFBind8(r) except that the goal is to find a length-10 DNA sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' Both DNA tasks measure the entire search space and we adopt these measurements as the approximate ground-truth evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' Protein Task 1 avGFP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' This task aims to find a protein sequence with approximately 239 amino acids to maximize the fluorescence level of Green Fluorescent Proteins (Sarkisyan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' The task oracle is constructed by using the full unobserved dataset (around 52,000 points) following (Ren et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' The oracle passes the average of the residue embeddings from the pre-trained Prot-T5 12 Under review Table 4: Dataset details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' Task Metric Min of D Max of D Min of Dentire Max of Dentire TFBind8(r) binding activity 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='242 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='000 TFBind10(r) binding activity −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='859 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='869 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='859 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='129 avGFP fluorescence level 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='283 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='175 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='283 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='123 AAV viruses viability −11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='176 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='814 −11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='176 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='536 E4B ubiquitination rate −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='589 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='770 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='589 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='998 Table 5: Experimental results on different pre-trained LMs for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' Pre-trained LM avGFP AAV E4B ProtAlbert 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='907 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='478 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='552 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='023 ProtBert(adopted) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='060 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='016 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='501 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='007 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='255 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='029 ProtBert-BFD 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='119 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='116 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='549 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='009 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='880 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='054 (Elnaggar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=', 2021) into a linear layer and then fits the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' The following two task oracles take the same form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' The offline algorithms can only access the lowest-scoring 26,000 data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' Protein Task 2 AAV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' The goal is to engineer a 28-amino acid segment (positions 561–588) of the VP1 protein to remain viable for gene therapy (Bryant et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' We use the entire 284, 000 data points to build the oracle and the lowest-scoring 142, 000 points for the offline algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' Protein Task 3 E4B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' This task aims to design a protein (around 102 amino acids) to maximize the ubiquitination rate to the target protein (Starita et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=', 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' The full dataset consisting of around 100, 000 points is used to build the oracle and the bottom half is used for the offline algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' The parameterization of the oracle is different from that of the regression model from two aspects: 1) model architecture;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' 2) pre-trained information source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' First, the oracle adopts the Prot-T5 model which consists of an encoder and a decoder, while the regression model adopts the Prot-BERT model which only has an encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' Second, Prot-T5 is trained on the BFD and UniRef100 datasets and ProtBert is trained on the UniRef50 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' These two points demonstrate that the oracle and the regression model are different function classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' We choose the Prot-T5 model as the oracle because this is the state-of-the-art protein LM to extract features and recent work Elnaggar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' (2021) has demonstrated its effectiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' In order to test how related the Prot-T5 (oracle)/Prot-BERT(proxy) models are, we trained them on a sampled training dataset and compared the test predictions of the testing set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' By evaluating the Pearson correlation coefficient (PCC) between the two prediction errors PCC(ProtT5 predictions - test labels, ProtBERT predictions -test labels), we obtain 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='0104 on avGFP, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='0005 on AAV, and −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='0062 on E4B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' These results suggest that the two models are not strongly related in terms of the predictions they form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' Following (Trabucco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=', 2021), we select the top N = 128 most promising sequences for each comparison method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' Among these sequences, we report the maximum normalized ground truth score as the evaluation metric following (Ren et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='3 TRAINING DETAILS We use Pytorch (Paszke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=', 2019) to run all experiments on one V100 GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' Following the setting in Norn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' (2021), we introduce a length-L protein sequence as a continuous random matrix Xh ∈ RL×20 (Xh ∈ RL×4 for DNA), initialized using a normal distribution with the mean 0 and the standard deviation of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' To make this sequence correspond correctly to the candidate sequence, we exchange the largest value in X[l, :] with the value in the amino acid index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='4 DIFFERENT PRETRAINED LMS As shown in Table 5, we have tested the ProtBERT, ProtAlbert, and ProtBert-BFD models and found that better-quality models generally work better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' The publicly available pre-trained DNA models are limited and thus we only perform experiments on the protein tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' Elnaggar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' (2021) demonstrate that the language model performances follow the ordering: ProtBert-BFD > ProtBert > ProtAlbert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' We can see that the performance ranks over the three protein tasks avGFP, AAV, and E4B are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' 13 Under review Table 6: Experimental results on different size datasets for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' Dataset size 20 40 60 80 100 TFBind8(r) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='849 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='027 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='883 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='036 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='890 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='033 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='911 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='042 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='923 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='049 TFBind10(r) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='248 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='596 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='035 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='602 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='023 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='616 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='024 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='632 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='036 Table 7: Mean squared prediction losses for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' Method TFBind8(r) TFBind10(r) avGFP AAV E4B Finetuned NN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='101 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='001 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='130 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='041 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='323 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='006 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='148 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='074 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='683 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='012 Linearized NN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='107 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='618 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='000 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='956 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='000 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='041 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='050 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='000 NTK 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='111 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='840 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='000 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='866 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='000 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='451 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='075 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='000 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='5 DIFFERENT DATASET SIZE As shown in Table 6, we have tested the performance of BDI as a function of dataset size (N= 20, 40, 60, 80, 100) in TFBind8(r) and TFBind10(r) since they have exact oracle evaluations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' We see that performance is already good for N=20 for TFBind8(r) and N=40 for TFBind10(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='6 RANKING PERFORMANCE As for prediction performances, the rank should be: a NN > linearized pre-trained LM > NTK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' We have conducted experiments to verify this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' We sample half of the data, train a model to predict another half data, and report the mean squared loss here and in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content='6 Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' A small mean squared loss indicates a good prediction performance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' thus, we have verified the above ranking order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE1T4oBgHgl3EQfHwPN/content/2301.02931v1.pdf'} +page_content=' 14' metadata={'source': 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Turin, Italy +4Department of Sociology and Social Research, University of Trento, Trento, Italy +∗To whom correspondence should be addressed: michele.starnini@gmail.com, +†To whom correspondence should be addressed: michele.tizzoni@unitn.it +Abstract +The course of an epidemic can be drastically altered by changes in human behav- +ior. Incorporating the dynamics of individual decision-making during an outbreak +represents a key challenge of epidemiology, faced by several modeling approaches +siloed by different disciplines. Here, we propose an epi-economic model including +adaptive, forward-looking behavioral response on a heterogeneous networked sub- +strate, where individuals tune their social activity based on future health expecta- +tions. Under basic assumptions, we show that it is possible to derive an analytical +expression of the optimal value of the social activity that matches the traditional +assumptions of classic epidemic models. Through numerical simulations, we con- +trast the settings of global awareness – individuals only know the prevalence of the +disease in the population – with local awareness, where individuals explicitly know +which of their contacts are infected. We show that behavior change can flatten the +epidemic curve by lowering the peak prevalence, but local awareness is much more +effective in curbing the disease early with respect to global awareness. Our work +bridges classical epidemic modeling with the epi-economic approach, and sheds light +on the effects of heterogeneous behavioral responses in curbing the epidemic spread. +1 +Introduction +Behavioral adaptation in response to infectious disease outbreaks is one of the key factors +that shape the course of an epidemic [1]. +Individual choices regarding the adoption +of self-protective measures, such as wearing masks, avoiding close social contacts, or +vaccinating, contribute to reducing disease transmission in the population. In turn, such +choices depend on the perceived severity of the epidemic, the available information about +1 +arXiv:2301.04947v1 [physics.soc-ph] 12 Jan 2023 + +it, and individual risk perception, thus they vary over time as the epidemic progresses. +As the epidemic fades out, behavioral responses may relax, with the consequence of +leading to disease resurgence. +Self-initiated behavioral responses have been observed +across all kinds of epidemics, from small-scale outbreaks, involving few individuals, such +as the 2015 Middle East respiratory syndrome outbreak in South Korea [2], to worldwide +pandemics, such as the 1918 pandemic [3], the 2009 A/H1N1 pandemic [4, 5, 6], as well +as the current COVID-19 pandemic [7]. +Incorporating the dynamics of individual decision-making during an outbreak repre- +sents a key challenge of epidemic modeling [8]. To this aim, a wide variety of mathemat- +ical epidemic models that capture the effects of behavioral changes have been proposed +[9, 10, 11]. Generally speaking, these models can be classified into three broad classes. +In the simplest case, classical compartmental models have been expanded to consider +additional behavioral classes in the population, characterized by different behavioral re- +sponses to the disease prevalence, whose transmission parameters do not change over time +[12, 13]. The second class of models includes those that aim at capturing the interplay +between individual adaptation and individual knowledge of the disease, often represented +as two coupled dynamical processes [14, 15]. Information about the disease can be local +or global, and sometimes it is assumed to spread through the population [16]. Finally, a +distinct class of models aims to explicitly describe the individual decision-making process +using approaches of behavioral economics, where individuals evaluate their payoffs and +adopt the behavior that optimally increases the payoff [17, 18]. These “epi-economic" +models simulate the process by which people choose the best course of action by ad- +justing to the current epidemic state and seeking the best possible future outcome via +an optimization process [19, 20, 21, 22]. Such optimization processes typically rely on +numerical methods. +The COVID-19 pandemic reopened the debate on how human behavior should be +included in epidemic models, with researchers with many different backgrounds con- +tributing to the modeling effort, including for instance game-theoretical models [23, 24]. +However, different approaches have often been siloed within the boundaries of the re- +lated discipline and consequent limitations. On the one hand, agent-based models with +additional behavioral classes require a behavioral response to be defined a-priori. On +the other hand, epi-economic models do consider an adaptive response but generally rely +on the homogeneous mixing hypothesis, under which all susceptible individuals have an +equal chance of contracting the infection. The great degree of heterogeneity in human +behavior [25, 26, 27, 28] makes it clear that different individuals have varying chances +of contracting the disease. As a consequence, heterogeneous behavioral responses in the +population should arise also under the epi-economic approach. +In this work, we try to fill this gap by proposing an epi-economic model including +adaptive, forward-looking behavioral response on a heterogeneous networked substrate. +Individuals choose their social activity to maximize their future expected utility, by +balancing the risk of being infected in the future while maintaining the highest possible +social activity. We rely on some simplifying yet realistic assumptions that allow us to +describe the optimal behavior by an analytical expression depending on the epidemic +2 + +conditions (prevalence and disease parameters) and the behavior of the population itself. +We explicitly contrast the cases of global awareness, in which individuals have a bird-eye +view of the epidemic unfolding on the whole population, with a local awareness setting, +where individuals only know which of their contacts are infected. The latter case triggers +a highly heterogeneous behavioral response. We show that behavior change can flatten +the epidemic curve by lowering the peak prevalence, thus potentially reducing the load on +the health system at the epidemic peak. However, local awareness is much more effective +in curbing early the disease with respect to global awareness, thus shrinking the overall +number of infections. +2 +Results +2.1 +A model of forward-looking adaptive behavior +We propose an analytically tractable epidemic model including a feedback loop between +the social activity of individuals and the spreading of the epidemic. To this aim, we +consider a SIR model in which individuals change their behavior, reducing or increasing +their social activity, depending on the prevalence of the disease. Crucially, susceptible +individuals tune their social activity with a forward-looking approach, i.e., they balance +the risk of being infected in the future while maintaining the highest possible social +activity. +The base of our model will be the SIR model [29] in discrete time steps. We con- +sider a population of N individuals, each one belonging to one of three compartments: +susceptible (S), infected (I), or recovered (R). A S individual who is in contact with a +I individual has a probability β of becoming infected in a day. Each I individual has a +probability µ of recovering. R individuals can not be infected anymore. +Next, we specify how changing behavior affects the probability of infection. Inspired +by Ref. [21], we define a time-dependent social activity at for individuals, representing +their propensity of engaging in social interactions with peers. Social activity is bounded +0 ≤ at ≤ 1, with a = 1 corresponding to normal behavior in absence of the disease, and +a = 0 corresponding to a situation in which disease transmission is not possible, equiva- +lent to quarantine. We assume that disease transmission between individuals depends on +their social activity. If a S individual with social activity aS +t comes into contact at time t +with an I individual with social activity aI +t , we assume the probability of the S individ- +ual getting infected to depend linearly on both aS +t and aI +t , so that the S individual gets +infected with probability βaS +t aI +t . We note that the social activity at can have multiple +interpretations as long as they result in a decreased probability of disease transmission. +For instance, a lower social activity could represent less frequent contacts with other +individuals or it could represent the adoption of prophylactic measures, such as the use +of face masks in the case of airborne disease ([30]), that limit the probability of infection +on contact without actually reducing the contact rate. Intuitively, individuals limit their +social activity in order to reduce the probability to be infected: when the prevalence is +high, they will adopt prudent behavior. The feedback loop between the disease spread +3 + +and social activity is illustrated in Figure 1 (a). +Following the epi-economic approach, we model the dynamics of at as an optimization +process in which S individuals balance the risk of infection and the benefits of social +activity, which are represented by a utility function. At each time step, each individual +is assigned a score, named utility, based on their behavior and health status. Adopting +a behavior that reduces the probability of infection (i.e. lower social activity) results in +lower utility, but also reduces the probability of getting infected which would also result +in a penalty. At each time step, each individual deals with this trade-off by choosing +the behavior that maximizes its future expected utility (objective function), see Fig. 1 +(b). A common strategy to approach this problem is dynamic programming, which can +be solved via the Bellman Equation [19, 31, 32]. Here, we make a series of simplifying +assumptions that allow us to find an analytical expression for the optimal value of the +social activity. +Our first simplifying assumption is that all individuals optimize social activity in the +same way. This is equivalent to assuming that individuals ignore their health status. We +stress that this does not necessarily mean that all individuals behave in the same way. +For instance, prophylactic measures that are taken by I individuals regardless of the +current state of the epidemic (i.e. behavior that is not adaptive), such as always wearing +a face mask when in public or reducing their contact by a fixed amount by not going to +work, can be modeled by changing the value of β. Also, the behavior of R individuals +does not affect the dynamics of the disease in our model. Second, we assume individuals +to believe that current conditions (i.e. the sizes of the S, I, and R compartments, as well +as their social activity at and that of the rest of the population) will remain unchanged. +The validity of these assumptions is discussed in Section 2.4 (Model calibration). +2.2 +Utility function +We assume the utility function to be the sum of two terms Ua and UH: the first one +depends only on social activity and the second one depends only on the health status +H = {S, I, R}. We choose the state-dependent term UH of the utility function to corre- +spond to a fixed penalty of magnitude UI for each time step in the infected state and to +be UH = 0 otherwise (H ̸= I): +UH(t) = +� +−UI +(if H = I at time t) +0 +(otherwise). +(1) +To determine the functional form of the utility function related to social activity, Ua, we +use the following argument. On the one hand, decreasing social activity should require +a higher cost in terms of utility when social activity is already low, so we assume the +derivative of Ua with respect to a to be inversely proportional to a. On the other hand, +increasing social activity also comes with a cost, otherwise, there would be no finite +optimal value of social activity (which we assumed to be a = 1) when there is no risk +of getting infected. The simplest hypothesis that we can make about this cost is that +it does not depend on social activity, so we add a constant term to the derivative of Ua +4 + +Figure 1: +Schematic illustration of the model. (a) Feedback loop involving epidemic +spreading and social activity. If the prevalence is high, individuals can decrease their +social activity, which in turn reduces the transmissibility of the disease, decreasing preva- +lence. With low prevalence, individuals can choose high levels of social activity, increasing +the prevalence. (b) Individuals choose their social activity to maximize their future ex- +pected utility. +They can choose a low value of social activity to delay the expected +infection (and the expected penalty UI), or benefit from high social activity at a higher +infection risk. Awareness and behavioral response can be of two kinds. Global awareness +(c): individuals know the prevalence of the disease in the population but do not know +who is infected, so social activity is homogeneous across the population. Local awareness +(d): individuals know which of their contacts are infected, resulting in different values +of social activity for each individual. +5 + +a +Disease-behaviour feedback +high +high +disease +social +prevalence +activity +reduced transmissibility +low +low +b +Forward-looking +UI +UI +choose low +S +S +social activity +UI +choose high +S +social activity +t +t+1 +t+2 +t+3 +t+4 +t+5 +Global awareness +c +d +Local awareness +disease prevalence +low social activity +O + social activity +high social activitywith respect to a, resulting in dUa +da = A +a − B, which has solution Ua = A log(a) − Ba + C. +Requiring that a = 1 is a maximum of Ua results in A = B. Since we are only interested in +the position of the maximum and not in the actual value of Ua, we can fix A = B = C = 1 +without loss of generality. +From which follows that at time t individuals with social +activity at get a utility score equal to: +Ua(t) = log(at) − at + 1. +(2) +We note that this choice of a concave form for the utility function is common in the +economic literature, as well as in epi-economic models [21]. +The optimal value of social activity at each time t is then given by the maximum +with respect to at of the following objective function: +J(t) = E +∞ +� +∆t=0 +δ∆t [Ua(t + ∆t) + UH(t + ∆t)] . +(3) +The objective function in Eq. (3) represents the expected value, over all possible future +health states, starting from the S state at time t, of the utility function, in which utility +corresponding to ∆t days into the future gets discounted by a factor δ∆t, where δ is the +discount factor. The discount factor 0 < δ < 1 implies that behavior resulting in high +values of utility in the near future is preferred to behavior that pays off in the distant +future. It also ensures that the series in Eq. (3) converges. +Under our assumptions, it is possible to express the objective function at time t, Eq. +(3), associated with a choice of social activity at in a closed-form, see Methods: +J(t) = +1 +1 − δ [Ua(t) − αδPI(t)] , +(4) +where the first term represents the benefits of social activity and the second term repre- +sents the risk of getting infected. The strength of the behavioral response to the infection +risk is determined by the probability of becoming infected at time t, PI(t), the discount +factor δ, and the average utility loss caused by an infection, α. The larger the average +infection cost α, the infection probability PI, or the discount factor δ, the stronger the +behavioral response. The probability of infection depends on the choice of the underlying +network substrate, and it will be discussed in the next Section. The parameter α instead +can be calculated (see Methods) as +α = +UI +1 − δ(1 − µ). +(5) +The average cost of infection is proportional to the penalty UI and it becomes smaller +for increasing discount factor δ (the duration of the infection is discounted by δ) and the +recovery rate µ (the larger µ, the shorter the duration of the infection). +The optimal value of the social activity can be determined as the maximum with re- +spect to a of the objective function in Eq. (4). In particular, if the behavioral component +6 + +of the utility function is given by Eq. (2), then the optimal value of social activity at +time t is: +a∗ +t = +1 +1 + αδ ∂PI(t) +∂a∗ +t +. +(6) +Therefore, individuals will adopt the optimal value of social activity a∗ +t at each time step +t, depending on the constant parameters α and δ, and on the probability of becoming +infected at time t, PI(t). Note that the infection probability does depend on the un- +derlying network’s structure and on the optimal social activity chosen by the involved +individuals. From now on, we will always refer to the optimal social activity, therefore, +for the sake of simplicity, we will denote it as at in the remainder of the paper. +2.3 +Local vs global awareness +Next, we specify the substrate over which the disease spreads. We assume that disease +transmission is mediated through social interactions, represented by a contact network +where nodes represent individuals and links represent contacts between them. We assume +that each active link connecting a S node (labelled i) to a I neighbor (labelled j) can +carry the disease in the unit time with probability βaiaj, where ai and aj are the social +activities of the two nodes (we omit the time dependence here). By assuming independent +infection processes for each active link, the probability of the S node being infected by +any of its I neighbors is PI = 1 − � +j(1 − βaiaj), where the product runs on the infected +neighbors of node i. If we neglect terms of order β2, we obtain PI = βai +� +j aj, which +means that the optimal social activity (Eq. (6)) is: +ai,t+1 = +1 +1 + αδβ � +j aj,t +, +(7) +where ai,t is the social activity of the node i at time t and the sum is on the infected +neighbors of the node i. +This approach, which we refer to as local awareness (Fig. +1c), requires detailed knowledge of the health state of each node and thus can only be +used in an individual-based approach in which contagion occurs on a quenched network +[33]. According to Eq. (7), individuals update their social activity depending on the +infection cost α, the discount factor δ, their local prevalence and the social activity of +their neighbors. +We can simplify the expression for PI if we adopt a degree-based mean-field approach +(annealed network) [34]. Within this formalism, all nodes with the same degree k (where +ki represents the number of contacts of node i) are considered statistically equivalent. +Under this assumption a susceptible node with degree k (labelled Sk) has a probability +of becoming infected (Ik) given by PIk = βkakθ, where ak is the social activity of nodes +with degree k (they will all make the same choice for social activity). +Furthermore, +we defined the “weighted density of infected neighbors" as θ = � +k′ ak′(k′ − 1)pk′ik′/⟨k⟩, +where the prevalence in the degree class k is weighted by their social activity ak. Here, +we indicate by pk the fraction of nodes with degree k, by ik (sk, rk) the fraction of nodes +7 + +with degree k that are in the I (S,R) health state, while ⟨k⟩ is the average degree of the +network, ⟨k⟩ = � +k kpk [35]. If ak = 1 for all k (no behavioral change), then θ becomes +the fraction of the neighbors of any node that are in the infected state (assuming no +degree correlations in the contact network). +This results in the following set of equations describing the heterogenous mean-field +(HMF) model: +sk,t+1 = sk,t − βkak,tsk,tθt +(8a) +ik,t+1 = ik,t + βkak,tsk,tθt − µik,t +(8b) +rk,t+1 = rk,t + µik,t +(8c) +ak,t+1 = +1 +1 + αδβkθt +(8d) +θt = +� +k′ +ak′,t(k′ − 1)pk′ik′,t +⟨k⟩ +, +(8e) +where we added the time dependency to the state prevalence sk,t, ik,t and rk,t, social +activity ak,t, and weighted density of infected neighbors θt. Note that each individual +can only act on their social activity, therefore θt should be considered constant during the +optimization performed at time t. We refer to this approach as global awareness because +social activity only depends on the epidemic conditions in the whole population. Eq. +(8d) is the equivalent of Eq. (7) for global awareness: at each time step t, individuals +with degree k choose their optimal social activity depending on the weighted density of +infected neighbors θt. Note that highly-connected individuals (large degree k) will adopt +a smaller social activity, other conditions being equal, than individuals with few social +interactions (small degree k). +Fig. +2 shows the optimal social activity as a function of the weighted density of +infected neighbors θ, for different choices of the parameter α. One can see that optimal +social activity always decreases as the weighted density of infected neighbors increases, +but its functional form depends on α: optimal social activity decreases slowly or more +abruptly when the average cost of the infection is small or large, respectively. In par- +ticular, when α is small, the optimal social activity decreases linearly with α. We also +tested the effect of the discount factor δ (not shown), which, in the range of values we +consider for δ (see next Section) is negligible. +Finally, one can assume a homogeneous mixing hypothesis, meaning that all nodes +are equal and one can approximate the degree of each node with the average degree of +the network ⟨k⟩. Under this assumption, all individuals adopt the same social activity at +and the weighted density of infected neighbors becomes θt = atit where it is the fraction +of infected individuals in the population (prevalence) at time t. The homogeneous mean- +8 + +Figure 2: Optimal value of the social activity at (Eq. 9d) as a function of the weighted +density of infected neighbors θt. We consider δ = 0.9, µ = 0.1, β = 0.017 and ⟨k⟩ = 17.4 +(i.e. β⟨k⟩ = 0.3). Different colors correspond to different choices of α. +field (MF) model is summarized by the following set of equations: +st+1 = st − β⟨k⟩a2 +t stit +(9a) +it+1 = it + β⟨k⟩a2 +t stit − µit +(9b) +rt+1 = rt + µit +(9c) +at+1 = +1 +1 + αδβ⟨k⟩θt +(9d) +θt = atit +(9e) +Again, Eq. +(9d) is the equivalent of Eqs. +(7) and (8d) in the homogeneous mixing +hypothesis: at each time step t, individuals choose their optimal social activity depending +only on the prevalence and other individuals’ social activity. +Therefore, we consider three different scenarios of awareness: i) In the homogeneous +mixing case, individuals know the fraction of infected individuals in the population, ii) in +the global (heterogeneous mean-field) case, individuals know how likely an individual is +to be infected based on their degree, and iii) in the local awareness (quenched network) +case, individuals know which of their contacts are infected on a per-individual basis. +2.4 +Model calibration +We consider a time step to be equal to one day. +We calibrate the MF model on a +disease having basic reproduction number R0 = 3 in absence of any mitigation measure, +implying that a single infection case is expected, on average, to generate three new cases +9 + +1.0 +α= 0 +0.8 +α=1 +α= 2 +α= 5 +0.6 +α=10 +at(θt) +α= 20 +0.4 +α=50 +α=100 +0.2 +0.0. +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +tin a population of fully susceptible individuals. We set µ = 1/10, thus assuming an +average infectious period equal to 10 days. +The choices of R0 and µ imply, for the +MF model, a value of the infection rate β = 0.3/⟨k⟩, where ⟨k⟩ represents the average +number of contacts per day. We fix ⟨k⟩ = 17.4 (see Methods for more information on +network generation), thus obtaining β = 0.017. The choice of the discount factor δ is +more challenging and will be addressed later in this Section, after discussing the two +main simplifying assumptions of our model in detail. Our choice for the epidemiological +parameters is compatible with estimates of R0 and µ for SARS-CoV-2 [36, 37, 38], and, +more in general, with a typical rapidly transmitted respiratory infection, such as influenza +[39]. +A first simplifying assumption is to consider behavior to be homogeneous across +all health states. +While this assumption may not be realistic, other choices are also +problematic. +For instance, one could assume that infected individuals would always +choose a maximum social activity a = 1, since they can no longer get infected. Our +assumption of considering that S and I individuals optimize in the same way their +behavior is compatible with either individuals not knowing their health state (a concrete +possibility in case of asymptomatic infections), or altruistic behavior. We also stress +that assuming at to be the same for all individuals does not necessarily mean that all +individuals behave in the same way. Finally, we remark that the behavior of R individuals +is irrelevant since they do not participate in active links. +Second, individuals believe that current conditions (i.e. the sizes of the S, I and R +compartments and the present value of at) will remain unchanged when planning. One +might argue that this assumption is only realistic in the near future, however, in our +model, the future utility gets exponentially discounted because of the term δ∆t in Eq. +(3). +So although predictions are based on conditions that might not be valid in the +distant future, they have less and less impact as they get further apart in time. On the +other hand, assuming that individuals predict the future prevalence of the disease (for +instance by means of the SIR model itself, [21]) can be equally unrealistic, since they +may not know, for instance, the transmission and recovery rates. +Moreover, it seems unlikely that individuals take extended periods of time into ac- +count when planning. Some models address this issue by fixing a finite planning horizon, +i.e. choosing the number of future days that are taken into account when planning [40]. +However, this choice has two drawbacks: i) it adds another parameter to the model (the +planning horizon), making it more difficult to fit with empirical data, and ii) it implies +that future expected utility abruptly drops to zero when the planning horizon is reached. +Instead, the discount factor δ ensures that future expected utility gradually decreases +over time. Also, we emphasize that, since we consider an infinite planning horizon, in +our model behavioral response is ultimately caused by discounting. In fact, if there was +no discounting, limiting social activity would delay the infection, but the cost of infection +would not change with time, so there would be no incentive to react to the spread of the +disease. +This assumption has thus an impact on the plausible values of the discount factor +δ, which we choose to be rather small with respect to previous modeling approaches +10 + +Figure 3: Equivalent social activity aeq corresponding to different values of α (dashed +line). The dashed line represents the solution of Eq. (10). We set δ = 0.9. The color +gradient is based on the values of aeq on the y-axis. +[19, 21, 32]. +Note that in our model the discount factor includes both the standard +economic discounting of the future and the expectations for a cure or a vaccine. +In +general, we set δ = 0.9, and test the robustness of the model in the range δ ∈ [0.8, 0.95], +observing no qualitative changes. With δ = 0.9, the utility one week from now weighs +about 48% of today’s utility, this becomes 23% after two weeks and 4% after a month. +This choice ensures that the planning horizon of individuals is actually finite so that they +do not speculate about the distant future. +Finally, we address the issue of interpretability of the α parameter, which currently +lacks a scale to be compared to (i.e. we have not yet determined which values of α should +be associated with minor illnesses and which with more serious ones). To this aim, we +define the equivalent social activity aeq as the value of the social activity for which the +utility lost in one day due to the reduction of social activity Ua(aeq) (given by Eq. (2)) +is equal to UI. To put it in other terms, aeq is the value of social activity for which there +is no preferable choice in terms of utility between getting infected and spending a period +corresponding to the average infectious period with social activity aeq. This means that, +with our choice of the utility function, aeq can be determined numerically as the root of +the equation log(at) − at + 1 + UI = 0. By using Eq. (5), we can express the previous +equation in terms of α, obtaining +log(at) − at + 1 + [1 − δ(1 − µ)] α = 0. +(10) +Fig. 3 shows the equivalent social activity corresponding to different values of α with +δ = 0.9. Intuitively, aeq decreases as the average infection cost increases: individuals +11 + +1.0 +0.8 +Strong +behavioural +0.6 +response +aeq +0.4 +Weak +behavioural +0.2 +response +0.0 +i0-3 +10-2 +10-1 +100 +101 +102 +103 +αare not willing to reduce their social activity if the infection cost is small. However, it is +interesting to note that the dependency on α is weak, due to the choice of the logarithmic +form of the utility function. Individuals are not willing to significantly decrease their +social activity for α ⪅ 10−2, regardless of the infection probability (prevalence). They +gradually decrease their social activity as a function of α, until they are willing to reduce +it to almost zero for α ⪆ 10. For these values of α, individuals consider becoming infected +as serious as having virtually no social activity. We stress that this does not necessarily +mean that individuals will choose to isolate (i.e., choose zero social activity) to avoid +infection since the probability of getting infected is usually small (PI ≪ 1). +Here, we are interested in the regime for which there is a strong behavioral response +(highlighted in red in Fig. 3), since behavior only becomes relevant when aeq ≪ 1. Also, +we observe that only for high values of α (about α ≃ 100 in the simulations presented in +the next section) behavioral response based on local awareness is strong enough to prevent +the epidemic from spreading. Therefore we will focus on the range 0.1 < α < 100. +2.5 +Numerical simulations +We will now show the results of numerical simulations of the model. We will consider the +following four settings for the substrate of the epidemic: homogeneous mean-field (MF), +heterogeneous mean-field (HMF) on a scale-free network, quenched random networks, +and quenched scale-free networks, see Methods for details of the numerical simulations. +The former two cases (annealed networks) represent global awareness, and the latter two +cases (quenched networks) stand for local awareness. +We quantify the outcome of the epidemic spread by two key quantities: the peak +prevalence imax, corresponding to the maximum number of infected individuals at the +same time, and the final attack rate r∞, which corresponds to the limit t → ∞ of the +fraction of recovered individual rt and represents the total fraction of the population +that has been infected by the disease. Notice that the peak prevalence can be related to +the maximum capacity of the health system at the peak of the epidemic, while the final +attack rate can be directly related to the number of deceased individuals. Since we will +consider the ratio between different settings (corresponding to different values of α), our +results can be directly interpreted in this sense. +The main effect of behavioral change is to flatten the epidemic curve, lowering the +peak prevalence and delaying the moment when the peak is reached. This effect is clearly +shown in Fig. 4(a), in which we plot the peak prevalence imax as a function of α. One +can see that with a stronger behavioral response (larger α) the peak prevalence decreases, +thus indicating that behavioral responses flatten the epidemic curve. We also note no +significant differences between global and local awareness. +In Fig. 4(b) we plot how the final attack rate r∞ changes with α, showing that for +both global and local awareness, r∞ decreases as the infection cost increases. However, +the effects of behavioral change on r∞ are much stronger for the quenched case (local +awareness) than the mean-field one (global awareness). The effect is particularly evident +in the regime of strong behavioral response (large α for which aeq ≪ 1)), where the +epidemic is almost suppressed when the awareness is local, while if individuals have only +12 + +Figure 4: Ratio between the peak prevalence imax (a) or the final attack rate r∞ (b) +obtained for a given value of α and that corresponding to α = 0 (no behavioral response). +We set δ = 0.9. Error bars are calculated as the standard error of the mean, not shown +as smaller than points in the plot. +13 + +1.0 +a +0.8 +imax(α) +0.6 +imax(0) +0.4 +0.2 +0.0 +1.0 +b +0.8 +r(α) +0.6 +r。(0) +0.4 +Global awareness (MF) +Global awareness (HMF) +0.2 +Local awareness (Random) +Local awareness (Scale-Free) +100 +101 +102 +αglobal awareness of the prevalence, the reduction in the final attack rate is relatively +small. +The stronger effect for local awareness can be attributed to the more fine-grained +behavioral response: individuals in direct contact with local outbreaks reduce their social +activity, hampering early disease propagation. Therefore, even when the prevalence is +low in the population and localized in a few individuals, the behavioral response of +the neighbors of the infected individuals is sufficient to curb the disease spreading. In +contrast, within the global setting, the prevalence has to grow enough in the population +(in a specific degree class in the HMF case) in order to trigger a behavioral response +from individuals. The possible presence of clusters (groups of highly-connected nodes) +in the quenched networks acts similarly: when the behavioral response is strong (large +α regime), the epidemic can not escape from the cluster where it started, but it dies out +after exhausting the reservoir of susceptible individuals inside the cluster. The difference +between local and global awareness instead disappears when the behavioral response is +weak (small α values). In this regime, indeed, it is necessary that a considerable fraction +of nodes is infected before triggering the behavioral response, so that many susceptible +individuals will likely be in contact with several infected ones, a situation similar to the +annealed network scenario. +3 +Discussion +Our study aimed at bridging classical epidemic modelling, in which the transmission rate +is modulated by some nonlinear function of the prevalence, and epi-economic models. +These two approaches have been reported to largely operate in isolation and even to dis- +play, to some extent, a lack of consensus [41, 42]. We adopt the epi-economic framework, +in which behavior is adopted by following a forward-looking approach, but we made some +simplifying assumptions that allow us to describe the optimal behavior (i.e., the optimal +degree of social activity) by analytical expressions. Crucially, we go beyond the homo- +geneous mixing hypothesis usually assumed in epi-economic models and test different +degrees of heterogeneity in the population: local and global (degree-based) awareness. +We show that, if individuals expect current conditions to be stationary when planning, +it is possible to find an analytical expression for the optimal social activity at (Eq. (6)), +that depends on the infection cost α, the discount factor δ, and infection probability +PI(t). We note that the functional form of at is not surprising. In fact, behavioral change +models based on prevalence-dependent transmission rates have been using transmission +rates of the form A/(1 + Bit) for decades [43, 44]. However, in our model, the social +activity does not depend solely on prevalence, but also on the current behavioral choices +of the population, so changes in the prevalence are more relevant when collective behavior +favors disease transmission (at ≃ 1) and become less and less relevant as behavior changes +limiting the probability of infection (at ≪ 1). +Moreover, we operationalize the infection probability in three different settings, local, +global (HMF case), and homogeneous behavioral responses. We provide a simple inter- +pretation for the infection cost α in terms of an equivalent social activity aeq, defined as +14 + +the acceptable social activity equivalent to the risk of infection. Such equivalence allows +us to distinguish regimes of weak and strong behavioral responses. Finally, we quantify +the effect of behavior change on the final attack rate of the disease in the regime of strong +behavioral response, showing that local awareness allows for a much stronger outbreak +reduction than global awareness. +Our model is not exempt from limitations. We discuss the validity of our two simplify- +ing assumptions (homogeneous behavior across all health states and individuals believing +current epidemic conditions will remain unchanged) in Section 2.4 (Model calibration). +Furthermore, we assume that individuals are immediately aware of the health status of +their peers (the global heterogeneous case or local prevalence). This is certainly unreal- +istic since some delay is to be expected between the moment an individual gets infected +and the moment other individuals learn about it. However, we do not expect this to be +particularly relevant for the mean-field versions of the model as that would just result +in a different value of the weighted density of infected θ. On the other hand, we believe +that in a quenched setting the introduction of a delay between the moment a node gets +infected and the moment its neighbors discover it would have a far stronger effect. In +fact, since behavior is determined on a per-individual basis for simulations on a quenched +network, nodes would not immediately limit their social activity when one of their neigh- +bors gets infected, potentially resulting in nodes whose local prevalence is still low not +reacting in time to the new infection even if global prevalence had already been high for +a while. While the effects of delayed information in an epi-economic model that assumes +homogeneous mixing have already been studied [45], future work could be devoted to +investigating the effects of delayed information in a quenched setting. +Finally, we do not calibrate our model with empirical data. During the COVID-19 +pandemic, a vast amount of empirical measurements of human behavior has become +accessible, especially through mobile phone data [46]. These have been often used to +incorporate human behavior into epidemic models in an effective manner, that is, by ret- +rospectively integrating the observed changes in behavior, for instance, the reduction in +movements, into disease dynamics [37, 47, 48]. However, epi-economic models would re- +quire rather different, and more granular, empirical measures of human behavior, aimed +at quantifying individual future expectations and their heterogeneity in a population. +Previous studies including game-theoretic behavioral changes have typically explored +different scenarios, relying on assumed values for the parameters that regulate the be- +havioral responses [24, 21]. However, empirical measurements for these parameters, such +as the expected cost of infection, remain scarce. +Our model depends only on two key parameters, the infection cost α and the discount +factor δ, which makes our model very parsimonious in terms of parameterization needs. +To this aim, thanks to the equivalence between infection cost α and social activity we +developed, the former could be estimated by means of surveys, by asking individuals how +many days they would accept to isolate in order to avoid the infection [32]. The discount +factor δ could be more difficult to estimate since it includes the expectations for a future +cure. +Previous studies have assumed standard values of δ but more work is needed +to understand how δ may vary across epidemic scenarios, depending on the severity of +15 + +the disease. +In general, we remark that data quality is crucial to assess parameters +of epidemiological models even in very simple settings, such as the basic reproduction +number R0 [49]. +In conclusion, our study provides the description of a simple, yet a realistic model of +forward-looking behavior that can be integrated into large-scale network epidemic models +[50], contributing an additional layer of realism to models used to inform policymakers. +4 +Methods +4.1 +Derivation of closed-form objective function +In this section, we will present the derivation of the closed-form expression of the objective +function J, defined in Eq. (3). The objective function is composed of two terms. The +term involving Ua depends neither on time (we assumed individuals to believe current +conditions to persist in the future) nor on the health state (we assumed social activity +to be homogeneous across all health states), so it is just a geometric series: +E +∞ +� +∆t=0 +δ∆tUa(t) = Ua(t) +1 − δ +(11) +Let’s now focus on the term involving UH. First, we will calculate the average cost of +infection, which we indicate by α. Considering an individual that has just transitioned +from the S state to the I state, for each day spent in the infected state the individual +gets a penalty of magnitude UI, as for Eq. (1). Each day they recover from the disease +with probability µ, meaning that the probability of being still infected after ∆t days is +(1 − µ)∆t. Since after ∆t days the utility gets discounted by a factor δ∆t, the average +total loss of utility caused by an infection is: +α ≡ UI +∞ +� +∆t=0 +[δ(1 − µ)]∆t = +UI +1 − δ(1 − µ). +(12) +The average cost of infection (here defined positive, to be subtracted in the expected +utility) is thus proportional to the penalty UI and it becomes smaller for increasing +discount factor δ and recovery rate µ. +Let’s now consider an S individual that is evaluating their expected utility loss be- +cause of the risk of infection at time t. If an S individual gets infected ∆t days from t, +the expected penalty (given by Eq. (12)) gets discounted by a factor δ∆t. The remaining +term to close the calculation is the probability of becoming infected after ∆t time steps. +Let PI(t) be the probability of becoming infected in one time step at time t (the time in +which the expected utility is evaluated), which is assumed to remain constant in the fu- +ture (since individuals believe the current prevalence, as well as other individuals’ social +activity, will persist). The probability of becoming infected after ∆t time steps is thus +equal to PI(t)(1 − PI(t))∆t−1 (individuals must not have already been infected for the +first ∆t − 1 time steps). Therefore, the expected utility loss due to the risk of infection +16 + +is equal to the average loss for infection α, multiplied by the probability it happens after +∆t time steps, and thus discounted by δ∆t: +E +∞ +� +∆t=0 +δ∆tUH(t) = −δPI(t) +∞ +� +∆t=1 +[δ(1 − PI(t))]∆t−1α. +(13) +By solving the geometric series and assuming that the probability of an S individual +getting infected at time t is small, PI(t) ≪ 1, one obtains +E +∞ +� +∆t=0 +δ∆tUH(t) = − αδ +1 − δPI(t). +(14) +Combining Eq. (11) and Eq. (14) into Eq. (3), the closed-form expression for J is +obtained: +J(t) = +1 +1 − δ [Ua(t) − αδPI(t)] . +(15) +4.2 +Details of numerical simulations +Here we provide details about how we carried numerical simulations out. Simulations on +quenched networks (both the random and the scale-free case) are performed by looping +over infected nodes and their neighbors and choosing whether transitions happen or not +based on the generation of a random variable. At each time step contagion occurs in +each connected pair of one S node and I node with probability β and each I individual +has a probability µ of recovery. Mean-field simulations are based on equations 8a - 8e +(HMF) and 9a - 9d (MF). +All simulations whose results are presented in Fig. 4 are performed with β⟨k⟩ = 0.3, +µ = 0.1, and initial conditions i0 = 0.01 and ai,0 = 1 for all nodes. For simulations on +quenched networks, results correspond to ensemble averages over 103 simulations (and +thus 103 quenched networks). All networks have N = 105 nodes. Scale-free networks +are generated using the configuration model [51] implemented in the NetworkX Python +package [52] with minimum degree kmin = 5, maximum degree kmax ≃ +√ +N = 316 and +exponent γ = 2.1. The resulting expected average degree is ⟨k⟩ = 17.4. The same value +of ⟨k⟩ is used for random networks too. +5 +Funding +This work did not receive any funding. +6 +Author contributions statement +L.A.N.F. developed the epidemic model, performed analytical and numerical experi- +ments, interpreted the results, and wrote the initial draft of the manuscript. M.S. and +17 + +M.T. conceived and designed the study, supervised the research, interpreted the re- +sults, and wrote the manuscript. All authors read and approved the final version of the +manuscript. +7 +Code availability +The code to reproduce the results of the manuscript is available at https://github. +com/lorenzoamir/EpiNetworkPaper +References +[1] Neil M. Ferguson. Capturing human behaviour. Nature, 446:733–733, 2007. +[2] Chansung Kim, Seung Hoon Cheon, Keechoo Choi, Chang-Hyeon Joh, and Hyuk- +Jin Lee. Exposure to fear: Changes in travel behavior during mers outbreak in seoul. +KSCE Journal of Civil Engineering, 21(7):2888–2895, 2017. +[3] Alfred W. Crosby. 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In Gaël Varoquaux, Travis Vaught, and +Jarrod Millman, editors, Proceedings of the 7th Python in Science Conference, pages +11 – 15, Pasadena, CA USA, 2008. +22 + diff --git a/VNE4T4oBgHgl3EQfMgxJ/content/tmp_files/load_file.txt b/VNE4T4oBgHgl3EQfMgxJ/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..748070c5afdd5ab435f8aaa0969b5b7c42d33f09 --- /dev/null +++ b/VNE4T4oBgHgl3EQfMgxJ/content/tmp_files/load_file.txt @@ -0,0 +1,606 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf,len=605 +page_content='Modeling adaptive forward-looking behavior in epidemics on networks Lorenzo Amir Nemati Fard1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' Michele Starnini2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content='3 ∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' and Michele Tizzoni4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content='† 1Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' University of Pisa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' Largo Bruno Pontecorvo 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' 56127,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' Pisa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' Italy 2Departament de Fisica,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' Universitat Politecnica de Catalunya,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' Campus Nord,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' 08034,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' Barcelona,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' Spain 3CENTAI Institute,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' Turin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' Italy 4Department of Sociology and Social Research,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' University of Trento,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' Trento,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' Italy ∗To whom correspondence should be addressed: michele.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content='starnini@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content='com, †To whom correspondence should be addressed: michele.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content='tizzoni@unitn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content='it Abstract The course of an epidemic can be drastically altered by changes in human behav- ior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' Incorporating the dynamics of individual decision-making during an outbreak represents a key challenge of epidemiology, faced by several modeling approaches siloed by different disciplines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' Here, we propose an epi-economic model including adaptive, forward-looking behavioral response on a heterogeneous networked sub- strate, where individuals tune their social activity based on future health expecta- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' Under basic assumptions, we show that it is possible to derive an analytical expression of the optimal value of the social activity that matches the traditional assumptions of classic epidemic models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' Through numerical simulations, we con- trast the settings of global awareness – individuals only know the prevalence of the disease in the population – with local awareness, where individuals explicitly know which of their contacts are infected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' We show that behavior change can flatten the epidemic curve by lowering the peak prevalence, but local awareness is much more effective in curbing the disease early with respect to global awareness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' Our work bridges classical epidemic modeling with the epi-economic approach, and sheds light on the effects of heterogeneous behavioral responses in curbing the epidemic spread.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' 1 Introduction Behavioral adaptation in response to infectious disease outbreaks is one of the key factors that shape the course of an epidemic [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' Individual choices regarding the adoption of self-protective measures, such as wearing masks, avoiding close social contacts, or vaccinating, contribute to reducing disease transmission in the population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' In turn, such choices depend on the perceived severity of the epidemic, the available information about 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content='04947v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content='soc-ph] 12 Jan 2023 it, and individual risk perception, thus they vary over time as the epidemic progresses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' As the epidemic fades out, behavioral responses may relax, with the consequence of leading to disease resurgence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' Self-initiated behavioral responses have been observed across all kinds of epidemics, from small-scale outbreaks, involving few individuals, such as the 2015 Middle East respiratory syndrome outbreak in South Korea [2], to worldwide pandemics, such as the 1918 pandemic [3], the 2009 A/H1N1 pandemic [4, 5, 6], as well as the current COVID-19 pandemic [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' Incorporating the dynamics of individual decision-making during an outbreak repre- sents a key challenge of epidemic modeling [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' To this aim, a wide variety of mathemat- ical epidemic models that capture the effects of behavioral changes have been proposed [9, 10, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' Generally speaking, these models can be classified into three broad classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' In the simplest case, classical compartmental models have been expanded to consider additional behavioral classes in the population, characterized by different behavioral re- sponses to the disease prevalence, whose transmission parameters do not change over time [12, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' The second class of models includes those that aim at capturing the interplay between individual adaptation and individual knowledge of the disease, often represented as two coupled dynamical processes [14, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' Information about the disease can be local or global, and sometimes it is assumed to spread through the population [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' Finally, a distinct class of models aims to explicitly describe the individual decision-making process using approaches of behavioral economics, where individuals evaluate their payoffs and adopt the behavior that optimally increases the payoff [17, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' These “epi-economic" models simulate the process by which people choose the best course of action by ad- justing to the current epidemic state and seeking the best possible future outcome via an optimization process [19, 20, 21, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' Such optimization processes typically rely on numerical methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' The COVID-19 pandemic reopened the debate on how human behavior should be included in epidemic models, with researchers with many different backgrounds con- tributing to the modeling effort, including for instance game-theoretical models [23, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' However, different approaches have often been siloed within the boundaries of the re- lated discipline and consequent limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' On the one hand, agent-based models with additional behavioral classes require a behavioral response to be defined a-priori.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' On the other hand, epi-economic models do consider an adaptive response but generally rely on the homogeneous mixing hypothesis, under which all susceptible individuals have an equal chance of contracting the infection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' The great degree of heterogeneity in human behavior [25, 26, 27, 28] makes it clear that different individuals have varying chances of contracting the disease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' As a consequence, heterogeneous behavioral responses in the population should arise also under the epi-economic approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' In this work, we try to fill this gap by proposing an epi-economic model including adaptive, forward-looking behavioral response on a heterogeneous networked substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' Individuals choose their social activity to maximize their future expected utility, by balancing the risk of being infected in the future while maintaining the highest possible social activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' We rely on some simplifying yet realistic assumptions that allow us to describe the optimal behavior by an analytical expression depending on the epidemic 2 conditions (prevalence and disease parameters) and the behavior of the population itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' We explicitly contrast the cases of global awareness, in which individuals have a bird-eye view of the epidemic unfolding on the whole population, with a local awareness setting, where individuals only know which of their contacts are infected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' The latter case triggers a highly heterogeneous behavioral response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' We show that behavior change can flatten the epidemic curve by lowering the peak prevalence, thus potentially reducing the load on the health system at the epidemic peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' However, local awareness is much more effective in curbing early the disease with respect to global awareness, thus shrinking the overall number of infections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' 2 Results 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content='1 A model of forward-looking adaptive behavior We propose an analytically tractable epidemic model including a feedback loop between the social activity of individuals and the spreading of the epidemic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' To this aim, we consider a SIR model in which individuals change their behavior, reducing or increasing their social activity, depending on the prevalence of the disease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' Crucially, susceptible individuals tune their social activity with a forward-looking approach, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=', they balance the risk of being infected in the future while maintaining the highest possible social activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' The base of our model will be the SIR model [29] in discrete time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' We con- sider a population of N individuals, each one belonging to one of three compartments: susceptible (S), infected (I), or recovered (R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' A S individual who is in contact with a I individual has a probability β of becoming infected in a day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' Each I individual has a probability µ of recovering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' R individuals can not be infected anymore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' Next, we specify how changing behavior affects the probability of infection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' Inspired by Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' [21], we define a time-dependent social activity at for individuals, representing their propensity of engaging in social interactions with peers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' Social activity is bounded 0 ≤ at ≤ 1, with a = 1 corresponding to normal behavior in absence of the disease, and a = 0 corresponding to a situation in which disease transmission is not possible, equiva- lent to quarantine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' We assume that disease transmission between individuals depends on their social activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' If a S individual with social activity aS t comes into contact at time t with an I individual with social activity aI t , we assume the probability of the S individ- ual getting infected to depend linearly on both aS t and aI t , so that the S individual gets infected with probability βaS t aI t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' We note that the social activity at can have multiple interpretations as long as they result in a decreased probability of disease transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' For instance, a lower social activity could represent less frequent contacts with other individuals or it could represent the adoption of prophylactic measures, such as the use of face masks in the case of airborne disease ([30]), that limit the probability of infection on contact without actually reducing the contact rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' Intuitively, individuals limit their social activity in order to reduce the probability to be infected: when the prevalence is high, they will adopt prudent behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' The feedback loop between the disease spread 3 and social activity is illustrated in Figure 1 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' Following the epi-economic approach, we model the dynamics of at as an optimization process in which S individuals balance the risk of infection and the benefits of social activity, which are represented by a utility function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' At each time step, each individual is assigned a score, named utility, based on their behavior and health status.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' Adopting a behavior that reduces the probability of infection (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' lower social activity) results in lower utility, but also reduces the probability of getting infected which would also result in a penalty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' At each time step, each individual deals with this trade-off by choosing the behavior that maximizes its future expected utility (objective function), see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' 1 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' A common strategy to approach this problem is dynamic programming, which can be solved via the Bellman Equation [19, 31, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' Here, we make a series of simplifying assumptions that allow us to find an analytical expression for the optimal value of the social activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' Our first simplifying assumption is that all individuals optimize social activity in the same way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' This is equivalent to assuming that individuals ignore their health status.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' We stress that this does not necessarily mean that all individuals behave in the same way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' For instance, prophylactic measures that are taken by I individuals regardless of the current state of the epidemic (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' behavior that is not adaptive), such as always wearing a face mask when in public or reducing their contact by a fixed amount by not going to work, can be modeled by changing the value of β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' Also, the behavior of R individuals does not affect the dynamics of the disease in our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' Second, we assume individuals to believe that current conditions (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' the sizes of the S, I, and R compartments, as well as their social activity at and that of the rest of the population) will remain unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' The validity of these assumptions is discussed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content='4 (Model calibration).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content='2 Utility function We assume the utility function to be the sum of two terms Ua and UH: the first one depends only on social activity and the second one depends only on the health status H = {S, I, R}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' We choose the state-dependent term UH of the utility function to corre- spond to a fixed penalty of magnitude UI for each time step in the infected state and to be UH = 0 otherwise (H ̸= I): UH(t) = � −UI (if H = I at time t) 0 (otherwise).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' (1) To determine the functional form of the utility function related to social activity, Ua, we use the following argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' On the one hand, decreasing social activity should require a higher cost in terms of utility when social activity is already low, so we assume the derivative of Ua with respect to a to be inversely proportional to a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' On the other hand, increasing social activity also comes with a cost, otherwise, there would be no finite optimal value of social activity (which we assumed to be a = 1) when there is no risk of getting infected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' The simplest hypothesis that we can make about this cost is that it does not depend on social activity, so we add a constant term to the derivative of Ua 4 Figure 1: Schematic illustration of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' (a) Feedback loop involving epidemic spreading and social activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' If the prevalence is high, individuals can decrease their social activity, which in turn reduces the transmissibility of the disease, decreasing preva- lence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' With low prevalence, individuals can choose high levels of social activity, increasing the prevalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' (b) Individuals choose their social activity to maximize their future ex- pected utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' They can choose a low value of social activity to delay the expected infection (and the expected penalty UI), or benefit from high social activity at a higher infection risk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' Awareness and behavioral response can be of two kinds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' Global awareness (c): individuals know the prevalence of the disease in the population but do not know who is infected, so social activity is homogeneous across the population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' Local awareness (d): individuals know which of their contacts are infected, resulting in different values of social activity for each individual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' 5 a Disease-behaviour feedback high high disease social prevalence activity reduced transmissibility low low b Forward-looking UI UI choose low S S social activity UI choose high S social activity t t+1 t+2 t+3 t+4 t+5 Global awareness c d Local awareness disease prevalence low social activity O social activity high social activitywith respect to a, resulting in dUa da = A a − B, which has solution Ua = A log(a) − Ba + C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' Requiring that a = 1 is a maximum of Ua results in A = B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' Since we are only interested in the position of the maximum and not in the actual value of Ua, we can fix A = B = C = 1 without loss of generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' From which follows that at time t individuals with social activity at get a utility score equal to: Ua(t) = log(at) − at + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' (2) We note that this choice of a concave form for the utility function is common in the economic literature, as well as in epi-economic models [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' The optimal value of social activity at each time t is then given by the maximum with respect to at of the following objective function: J(t) = E ∞ � ∆t=0 δ∆t [Ua(t + ∆t) + UH(t + ∆t)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' (3) The objective function in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' (3) represents the expected value, over all possible future health states, starting from the S state at time t, of the utility function, in which utility corresponding to ∆t days into the future gets discounted by a factor δ∆t, where δ is the discount factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' The discount factor 0 < δ < 1 implies that behavior resulting in high values of utility in the near future is preferred to behavior that pays off in the distant future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' It also ensures that the series in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' (3) converges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' Under our assumptions, it is possible to express the objective function at time t, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' (3), associated with a choice of social activity at in a closed-form, see Methods: J(t) = 1 1 − δ [Ua(t) − αδPI(t)] , (4) where the first term represents the benefits of social activity and the second term repre- sents the risk of getting infected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' The strength of the behavioral response to the infection risk is determined by the probability of becoming infected at time t, PI(t), the discount factor δ, and the average utility loss caused by an infection, α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' The larger the average infection cost α, the infection probability PI, or the discount factor δ, the stronger the behavioral response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' The probability of infection depends on the choice of the underlying network substrate, and it will be discussed in the next Section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' The parameter α instead can be calculated (see Methods) as α = UI 1 − δ(1 − µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' (5) The average cost of infection is proportional to the penalty UI and it becomes smaller for increasing discount factor δ (the duration of the infection is discounted by δ) and the recovery rate µ (the larger µ, the shorter the duration of the infection).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' The optimal value of the social activity can be determined as the maximum with re- spect to a of the objective function in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' In particular, if the behavioral component 6 of the utility function is given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' (2), then the optimal value of social activity at time t is: a∗ t = 1 1 + αδ ∂PI(t) ∂a∗ t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' (6) Therefore, individuals will adopt the optimal value of social activity a∗ t at each time step t, depending on the constant parameters α and δ, and on the probability of becoming infected at time t, PI(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' Note that the infection probability does depend on the un- derlying network’s structure and on the optimal social activity chosen by the involved individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' From now on, we will always refer to the optimal social activity, therefore, for the sake of simplicity, we will denote it as at in the remainder of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content='3 Local vs global awareness Next, we specify the substrate over which the disease spreads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' We assume that disease transmission is mediated through social interactions, represented by a contact network where nodes represent individuals and links represent contacts between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' We assume that each active link connecting a S node (labelled i) to a I neighbor (labelled j) can carry the disease in the unit time with probability βaiaj, where ai and aj are the social activities of the two nodes (we omit the time dependence here).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' By assuming independent infection processes for each active link, the probability of the S node being infected by any of its I neighbors is PI = 1 − � j(1 − βaiaj), where the product runs on the infected neighbors of node i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' If we neglect terms of order β2, we obtain PI = βai � j aj, which means that the optimal social activity (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' (6)) is: ai,t+1 = 1 1 + αδβ � j aj,t , (7) where ai,t is the social activity of the node i at time t and the sum is on the infected neighbors of the node i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' This approach, which we refer to as local awareness (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' 1c), requires detailed knowledge of the health state of each node and thus can only be used in an individual-based approach in which contagion occurs on a quenched network [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' According to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' (7), individuals update their social activity depending on the infection cost α, the discount factor δ, their local prevalence and the social activity of their neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' We can simplify the expression for PI if we adopt a degree-based mean-field approach (annealed network) [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' Within this formalism, all nodes with the same degree k (where ki represents the number of contacts of node i) are considered statistically equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' Under this assumption a susceptible node with degree k (labelled Sk) has a probability of becoming infected (Ik) given by PIk = βkakθ, where ak is the social activity of nodes with degree k (they will all make the same choice for social activity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' Furthermore, we defined the “weighted density of infected neighbors" as θ = � k′ ak′(k′ − 1)pk′ik′/⟨k⟩, where the prevalence in the degree class k is weighted by their social activity ak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' Here, we indicate by pk the fraction of nodes with degree k, by ik (sk, rk) the fraction of nodes 7 with degree k that are in the I (S,R) health state, while ⟨k⟩ is the average degree of the network, ⟨k⟩ = � k kpk [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' If ak = 1 for all k (no behavioral change), then θ becomes the fraction of the neighbors of any node that are in the infected state (assuming no degree correlations in the contact network).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' This results in the following set of equations describing the heterogenous mean-field (HMF) model: sk,t+1 = sk,t − βkak,tsk,tθt (8a) ik,t+1 = ik,t + βkak,tsk,tθt − µik,t (8b) rk,t+1 = rk,t + µik,t (8c) ak,t+1 = 1 1 + αδβkθt (8d) θt = � k′ ak′,t(k′ − 1)pk′ik′,t ⟨k⟩ , (8e) where we added the time dependency to the state prevalence sk,t, ik,t and rk,t, social activity ak,t, and weighted density of infected neighbors θt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' Note that each individual can only act on their social activity, therefore θt should be considered constant during the optimization performed at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' We refer to this approach as global awareness because social activity only depends on the epidemic conditions in the whole population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' (8d) is the equivalent of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' (7) for global awareness: at each time step t, individuals with degree k choose their optimal social activity depending on the weighted density of infected neighbors θt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' Note that highly-connected individuals (large degree k) will adopt a smaller social activity, other conditions being equal, than individuals with few social interactions (small degree k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' 2 shows the optimal social activity as a function of the weighted density of infected neighbors θ, for different choices of the parameter α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' One can see that optimal social activity always decreases as the weighted density of infected neighbors increases, but its functional form depends on α: optimal social activity decreases slowly or more abruptly when the average cost of the infection is small or large, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' In par- ticular, when α is small, the optimal social activity decreases linearly with α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' We also tested the effect of the discount factor δ (not shown), which, in the range of values we consider for δ (see next Section) is negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' Finally, one can assume a homogeneous mixing hypothesis, meaning that all nodes are equal and one can approximate the degree of each node with the average degree of the network ⟨k⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' Under this assumption, all individuals adopt the same social activity at and the weighted density of infected neighbors becomes θt = atit where it is the fraction of infected individuals in the population (prevalence) at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' The homogeneous mean- 8 Figure 2: Optimal value of the social activity at (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' 9d) as a function of the weighted density of infected neighbors θt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' We consider δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content='9, µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content='1, β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content='017 and ⟨k⟩ = 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content='4 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' β⟨k⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' Different colors correspond to different choices of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' field (MF) model is summarized by the following set of equations: st+1 = st − β⟨k⟩a2 t stit (9a) it+1 = it + β⟨k⟩a2 t stit − µit (9b) rt+1 = rt + µit (9c) at+1 = 1 1 + αδβ⟨k⟩θt (9d) θt = atit (9e) Again, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' (9d) is the equivalent of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' (7) and (8d) in the homogeneous mixing hypothesis: at each time step t, individuals choose their optimal social activity depending only on the prevalence and other individuals’ social activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' Therefore, we consider three different scenarios of awareness: i) In the homogeneous mixing case, individuals know the fraction of infected individuals in the population, ii) in the global (heterogeneous mean-field) case, individuals know how likely an individual is to be infected based on their degree, and iii) in the local awareness (quenched network) case, individuals know which of their contacts are infected on a per-individual basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content='4 Model calibration We consider a time step to be equal to one day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' We calibrate the MF model on a disease having basic reproduction number R0 = 3 in absence of any mitigation measure, implying that a single infection case is expected, on average, to generate three new cases 9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content='0 α= 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content='8 α=1 α= 2 α= 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content='6 α=10 at(θt) α= 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content='4 α=50 α=100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content='0 tin a population of fully susceptible individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' We set µ = 1/10, thus assuming an average infectious period equal to 10 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' The choices of R0 and µ imply, for the MF model, a value of the infection rate β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content='3/⟨k⟩, where ⟨k⟩ represents the average number of contacts per day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' We fix ⟨k⟩ = 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content='4 (see Methods for more information on network generation), thus obtaining β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content='017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' The choice of the discount factor δ is more challenging and will be addressed later in this Section, after discussing the two main simplifying assumptions of our model in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' Our choice for the epidemiological parameters is compatible with estimates of R0 and µ for SARS-CoV-2 [36, 37, 38], and, more in general, with a typical rapidly transmitted respiratory infection, such as influenza [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' A first simplifying assumption is to consider behavior to be homogeneous across all health states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' While this assumption may not be realistic, other choices are also problematic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' For instance, one could assume that infected individuals would always choose a maximum social activity a = 1, since they can no longer get infected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' Our assumption of considering that S and I individuals optimize in the same way their behavior is compatible with either individuals not knowing their health state (a concrete possibility in case of asymptomatic infections), or altruistic behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' We also stress that assuming at to be the same for all individuals does not necessarily mean that all individuals behave in the same way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' Finally, we remark that the behavior of R individuals is irrelevant since they do not participate in active links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' Second, individuals believe that current conditions (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' the sizes of the S, I and R compartments and the present value of at) will remain unchanged when planning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' One might argue that this assumption is only realistic in the near future, however, in our model, the future utility gets exponentially discounted because of the term δ∆t in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' So although predictions are based on conditions that might not be valid in the distant future, they have less and less impact as they get further apart in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' On the other hand, assuming that individuals predict the future prevalence of the disease (for instance by means of the SIR model itself, [21]) can be equally unrealistic, since they may not know, for instance, the transmission and recovery rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' Moreover, it seems unlikely that individuals take extended periods of time into ac- count when planning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' Some models address this issue by fixing a finite planning horizon, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' choosing the number of future days that are taken into account when planning [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' However, this choice has two drawbacks: i) it adds another parameter to the model (the planning horizon), making it more difficult to fit with empirical data, and ii) it implies that future expected utility abruptly drops to zero when the planning horizon is reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' Instead, the discount factor δ ensures that future expected utility gradually decreases over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' Also, we emphasize that, since we consider an infinite planning horizon, in our model behavioral response is ultimately caused by discounting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' In fact, if there was no discounting, limiting social activity would delay the infection, but the cost of infection would not change with time, so there would be no incentive to react to the spread of the disease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' This assumption has thus an impact on the plausible values of the discount factor δ, which we choose to be rather small with respect to previous modeling approaches 10 Figure 3: Equivalent social activity aeq corresponding to different values of α (dashed line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' The dashed line represents the solution of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' We set δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' The color gradient is based on the values of aeq on the y-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' [19, 21, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' Note that in our model the discount factor includes both the standard economic discounting of the future and the expectations for a cure or a vaccine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' In general, we set δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content='9, and test the robustness of the model in the range δ ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content='8, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content='95], observing no qualitative changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' With δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content='9, the utility one week from now weighs about 48% of today’s utility, this becomes 23% after two weeks and 4% after a month.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' This choice ensures that the planning horizon of individuals is actually finite so that they do not speculate about the distant future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' Finally, we address the issue of interpretability of the α parameter, which currently lacks a scale to be compared to (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' we have not yet determined which values of α should be associated with minor illnesses and which with more serious ones).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' To this aim, we define the equivalent social activity aeq as the value of the social activity for which the utility lost in one day due to the reduction of social activity Ua(aeq) (given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' (2)) is equal to UI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' To put it in other terms, aeq is the value of social activity for which there is no preferable choice in terms of utility between getting infected and spending a period corresponding to the average infectious period with social activity aeq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' This means that, with our choice of the utility function, aeq can be determined numerically as the root of the equation log(at) − at + 1 + UI = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' By using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' (5), we can express the previous equation in terms of α, obtaining log(at) − at + 1 + [1 − δ(1 − µ)] α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' (10) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' 3 shows the equivalent social activity corresponding to different values of α with δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' Intuitively, aeq decreases as the average infection cost increases: individuals 11 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content='8 Strong behavioural 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content='6 response aeq 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content='4 Weak behavioural 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content='2 response 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content='0 i0-3 10-2 10-1 100 101 102 103 αare not willing to reduce their social activity if the infection cost is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' However, it is interesting to note that the dependency on α is weak, due to the choice of the logarithmic form of the utility function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' Individuals are not willing to significantly decrease their social activity for α ⪅ 10−2, regardless of the infection probability (prevalence).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' They gradually decrease their social activity as a function of α, until they are willing to reduce it to almost zero for α ⪆ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' For these values of α, individuals consider becoming infected as serious as having virtually no social activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' We stress that this does not necessarily mean that individuals will choose to isolate (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=', choose zero social activity) to avoid infection since the probability of getting infected is usually small (PI ≪ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' Here, we are interested in the regime for which there is a strong behavioral response (highlighted in red in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' 3), since behavior only becomes relevant when aeq ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' Also, we observe that only for high values of α (about α ≃ 100 in the simulations presented in the next section) behavioral response based on local awareness is strong enough to prevent the epidemic from spreading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' Therefore we will focus on the range 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content='1 < α < 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content='5 Numerical simulations We will now show the results of numerical simulations of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' We will consider the following four settings for the substrate of the epidemic: homogeneous mean-field (MF), heterogeneous mean-field (HMF) on a scale-free network, quenched random networks, and quenched scale-free networks, see Methods for details of the numerical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' The former two cases (annealed networks) represent global awareness, and the latter two cases (quenched networks) stand for local awareness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' We quantify the outcome of the epidemic spread by two key quantities: the peak prevalence imax, corresponding to the maximum number of infected individuals at the same time, and the final attack rate r∞, which corresponds to the limit t → ∞ of the fraction of recovered individual rt and represents the total fraction of the population that has been infected by the disease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' Notice that the peak prevalence can be related to the maximum capacity of the health system at the peak of the epidemic, while the final attack rate can be directly related to the number of deceased individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' Since we will consider the ratio between different settings (corresponding to different values of α), our results can be directly interpreted in this sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' The main effect of behavioral change is to flatten the epidemic curve, lowering the peak prevalence and delaying the moment when the peak is reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' This effect is clearly shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' 4(a), in which we plot the peak prevalence imax as a function of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' One can see that with a stronger behavioral response (larger α) the peak prevalence decreases, thus indicating that behavioral responses flatten the epidemic curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' We also note no significant differences between global and local awareness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' 4(b) we plot how the final attack rate r∞ changes with α, showing that for both global and local awareness, r∞ decreases as the infection cost increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' However, the effects of behavioral change on r∞ are much stronger for the quenched case (local awareness) than the mean-field one (global awareness).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' The effect is particularly evident in the regime of strong behavioral response (large α for which aeq ≪ 1)), where the epidemic is almost suppressed when the awareness is local, while if individuals have only 12 Figure 4: Ratio between the peak prevalence imax (a) or the final attack rate r∞ (b) obtained for a given value of α and that corresponding to α = 0 (no behavioral response).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' We set δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' Error bars are calculated as the standard error of the mean, not shown as smaller than points in the plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' 13 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content='0 a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content='8 imax(α) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content='6 imax(0) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content='0 b 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content='8 r(α) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content='6 r。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content='(0) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content='4 Global awareness (MF) Global awareness (HMF) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content='2 Local awareness (Random) Local awareness (Scale-Free) 100 101 102 αglobal awareness of the prevalence, the reduction in the final attack rate is relatively small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' The stronger effect for local awareness can be attributed to the more fine-grained behavioral response: individuals in direct contact with local outbreaks reduce their social activity, hampering early disease propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' Therefore, even when the prevalence is low in the population and localized in a few individuals, the behavioral response of the neighbors of the infected individuals is sufficient to curb the disease spreading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' In contrast, within the global setting, the prevalence has to grow enough in the population (in a specific degree class in the HMF case) in order to trigger a behavioral response from individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' The possible presence of clusters (groups of highly-connected nodes) in the quenched networks acts similarly: when the behavioral response is strong (large α regime), the epidemic can not escape from the cluster where it started, but it dies out after exhausting the reservoir of susceptible individuals inside the cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' The difference between local and global awareness instead disappears when the behavioral response is weak (small α values).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' In this regime, indeed, it is necessary that a considerable fraction of nodes is infected before triggering the behavioral response, so that many susceptible individuals will likely be in contact with several infected ones, a situation similar to the annealed network scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' 3 Discussion Our study aimed at bridging classical epidemic modelling, in which the transmission rate is modulated by some nonlinear function of the prevalence, and epi-economic models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' These two approaches have been reported to largely operate in isolation and even to dis- play, to some extent, a lack of consensus [41, 42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' We adopt the epi-economic framework, in which behavior is adopted by following a forward-looking approach, but we made some simplifying assumptions that allow us to describe the optimal behavior (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=', the optimal degree of social activity) by analytical expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' Crucially, we go beyond the homo- geneous mixing hypothesis usually assumed in epi-economic models and test different degrees of heterogeneity in the population: local and global (degree-based) awareness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' We show that, if individuals expect current conditions to be stationary when planning, it is possible to find an analytical expression for the optimal social activity at (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' (6)), that depends on the infection cost α, the discount factor δ, and infection probability PI(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' We note that the functional form of at is not surprising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' In fact, behavioral change models based on prevalence-dependent transmission rates have been using transmission rates of the form A/(1 + Bit) for decades [43, 44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' However, in our model, the social activity does not depend solely on prevalence, but also on the current behavioral choices of the population, so changes in the prevalence are more relevant when collective behavior favors disease transmission (at ≃ 1) and become less and less relevant as behavior changes limiting the probability of infection (at ≪ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' Moreover, we operationalize the infection probability in three different settings, local, global (HMF case), and homogeneous behavioral responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' We provide a simple inter- pretation for the infection cost α in terms of an equivalent social activity aeq, defined as 14 the acceptable social activity equivalent to the risk of infection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' Such equivalence allows us to distinguish regimes of weak and strong behavioral responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' Finally, we quantify the effect of behavior change on the final attack rate of the disease in the regime of strong behavioral response, showing that local awareness allows for a much stronger outbreak reduction than global awareness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' Our model is not exempt from limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' We discuss the validity of our two simplify- ing assumptions (homogeneous behavior across all health states and individuals believing current epidemic conditions will remain unchanged) in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content='4 (Model calibration).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' Furthermore, we assume that individuals are immediately aware of the health status of their peers (the global heterogeneous case or local prevalence).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' This is certainly unreal- istic since some delay is to be expected between the moment an individual gets infected and the moment other individuals learn about it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' However, we do not expect this to be particularly relevant for the mean-field versions of the model as that would just result in a different value of the weighted density of infected θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' On the other hand, we believe that in a quenched setting the introduction of a delay between the moment a node gets infected and the moment its neighbors discover it would have a far stronger effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' In fact, since behavior is determined on a per-individual basis for simulations on a quenched network, nodes would not immediately limit their social activity when one of their neigh- bors gets infected, potentially resulting in nodes whose local prevalence is still low not reacting in time to the new infection even if global prevalence had already been high for a while.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' While the effects of delayed information in an epi-economic model that assumes homogeneous mixing have already been studied [45], future work could be devoted to investigating the effects of delayed information in a quenched setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' Finally, we do not calibrate our model with empirical data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' During the COVID-19 pandemic, a vast amount of empirical measurements of human behavior has become accessible, especially through mobile phone data [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' These have been often used to incorporate human behavior into epidemic models in an effective manner, that is, by ret- rospectively integrating the observed changes in behavior, for instance, the reduction in movements, into disease dynamics [37, 47, 48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' However, epi-economic models would re- quire rather different, and more granular, empirical measures of human behavior, aimed at quantifying individual future expectations and their heterogeneity in a population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' Previous studies including game-theoretic behavioral changes have typically explored different scenarios, relying on assumed values for the parameters that regulate the be- havioral responses [24, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' However, empirical measurements for these parameters, such as the expected cost of infection, remain scarce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' Our model depends only on two key parameters, the infection cost α and the discount factor δ, which makes our model very parsimonious in terms of parameterization needs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' To this aim, thanks to the equivalence between infection cost α and social activity we developed, the former could be estimated by means of surveys, by asking individuals how many days they would accept to isolate in order to avoid the infection [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' The discount factor δ could be more difficult to estimate since it includes the expectations for a future cure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' Previous studies have assumed standard values of δ but more work is needed to understand how δ may vary across epidemic scenarios, depending on the severity of 15 the disease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' In general, we remark that data quality is crucial to assess parameters of epidemiological models even in very simple settings, such as the basic reproduction number R0 [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' In conclusion, our study provides the description of a simple, yet a realistic model of forward-looking behavior that can be integrated into large-scale network epidemic models [50], contributing an additional layer of realism to models used to inform policymakers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' 4 Methods 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content='1 Derivation of closed-form objective function In this section, we will present the derivation of the closed-form expression of the objective function J, defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' The objective function is composed of two terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' The term involving Ua depends neither on time (we assumed individuals to believe current conditions to persist in the future) nor on the health state (we assumed social activity to be homogeneous across all health states), so it is just a geometric series: E ∞ � ∆t=0 δ∆tUa(t) = Ua(t) 1 − δ (11) Let’s now focus on the term involving UH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' First, we will calculate the average cost of infection, which we indicate by α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' Considering an individual that has just transitioned from the S state to the I state, for each day spent in the infected state the individual gets a penalty of magnitude UI, as for Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' Each day they recover from the disease with probability µ, meaning that the probability of being still infected after ∆t days is (1 − µ)∆t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' Since after ∆t days the utility gets discounted by a factor δ∆t, the average total loss of utility caused by an infection is: α ≡ UI ∞ � ∆t=0 [δ(1 − µ)]∆t = UI 1 − δ(1 − µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' (12) The average cost of infection (here defined positive, to be subtracted in the expected utility) is thus proportional to the penalty UI and it becomes smaller for increasing discount factor δ and recovery rate µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' Let’s now consider an S individual that is evaluating their expected utility loss be- cause of the risk of infection at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' If an S individual gets infected ∆t days from t, the expected penalty (given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' (12)) gets discounted by a factor δ∆t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' The remaining term to close the calculation is the probability of becoming infected after ∆t time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' Let PI(t) be the probability of becoming infected in one time step at time t (the time in which the expected utility is evaluated), which is assumed to remain constant in the fu- ture (since individuals believe the current prevalence, as well as other individuals’ social activity, will persist).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' The probability of becoming infected after ∆t time steps is thus equal to PI(t)(1 − PI(t))∆t−1 (individuals must not have already been infected for the first ∆t − 1 time steps).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' Therefore, the expected utility loss due to the risk of infection 16 is equal to the average loss for infection α, multiplied by the probability it happens after ∆t time steps, and thus discounted by δ∆t: E ∞ � ∆t=0 δ∆tUH(t) = −δPI(t) ∞ � ∆t=1 [δ(1 − PI(t))]∆t−1α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' (13) By solving the geometric series and assuming that the probability of an S individual getting infected at time t is small, PI(t) ≪ 1, one obtains E ∞ � ∆t=0 δ∆tUH(t) = − αδ 1 − δPI(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' (14) Combining Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' (11) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' (14) into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' (3), the closed-form expression for J is obtained: J(t) = 1 1 − δ [Ua(t) − αδPI(t)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' (15) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content='2 Details of numerical simulations Here we provide details about how we carried numerical simulations out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' Simulations on quenched networks (both the random and the scale-free case) are performed by looping over infected nodes and their neighbors and choosing whether transitions happen or not based on the generation of a random variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' At each time step contagion occurs in each connected pair of one S node and I node with probability β and each I individual has a probability µ of recovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' Mean-field simulations are based on equations 8a - 8e (HMF) and 9a - 9d (MF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' All simulations whose results are presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' 4 are performed with β⟨k⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content='3, µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content='1, and initial conditions i0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content='01 and ai,0 = 1 for all nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' For simulations on quenched networks, results correspond to ensemble averages over 103 simulations (and thus 103 quenched networks).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' All networks have N = 105 nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' Scale-free networks are generated using the configuration model [51] implemented in the NetworkX Python package [52] with minimum degree kmin = 5, maximum degree kmax ≃ √ N = 316 and exponent γ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' The resulting expected average degree is ⟨k⟩ = 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' The same value of ⟨k⟩ is used for random networks too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' 5 Funding This work did not receive any funding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' 6 Author contributions statement L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' developed the epidemic model, performed analytical and numerical experi- ments, interpreted the results, and wrote the initial draft of the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' and 17 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' conceived and designed the study, supervised the research, interpreted the re- sults, and wrote the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' All authors read and approved the final version of the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' 7 Code availability The 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' In Gaël Varoquaux, Travis Vaught, and Jarrod Millman, editors, Proceedings of the 7th Python in Science Conference, pages 11 – 15, Pasadena, CA USA, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} +page_content=' 22' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE4T4oBgHgl3EQfMgxJ/content/2301.04947v1.pdf'} diff --git a/VdFKT4oBgHgl3EQfmi5y/vector_store/index.faiss b/VdFKT4oBgHgl3EQfmi5y/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..d93c239d42062eaa661476c284e3424982a6051e --- /dev/null +++ b/VdFKT4oBgHgl3EQfmi5y/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:75ad5c7c800911af7aa7ea86726a7c43b03c107db1342c242ca64d6f1eb6e62c +size 2424877 diff --git 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adaptation methods reduce domain shift +typically by learning domain-invariant features. Most existing +methods are built on distribution matching, e.g., adversarial +domain adaptation, which tends to corrupt feature discriminabil- +ity. In this paper, we propose Discriminative Radial Domain +Adaptation (DRDR) which bridges source and target domains via +a shared radial structure. It’s motivated by the observation that +as the model is trained to be progressively discriminative, features +of different categories expand outwards in different directions, +forming a radial structure. We show that transferring such an in- +herently discriminative structure would enable to enhance feature +transferability and discriminability simultaneously. Specifically, +we represent each domain with a global anchor and each category +a local anchor to form a radial structure and reduce domain shift +via structure matching. It consists of two parts, namely isometric +transformation to align the structure globally and local refine- +ment to match each category. To enhance the discriminability +of the structure, we further encourage samples to cluster close +to the corresponding local anchors based on optimal-transport +assignment. Extensively experimenting on multiple benchmarks, +our method is shown to consistently outperforms state-of-the-art +approaches on varied tasks, including the typical unsupervised +domain adaptation, multi-source domain adaptation, domain- +agnostic learning, and domain generalization. +Index Terms—Domain Adaptation, Transfer Learning, Radial +Structure Matching +I. INTRODUCTION +Machine learning methods generally assume that training +and test data come from the same data distribution. However, +such an assumption may not hold in practice, since a model +trained on one distribution or one domain may need to be +applied to data from another distribution or domain. Typically, +such distribution shifts or domain shifts would cause signifi- +cant performance drop [1], [2]. To address this issue, domain +adaptation methods are proposed, which aim to generalize the +learned knowledge from source domain to target domains. +Zenan Huang is with the Qiushi Academy for Advanced Studies, Zhejiang +University, Hangzhou, Zhejiang 310007, China and also with the College of +Computer Science and Technology, Zhejiang University, Hangzhou, Zhejiang +310007, China. (E-mail: lccurious@zju.edu.cn). +Jun Wen was with the Qiushi Academy for Advanced Studies, Zhejiang +University, Hangzhou, Zhejiang 310007, China and also with the College of +Computer Science and Technology, Zhejiang University, Hangzhou, Zhejiang +310007, China. (E-mail: jungel2star@gmail.com). +Siheng +Chen +is +with +the +Shanghai +Jiao +Tong +University +and +also +with +Shanghai +AI +Laboratory, +Shanghai, +200240, +China. +(E- +mail:sihengc@sjut.edu.cn). +Linchao Zhu is with College of Computer Science and Technology, Zhe- +jiang University, Hangzhou, China. (E-mail: zhulinchao@zju.edu.cn). +Nenggan Zheng is with the Qiushi Academy for Advanced Studies, Zhe- +jiang University, Hangzhou, Zhejiang, 310007, China, also with Collaborative +Innovation Center for Artificial Intelligence by MOE and Zhejiang Provincial +Government (ZJU) and Zhejiang Lab, Hangzhou, Zhejiang, 311121, China +(E-mail: zng@cs.zju.edu.cn). +Nenggan Zheng is the corresponding author. +Feature Space +Unaligned features +Aligned features +source category: 1 +source category: 2 +source category: 3 +source category: 4 +target category: 1 +target category: 2 +target category: 3 +target category: 4 +Radial Structure +Radial structure alignment +Fig. 1. +Illustration of the proposed method which represents each domain +using a radial structure and reduce domain shift via structure matching (source: +red; target: blue; best viewed in color). +Domain adaptation methods reduce domain shift typically +by learning domain-invariant representations [2]. Previously, +shallow features from both the source and target domains are +mapped into a shared subspace [3]. With the success of deep +learning, domain-invariant features are learned using deep +neural networks [4], upon which various domain discrepancy +measures are proposed, e.g., Maximum Mean Discrepancy +(MMD) [5], second-order correlations [6], and moments [7]. +Recently, adversarial domain adaptation methods [8], [9] have +achieved excellent performances and became the most popular +approach by training an additional discriminator network to +distinguish the features from different domains [10]–[13]. +Domain-invariant features are expected to be learned by +training the feature extractor to produce features that are +indistinguishable by the discriminator. +Though the prevalent adversarial domain adaptation has +shown success in many areas, there are still two limitations. +Firstly, the minmax game of adversarial learning is notoriously +known to be difficult to optimize, requiring lots of training +tricks [14]. When the domain gap is large or the data distri- +bution is complicated with multi modes, these models tend +to collapse with false feature alignment [15], [16], especially +when trained from scratch. Secondly, adversarial training is +shown to damage the learned feature discriminability [13], +[17]. While shown to be alleviated by either balancing the +feature singular values [13] or lifting feature norms [17], such +a discriminability corruption still persists because of the con- +flicts between transferability and discriminablity which tends +to be biased by source labels and with weaker transferability. +One more promising approach is to learn a discriminative +arXiv:2301.00383v1 [cs.LG] 1 Jan 2023 + +IEEE TRANSACTIONS ON IMAGE PROCESSING +2 +structure that is inherently transferable across domains. +In this paper, we propose Discriminative Radial Domain +Adaptation, that gets rid of the typical adversarial learning and +bridges the source and target domains via a shared discrimina- +tive radial structure. It’s motivated by the observation that the +features initially all cluster together and as the model is trained +to be more discriminative, features of different categories +expand outwards in different directions to be more separated in +the feature space, forming a radial structure, as also observed +in [18], [19]. We bridge the source and target domains by +aligning the radial structures. Specifically, we first build a +radial structure for each domain that consists of a global +domain anchor, which is the centroid of the domain data, and +a set of local category anchors. Brute-force matching tends to +twist the radial structure and damage its discriminability. To +alleviate this, we decompose the structure matching into two +components, namely global isometric transformation and local +refinement as shown in Fig.1. Global isometric transformation +aims to align the global shape of the two radial structures by +bringing close the domain anchors and rotating the overall +radial structure using a Sitefel layer [20]. To achieve fine- +grained alignment of each category, local refinement further +matches the angles and norms of local category anchors across +domains. +To enhance the discriminablity of the radial-like feature +distribution, we encourage local features to cluster close to +the corresponding local anchors. This is achieved by first +assigning each local feature to the optimal local anchors +via optimal transport, which prevents false assignments, and +then minimizing a optimal-transport distance. Meanwhile, we +enforce a prediction consistency between the radial structure +and classifier the prevent conditional shift of the classifier. We +observe such a consistency also promotes the radial structure +to be more discriminative. +The main contributions of this work can be summarized as +follows: +• We propose a novel domain adaptation approach, called +Discriminative Radial Domain Adaptation, that gets rid +of the typical adversarial training and reduces domain +shift by matching radial structures that are inherently +discriminative. +• We propose to decompose the alignment of radial struc- +ture into global isometric transformation and local anchor +refinement to prevent damage to the discriminablity of the +radial structure. +• We enhance the discriminablity of the radial structure +by minimizing a optimal-transport distance that optimally +assigns each feature to the corresponding local anchors to +combat false alignment. Further, we perform a prediction +consistency between the radical structure and classifier to +alleviate conditional shift. +• Extensively +experimenting +on +several +benchmark +datasets, our method outperforms the state-of-the-art +approaches not only on the typical single-to-single +unsupervised domain adaptation but also on multi-source +domain +adaptation, +domain-agnostic +adaptation +and +domain generalization. +II. RELATED WORKS +In this part, we first review domain adaptation methods and +then introduce discriminative structure learning. +A. Domain Adaptation +To alleviate the domain shift, typical solutions include +minimizing domain discrepancy and learning domain invariant +features. In the context of domain discrepancy minimization, +approaches can be classified according to their discrepancy +metrics and their ways of extracting features. Discrepancy met- +rics include the Proxy A-distance [21], the Kullback-Leibler +(KL) divergence [22], the Mean Maximum Mean Discrepancy +[5], [23], other higher order statistical moments based distance +measures [24], and Optimal Transport distance [25], [26]. +Many types of feature extraction have also been considered for +domain alignment, including handcrafted features [5], shallow +features at the pixel level [27], and bottleneck features of +deep neural networks [10], [28]. Along with their efficiency in +reducing marginal domain discrepancies, these methods were +found potentially hinder the learning of feature discriminant +information [29]. Therefore, recent advances have focused on +the discrepancy in conditional distribution by using labels +or soft labels. Such approaches include conditional variants +of MMD, Joint Distribution Optimal Transport (JDOT) [30], +Moving Semantic Transfer Network (MSTN) [31], Robust +Spherical Domain Adaptation (RSDA) [32], Category-Level +Adversarial Network (CLAN) [33], Enhanced Transport Dis- +tance [29], Discriminative Manifold Propagation [34], and +Conditional Kernel Bures (CKB) metric [29]. These improve- +ments resulting from conditional alignment are evident; how- +ever, the changes in the class prior distribution and the noise +of estimated target labels also pose risks of misalignment. In +order to solve these issues, we propose to simultaneously learn +the structure of the source and target distributions and align +the two domains based on this structure. That is inspired by +factorized optimal transport [35], which highlights the benefits +of using low-dimensional structures to align data. In our +framework, domain adaptation is carried out by aligning these +radial structures learned from each domain, without relying on +sample-level distribution. +Another approach mainly aims at learning domain invariant +features so that the target can share the classifier trained +from the labeled source. An effective method to guarantee +features transferability is to train the generator to produce +indistinguishable features, which can deceive the domain +discriminator as a whole, i.e. Domain Adversarial Neural +Networks (DANN) [9] and Adversarial Discriminative Domain +Adaptation (ADDA) [8]. In addition, a number of studies +have been published that examine ways to improve training +strategies in order to create better transferable features from +pixel-level features [36]–[38] or high-level features [39]. A +further effort on producing transferable features is conditional +adversarial training discriminator on features and class pre- +diction jointly [11], [12]. Later, more works focus on disen- +tangling original features into domain invariant and domain- +specific parts [40], [41]. Nevertheless, these models seek +to intensify feature transferability at the expense of feature + +IEEE TRANSACTIONS ON IMAGE PROCESSING +3 +discriminability. In contrast, DRDA applies domain adaptation +based on established discriminative radial structures, so the +discriminability of features can be well maintained. +B. Discriminative Structure Learning +Discriminative learning is aimed at pushing dissimilar fea- +tures away from each other and enclosing the similar ones +to be compact. Many efforts have been made to minimize +intra-class feature distances and maximize inter-class feature +distances, such as contrastive loss [18] and center loss [19], +originally proposed in face recognition tasks. Inspired by +softmax objective, L-Softmax (large margin softmax) [42] is +introduced as another extension of enhancing discriminability +by lifting angular separability between learned features. The +principle of discriminative learning also enhances the perfor- +mance of domain adaptation tasks. Follow the discriminative +clustering, entropy minimization is introduced into domain +adaptation [12], [43] to encourage classifier to produce ideal +one-hot predictions are promising method. More recent stud- +ies in adversarial-based domain adaptation have shown that +the discriminability of target features can be damaged by +adversarial feature alignment. Based on the observations of +close relation between the singular values of learned features +and discriminative power, [13] correct the degeneration of +discriminability by adding the penalties corresponding to these +singular values. Also, [17] identified the connection between +norm values and discriminatory power, and then lift the norm +values for target features in order to increase the discriminatory +power. +As well as features being discriminative during the learning +process, the feature distribution is likely to perform in a +particular low-dimensional format. The whole domain can +therefore be well sketched by several clusters rather than +using entire samples, allowing for a more robust approach +to domain adaptation. In line with this idea, MSTN [31] +is proposed to align the centroids of each category across +domains, which reduces the noise influence of false pseudo +labels compared to direct matching distributions. Prototypical +networks [44] is proposed to learn prototypes of each category +region and reduce conditional domain discrepancy by learning +similar prototypes across domains. And [45] recognizes the +importance of structure, connects the statistical property to +geometric structures of data, and integrates feature selection +and structure preservation into a unified optimization process. +Moreover, [46] considered unsupervised domain adaptation as +a clustering problem with missing labels using the structure +preserve framework. Compared to these methods, our ap- +proach direct models feature distribution with a radial structure +which maintains the intrinsic structure of the data while +increasing the feature discriminability. +III. DISCRIMINATIVE RADIAL DOMAIN ADAPTATION +In this section, we first introduce the construction of the +radial structure. Then, we describe a proposed structure align- +ment strategy which decouple alignment into two independent +components, namely global isometric transformation and local +anchor refinement. +A. Notations and Overview +In an unsupervised domain adaptation task, we are given +labeled source domain Ds = {(xs +i, ys +i )}ns +i=1 of ns labeled +examples and unlabeled target domain Dt = {xt +j}nt +j=1 of +nt unlabeled examples. Our model mainly contains a shared +backbone G(·) with parameter θ, a shared classifier F(·) with +parameter ϕ, and a Stiefel layer S(·) whose parameters ∆ are +defined on Stiefel manifold Vk(Rd) = {∆ ∈ Rd×k|∆⊤∆ = +Ik}. Let zs +i += G(xs +i) and �ys +i += F(zs +i) be the feature +representation and the estimated label of the i-th sample in +the source domain, respectively. +With insights that linear classification output probability +pik ∝ exp(∆T +k zi + b) = exp(∥Wk∥∥zi∥ cos(Wk, zi) + b) +supposing importance of feature direction and norm in dis- +crimination, we suggest a radial expansion-like structure for +modeling features. Therefore, our framework is aiming to learn +and align radial structures Gs = {as, N s} and Gt = {at, N t} +from source and target domains, each structure containing a +global anchor as/t and a set N s/t = {as/t +i +}k +i=1 of k local +anchors as/t +i +∈ Rd. From an intuitive viewpoint, a radial +structure in latent space can be understood as a structure +with a group of arrows that point from a global anchor to +local anchors. Thus, for emphasizing the radial chararistic of +structures, we also use the egocentric representation version +Vs/t := {vs/t +i += (as/t +i +− as/t)|as/t ∈ N s/t} of radial struc- +ture when comparing the shape differences of the structures. +Finally, domain shifts and class prior differences are then +manifested in terms of the isometric transformation and shape +differences between two radial structures. We align the Gs +and Gt by reducing isometric transformation to match each +other globally in latent space and then refine them into the +same shape. Where the isometric transformation and shape +refinement are applied in a non-interfering manner for avoiding +negative alignment. The DRDA approach can be viewed as +an alternative optimization strategy that iteratively updates +the radial structures Gs, Gt to be more representative and +aligns radial structures in order to obtain more accurate label +predictions in the target domain. +B. Discriminative Radial Structure +Extraction of radial structure Gs/t includes aggregating +global anchor as/t and a collection of local anchors N s/t. +We represent the global and local anchors using vectorial +embeddings, and iteratively update the anchors and model +parameters. +1) Global anchors: For each domain, we define the global +anchor as the centroid of overall features extracted by the +shared feature extractor Gθ(·). Formally, the global anchors +as, at in the source and target domains are: +as = 1 +ns +ns +� +i=1 +Gθ (xs +i) , +at = 1 +nt +nt +� +j=1 +Gθ +� +xt +j +� +. +(1) +As an indicator of the mean position of features, global anchor +is ideal reference point for contrasting two feature vector under +the context of linear classification. They are also the reference +points for comparing the radial structures. 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+JyUIUe9V/rqRopmCZNIBTGm43spBmOikVPBJsVuZlhK6JD0WcdSRJmgvHs3ol7apXIjZW2JdGdqb8nxiQxZpSEtjMhODCL3lT8z+tkGF +8HYy7TDJmk80VxJlxU7vR5N+KaURQjSwjV3N7q0gHRhKNqGhD8BdfXibN86p/WfXvLso1L4+jAMdwAhXw4QpqcAt1aAFAc/wCm/Oo/Pi +vDsf89YVJ585gj9wPn8AQkuPaA=S(·) : Stiefel layer +Fig. 2. +Architecture of Discriminative Radial Domain Adaptation (DRDA). It bridges source and target domains by matching the radial structure which +consists of a global domain anchor and a set of local category anchors. DRDA aligns the radial structures across domains via global isometric transformation +and local anchor refinement (best viewed in color). +between global anchors naturally represent the mean feature +shift E[zs] − E[zt]. We then use the distance between as and +as as the global translation distance between two domains. +2) Local anchors: Source and the target radial structures +contains ks and kt local anchors, respectively. Local anchors +are located within high-density regions in each domain, each +region containing a set of semantically related features. +For the general UDA task, it is straightforward to set ks, kt +equal to the number of categories to be classified. Then, the +local anchor is equivalent to the centroid of the features with +same category. In labeled source data, such local anchors can +be obtained directly from labels, while in the target domain +local anchors can be obtained from pseudo-labels: +as +k = +1 +Mk +ns +� +i=1 +ziI[yi = k], +at +k = +1 +Mk +nt +� +i=1 +ziI[ˆyi = k], (2) +where I[·] is indicator function, Mk = � I[yi = k] is a +normalize constant. +C. Radial Structure Alignment +We disentangle the structure alignment into two parts, +namely global isometric transformation and local anchor re- +finement, to prevent its corruption to the discriminablity of +the radical structure. It’s shown that when Ds and Dt are +not aligned, the learned features would be arbitrarily rotated, +translated or permuted [47]. By disentangling the alignment +process into two independent processes, it is hopefully to best +prevent false feature alignment. +1) Isometric transformation: We first globally align the +shape of the radial structures to reduce the isometric transfor- +mation ˜T(·) between source Gs and target Gt, It is equivalent +to minimize the isometric transformation ˜T(·) between the +source and target, which is defined as: +˜T := arg minT ∥Gs − T(Gt)∥. +We seek to optimize the backbone Gθ(·), thereby making ˜T(·) +to be an identical transformation I(·) : I(G) = G. +We disentangle the objective into translation and rotation +parts in order to optimize feature extractor to achieve isometric +alignment. The translation reduction is performed by minimiz- +ing the distance between global anchors among domains as +follows: +Lglobal = d(as, at) = ∥as − at∥F , +(3) +where the global distance measures the common distribution +difference between source and target. By applying such global +distance minimization, we force two global anchors to align +and, consequently, we shift two feature distributions so that +they share the same centroid. In addition, it is intuitively +possible to make the entire radial structures (as the structures +shown in Fig.1) on which the feature points lie roughly +coincide. +The rotation reduction is accomplished by adding a Stiefel +layer S(·) to rotate the target features. According to a +strict definition of the Stiefel manifold Vk(Rd) = {∆ ∈ +Rd×k|∆⊤∆ = Ik}, the Stiefel layer would perform rotation +transform without causing any side effects to the features. In +the backbone networks, we embed the Stiefel layer for target- +specific use. In addition, for emphasizing the radial chararistic, +we shift the radial structures Gs, Gt with respect to the global +anchors, so that they share the same center and are referred +to as egocentric version Vs, Vt, respectively. Thanks to the +benefits of manifold optimization methods [48], it is easy to +optimize the Stiefel layer as following objective: +∆∗ = min +Vk(Rd) d(Vs, S(Vt)), +(4) +where the parameter ∆ is optimized with respect to a shape +difference metric d(·, ·) (induced from local alignment) be- +tween the radial structures Gs, Gt of the target and the source. +It is noteworthy that the backbone is shared by the source and + +IEEE TRANSACTIONS ON IMAGE PROCESSING +5 +target, but ∆∗ is embed in backbone network for target-specific +use. +Optimization on Eq.(3) and Eq.(4) enables a coarse align- +ment among domains, and increases the reliability of target +pseudo labels given by the classifier, as discriminative radial +structures achieve more and more overlapping as shown in +Fig.2. +2) Local refinement: Global alignment is intended to elimi- +nate isometric discrepancies between the source and the target, +whereas local alignment involves refining the two structures +to be identical in shape. In order to avoid the occurrence of +a contradiction between global and local alignments, the fine- +grained structure difference measured here need to be indepen- +dent of global alignment. In light of this, we apply Gromov- +Wasserstein (GW) [49] distance to compare the shapes of two +radial structures. According to the definition, GW distance +is solely based on intra-space measurements, it has many +desirable properties, especially in terms of invariances. And +it terms out the invariant of translations, permutations, and +rotations when Euclidean distance is used for intra-space +measurement. Accordingly, whenever two structures Gs and +Gt are shifted with different offsets or rotations, GW distance +only determines the shape difference between them. For em- +phasizing the radial chararistic differences of the source and +target structures we use the egocentric representation Vs, Vt +instead of standard form Gs, Gt. Accordingly, the GW distance +is defined as follows: +GW 2 +2 (cs, ct, µ, ν) = +min +π∈Π(Vs,Vt) J(cs, ct, π), +(5) +where +J(cs, ct, π) = +� +i,j,k,l +|cs(vs +i , vs +k) − ct(vt +j, vt +l)|2πi,jπk,l, +where, µ = �ks +i=1 δas +i , ν = �kt +i=1 δat +i, are the measure of +anchors. π is the transport plan, Π(·, ·) represent set of total +transport permutation combination. cs and ct are specific intra- +distance metrics defined on the radial structures of the source +and the target, respectively. To incorporate the discriminative +information, we recall classifier formulation p(y = k|zi) ∝ +exp(∥Wk∥∥zi∥ cos(Wk, zi) + b) suggests angular and norm +value is critical for vectors discrimination, we combine the +both information in intra-distance function for cs, ct, and +define them with same formulation: +c(vi, vj) = [1 − ⟨vi, vj⟩ +∥vi∥∥vj∥] + λdist +1 +2∥vi − vj∥2 +2, +(6) +with λdist weight parameter to tradeoff angular difference loss +between the structures. The first term calculates the cosine +distances between the corresponding pairs of displacement +vectors as angular difference. The second term captures the +length difference by calculating the ℓ2 distances between the +corresponding pairs of displacement vectors. Furthermore, the +transport plan π can be fixed due to one-to-one correspon- +dences of discriminative vectors Vs and Vt from source and +target are known, i.e. we force πi,j = 0 when i ̸= j which +gives: +GW(Vs, Vt) = +� +ij +|c(vs +i , vs +j) − c(vt +i, vt +j)|2, +where the GW distance with a fixed transport plan implies that +a certain property of GW are lost, that is rotational invariance. +However, a loss of rotational difference yields a metric of +shape difference that is useful for optimizing the Stiefel layer. +This completed the distance metric d(·, ·) in Eq.(4). Which +has the advantage of providing a more efficient formula for +the loss of local alignment and we defined it as φ(Vs, Vt). +Where the φ(Vs, Vt) can be expressed by the expectation of +pairs of elements difference across two domains: +φ(Vs, Vt) = E(vs +k,vt +k)∈(Vs,Vt)[c(vs +k, vt +k)]. +(7) +The simplified objective Eq.(7) gradually forces corresponding +vectors being the same length and pointing to the same +direction, thus ensuring the local structure alignment. +With synchronously minimizing the discrepancy among +domains based on the radial structures by isometric trans- +formation and structure refinement, data distributions of the +source and target will move towards to each other and finally +present the identical radial structure. In this way, the posterior +probability expectations of each category in the source and the +target domains can also be consistent. +D. Radial Structure Enhancement +We further improve the learning of radial structures from +the following two aspects; 1) First one is structure faithful- +ness requirement, which encourages samples to enclose their +corresponding local anchors. 2) Second one is semantic mean- +ingfulness requirement, extracted radial structure should be +informative for the semantic information of data distribution, +i.e., the consensus between geometrical assignment labels and +classifier labels. +1) Enclose features to local anchors: According to the +structure faithfulness requirement, features are expected to be +located near desired anchors. Since the distribution of features +is unknown, we model the assignment of features to desired +anchors by optimal transport plan [50]. In the case where the +distances between features and anchors determine the transport +cost, the optimal transport plan is the one which has the +lowest total cost (also referred as optimal transport distance +or Wasserstein distance) for moving features to corresponding +anchors. The optimal transport plan can also be viewed as an +adaptive distribution model allowing different anchors corre- +spond to different probability densities. Then, to fairly push +instances toward the desired local anchors, shared backbone +network is learned to minimize optimal transport. Further, for +relaxing the objective and stablizing the end-to-end training +we use entropic optimal transport [51] distance defined by: +OTϵ +θ(X, N) = +min +π∈Π(µ,µa) +� +i,j +d(Gθ(xi), aj)πi,j + ϵKL(π|µ ⊗ µa), (8) +where µ = �n +i=1 δxi the measure of data instances and +µa = �k +j=1 δaj the measure of anchors, d(·, ·) is euclidean +distance metric, π is the transport plan, Π(·, ·) represent set +of total transport permutation combination, ϵ ≥ 0 is the +regularization coefficient. As a relevant metric for assigning +samples to the best-fitted anchors, optimal transport distance + +IEEE TRANSACTIONS ON IMAGE PROCESSING +6 +can lead to a more reliable assignment than nearest neighbor +search [52]. Therefore, by optimizing Gθ(·) in minimizing +Lot = OTϵ +θ(X s, N s)+OTϵ +θ(X t, N t) in both source and target +domain independently, the extracted features in both domains +are more compactly arranged around their radial structures, the +structure faithfulness requirement can be indirectly achieved. +2) Consensus regularization: For semantic meaningfulness +requirement, we regard that instances assigned to the same +local anchor have the same label, and for each instance, the +label assigned by the classifier must match the label assigned +by the radial structure. Hence, to train a network meets seman- +tic meaningfulness requirements, a consensus regularization is +designed to force the labels assigned by the classifier match +the labels assigned by the radial structure, +Rϕ(Q, P) = KL(Q||P) + H(P), +(9) +where regularization is performed at classifier parameter ϕ, +KL(·||·) is Kullback-Leibler divergence, H(·) is entropy that +balances the discriminability negative effects in this regular- +ization, Q = {qi,k} is soft-assignments given by the transport +plan π, P = {(pi,1, . . . , pi,k)} ∈ [0, 1]K×N indicates the pos- +teriors given by classifier. Consensus between data distribution +structures and classifications can be improved by minimizing +terms of regularization LR = Rϕ(Qs, Ps) + Rϕ(Qt, Pt). +Intuitively, the objective based on Eq.(8) and Eq.(9) grad- +ually enhances the representative and discriminative of radial +structures in each domain through minimizing optimal trans- +port distance from samples to local anchors and consensus +regularization between radial structure assignment and classi- +fication. +E. Optimization +The optimization is conducted in two steps, i.e., radial +structures extraction and alignment. +a) Radial structure update: The ideal implementation +of calculating local anchors in Eq.(2) requires iterating over +the entire dataset, which is computationally expensive. By +employing an appropriate exponential moving average update +strategy, we can easily perform end-to-end training: +ak = η 1 +Mk +B +� +i=1 +ziI[yi = k] + (1 − η)a′ +k, +(10) +where B indicates the batch size and Mk = �B +i I[yi = k] is a +normalization constant, a′ +k indicates the last updated anchors +and ak indicates new anchors computed in current iteration. +b) Network update: +Recall objective of Eq.(8) and +Eq.(9), a critical insights on behind successful optimization +is similar to Expectation–Maximization (EM) algorithm. To +optimize optimal transport distance from samples to local +anchors, we fixed local anchors and update θ according to +Eq.(8), then update local anchors make use of updated θ +next iteration according to Eq.(2). To optimize the consensus +between geometrical assignments and classifier assignments, +we fixed Q and only update classifier ϕ according to Eq.(9) +with insights that classifier shall make trade off to respect +intrinsic data distribution. Finally, alternative network update +Algorithm 1: DRDA Training +Data: Labeled source Ds, Unlabeled target Dt +Result: θ, ϕ, ∆ +Initialization: θ ← θ0, ϕ ← ϕ0, ∆ ← I; +while Not Converge do +Sample {(X s, Ys)} and {X t} from Ds and Dt; +(ˆPs, ˆPt) ← (fϕ(G(X s), fϕ(S(G(X t)); +Update radial structures Gs, Gt according to +Eq.(2),Eq.(1); +Calculate source classification loss Lce(ˆPs, Ys) ; +Calculate alignment loss Lglobal, φ(Vs, Vt) +according to Eq.(3), Eq.(7) ; +Calculate OT distance Lot by Eq.(8); +Calculate prediction discrepancy LR between +classifier and radial structure in Eq.(9); +// Update parameters according to gradients; +∆ ++ +← −∇∆λφφ(Vs, Vt); +ϕ ++ +← −∇ϕ(Lce + λRLR); +θ ++ +← −∇θ(Lce+λotLot+λTLglobal+λφφ(Vs, Vt)); +end +return θ, ϕ, ∆ +approach can be easily implemented by stop gradient tricks, +then the overall objective respectively: +min +θ,ϕ,∆ Lce + λT Lglobal + λφφ(Vs, Vt) ++ λot[OT ϵ +θ(X s, SG[N s]) + OT ϵ +θ(X t, SG[N t])] ++ λR[Rϕ(SG[Qs], Ps) + Rϕ(SG[Qt], Pt)], +(11) +where SG[·] indicates the stop-gradient operation. This opera- +tion prevents parameters from being updated by the gradients. +In the light of alternative network update approach, stop- +gradient operation is critical for preventing degeneration of +the structure during learning. Specifically, in Eq.(11), first +term Lce is classification error; the second term Lglobal and +third term φ(Vs, Vt) jointly perform isometric transformation +and structure refinement for aligning feature distributions of +different domains; the rest terms enhance the representativity +and discriminability of radial structures. To balance the scale +of terms in overall objective, the global loss (i.e. global +translation distance) is scaled by λT , intra-structures difference +is scaled by λφ, OT distance is scaled by λot and consensus +regularization is scaled by λR. Notice, based upon differences +Eq.(7) in the radial structures, the global rotation transforma- +tion distance minimization is implicitly optimized with respect +to Stiefel layer parameters. +IV. EXPERIMENTS +We compare the proposed method with several state-of- +art methods on three types of UDA tasks, including single +UDA, Domain-Agnostic UDA and Multi-Source UDA. The +experimental results show that our method outperforms the +other methods in terms of the average classification accuracy. +In addition, we present a series of visualization results and +ablation studies to demonstrate the insights of our method and +the effectiveness of each component in our model. + +IEEE TRANSACTIONS ON IMAGE PROCESSING +7 +TABLE I +ACCURACY (%) ON OFFICE-31 FOR UNSUPERVISED DOMAIN ADAPTATION (RESNET-50) +Method +A → W +D → W +W → D +A → D +D → A +W → A +Average +ResNet-50 +68.4 ± 0.2 +96.7 ± 0.1 +99.3 ± 0.1 +68.9 ± 0.2 +62.5 ± 0.3 +60.7 ± 0.3 +76.1 +RevGrad [53] +82.0 ± 0.4 +96.9 ± 0.2 +99.1 ± 0.1 +79.7 ± 0.4 +68.2 ± 0.4 +67.4 ± 0.5 +82.2 +DAN [10] +80.5 ± 0.4 +97.1 ± 0.2 +99.6 ± 0.1 +78.6 ± 0.2 +63.6 ± 0.3 +62.8 ± 0.2 +80.4 +JAN [54] +85.4 ± 0.3 +97.4 ± 0.2 +99.8 ± 0.2 +84.7 ± 0.3 +68.6 ± 0.3 +70.0 ± 0.4 +84.3 +MADA [55] +90.0 ± 0.2 +97.4 ± 0.1 +99.6 ± 0.1 +87.8 ± 0.2 +70.3 ± 0.3 +66.4 ± 0.3 +85.2 +CDAN+E* [12] +94.1 ± 0.1 +98.6 ± 0.1 +100.0 ± .0 +92.9 ± 0.2 +71.0 ± 0.3 +69.3 ± 0.3 +87.7 +ALDA [56] +95.6 ± 0.5 +97.7 ± 0.5 +100.0 ± .0 +94.0 ± 0.4 +72.2 ± 0.4 +72.5 ± 0.2 +88.7 +DRDA (w/o Angular) +92.6 ± 0.5 +98.3 ± 0.2 +100.0 ± .0 +91.9 ± 0.5 +71.0 ± 0.3 +70.6 ± 0.1 +87.4 +DRDA (w/o Stiefel) +92.3 ± 0.5 +98.7 ± 0.1 +100.0 ± .0 +92.1 ± 0.5 +74.7 ± 0.2 +75.3 ± 0.2 +88.8 +DRDA (w/o Rϕ) +94.9 ± 0.3 +98.2 ± 0.1 +100.0 ± .0 +93.8 ± 0.3 +74.2 ± 0.1 +75.8 ± 0.1 +89.4 +DRDA (w/o OTϵ +θ) +94.8 ± 0.2 +98.0 ± 0.1 +100.0 ± .0 +94.0 ± 0.2 +74.8 ± 0.1 +75.4 ± 0.1 +89.5 +DRDA +95.8 ± 0.4 +98.8 ± 0.4 +100.0 ± .0 +94.5 ± 0.3 +75.6 ± 0.2 +76.6 ± 0.4 +90.2 +TABLE II +ACCURACY (%) ON OFFICE-HOME FOR UNSUPERVISED DOMAIN ADAPTATION (RESNET-50) +Method +Ar→Cl +Ar→Pr +Ar→Rw +Cl→Ar +Cl→Pr +Cl→Rw +Pr→Ar +Pr→Cl +Pr→Rw +Rw→Ar +Rw→Cl +Rw→Pr +Average +ResNet-50 +34.9 +50.0 +58.0 +37.4 +41.9 +46.2 +38.5 +31.2 +60.4 +53.9 +41.2 +59.9 +46.1 +DANN [9] +45.6 +59.3 +70.1 +47.0 +58.5 +60.9 +46.1 +43.7 +68.5 +63.2 +51.8 +76.8 +57.6 +JAN [54] +45.9 +61.2 +68.9 +50.4 +59.7 +61.0 +45.8 +43.4 +70.3 +63.9 +52.4 +76.8 +58.3 +CDAN+E [12] +50.7 +70.6 +76.0 +57.6 +70.0 +70.0 +57.4 +50.9 +77.3 +70.9 +56.7 +81.6 +65.8 +ALDA [56] +53.7 +70.1 +76.4 +60.2 +72.6 +71.5 +56.8 +51.9 +77.1 +70.2 +56.3 +82.1 +66.6 +MDD [57] +54.9 +73.7 +77.8 +60.0 +71.4 +71.8 +61.2 +53.6 +78.1 +72.5 +60.2 +82.3 +68.1 +DRDA (w/o Angular) +54.3 +70.3 +74.8 +60.7 +69.2 +69.8 +59.1 +52.8 +76.4 +70.9 +58.3 +82.0 +66.5 +DRDA (w/o Stiefel) +57.4 +74.5 +79.3 +64.8 +75.6 +74.0 +62.9 +56.2 +79.7 +72.0 +62.9 +84.1 +70.4 +DRDA (w/o Rϕ) +57.3 +74.3 +80.4 +64.7 +74.3 +73.0 +64.9 +55.8 +79.7 +74.5 +63.0 +84.3 +70.5 +DRDA (w/o OTϵ +θ) +57.0 +73.9 +80.2 +64.1 +73.8 +73.1 +64.4 +56.1 +78.9 +73.4 +62.8 +84.1 +70.1 +DRDA +58.2 +74.2 +81.2 +65.6 +75.1 +73.3 +65.8 +57.1 +80.4 +75.6 +63.2 +85.1 +71.2 +A. Experimental Setup +1) Office-31: +[58] is a widely used dataset for visual +domain adaptation, which consists of 31 categories count up to +4,652 images from three distinct domains: 2,817 Amazon(A) +images, 795 Webcam(W) images, and 498 DSLR(D) images. +We evaluate methods upon all 6 in pairs of transfer tasks. +2) Office-Home: +[59] is a better organized dataset and +more difficult dataset compared to Office-31, which consists +of 65 categories count up to 15,500 images in office and +home setting, formed with four extremely dissimilar domains: +Artistic images (Ar), Clipart images (Cl), Product images (Pr), +and Real-World images (Rw). +3) Office-Caltech10: +[60] is collected from Office31 and +Caltech formed with four domains: A (Amazon), C (Caltech), +W (Webcam), and D (DSLR). It consists of 10 object cate- +gories, each domain includes 958, 295, 157, and 1,123 images, +respectively. +We compared the proposed (DRDA) with state-of-the- +art domain adaptation methods: Domain Adversarial Neu- +ral Network (DANN) [9], Joint Adaptation Network (JAN) +[54], Conditional Domain Adversarial Network with Entropy +(CDAN+E) [12], Adversarial-Learned Loss for Domain Adap- +tation (ALDA) [56] and Margin Disparity Discrepancy (MDD) +[57]. For multi-source domain adaptation we compared +our model with state-of-the-art domain adaptation methods: +Deep Alignment Network (DAN) [11], Domain Adversarial +Neural Network (DANN) [9], Manifold Embedded Distribu- +tion Alignment (MEDA) [61], Maximum Classifier Discrep- +ancy (MCD) [28] and Moment Matching for Multi-Source +Domain Adaptation (M3SDA) [24]. For domain agnostic +domain adaptation, we compared our model with state-of- +the-art methods: Self-Ensembling (SE) [37], Maximum Clas- +sifier Discrepancy (MCD) [28], Domain Adversarial Neural +Network (DANN) [9] and Deep Adversarial Disentangled +Autoencoder (DADA) [41]. +We follow the standard protocols of unsupervised domain +adaptation. We use all labeled source samples and unlabeled +target samples and compare the average classification accuracy +based on three experiments. The overall architecture consists +of a backbone, ResNet-50, a bottleneck layer with 256 units +and a full-connected layer. The Stiefel layer is a simple full- +connected layer whose parameters are manipulated on Stiefel +Manifold implemented with geoopt [48]. And this Stiefel layer +is used for processing target features only. We implement our +method in Pytorch. We finetune from ImageNet pre-trained +models as the feature extract backbone. We essentially tune +the hyper-parameters in Eq.(11), λT ∼ 200, λφ ∼ 0.6, λot ∼ +0.0005, λR ∼ 1, they control the scaling of each loss term +in overall objective. Both backbone layers and task-specific +layers are trained through back-propagation using Stochastic +Gradient Descent (SGD). The Stiefel layer is optimized using +Riemannian SGD [48]. The backbone layers is finetuned based +on pre-trained ResNet-50 on ImageNet, while the task-specific +layers are trained from scratch whose learning rate is 10 times + +IEEE TRANSACTIONS ON IMAGE PROCESSING +8 +that of backbone layers. +We use mini-batch stochastic gradient descent as the opti- +mizer and apply momentum of 0.9 and learning rate schedule +rule [9] with ηp = η0(1 + γp)−β, where p is within the range +of [0, 1], and η0 = 0.01, γ = 10, β = 0.75. We conduct +the grid hyper-parameter selection base on loss curve fitting +to obtain optimal weighted combination of objective in the +experiments. To reduce noise influence and stable optimization +convergence, we adopt progressive domain transfer weights +with factor λp = +2 +1+exp(−αp) −1 increasing from 0 to 1 where +α = 10 as a default setting. Especially, for Office31 dataset, +due to the small number of samples in DSLR and Webcam +domain, we add temperature factor t = 0.85 to adjust the +convergence speed of a cross-entropy loss. +For the experiments on Multiple source unsupervised do- +main adaptation using Office-Caltech and OfficeHome, we +sample instances uniformly from the combined source do- +mains as source inputs. For the experiments on Domain agnos- +tic unsupervised domain adaptation using Office-Caltech we +sample instances uniformly from the combined target domains +as target inputs. For the experiments on domain-generalized +unsupervised domain adaptation, we train models similar to +the setting of one-to-one UDA, and demonstrate the validation +accuracies in domains which were not the sources or targets. +B. Comparison with the State-of-The-Art Methods +1) Single source to single target UDA: To testify the +effectiveness of the proposed DRDA, we first compare our +method with state-of-art single domain UDA tasks. For a fair +comparison, we report previous domain adaptation methods +whose results are based on ResNet-50 and test in same +validation setting. +The results on Office-31 are reported in Table I. In most +transfer sub-tasks, DRDA attains the highest classification +accuracy and improves the average accuracy over state-of-the- +art methods. Our method works particularly well for small- +to-large transfer tasks, such as D→A,W→A. Even though +the sample size in the source domain is small, the proposed +discriminative radial structure is sufficiently representative and +discriminative to serve as a guide to domain alignment and +more robust to noise. +The results on Office-Home are reported in Table II. The +proposed DRDA achieves the best accuracy in all transfer +tasks and improves the average accuracy over the state- +of-the-art methods by 3%. Compared to Office-31, Office- +Home is more challenging because it has more categories +and greater discrepancies between domains. As the task be- +comes increasingly challenging, our approach outperforms +our competitors by a greater margin. As explained in the +hypothesis, radial-like structures are beneficial for sketching +and preserving discriminative structures, and this structure is +well suited for domain alignment. Therefore, in the case of +more categories and greater discrepancies between domains, +the radial-like structure shows greater superior performance in +domain alignment than the other methods. In the context of +domain alignment tasks, these results illustrate the importance +of discrimination preservation and low-dimensional structures +(i.e. the proposed radial-like structure). +TABLE III +ACCURACY (%) ON OFFICE-CALTECH FOR MULTI-SOURCE +UNSUPERVISED DOMAIN ADAPTATION (RESNET-50) +Method +A,C,D→W +A,C,W→D +A,D,W→C +C,D,W→A +Avg +ResNet-50 +97.1 +99.2 +89.4 +94.7 +95.6 +DANN [9] +96.5 +99.1 +89.2 +94.7 +94.8 +MEDA [61] +99.3 +99.2 +91.4 +92.9 +95.7 +MCD [28] +99.5 +99.1 +91.5 +92.1 +95.6 +M3SDA-β [24] +99.5 +99.2 +92.2 +94.5 +96.4 +DRDA (w/o Angular) +100.0 +100.0 +95.7 +96.5 +98.1 +DRDA (w/o Stiefel) +100.0 +100.0 +95.7 +96.8 +98.1 +DRDA (w/o Rϕ) +100.0 +100.0 +95.8 +96.5 +98.0 +DRDA (w/o OTϵ +θ) +100.0 +100.0 +96.0 +96.8 +98.2 +DRDA (ours) +100.0 +100.0 +96.4 +96.9 +98.3 +2) Multi-source to single target UDA: multi-source unsu- +pervised domain adaptation [24], which transfers knowledge +from multiple source domains to one unlabeled target domain. +Compared to one-to-one unsupervised domain adaptation, this +task is much more difficult as the source domain is a mixture +of multiple domains. In this task, we merge the multiple +source domain into a single one. The results on the Office- +Caltech10 dataset are reported in Table III. According to +the observation, the proposed DRDA surpasses state-of-the +art methods even those developed for such tasks specifically. +Recall the conception of radial-like structure, the key idea is +sketching the discriminative structure of data distributions. In +this respect, the more domains the algorithm uses as sources, +the greater the generalization power the source structure has. +Therefore, the proposed DRDA is naturally fit for UDA tasks +involving multiple source domains. +TABLE IV +ACCURACY (%) ON OFFICE-CALTECH FOR DOMAIN-AGNOSTIC +UNSUPERVISED DOMAIN ADAPTATION (RESNET-50) +Method +A→ C,D,W +C→ A,D,W +D→ A,C,W +W→ A,C,D +Avg +ResNet-50 +90.5±0.3 +94.3±0.2 +88.7±0.4 +82.5±0.3 +89 +SE [37] +90.3±0.4 +94.7±0.4 +88.5±0.3 +85.5±0.4 +89.7 +MCD [28] +91.7±0.4 +95.3±0.3 +89.5±0.2 +84.3±0.2 +90.2 +DANN [9] +91.5±0.4 +94.3±0.4 +90.5±0.3 +86.3±0.3 +90.6 +DADA [41] +92.0±0.4 +95.1±0.3 +91.3±0.4 +93.1±0.3 +92.9 +DRDA (w/o Angular) +97.6±0.4 +97.8±0.5 +96.2±0.4 +96.7±0.1 +97.2 +DRDA (w/o Stiefel) +97.7±0.3 +97.1±0.7 +96.8±0.1 +97.0±0.2 +97.2 +DRDA (w/o Rϕ) +97.6±0.5 +97.6±0.3 +96.6±0.5 +96.7±0.2 +97.1 +DRDA (w/o OTϵ +θ) +97.8±0.5 +98.0±0.2 +97.1±0.5 +96.8±0.3 +97.5 +DRDA (ours) +98.1±0.2 +97.5±0.2 +96.6±0.4 +96.8±0.2 +97.3 +3) Single source to agnostic multi-target UDA: We also +consider another type of unsupervised domain adaptation +task: domain agnostic unsupervised domain adaptation +[41], which transfers knowledge from a labeled source domain +to unlabeled data in one of multiple target domains. In this +task, we regard the mixture target domain as a single one. The +Table IV shows that our model gets a 97.3% average accuracy +and improves the other methods by 4.4% in the domain +agnostic unsupervised domain adaptation task. It appears that +the radial-like structure is consistently effective at representing +discriminative structures regardless of domain heterogeneity. +Hence, the domain alignment can be well assured with the +help of the radial structure. +4) Domain generalize UDA: A further extension to illus- +trate the utility of DRDA for knowledge abstraction is to + +IEEE TRANSACTIONS ON IMAGE PROCESSING +9 +TABLE V +ACCURACY (%) ON OFFICE-HOME FOR DOMAIN GENERALIZE UNSUPERVISED DOMAIN ADAPTATION (RESNET-50) +Train +Ar→Cl +Ar→Pr +Ar→Rw +Cl→Ar +Cl→Pr +Cl→Rw +Pr→Ar +Pr→Cl +Pr→Rw +Rw→Ar +Rw→Cl +Rw→Pr +Avg +Test +Pr +Rw +Cl +Rw +Cl +Pr +Rw +Pr +Ar +Rw +Ar +Pr +Cl +Rw +Ar +Rw +Ar +Cl +Pr +Cl +Ar +Pr +Ar +Cl +ResNet-50 +50.0 +58.0 +34.9 +58.0 +34.9 +50.0 +46.2 +41.9 +37.4 +41.9 +37.4 +41.9 +31.2 +60.4 +38.5 +60.4 +38.5 +31.2 +59.9 +41.2 +53.9 +59.9 +53.9 +41.2 +46.1 +DANN [9] +60.0 +68.7 +37.8 +70.0 +42.3 +66.3 +63.0 +59.1 +49.4 +64.3 +54.8 +63.2 +37.4 +71.9 +48.0 +68.0 +58.6 +39.7 +75.1 +41.2 +59.2 +73.3 +63.1 +43.4 +57.4 +MDD [57] +61.6 +68.5 +38.1 +71.4 +40.7 +67.0 +63.9 +60.2 +47.1 +62.3 +53.7 +61.9 +34.7 +70.1 +46.7 +67.9 +54.6 +35.9 +74.5 +40.5 +61.5 +74.3 +60.6 +40.0 +56.5 +DRDA +65.6 +74.1 +50.6 +75.8 +50.3 +72.1 +71.6 +68.9 +60.1 +71.4 +61.7 +68.9 +48.2 +76.9 +59.7 +75.7 +64.4 +51.2 +80.7 +52.9 +70.8 +79.3 +72.0 +52.2 +65.6 +Fig. 3. Training curve and latent features visualization at difference stages, the colors of points indicate instances colors. We can see a clear structure evolution +progress while structure loss decreasing. Where φ(Vs, Vt +GT ) is computed every validation, Vs +GT indicates the structure calculated with ground truth labels +(resp φ(Vt, Vt +GT )). φ(Vs, Vt) is passed through a median filter (original with light blue color) for visualization purpose, best view in color. +extend it to domain-generalized UDA problems, that is, to +train the model on one task and to test it on another domain +that is different from the source and target domains. +The detailed results on OfficeHome were reported in Ta- +ble V, where the first row indicates the standard single-to- +single UDA task that the models are trained for, and the second +row indicates the test domain used only for evaluation. These +results indicate that our method is capable of generalizing to +domain generalization tasks. The DRDA method performed +well in a number of subtasks. In many domains of the test, +our method was superior to those classical UDA methods that +directly optimize adaptation performance on those domains, +even though our method did not incorporate this domain +information for training. These results presented here pro- +vide evidence for the effectiveness of radial-like structures in +discriminative feature modeling. The in-depth explanation is +that the alignment using discriminative radial structures forces +the network to learn more meaningful features as a result of +regularizing its optimization pathway. As a consequence, when +the trained model encounters instances from domains that have +not previously been encountered, they can also be classified +on the basis of their semantic features. +C. Analysis +In this subsection, we present various experiments that illus- +trate the intuition behind radial-like structures and demonstrate +the effectiveness of our approach. +1) Low-dimensional radial structures: To illustrate the con- +vergence of the structure, we built a simplified LetNet similar +to [56] while reducing the bottleneck dimension to 2 and +training it with the task MNIST→USPS. We visualize the +two dimensional features from different stages of the overall +training procedure, with different colored points showing +different categories. To facilitate visual understanding, the +source domain features and the target domain features were +plotted side-by-side in separate plots, and instance points were +uniformly sampled from entire datasets with stride 10. In +Fig.3, we observe that the features coming from the source and +target domains gradually evolve the radial-like structures. It is +worth noting that, at the beginning of the process, there is a +significant structure discrepancy between the source and target. +Then, as the training progresses, the structures of two domains +become more and more discriminative (i.e. the features in +latent space present a more and more clear radial-like manner). +Also, the decreasing of both source structure error φ(Vs, Vs +GT ) +and target structure error φ(Vt, Vt +GT ) indicates these radial-like +structures become more and more close to ground-truth ones. +As radial-like structures become more reliable and discrimi- +native, domain alignment is expected to be more accurate as +well. +2) Global isometric effects: +To evaluate the proposed +Stiefel layer, we implement a network without the Stiefel layer +for comparison, denoting as DRDA (w/o Stiefel). The detailed +performance results presented in Table I,II,III,IV indicate the +importance of the Stiefel layer to the final accuracy. Based +on the results of the performance drop, we can confirm the +hypothesis that features will be rotated globally and that +the Stiefel layer can reduce the negative effect on domain +alignment. Furthermore, we tried to elaborate on the in-depth +explanations in Fig.6. As we can see, when we use the Stiefel +layer, we can reduce the structure loss to a small range quickly, + +IEEE TRANSACTIONS ON IMAGE PROCESSING +10 +0 +5000 +10000 +15000 +20000 +Iterations +0.40 +0.45 +0.50 +0.55 +0.60 +Error +burn_in:0.5 +burn_in:0.2 +burn_in:0.1 +burn_in:0.02 +(a) Accuracy Convergence +0 +5000 +10000 +15000 +20000 +Iterations +0.5 +1.0 +1.5 +2.0 +2.5 +Local Loss +burn_in:0.5 +burn_in:0.2 +burn_in:0.1 +burn_in:0.02 +(b) Structure Loss +0 +5000 +10000 +15000 +20000 +Iterations +10 +20 +30 +40 +50 +60 +70 +Wasserstein Loss +burn_in:0.5 +burn_in:0.2 +burn_in:0.1 +burn_in:0.02 +(c) Wasserstein +0 +5000 +10000 +15000 +20000 +Iterations +0 +1 +2 +3 +4 +5 +KL Divergence +burn_in:0.5 +burn_in:0.2 +burn_in:0.1 +burn_in:0.02 +(d) KL Divergence +Fig. 4. +The component loss along with the train iterations. We use ‘burn in’ to indicate the percentage of training progress before adding the structure +alignment operations. (a). validation accuracies of Clipart along with training iterations (b). φ(Vs, Vt) along with training iterations. (c). wasserstein distance +between instances and local anchors in the target domain, along with training iterations. (d). The KL divergence between target geometrical label assignments +and the classifier label assignments. (best view in color) +100 +150 +200 +250 +T +0 +10 +20 +30 +40 +50 +60 +70 +80 +Accuracy +(a) λT +0.0001 +0.0005 +0.001 +0.01 +ot +0 +10 +20 +30 +40 +50 +60 +70 +80 +Accuracy +(b) λot +0.6 +0.8 +1 +1.2 +0 +10 +20 +30 +40 +50 +60 +70 +80 +Accuracy +(c) λφ +1 +5 +10 +20 +R +0 +10 +20 +30 +40 +50 +60 +70 +80 +Accuracy +(d) λR +Fig. 5. The effect of different hyperparameters on the performance of OfficeHome dataset. Here we report subtask results on Art→Clipart. +0 +5000 +10000 +15000 +20000 +Iterations +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +Local Loss +DRDA +DRDA (w/o Stiefel Layer) +Local Alignment Start +Fig. 6. Global translation distance Lglobal and local structure loss φ(Vs, Vt). +and the maximum loss values are significantly less than those +obtained without the Stiefel layer. This is partly due to the +fact that the global rotation difference between the source +and target domains is particularly a misleading factor in the +calculation of domain discrepancies in the early stages of do- +main alignment. Furthermore, this misleading factor can have +irreversible negative impacts on overall domain alignment. +The final accuracy drops for the models without the Stiefel +layer also confirmed these irreversible negative impacts. The +global rotational component can be easily extracted from the +difference between two radial structures when there is a Stiefel +layer, which would naturally mitigate such negative impacts in +the early stages of domain alignment. Additionally, the results +of this study demonstrate the necessity of decoupling global +and local transformations when performing alignment and the +Stiefel layer is a suitable choice. +3) Effects of structures alignment: To better understand +the effects of radial structure local alignment on domain +alignment, we perform an ablation study that does not op- +timize for local structure loss at the beginning of the training +period. The detailed results are shown in Fig.4. As we can +see accuracy on target increasing during the early learning +stage and decreases while training moves on. As we can see +obviously when ‘burn in:0.5’ the accuracy on target increases +during the early learning stage and then gets stuck. Meanwhile, +the structure loss, the Wasserstein distance from instances to +local anchors, and the KL divergence between geometrical +labels and classifier labels were increased. It is clear from +these simultaneous losses increasing that the structure of +the target domain is crumbling. This is because, with the +progress of training, the network gradually learns the common +semantic information at the beginning, and then begins to +over-fit the data in the source domain. Moreover, this over- +fit phenomenon is accompanied by arbitrary distortions to +discriminative structures. As shown in Fig.4, stretching the +‘burn in’ results in irreversible damage to the final accuracy. +By comparing the influences of different ‘burn in’ on the rest +component losses in Fig.4, we notice that once the structure +alignment operations are restored, the corresponding losses +drop rapidly. Correspondingly, the test errors on the target +domain also decreased rapidly after structure alignment was +restored. The results show that our structure alignment can +always reconstruct and align discriminative structures, which +supports the validity of our model in the domain alignment. +4) Angular term effects in intra-structure comparison: +When the angular losses are removed from the intra-structure +comparison loss function, denoting as DRDA (w/o Angular), +the performance returns to baseline, which indicates that +discrepancy based on angular distance between discriminative +vectors is very critical. The reason can be two folds. First, + +IEEE TRANSACTIONS ON IMAGE PROCESSING +11 +the angular loss is more consistent with the formulation of +classification. Secondly, in high-dimensional space, the mass +of the sphere is primarily concentrated on the shell and the +distance between any two point pairs becomes even smaller. +Therefore, angular loss confirms that modeling data distribu- +tion with radial-like structures that are well suited to angular +comparison is an effective strategy. +5) Effects of optimal transport distance minimization: To +verify the effectiveness of optimal transport distance min- +imization between instance and local anchors, we conduct +the ablation studies by removing this minimization term in +training, denoting as DRDA (w/o OT). The detailed results +reported in Table I and II illustrate the performance drop +when instances are not restricted to being located nearby local +anchors. We note that the performance degradation of DRDA +(w/o OT) is much smaller than that of DRDA (w/o Angular), +indicating that our proposed radial structure based alignment is +efficient and robust in domain adaptation even if the structures +are not forced to be compact. +6) Effects of consensus regularization: We evaluate the +proposed classifier regularization terms by implementing the +model without regularization loss R(P, Q) denoting as DRDA +(w/o Rϕ). The results reported in Table I and II indicate +that such a regularization term enhances the performance of +domain alignment across all subtasks. It is shown that the +consensus regularization between geometrical assignments and +classifier assignments can enhance the classification perfor- +mance of the classifier by encouraging the classifier to admit +geometrical assignments. +7) Parameters sensitivity analysis: +In this section, we +conduct sensitivity analysis on the hyper-parameters for our +proposed method. The detailed results are shown in Fig.5. +Parameters λT , λφ, λot and λR are mainly for scaling the +loss value. From the observation, we choose λT = 150 which +controls the impacts of global translation between domains. +The parameters λot balance the radial-like structure compact- +ness and alignment effects, and we find when λot is around +0.0001-0.005 the model performance reaches the peaks. The +parameters λR regularize the agreement between classifier +and geometric assignments, when λR = 20.0 the model +performance reaches the peaks. As for structure alignment, +λφ = 3 provides the best performance. It is also obvious that +the performance of the system is quite stable across a wide +range of transfer losses when they are ranged in respective +orders of magnitude. +8) The evolution of structures discrepancies: +We also +demonstrate the differences of the radial structure between +domains with increasing iteration numbers. As illustrated in +Fig.7, after adding the structure alignment, the discrepancies +φ(Vs +GT , Vt +GT ) of structures derived from ground truth labels +appear to consistently decrease, as well as local differences +between structures being minimized. It is evident from the +results that our method is able to consistently produce positive +alignment with the increasing number of training iterations. +V. CONCLUSION +This paper presents a new structure-preserved domain adap- +tation method, which has two key features: a new discrimina- +0 +2500 +5000 +7500 +10000 +12500 +15000 +17500 +20000 +Iterations +0 +1 +2 +3 +4 +5 +Local Loss +Compare different structure discrepancies in training +Local Loss +( +s +GT, +t +GT) +( +s, +s +GT) +( +t, +t +GT) +Local Alignment Start +Fig. 7. Radial structures alignment convergence visualization with an increas- +ing number of iterations, where φ(Vs,Vs +GT ),φ(Vt, Vt +GT ), φ(Vs +GT , Vt +GT ) are +computed every validation, Vs +GT indicates the structure calculated with ground +truth labels (resp Vt +GT ). +tive radial structure and a new alignment strategy based on +radial structure. The discriminative radial structure preserves +both representative and discriminative information in feature +distribution. The decoupled global alignment and fine-grained +morphological alignment reduce the common domain shifts +and conditional domain shifts. Experimental results on several +benchmark datasets showed that i) our method consistently +outperforms state-of-the-art methods on four types of unsu- +pervised domain adaptation tasks, and ii) our method leads to +more superiority when the task is more challenging. +ACKNOWLEDGMENT +This work is supported by the National Key R&D Program +of China(2020YFB1313501), Zhejiang Provincial Natural Sci- +ence Foundation (LR19F020005) , National Natural Science +Foundation of China (61972347, T2293723) and the Funda- +mental Research Funds for the Central Universities (No. 226- +2022-00051). +REFERENCES +[1] S. J. Pan and Q. Yang, “A survey on transfer learning,” IEEE Transac- +tions on Knowledge and Data Engineering, vol. 22, no. 10, pp. 1345– +1359, Oct. 2010. +[2] S. Ben-David, J. Blitzer, K. Crammer, A. Kulesza, F. Pereira, and +J. W. Vaughan, “A theory of learning from different domains,” Machine +Learning, vol. 79, no. 1-2, pp. 151–175, May 2010. +[3] B. Gong, K. Grauman, and F. 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Grauman, “Geodesic flow kernel +for unsupervised domain adaptation,” in 2012 IEEE Conference on +Computer Vision and Pattern Recognition, Jun. 2012, pp. 2066–2073. +[61] J. Wang, W. Feng, Y. Chen, H. Yu, M. Huang, and P. S. Yu, “Visual +domain adaptation with manifold embedded distribution alignment,” in +Proceedings of the 26th ACM International Conference on Multimedia, +ser. MM ’18. +New York, NY, USA: Association for Computing +Machinery, Oct. 2018, pp. 402–410. +Zenan Huang received B.E. degree in the Computer +Science from Zhejiang University of Technology, in +2018. He is currently pursuing the Ph.D degree in +the College of Computer Science and Technology, +Zhejiang University. His research interests include +computer vision, causalility, and machine learning. +Jun Wen received the Ph.D. degree in computer +science from Zhejiang University, Hangzhou, China, +in 2020. He is currently a Postdoctoral Research +Fellow at the Harvard Medical School. His research +interests include transfer learning and biomedical +informatics. +Siheng Chen is a tenure-track associate professor of +Shanghai Jiao Tong University and Co-PI at Shang- +hai AI Laboratory. Dr. Chen received his doctorate +from Carnegie Mellon University. Dr. Chen’s work +on sampling theory of graph data received the 2018 +IEEE Signal Processing Society Young Author Best +Paper Award. His co-authored paper on structural +health monitoring received ASME SHM/NDE 2020 +Best Journal Paper Runner-Up Award and another +paper on 3D point cloud processing received the +Best Student Paper Award at 2018 IEEE Global +Conference on Signal and Information Processing. Dr. Chen contributed to +the project of scene-aware interaction, winning MERL President’s Award. +His research interests include collective intelligence, autonomous driving and +graph neural networks. +Linchao Zhu (Member, IEEE) received the B.E. +degree from Zhejiang University, China, in 2015, +and the Ph.D. degree in computer science from +the University of Technology Sydney, Australia, in +2019. He is a Research Professor with the College of +Computer Science and Technology, Zhejiang Univer- +sity, China. His research interests are video analysis +and understanding. +Nenggan Zheng received the bachelor’s and Ph.D. +degrees from Zhejiang University, Hangzhou, China, +in 2002 and 2009, respectively. He is currently +a Full Professor in computer science with the +Academy for Advanced Studies, Zhejiang Univer- +sity. His research interests include artificial intel- +ligence, brain–computer interface, and embedded +systems. + diff --git a/WNAyT4oBgHgl3EQfhvic/content/tmp_files/load_file.txt b/WNAyT4oBgHgl3EQfhvic/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..f7d18992c98c752c048123cbebaa4efcfb059247 --- /dev/null +++ b/WNAyT4oBgHgl3EQfhvic/content/tmp_files/load_file.txt @@ -0,0 +1,1724 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf,len=1723 +page_content='IEEE TRANSACTIONS ON IMAGE PROCESSING 1 Discriminative Radial Domain Adaptation Zenan Huang, Jun Wen, Member, IEEE, Siheng Chen, Member, IEEE, Linchao Zhu, Member, IEEE, and Nenggan Zheng, Senior Member, IEEE Abstract—Domain adaptation methods reduce domain shift typically by learning domain-invariant features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content=' Most existing methods are built on distribution matching, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content=', adversarial domain adaptation, which tends to corrupt feature discriminabil- ity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content=' In this paper, we propose Discriminative Radial Domain Adaptation (DRDR) which bridges source and target domains via a shared radial structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content=' It’s motivated by the observation that as the model is trained to be progressively discriminative, features of different categories expand outwards in different directions, forming a radial structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content=' We show that transferring such an in- herently discriminative structure would enable to enhance feature transferability and discriminability simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content=' Specifically, we represent each domain with a global anchor and each category a local anchor to form a radial structure and reduce domain shift via structure matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content=' It consists of two parts, namely isometric transformation to align the structure globally and local refine- ment to match each category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content=' To enhance the discriminability of the structure, we further encourage samples to cluster close to the corresponding local anchors based on optimal-transport assignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content=' Extensively experimenting on multiple benchmarks, our method is shown to consistently outperforms state-of-the-art approaches on varied tasks, including the typical unsupervised domain adaptation, multi-source domain adaptation, domain- agnostic learning, and domain generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content=' Index Terms—Domain Adaptation, Transfer Learning, Radial Structure Matching I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content=' INTRODUCTION Machine learning methods generally assume that training and test data come from the same data distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content=' However, such an assumption may not hold in practice, since a model trained on one distribution or one domain may need to be applied to data from another distribution or domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content=' Typically, such distribution shifts or domain shifts would cause signifi- cant performance drop [1], [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content=' To address this issue, domain adaptation methods are proposed, which aim to generalize the learned knowledge from source domain to target domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content=' Zenan Huang is with the Qiushi Academy for Advanced Studies, Zhejiang University, Hangzhou, Zhejiang 310007, China and also with the College of Computer Science and Technology, Zhejiang University, Hangzhou, Zhejiang 310007, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content=' (E-mail: lccurious@zju.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content='cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content=' Jun Wen was with the Qiushi Academy for Advanced Studies, Zhejiang University, Hangzhou, Zhejiang 310007, China and also with the College of Computer Science and Technology, Zhejiang University, Hangzhou, Zhejiang 310007, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content=' (E-mail: jungel2star@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content='com).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content=' Siheng Chen is with the Shanghai Jiao Tong University and also with Shanghai AI Laboratory, Shanghai, 200240, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content=' (E- mail:sihengc@sjut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content='cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content=' Linchao Zhu is with College of Computer Science and Technology, Zhe- jiang University, Hangzhou, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content=' (E-mail: zhulinchao@zju.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content='cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content=' Nenggan Zheng is with the Qiushi Academy for Advanced Studies, Zhe- jiang University, Hangzhou, Zhejiang, 310007, China, also with Collaborative Innovation Center for Artificial Intelligence by MOE and Zhejiang Provincial Government (ZJU) and Zhejiang Lab, Hangzhou, Zhejiang, 311121, China (E-mail: zng@cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content='zju.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content='cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content=' Nenggan Zheng is the corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content=' Feature Space Unaligned features Aligned features source category: 1 source category: 2 source category: 3 source category: 4 target category: 1 target category: 2 target category: 3 target category: 4 Radial Structure Radial structure alignment Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content=' Illustration of the proposed method which represents each domain using a radial structure and reduce domain shift via structure matching (source: red;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content=' target: blue;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content=' best viewed in color).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content=' Domain adaptation methods reduce domain shift typically by learning domain-invariant representations [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content=' Previously, shallow features from both the source and target domains are mapped into a shared subspace [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content=' With the success of deep learning, domain-invariant features are learned using deep neural networks [4], upon which various domain discrepancy measures are proposed, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content=', Maximum Mean Discrepancy (MMD) [5], second-order correlations [6], and moments [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content=' Recently, adversarial domain adaptation methods [8], [9] have achieved excellent performances and became the most popular approach by training an additional discriminator network to distinguish the features from different domains [10]–[13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content=' Domain-invariant features are expected to be learned by training the feature extractor to produce features that are indistinguishable by the discriminator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content=' Though the prevalent adversarial domain adaptation has shown success in many areas, there are still two limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content=' Firstly, the minmax game of adversarial learning is notoriously known to be difficult to optimize, requiring lots of training tricks [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content=' When the domain gap is large or the data distri- bution is complicated with multi modes, these models tend to collapse with false feature alignment [15], [16], especially when trained from scratch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content=' Secondly, adversarial training is shown to damage the learned feature discriminability [13], [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content=' While shown to be alleviated by either balancing the feature singular values [13] or lifting feature norms [17], such a discriminability corruption still persists because of the con- flicts between transferability and discriminablity which tends to be biased by source labels and with weaker transferability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content=' One more promising approach is to learn a discriminative arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content='00383v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content='LG] 1 Jan 2023 IEEE TRANSACTIONS ON IMAGE PROCESSING 2 structure that is inherently transferable across domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content=' In this paper, we propose Discriminative Radial Domain Adaptation, that gets rid of the typical adversarial learning and bridges the source and target domains via a shared discrimina- tive radial structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content=' It’s motivated by the observation that the features initially all cluster together and as the model is trained to be more discriminative, features of different categories expand outwards in different directions to be more separated in the feature space, forming a radial structure, as also observed in [18], [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content=' We bridge the source and target domains by aligning the radial structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content=' Specifically, we first build a radial structure for each domain that consists of a global domain anchor, which is the centroid of the domain data, and a set of local category anchors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content=' Brute-force matching tends to twist the radial structure and damage its discriminability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content=' To alleviate this, we decompose the structure matching into two components, namely global isometric transformation and local refinement as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content=' Global isometric transformation aims to align the global shape of the two radial structures by bringing close the domain anchors and rotating the overall radial structure using a Sitefel layer [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content=' To achieve fine- grained alignment of each category, local refinement further matches the angles and norms of local category anchors across domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content=' To enhance the discriminablity of the radial-like feature distribution, we encourage local features to cluster close to the corresponding local anchors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content=' This is achieved by first assigning each local feature to the optimal local anchors via optimal transport, which prevents false assignments, and then minimizing a optimal-transport distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content=' Meanwhile, we enforce a prediction consistency between the radial structure and classifier the prevent conditional shift of the classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content=' We observe such a consistency also promotes the radial structure to be more discriminative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content=' The main contributions of this work can be summarized as follows: We propose a novel domain adaptation approach, called Discriminative Radial Domain Adaptation, that gets rid of the typical adversarial training and reduces domain shift by matching radial structures that are inherently discriminative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content=' We propose to decompose the alignment of radial struc- ture into global isometric transformation and local anchor refinement to prevent damage to the discriminablity of the radial structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content=' We enhance the discriminablity of the radial structure by minimizing a optimal-transport distance that optimally assigns each feature to the corresponding local anchors to combat false alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content=' Further, we perform a prediction consistency between the radical structure and classifier to alleviate conditional shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content=' Extensively experimenting on several benchmark datasets, our method outperforms the state-of-the-art approaches not only on the typical single-to-single unsupervised domain adaptation but also on multi-source domain adaptation, domain-agnostic adaptation and domain generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content=' RELATED WORKS In this part, we first review domain adaptation methods and then introduce discriminative structure learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content=' Domain Adaptation To alleviate the domain shift, typical solutions include minimizing domain discrepancy and learning domain invariant features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content=' In the context of domain discrepancy minimization, approaches can be classified according to their discrepancy metrics and their ways of extracting features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content=' Discrepancy met- rics include the Proxy A-distance [21], the Kullback-Leibler (KL) divergence [22], the Mean Maximum Mean Discrepancy [5], [23], other higher order statistical moments based distance measures [24], and Optimal Transport distance [25], [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content=' Many types of feature extraction have also been considered for domain alignment, including handcrafted features [5], shallow features at the pixel level [27], and bottleneck features of deep neural networks [10], [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content=' Along with their efficiency in reducing marginal domain discrepancies, these methods were found potentially hinder the learning of feature discriminant information [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content=' Therefore, recent advances have focused on the discrepancy in conditional distribution by using labels or soft labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content=' Such approaches include conditional variants of MMD, Joint Distribution Optimal Transport (JDOT) [30], Moving Semantic Transfer Network (MSTN) [31], Robust Spherical Domain Adaptation (RSDA) [32], Category-Level Adversarial Network (CLAN) [33], Enhanced Transport Dis- tance [29], Discriminative Manifold Propagation [34], and Conditional Kernel Bures (CKB) metric [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content=' These improve- ments resulting from conditional alignment are evident;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content=' how- ever, the changes in the class prior distribution and the noise of estimated target labels also pose risks of misalignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content=' In order to solve these issues, we propose to simultaneously learn the structure of the source and target distributions and align the two domains based on this structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content=' That is inspired by factorized optimal transport [35], which highlights the benefits of using low-dimensional structures to align data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content=' In our framework, domain adaptation is carried out by aligning these radial structures learned from each domain, without relying on sample-level distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content=' Another approach mainly aims at learning domain invariant features so that the target can share the classifier trained from the labeled source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content=' An effective method to guarantee features transferability is to train the generator to produce indistinguishable features, which can deceive the domain discriminator as a whole, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content=' Domain Adversarial Neural Networks (DANN) [9] and Adversarial Discriminative Domain Adaptation (ADDA) [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content=' In addition, a number of studies have been published that examine ways to improve training strategies in order to create better transferable features from pixel-level features [36]–[38] or high-level features [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content=' A further effort on producing transferable features is conditional adversarial training discriminator on features and class pre- diction jointly [11], [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content=' Later, more works focus on disen- tangling original features into domain invariant and domain- specific parts [40], [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content=' Nevertheless, these models seek to intensify feature transferability at the expense of feature IEEE TRANSACTIONS ON IMAGE PROCESSING 3 discriminability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content=' In contrast, DRDA applies domain adaptation based on established discriminative radial structures, so the discriminability of features can be well maintained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content=' Discriminative Structure Learning Discriminative learning is aimed at pushing dissimilar fea- tures away from each other and enclosing the similar ones to be compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content=' Many efforts have been made to minimize intra-class feature distances and maximize inter-class feature distances, such as contrastive loss [18] and center loss [19], originally proposed in face recognition tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content=' Inspired by softmax objective, L-Softmax (large margin softmax) [42] is introduced as another extension of enhancing discriminability by lifting angular separability between learned features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content=' The principle of discriminative learning also enhances the perfor- mance of domain adaptation tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content=' Follow the discriminative clustering, entropy minimization is introduced into domain adaptation [12], [43] to encourage classifier to produce ideal one-hot predictions are promising method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content=' More recent stud- ies in adversarial-based domain adaptation have shown that the discriminability of target features can be damaged by adversarial feature alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content=' Based on the observations of close relation between the singular values of learned features and discriminative power, [13] correct the degeneration of discriminability by adding the penalties corresponding to these singular values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content=' Also, [17] identified the connection between norm values and discriminatory power, and then lift the norm values for target features in order to increase the discriminatory power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content=' As well as features being discriminative during the learning process, the feature distribution is likely to perform in a particular low-dimensional format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content=' The whole domain can therefore be well sketched by several clusters rather than using entire samples, allowing for a more robust approach to domain adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content=' In line with this idea, MSTN [31] is proposed to align the centroids of each category across domains, which reduces the noise influence of false pseudo labels compared to direct matching distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content=' Prototypical networks [44] is proposed to learn prototypes of each category region and reduce conditional domain discrepancy by learning similar prototypes across domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content=' And [45] recognizes the importance of structure, connects the statistical property to geometric structures of data, and integrates feature selection and structure preservation into a unified optimization process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content=' Moreover, [46] considered unsupervised domain adaptation as a clustering problem with missing labels using the structure preserve framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content=' Compared to these methods, our ap- proach direct models feature distribution with a radial structure which maintains the intrinsic structure of the data while increasing the feature discriminability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content=' DISCRIMINATIVE RADIAL DOMAIN ADAPTATION In this section, we first introduce the construction of the radial structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content=' Then, we describe a proposed structure align- ment strategy which decouple alignment into two independent components, namely global isometric transformation and local anchor refinement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content=' Notations and Overview In an unsupervised domain adaptation task, we are given labeled source domain Ds = {(xs i, ys i )}ns i=1 of ns labeled examples and unlabeled target domain Dt = {xt j}nt j=1 of nt unlabeled examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content=' Our model mainly contains a shared backbone G(·) with parameter θ, a shared classifier F(·) with parameter ϕ, and a Stiefel layer S(·) whose parameters ∆ are defined on Stiefel manifold Vk(Rd) = {∆ ∈ Rd×k|∆⊤∆ = Ik}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content=' Let zs i = G(xs i) and �ys i = F(zs i) be the feature representation and the estimated label of the i-th sample in the source domain, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content=' With insights that linear classification output probability pik ∝ exp(∆T k zi + b) = exp(∥Wk∥∥zi∥ cos(Wk, zi) + b) supposing importance of feature direction and norm in dis- crimination, we suggest a radial expansion-like structure for modeling features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content=' Therefore, our framework is aiming to learn and align radial structures Gs = {as, N s} and Gt = {at, N t} from source and target domains, each structure containing a global anchor as/t and a set N s/t = {as/t i }k i=1 of k local anchors as/t i ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content=' From an intuitive viewpoint, a radial structure in latent space can be understood as a structure with a group of arrows that point from a global anchor to local anchors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content=' Thus, for emphasizing the radial chararistic of structures, we also use the egocentric representation version Vs/t := {vs/t i = (as/t i − as/t)|as/t ∈ N s/t} of radial struc- ture when comparing the shape differences of the structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content=' Finally, domain shifts and class prior differences are then manifested in terms of the isometric transformation and shape differences between two radial structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content=' We align the Gs and Gt by reducing isometric transformation to match each other globally in latent space and then refine them into the same shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content=' Where the isometric transformation and shape refinement are applied in a non-interfering manner for avoiding negative alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content=' The DRDA approach can be viewed as an alternative optimization strategy that iteratively updates the radial structures Gs, Gt to be more representative and aligns radial structures in order to obtain more accurate label predictions in the target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content=' Discriminative Radial Structure Extraction of radial structure Gs/t includes aggregating global anchor as/t and a collection of local anchors N s/t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content=' We represent the global and local anchors using vectorial embeddings, and iteratively update the anchors and model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content=' 1) Global anchors: For each domain, we define the global anchor as the centroid of overall features extracted by the shared feature extractor Gθ(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content=' Formally, the global anchors as, at in the source and target domains are: as = 1 ns ns � i=1 Gθ (xs i) , at = 1 nt nt � j=1 Gθ � xt j � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content=' (1) As an indicator of the mean position of features, global anchor is ideal reference point for contrasting two feature vector under the context of linear classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content=' They are also the reference points for comparing the radial structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content=' And displacement ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content='IEEE TRANSACTIONS ON IMAGE PROCESSING ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content='Source ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content='Target ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content='Feature Extractor ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content='G ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content=': Source / 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+page_content='4RXerCfrxXq3PuajK1a5cwh/YH3+ACHhk1g=LR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content='Classification boundary ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content='Classifier F ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content='Locally Refined ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content='Globally Aligned ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfhvic/content/2301.00383v1.pdf'} +page_content=' max +˜y̸=y∗ pv,˜y(G) + ∆ +where y∗ ≜ gv(G) denotes the majority class, and ˜y the follow-up (second best) class. +Proof in Appendix A. We also provide a certificate for binary node classification in Appendix A. +2We consider simple paths (all nodes appear only once), since we only receive perturbed messages via more +complex paths iff we receive perturbed messages via the simple part of the complex path. +4 + +0 +.2 .4 .6 .8 +1 +pa +0 +.2 +.4 +.6 +.8 +1 +pd +(a) +0 +.2 +.4 +.6 +.8 +1 +∆i +0 +.2 .4 .6 .8 +1 +pa +0 +.2 +.4 +.6 +.8 +1 +pd +(b) +0 +.2 +.4 +.6 +.8 +1 +∆i +0 +.2 .4 .6 .8 +1 +pa +0 +.2 +.4 +.6 +.8 +1 +pd +(c) +0 +.2 +.4 +.6 +.8 +1 +∆i +Figure 2: Single source bounding constant ∆i for different edge deletion probabilities pd and node +feature ablation probabilities pa. White isolines indicate ∆i = 0.5 and separate the theoretically +certifiable region (∆i < 0.5) from the uncertifiable region (∆i ≥ 0.5). (a) For the target node, pd +does not affect ∆i. (b) Direct neighbor of target node, single edge. (c) Second-hop neighbor, single +path (two edges). (a-c) More distant nodes have larger theoretically certifiable regions. +5 +Practical Interception Smoothing Certificates +Message-interception certificates constitute two challenges in practice: (1) computing the bounding +constant ∆ for arbitrary graphs, and (2) computing the label probabilities pv,y∗(G) and pv,˜y(G). +We address the first problem by providing upper bounds on ∆ (i.e. lower bounds on the certifiable +robustness). For the second problem we follow existing literature and estimate the smoothed classifier. +Lower bound on certifiable robustness. Computing ∆ of Theorem 1 poses two problems: First, +finding the worst-case nodes in arbitrary graphs involves a challenging optimization over the powerset +of nodes in the receptive field. Second, computing the probability p(E(ρv)) to receive perturbed +messages is challenging even for fixed ρv, since in general, it involves evaluating the inclusion- +exclusion principle (Appendix C). We can compute ∆ exactly only for special cases such as small or +tree-structured receptive fields (Appendix D). Notwithstanding the challenges, we provide practical +upper bounds on ∆. Instead of assuming a fixed ρv, we solve both problems regarding ∆ at once and +directly bound the maximum over all possible ρv by assuming independence between paths. Due to +Corollary 1, any upper bound on ∆ result in lower bounds on the certifiable robustness. +We first derive an upper bound on ∆ for a single perturbed node, and then generalize to multiple nodes. +Let Ew denote the event that the target node v receives messages from node w, and ∆w ≜ p(Ew). +Note in the special case of the target node v = w we just have ∆w = 1 − pa, since the features xv of +the target node v are used for the prediction independent of any edges. For any w ̸= v in the receptive +field we can derive the following upper bound for single sources (Proof in Appendix E): +Theorem 2 (Single Source Multiplicative Bound). Given target node v and source node w ̸= v in +the receptive field of a k-layer message-passing GNN f with respect to v. Let P k +wv denote all simple +paths from w to v of length at most k in graph G. Then ∆w ≤ ∆w for: +∆w ≜ +� +�1 − +� +q∈P k +wv +� +1 − (1 − pd)|q|� +� +� (1 − pa) +where |q| denotes the number of edges on the simple path q ∈ P k +wv from w to v. +We visualize ∆w for different pd and pa in Figure 2. The upper bound for single sources is tight for +one- and two-layer GNNs (∆ = ∆w), since then all paths from a single source to the target node are +independent (Appendix E). The single source multiplicative bound on ∆w can only be used to certify +a radius of ρ = 1. For multiple nodes (ρ > 1), we generalize Theorem 2 as follows: +Theorem 3 (Generalized multiplicative bound). Assume an adversarial budget of ρ nodes and let +∆1, . . . , ∆ρ denote the ρ largest ∆i for nodes i in the receptive field. Then we have ∆ ≤ ∆M for +∆M ≜ 1 − +ρ +� +i=1 +(1 − ∆i) +Proof in Appendix E. Notably, the multiplicative bound is tighter than a union bound. We specifically +address the approximation error in detail in Appendix F. +5 + +Estimating the smoothed classifier in practice. +Computing the probabilities pv,y∗(G) and +pv,˜y(G) exactly is challenging in practice. We instead estimate them similar to previous work +by drawing Monte-Carlo samples from φ (Cohen et al., 2019; Levine and Feizi, 2020b; Bojchevski +et al., 2020). We first identify the majority class y∗ and follow-up class ˜y using a few samples. +We then draw more samples to estimate a lower bound pv,y∗(G) on pv,y∗(G) and an upper bound +pv,˜y(G) on pv,˜y(G). We use the Clopper-Pearson Bernoulli confidence interval and apply Bonferroni +correction to ensure that the bounds hold simultaneously with significance level α (with probability +of at least 1 − α). Moreover, our smoothed classifier abstains from predicting if pv,y∗(G) ≤ pv,˜y(G), +meaning if the estimated probabilities are too similar. We experimentally analyze abstained predic- +tions in Appendix H. +Practical robustness certificates. Finally, our robustness certificates also hold when bounding ∆ +and the label probabilities as the following Corollary shows (Proof in Appendix A): +Corollary 2. We guarantee gv(G) = gv(G′) with probability of at least 1 − α for any G′ ∈ Bρ(G) +if pv,y∗(G) − ∆ > pv,˜y(G) + ∆, where y∗ denotes the majority class, and ˜y the follow-up class. +6 +Discussion +Our certificates require knowledge about the graph structure A and can only account for structure +perturbations if the perturbed adjacency matrix A′ is known. While adversarial edge deletion +potentially increases robustness (due to less messages to intercept), adversaries could arbitrarily +increase the number of messages via edge insertion. Moreover, the number of simple paths in the +graph can be huge. We argue, however, that (1) graphs are usually sparse, (2) the number of paths +can be reduced via sparsification, and (3) we have to compute paths only once for each graph. +Limitations of ablation certificates. Since the probability to receive messages from perturbed nodes +increases the more nodes are adversarial, ∆ is monotonously increasing in ρ. Thus, the certifiable +radius is bounded independent of the label probabilities (uncertifiable region for ∆ ≥ 0.5 due to +Corollary 1). This bound depends on the graph structure and changes for each target node, but in the +case of node feature ablation smoothing we can directly determine the bound (Proof in Appendix I): +Proposition 3. Given fixed pa > 0 and pd = 0, it is impossible to certify a radius ρ if pa ≤ +ρ√ +0.5. +This bound is only determined by the parameters of the smoothing distribution (pd, pa) and does not +depend on the base GNN f. The existence of an upper bound is in stark contrast to certificates whose +largest certifiable radius depends on the inverse Gaussian CDF of the label probabilities (Cohen +et al., 2019). Such certificates are theoretically tighter than ablation certificates: For example, if +the base classifier f classifies all samples from φ the same (py∗ = 1), they would certify a radius +of ∞, whereas the radius of ablation-based certificates is bounded. We leave the development of even +stronger gray-box certificates for GNNs to future work. +Limitations of probabilistic certificates. Our certificates are probabilistic and hold with significance +level α. Notably, our method still yields strong guarantees for significantly smaller confidence levels +(we show additional experiments for varying α in Appendix H). We found that α has just a minor +effect on the certificate strength, since increasing it cannot increase the largest certifiable radius, +which is theoretically bounded. Recent works also “derandomize” probabilistic certificates, that is +they compute the label probabilities exactly (Levine and Feizi, 2020a, 2021). In Appendix J we +propose the first derandomization technique that leverages message-passing structures. We believe +future work can build upon it towards even more efficient derandomization schemes. +Threat model extensions. Notably, edge-deletion smoothing (pd > 0) also yields guarantees for +adversarial node insertion and deletion, as disconnected nodes cannot alter the prediction.3 As +discussed above, we can only evaluate such certificates with structural information, that is how +inserted/deleted nodes are connected to target nodes: Given clean graphs (as in our evaluation), we +know which nodes adversaries could delete. Given perturbed graphs, we know which nodes could +have been inserted. Note that although we can technically extend our method to certify adversarial +edge deletion, we focus on the novel problem of arbitrary feature manipulations of entire nodes since +there are already certificates against edge-modification attacks (Bojchevski et al., 2020). +3We cannot certify node insertion/deletion with feature ablation smoothing, since e.g. new nodes affect the +smoothed classifier independent of whether features are ablated or not (unless we delete nodes entirely). +6 + +7 +Experimental Evaluation +We evaluate our certificates for different GNN architectures trained on node classification datasets. +Our certificates work in standard transductive learning settings used throughout the literature and we +report such results in Appendix H. However, combining transductive learning with an evasion threat +model comes with serious shortcomings for the evaluation of certificates, since no separate test data is +available. For example, we can usually achieve high accuracy by overfitting a Multi-Layer Perceptron +(MLP) to labels predicted by GNNs during training. MLPs do not propagate information through +the graph at test time and are robust to adversarial messages. Instead, we evaluate our certificates in +semi-supervised inductive learning settings with hold-out test nodes: +Experimental setup. As labelled nodes, we draw 20 nodes per class for training and validation, and +10% of the nodes for testing. We use the labelled training nodes and all remaining unlabeled nodes as +training graph, and successively insert (hold-out) validation and test nodes. We train on the training +graph, optimize hyperparameters against validation nodes, assume adversaries control nodes at test +time, and compute certificates for all test nodes. We also delete edges and ablate node features during +training (Appendix G). We use n0 = 1,000 samples for estimating the majority class, n1 = 3,000 +samples for certification, and set α = 0.01. We conduct five experiments for random splits and model +initializations, and report averaged results including standard deviation (shaded areas in the plots). +When comparing settings (e.g. architectures), we run 1,000 experiments for each setting and draw +deletion and ablation probabilities from [0, 1] for each experiment (sampling separately for training +and inference). Then, we compute dominating points on the Pareto front for each setting. For brevity, +we only show points with clean accuracy of at most 5% below the maximally achieved performance. +Datasets and models. We train our models on citation datasets: Cora-ML (Bojchevski and Günne- +mann, 2018; McCallum et al., 2000) with 2,810 nodes, 7,981 edges and 7 classes; Citeseer (Sen et al., +2008) with 2,110 nodes, 3,668 edges and 6 classes; and PubMed (Namata et al., 2012) with 19,717 +nodes, 44,324 edges and 3 classes. We implement smoothed classifiers for four architectures with +two message-passing layers: Graph convolutional networks (GCN) (Kipf and Welling, 2017), graph +attention networks (GAT and GATv2) (Velickovic et al., 2018; Brody et al., 2022), and soft medoid +aggregation networks (SMA) (Geisler et al., 2020). More details in Appendix G. We also compute +certificates for the larger graph ogbn-arxiv (Hu et al., 2020) in Appendix H. +Evaluation metrics. We report the classification accuracy of the smoothed classifier on the test +set (clean accuracy), and the certified ratio, that is the number of test nodes whose predictions are +certifiable robust for a given radius. Since all nodes have different receptive field sizes, we also divide +the certifiable radius by the receptive field size. The resulting normalized robustness better reflects +how much percentage of the “attack surface” (that is the number of nodes the adversary could attack) +can be certified. Moreover, we report the area under this (normalized) certified ratio curve (AUCRC). +For completeness, we also report the certified accuracy in Appendix H, that is the number of test +nodes that are correctly classified (without abstaining) and certifiable robust for a given radius. +0 +1 +2 +3 +4 +5 +6 +7 +Perturbed nodes +25 +50 +75 +Cert. ratio (%) +(a) +distance ≥ 2 +distance ≥ 1 +0% +20% +40% +60% +Perturbed nodes +25 +50 +75 +Cert. ratio (%) +(b) +distance ≥ 2 +distance ≥ 1 +1 +500 +1,500 +2,500 +d +10 +20 +30 +40 +AUCRC (%) +(c) +ours +(Bojchevski et al., 2020) +Figure 3: Smoothed GAT on Cora-ML: (a) Robustness at different distances to target nodes (pd=0.31, +pa=0.794, with skip, ACC=0.79). (b) Robustness normalized by receptive field size (“attack surface”). +(c) Naïve baseline comparison (base certificate (Bojchevski et al., 2020), 105 samples, α=0.01). +Message-interception smoothing. In Figure 3 (a,b) we demonstrate our certificates for specific +edge deletion probabilities pd and node feature ablation probabilities pa. By making our certificates +message-passing aware, we can (1) certify robustness against arbitrary feature perturbations of entire +nodes, (2) analyze robustness locally in the receptive fields by incorporating the “attack surface”, and +(3) provide stronger guarantees for attacks against nodes at larger distances to target nodes. +7 + +First certificate for stronger adversaries. Experimentally we obtain significantly better robustness +guarantees compared to previous (smoothing-based) certificates for Graph Neural Networks. Specifi- +cally, existing certificates for GNNs only certify perturbations to a few attributes ˜ρ in the entire graph. +Our certificates are novel as they provide guarantees for much stronger adversaries that can arbitrarily +manipulate features of a multiple nodes in the graph. To compare these two approaches, consider a +naïve baseline that certifies ρ = ˜ρ/d nodes, where d is the number of attributes per node.4 If each +node in the graph had just a single feature, the number of certifiable nodes ρ is high. As the number +of features d per node increases, however, the baseline dramatically deteriorates. In contrast, our +certificates are entirely independent of the dimension d and hold regardless of how high-dimensional +the underlying node data might be. We demonstrate this comparison in Figure 3 (c) for the first +smoothing-based certificate for GNNs (Bojchevski et al., 2020), assuming attribute deletions against +second-hop nodes (p+=0, p−=0.6). However, the superiority of our certificate regarding robustness +against all features of entire nodes holds for any other GNN certificate proposed so far. +0 +1 +2 +3 +4 +5 +6 +Perturbed nodes +25 +50 +75 +Cert. ratio (%) +(a) +sparsification +w/o sparsific. +0% +20% +40% +60% +Perturbed nodes +25 +50 +75 +Cert. ratio (%) +(b) +sparsification +w/o sparsific. +0 +1 +2 +3 +4 +5 +Perturbed nodes +25 +50 +75 +Cert. ratio (%) +(c) +n1=10 +n1=20 +n1=50 +n1=100 +n1=300 +n1=1,000 +Figure 4: (a,b) Sparsification significantly improves certifiable robustness of our gray-box certificates +to second-hop attacks since sparsification reduces (a) messages to intercept, and (b) receptive field +sizes and thus the “attack surface” (Smoothed GAT, Cora-ML, pd = 0.31, pa = 0.71, with skip- +connection, ACC = 0.8). (c) Our certificate with largest certifiable radius of 4 with varying samples +for certification (Smoothed GAT, Cora-ML, pd = 0, pa = 0.85). Our certificates are more sample +efficient than existing smoothing-based certificates for GNNs. +Stronger certificates for sparser graphs. Notably, our gray-box certificates incorporate graph +structure and become stronger for sparser graphs. This is in contrast to black-box certificates that +ignore the underlying message-passing principles of GNNs. We demonstrate this by applying graph +sparsification, which significantly improves robustness while retaining high clean accuracy: First, +sparsification reduces the number of paths in the graph and thus reduces the number of messages +to intercept. Second, sparsification reduces the number of nodes in the receptive fields and thus the +“attack surface”, that is the number of nodes that send messages. In Figure 4 (a,b) we apply GDC +preprocessing (Gasteiger et al., 2019) to the Cora-ML graph at test time. GDC preprocessing yields +directed graphs and reduces the number of edges in the graph from 15,962 to 14,606 (we set the +sparsification threshold of GDC to ϵ = 0.022 and ignore resulting edge attributes). Interestingly, +evaluating the model on the sparsified graph yields significantly higher certifiable robustness, although +both approaches show high clean accuracy of 80%. Note that for the validity of our certificates we +assume adversaries perturb nodes after sparsification and cannot attack the sparsification itself. +Efficient message-interception smoothing. Drawing Monte-Carlo samples from φ to estimate the +smoothed classifier is usually the most costly part when computing smoothing-based certificates +(Cohen et al., 2019). In Figure 4 (c) we show that our certificates are much more sample efficient as +we do not benefit from more than a few thousand samples from φ. This is in stark contrast to existing +smoothing-based certificates for GNNs (Bojchevski et al., 2020). For a fair comparison, we adopt +their transductive setting and compute certificates for pd = 0.3 and pa = 0.85. Bojchevski et al. +(2020) use 106 Monte-Carlo samples for certifying test nodes on Cora-ML, which takes up to 25 +minutes. In contrast, our certificates saturate already for 2,000 Monte-Carlo samples in this setting, +which takes only 17 seconds (preprocessing Cora-ML takes 8 additional seconds). Our gray-box +certificates are significantly more sample-efficient while also providing guarantees against much +stronger adversaries. We hypotheise that our certificates saturate much faster as the certifiable radius +does not depend on the inverse Gaussian CDF of the label probabilities as discussed in Section 6. +4We are the first to certify such strong adversaries. Thus no baselines exist so far and we compare our method +against existing certificates for GNNs using the naïve baseline we propose above. +8 + +0% +10% +20% +30% +AUCRC +78 +80 +82 +Clean ACC (%) +(a) +GAT +GATv2 +GCN +SMA +0% +10% +20% +30% +40% +AUCRC +78 +80 +82 +Clean ACC (%) +(b) +with skip-connection +w/o skip-connection +0% +10% +20% +AUCRC +78 +80 +82 +Clean ACC (%) +(c) +pt = pe−0.1 +pt = pe +pt = pe+0.1 +Figure 5: Second-hop attacks on Cora-ML: (a) Robustness-accuracy tradeoffs for different GNN archi- +tectures. (b) Skip-connections yield improved robustness-accuracy tradeoffs for node feature ablation +smoothing. (c) Ablating less during training yields better robustness-accuracy tradeoffs (GAT). +Different classifiers. In Figure 5 (a) we compare robustness-accuracy tradeoffs for different GNNs +against second-hop attacks. Attention-based message-passing GNNs (Velickovic et al., 2018) are +dominating. We hypothesize that the degree-normalization of GCN (Kipf and Welling, 2017) may be +problematic for the performance under randomized edge deletion. Our approach may promote novel +message-passing architectures, specifically designed for smoothed classifiers. +Skip-connections. With higher node feature ablation probability, more messages from the target +node itself will be intercepted, which may be detrimental for the accuracy. Assuming adversaries do +not attack target nodes, we can modify the architecture for improved robustness-accuracy tradeoffs +(Figure 5b). To this end, we forward the non-ablated input graph through the GNN without edges, +and add the resulting final representation of each node to the final representation when forwarding the +(ablated) graph with graph structure. We use the same weights of the base GNN, but more complex +skip-connections are straightforward. Such skip-connections yield better robustness-accuracy trade- +offs against second-hop attacks, but we also loose guarantees for the target node itself. To account for +that, future work could deploy existing smoothing methods for features of target nodes separately: +e.g., if nodes represent images, we could deploy Gaussian smoothing (Cohen et al., 2019) on node +features send through the skip-connection and still obtain robustness guarantees for target nodes. +Training-time smoothing parameters. In Figure 5 (c) we show that ablating less during training +can improve the robustness-accuracy tradeoffs. Note that only inference-time smoothing parameters +determine the strength of our certificates, and the probabilities pd, pa during training are just hyperpa- +rameters that we can optimize to improve the robustness-accuracy tradeoffs. In detail, we experiment +with three different settings: Using the same ablation probabilities during training and inference +(pt = pe), ablating 10% more during training (pt = pe+0.1), and ablating 10% less during training +(pt=pe−0.1). Note that we use max(min(pt, 1), 0) to project the training-time parameters into [0, 1]. +0% +10% +20% +30% +AUCRC +78 +80 +82 +Clean ACC (%) +(a) +pd>0,pa>0 +pd>0,pa=0 +pd=0,pa>0 +0% +10% +20% +30% +40% +AUCRC +74 +76 +78 +Clean ACC (%) +(b) +pd>0,pa>0 +pd>0,pa=0 +pd=0,pa>0 +0% +10% +20% +30% +40% +AUCRC +72 +74 +76 +Clean ACC (%) +(c) +pd>0,pa>0 +pd>0,pa=0 +pd=0,pa>0 +Figure 6: Robustness-accuracy tradeoffs for second-hop attacks against smoothed GAT models +(without skip). Edge deletion and node ablation dominates on Cora-ML (a) and Citeseer (b). On +PubMed (c), edge deletion is stronger. Lines connect dominating points on the Pareto front. +Robustness-accuracy. We compare robustness-accuracy tradeoffs of three different settings: (1) edge +deletion and feature ablation (pd > 0, pa > 0), (2) edge deletion only (pd > 0, pa = 0), and +(3) feature ablation only (pd = 0, pa > 0). Our experiments show that edge deletion and feature +ablation smoothing achieves significantly better robustness-accuracy tradeoffs against attribute attacks +to the second-hop neighborhood and dominates on Cora-ML and Citeseer (Figure 6b,c). On PubMed, +edge deletion smoothing dominates. More results (e.g. with skip-connections) in Appendix H. +9 + +8 +Related Work +GNN robustness. The vast majority of GNN robustness works focus on heuristic defenses, including +adversarial graph detection (Zhang and Ma, 2020; Zhang et al., 2019a); architecture modifications +(Brody et al., 2022; Zhang et al., 2019b); robust aggregations (Geisler et al., 2020); robust training +procedures (Xu et al., 2019; Zügner and Günnemann, 2019), transfer learning (Tang et al., 2020); +and graph preprocessing techniques such as edge pruning (Zhang and Zitnik, 2020; Wu et al., 2019), +low-rank approximations (Entezari et al., 2020), and graph anomaly detection (Ma et al., 2021). +The effectiveness of such seemingly robust defenses on the adversarial robustness of GNNs can only +be assessed against existing adversarial attacks. Heuristic defenses do not guarantee robustness, and +may even be broken by stronger attacks later on (Mujkanovic et al., 2022). Instead, we are interested +in robustness certificates that provably guarantee the stability of predictions. However, robustness +certificates for GNNs are still in their infancy (Günnemann, 2022): +Certificates for GNNs. Most certificates for GNNs are designed for specific architectures (Zügner +and Günnemann, 2020; Jin et al., 2020; Bojchevski and Günnemann, 2019; Zügner and Günnemann, +2019). Despite providing provable robustness guarantees, their applicability is limited to specific +architectures. Bojchevski et al. (2020) present the first tight and efficient smoothing-based, model- +agnostic certificate for graph-structured data. However, their method comes with crucial limitations: +First, their method cannot certify robustness against arbitrary feature modifications of entire nodes. +Second, their black-box certificate deletes edges but completely ignores the underlying message- +passing principle. Third, their certificate requires an expensive evaluation of the smoothed classifier, +which questions the practicability of their certificate beyond theoretical robustness assessments. +Randomized ablation certificates for image classifiers (Levine and Feizi, 2020b) are another approach +for discrete data. Such certificates have already been applied to point cloud classifiers (Liu et al., +2021) and even for individual attribute perturbations in GNNs (Bojchevski et al., 2020). However, +Bojchevski et al. (2020) show that their method outperforms such ablation certificates for individual +attributes. In contrast, we propose to certify entire nodes, instead of only a few of their attributes. As +already discussed, applying their ablation certificates for image classifiers directly to GNNs comes +with serious shortcomings that we overcome (Section 4 and details in Appendix B). +Gray-box certificates. Exploiting model knowledge to derive tighter randomized smoothing certifi- +cates constitutes a widely unexplored research problem. The first works derive tighter guarantees +using information about the model’s gradients (Mohapatra et al., 2020; Levine et al., 2020). Recently +proposed collective certificates (Schuchardt et al., 2021) incorporate knowledge about the receptive +fields of GNNs. Their certificates are orthogonal to ours, and our certificates could lead to significant +improvements in such collective settings, as adversaries cannot attack first-hop neighbors of all +nodes simultaneously. Schuchardt and Günnemann (2022) propose tight gray-box certificates for +models that are invariant to spatial transformations. +9 +Conclusion +We propose novel gray-box, message-passing aware robustness certificates for GNNs against strong +threat models where adversaries can arbitrarily manipulate all features of multiple nodes in the +graph. The main idea of our certificates is to intercept adversarial messages by randomly deleting +edges and/or masking features of entire nodes. Our certificates are significantly stronger and more +sample-efficient than existing methods. Future enhancements could smooth specific edges and +nodes with different probabilities, for example to intercept messages from central nodes with higher +probability. Our gray-box certificates could lead to novel architectures, training techniques and graph +preprocessing techniques to further strengthen the robustness of GNNs against adversarial examples. +Acknowledgments and Disclosure of Funding +This work has been funded by the German Federal Ministry of Education and Research, the Bavarian +State Ministry for Science and the Arts, and the German Research Foundation, grant GU 1409/4-1. +The authors of this work take full responsibility for its content. +10 + +References +Aleksandar Bojchevski and Stephan Günnemann. 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For all authors... +(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s +contributions and scope? [Yes] All claims in abstract and introduction reflect the +contributions and scope of our paper. We also provide a list of our core contributions +directly in our introduction. +(b) Did you describe the limitations of your work? [Yes] We discuss the limitations of our +approach in Section 6. +(c) Did you discuss any potential negative societal impacts of your work? [Yes] Without +doubt, adversarial attacks can have negative impacts on the society. Particularly +alarming are recent attacks against GNNs for more realistic threat models (Ma et al., +2020) and attacks that scale to large graphs (Geisler et al., 2021). Robustness certificates +are tools to assess robustness and help to (1) better understand robustness, (2) build +more robust classifiers, and (3) eventually prevent adversarial attacks including their +negative consequences. Our certificates represent a contribution towards diminishing +and preventing potential negative impacts of adversarial attacks on the society. +(d) Have you read the ethics review guidelines and ensured that your paper conforms +to them? [Yes] We carefully reviewed the ethics guidelines (https://neurips. +cc/public/EthicsGuidelines). Our paper conforms to all ethic guidelines. Our +certificates represent a contribution to prevent negative social impacts of adversarial +attacks, as discussed above. +2. If you are including theoretical results... +(a) Did you state the full set of assumptions of all theoretical results? [Yes] We state all +assumptions of our theoretical results. +(b) Did you include complete proofs of all theoretical results? [Yes] We show all statements, +including additional elaborations, in the Appendix. We always link theoretical results +in the paper to the corresponding proofs in the Appendix. +3. If you ran experiments... +(a) Did you include the code, data, and instructions needed to reproduce the main experi- +mental results (either in the supplemental material or as a URL)? [Yes] We uploaded +the code required to reproduce our main results. All required datasets are publicly +available, and can be loaded for example with PyTorch Geometric (Fey and Lenssen, +2019). +(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they +were chosen)? [Yes] We describe important training details directly at the beginning +of our experiment section (Section 7). We further thoroughly list all training details +in Appendix G. +(c) Did you report error bars (e.g., with respect to the random seed after running ex- +periments multiple times)? [Yes] We repeat each experiment for five random splits +and model initializations, only report averaged results, and our plots explicitly show +the standard deviation over the five experiments (shaded areas in plots). In separate +experiments, we tested our method for another set of five randomly drawn seeds, but +we did not observe significant differences in the results. We also uploaded all seeds to +ensure reproducibility. +(d) Did you include the total amount of compute and the type of resources used (e.g., type +of GPUs, internal cluster, or cloud provider)? [Yes] We discuss the runtime for each +experiment in our experiment section (Section 7). We conduct all experiments in an +internal cluster with the following GPU type: NVIDIA GeForce GTX 1080 Ti. +4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets... +(a) If your work uses existing assets, did you cite the creators? [Yes] Yes, we cite all +authors of the assets we use. +(b) Did you mention the license of the assets? [N/A] Our datasets are well-established +research datasets with MIT License or public domain. +14 + +(c) Did you include any new assets either in the supplemental material or as a URL? +[Yes] We provide supplemental materials and code to reproduce our main results at +https://www.cs.cit.tum.de/daml/interception-smoothing. +(d) Did you discuss whether and how consent was obtained from people whose data you’re +using/curating? [N/A] Our assets do not require consent. +(e) Did you discuss whether the data you are using/curating contains personally identifiable +information or offensive content? [N/A] Our assets do not contain any personal, +protected or offensive data. +5. If you used crowdsourcing or conducted research with human subjects... +(a) Did you include the full text of instructions given to participants and screenshots, if +applicable? [N/A] We do not conduct research with human subjects. +(b) Did you describe any potential participant risks, with links to Institutional Review +Board (IRB) approvals, if applicable? [N/A] We do not conduct research with human +subjects. +(c) Did you include the estimated hourly wage paid to participants and the total amount +spent on participant compensation? [N/A] We do not conduct research with human +subjects. +15 + +A +Proofs Main Certificate (Section 4) +Proposition 1. Given target node v in graph G, and adversarial budget ρ. Let E denote the event +that the prediction fv(φ(G)) receives at least one message from perturbed nodes. Then the change +in label probability |pv,y(G) − pv,y(G′)| is bounded by the probability ∆ = p(E) for all classes +y ∈ {1, . . . , C} and graphs G′ with G′ ∈ Bρ(G): |pv,y(G) − pv,y(G′)| ≤ ∆. +Proof. For a thorough formal proof in the context of image classifiers see (Levine and Feizi, 2020b). +Here, we show the statement in the context of GNNs: Consider a fixed target node v. We exploit that +whenever we intercept all adversarial messages (i.e. nodes are disconnected or we mask out their +features), the adversary cannot alter the prediction. Let ¯E denote the event that v does not receive +any message from perturbed nodes. Then we have for any class y: +p(fv(φ(G)) = y | ¯E) = p(fv(φ(G′)) = y | ¯E) +since all input representations with respect to G and G′, which affect the prediction for v, are the same +if all perturbed nodes are ablated or disconnected (i.e. their messages are intercepted). Multiplying +with p( ¯E) yields: +p(fv(φ(G)) = y ∧ ¯E) = p(fv(φ(G′)) = y ∧ ¯E) +(1) +Following the arguments of (Levine and Feizi, 2020b): +pv,y(G) − pv,y(G′) +(1) += p(fv(φ(G)) = y ∧ E) + p(fv(φ(G)) = y ∧ ¯E) − pv,y(G′) +(2) += p(fv(φ(G)) = y ∧ E) + p(fv(φ(G′)) = y ∧ ¯E) − pv,y(G′) +(3) += p(fv(φ(G)) = y ∧ E) − p(fv(φ(G′)) = y ∧ E) +≤ p(fv(φ(G)) = y ∧ E) +(4) +≤ p(E) +where (1) and (3) follow from the law of total probability, (2) is due to inserting Equation 1, and (4) +follows from p(A ∩ B) ≤ p(B) for any events A and B. +Analogously, pv,y(G′) − pv,y(G) ≤ p(E). Thus: |pv,y(G) − pv,y(G′)| ≤ p(E) = ∆ +Lemma 1. Given a fixed target node v and perturbed nodes B in the graph with v /∈ B. Then +fv(φ(G)) = fv(φ(G′)) for any graph G′ ∈ Bρ(G) if +∀w ∈ B : +� +∀p ∈ P k +wv : ∃(i, j) ∈ p : φ1(A)ij = 0 +� +∨ (φ2(xw) = t) +Proof. The prediction fv(φ(G)) cannot differ from fv(φ(G′)) if for all perturbed nodes w ∈ B we +have (1) w is disconnected from the target node v, or (2) the features of w are ablated. If the smoothing +distribution φ1 deletes an edge (i, j) (that is φ(A)ij = 0), the neighborhood N(j) changes, and thus +messages from i to j get intercepted on all GNN layers. That is, the final hidden representation h(k) +v +of a target node v can only be changed by some non-ablated perturbed source node w if there is at +least one simple path from w to v of length at most k such that no edge on this path is deleted. +Theorem 1. The worst-case change in label probability |pv,y(G) − pv,y(G′)| is bounded by +∆ = +max +||ρv||0≤ρ p (E(ρv)) +for all classes y ∈ {1, . . . , C} and any graph G′ ∈ Bρ(G). +Proof. Note the difference: +• E denotes the event that at least one message from perturbed nodes reaches a target node v +• E(ρv) denotes the event that at least one message from nodes indicated by ρv reaches a +target node v +16 + +Put differently, the maximization amounts to the additional worst-case assumption that the adversary +selects those nodes whose messages have the highest chance of getting to the target node. Importantly, +we have to make this additional worst-case assumption to obtain valid robustness certificates for our +threat model. +Since the probability ∆ bounds the worst-case change |pv,y(G) − pv,y(G′)| for all classes y, we can +utilize ∆ to construct robustness certificates: Intuitively, ∆ bounds how much probability mass of +the distribution pv,y(G) over labels y is compromised by the worst-case adversary: If an adversary +cannot shift enough probability mass to change the majority class, our smoothed classifier is robust: +Corollary 3 (Binary Certificate). Given ∆ as defined in Then we can certify the robustness gv(G) = +gv(G′) for any graph G′ ∈ Bρ(G) if +pv,y∗(G) − ∆ > 1 +2 +where y∗ ≜ gv(G) denotes the majority class predicted by smoothed classifier g. +Proof. Recall that ∆ bounds how much probability mass of the distribution pv,y(G) over y is +compromised by the adversary. Let y∗ ≜ g(G) denote the majority class, that is pv,y∗(G) > 1 +2 in +this binary classification setting. Thus, to change the majority class, the adversary needs to shift +enough probability mass from the majority class y∗ to the other class 1 − y∗. This is impossible if +pv,y∗(G) − ∆ > 1 +2, meaning the adversary cannot shift enough probability mass for a successful +attack. Put differently, even in the worst-case that the adversary always changes the prediction +whenever adversarial messages reach the target node, the majority class cannot be altered. +Corollary 1 (Multi-class certificate). Given ∆ as defined in Proposition 1. Then we can certify the +robustness gv(G) = gv(G′) for any graph G′ ∈ Bρ(G) if +pv,y∗(G) − ∆ > max +˜y̸=y∗ pv,˜y(G) + ∆ +where y∗ ≜ gv(G) denotes the majority class, and ˜y the follow-up (second best) class. +Proof. To prove this, we utilize the same arguments as in the binary setting above. Here, given +pv,y∗(G) − ∆ > max˜y̸=y∗ pv,˜y(G) + ∆, the adversary does not control enough probability mass +of pv,y(G) over y to alter the second-best class ˜y into the new majority class when classifying the +perturbed graph G′. +Corollary 2. We guarantee gv(G) = gv(G′) with probability of at least 1 − α for any G′ ∈ Bρ(G) +if pv,y∗(G) − ∆ > pv,˜y(G) + ∆, where y∗ denotes the majority class, and ˜y the follow-up class. +Proof. We have pv,y∗(G)−∆ ≥ pv,y∗(G)−∆ > pv,˜y(G)+∆ ≥ pv,˜y(G)+∆ due to the assumption +pv,y∗(G) − ∆ > pv,˜y(G) + ∆. The remaining claim follows from Corollary 1 and from the fact that +both bounds hold with significance level α. +17 + +B +Theoretical Connection to Randomized Ablation for Image Classifiers +Our gray-box certificates for GNNs are theoretically related to the randomized ablation black-box +certificates for image classifiers. In this section we thoroughly analyze the differences with more +technical insights and carefully discuss how our certificates go beyond theirs. Specifically, we show +that our gray-box certificates yield stronger guarantees, and are provably tighter even in the special +case without additional edge deletion smoothing. In the following we introduce their certificate again, +discuss the differences to our certificate, and eventually prove that our guarantees are tighter. +Randomized Ablation. +Levine and Feizi (2020b) introduce randomized ablation for image clas- +sifiers as follows: They define the space B(n, k) ≜ {M : M ∈ P({1, . . . , n}) ∧ |M| = k} of all +pixel-subsets with exactly k of n total pixels (P denoting the power set here). Then, their smoothing +distribution ablates all but k pixels in a uniformly drawn subset M ∈ B(n, k). They define ∆L as +the probability to keep (not ablate) perturbed pixels in the image under this smoothing distribution. +Assuming ρ perturbed pixels in an image: +∆L = 1 − +�n−ρ +k +� +�n +k +� +Discussion. +There are various ways of applying such black-box certificates for image classifiers to +certify the robustness of GNNs. One way is to use them to certify threat models where adversaries +control individual attributes all over the graph (Bojchevski et al., 2020). We are interested in certifying +robustness to adversaries that control all features of entire nodes in the graph instead. However, +applying the smoothing distribution of Levine and Feizi (2020b) for certifying robustness to our threat +model (that is by ablating entire node vectors) comes with several deficiencies, as their smoothing +distribution is specifically designed for image classifiers. Most importantly, applying their certificate +for image classifiers to GNNs results in black-box certificates that completely ignore the underlying +message-passing principle. +In contrast, we propose gray-box certificates – we partially open the black-box and consider the +underlying message-passing principle and paths in the graph, that is A and A2. This comes with +two crucial advantages as we show experimentally in Section 7: First, additionally deleting edges +leads to significantly better robustness guarantees for attacks against more distant nodes. Second, our +certificates become increasingly stronger for sparser graphs (while their certificate applied to GNNs +remains unchanged as it ignores graph structure). +B.1 +Special Case of Node Feature Ablation Smoothing +Notably, our certificates are provably tighter even without edge deletion smoothing. Specifically, we +formally show the difference between our ∆ for node feature ablation smoothing and ∆L of Levine +and Feizi (2020b) when naively applying their approach to GNNs by randomly ablating features of +entire nodes (instead of pixels in an image). Specifically, while their smoothing distribution samples +exactly k out of n nodes not to ablate (to keep), our smoothing distribution samples k out of n nodes +in expectation. This eventually leads to ∆ < ∆L. We start by characterizing our certificate for node +ablation smoothing: +Proposition 2. For node feature ablation smoothing only (pd = 0), we have ∆ = 1 − pρ +a. +Proof. Recall the definition of the probability ∆: E denotes the event that at least one perturbed +message reaches a target node v, and ∆ ≜ p(E). When only ablating nodes (pd = 0), all nodes are +equally important for the prediction fv(φ(G)), since messages are only intercepted in the input layer, +not during the message-passing itself. +We therefore do not have an optimization problem as in Theorem 1. Instead, the probability ∆ to +receive perturbed messages is just the probability that at least one perturbed node is not ablated. +Further, the complementary event denotes that all ρ perturbed nodes are ablated, whose probability is +just pρ +a. Thus ∆ = 1 − pρ +a. +18 + +Moreover, the multiplicative bound is tight in the special case of node ablation smoothing: +Proposition 4. For pd = 0, the multiplicative bound is tight ∆M = ∆. +Proof. We have +∆i +(1) += +� +�1 − +� +q∈P k +wv +� +1 − (1 − pd)|q|� +� +� (1 − pa) +(2) += 1 − pa +where (1) is by definition, and (2) due to our assumption pd = 0. Therefore: +∆M = 1 − +ρ +� +i=1 +(1 − ∆i) = 1 − +ρ +� +i=1 +pa = 1 − pρ +a = ∆ +where the first equality is due to definition again, and the last equality follows from Proposition 2. +Proposition 5 (Tighter guarantees). Given adversarial budget ρ > 1. Further assume k > 0. Let ∆L +denote the bounding constant for the smoothing distribution proposed by Levine and Feizi (2020b). +Then ∆ < ∆L. +Proof. Recall that due to uniform ablation we have (compare Levine and Feizi (2020b)): +∆L = 1 − +�n−ρ +k +� +�n +k +� +To compare this to our ∆ = 1 − pρ +a of Proposition 2, we first need to introduce k and n. We note +that pa is the probability to ablate a single node. We thus have pa = 1 − k +n, where k +n amounts to the +probability to “keep” (not ablate) a node. In this setting, we keep n k +n = k nodes in expectation. We +therefore have: +∆ = 1 − pρ +a = 1 − +� +1 − k +n +�ρ +We observe: +�n−ρ +k +� +�n +k +� += (n − ρ)!(n − k)! +n!(n − ρ − k)! = +ρ−1 +� +i=0 +n − k − i +n − i +(1) +< +�n − k +n +�ρ += +� +1 − k +n +�ρ +where (1) is due to the mediant inequality (ρ > 1 and k > 0): +∀y < x ∀i > 0 : +y − i +x − i < y +x +We conclude that ∆ < ∆L. +The difference decreases for larger n, but our smoothing distribution is significantly better for small +graphs/receptive fields: For example, for n = 10 and k = 1 (i.e. pa = 0.9), the largest certifiable +radius with our method is 6, but only 4 using their certificate. +In detail, there are two ways of applying their method for image classifiers to certify robustness of +GNNs against adversaries that control all features of entire nodes in the graph: by ablating all features +of k out of n uniformly chosen nodes (1) in the entire graph, or (2) locally in each receptive field. +Global randomized ablation. +Assume we uniformly ablate all features of k out of n nodes in the +entire graph. If the number of nodes n in the graph is large, the difference between ∆ and ∆L is +small. Still, the resulting black-box certificates only hold globally, not locally in the receptive fields. +Such certificates ignore the receptive fields, specifically that most nodes in the graph may not even be +connected to the target node. For example, in the most extreme case of A = 0 (meaning receptive +fields only consist of target nodes), their certificate applied to GNNs remains entirely unchanged +due to the black-box nature. In contrast, our gray-box certificates guarantee robustness for any ρ +(excluding target nodes) in this case (cf. normalized robustness in Section 7). +19 + +Local randomized ablation. +To remedy the black-box nature of their approach, one can obtain +local guarantees by ablating all features of k out of the n nodes locally in the receptive field of a +target node. However, our message-interception certificates are significantly tighter even without +edge deletion smoothing as receptive fields are typically small. We demonstrate this in Figure 7 +where our approach yields significantly stronger guarantees in practice (since Proposition 2 makes a +significant difference). +Note that when applying their approach to GNNs by ablating nodes locally, one also needs to consider +each receptive field individually and cannot use full-batch training/inference as usually implemented +for GNNs. Our message-interception certificates are easier to implement and more efficient as we +obtain local guarantees without considering and processing all receptive fields separately. +0% +20% +40% +Perturbed nodes +25 +50 +75 +Cert. ratio (%) +ours +Levine’s method applied to GNNs +Figure 7: Given pa = 0.72, we compare our certificate against the certificate proposed by Levine and +Feizi (2020b) by applying their smoothing distribution for image classifiers to GNNs (distance ≥ 1, +with skip-connection). We locally choose k = ⌊(n − 1) ∗ pa⌋ nodes not to ablate – where n − 1 is the +number of nodes in each receptive field, excluding the target node. Our certificates are experimentally +stronger even without additional edge deletion. +20 + +C +Closed-form via Inclusion-exclusion Principle +Recall that E(ρv) describes the event that v receives messages from any attacked node indicated by +the adversarial budget vector ρv ∈ {0, 1}n. Computing the probability p (E(ρv)) using edge deletion +probability pd and node feature ablation probability pa is challenging as it involves evaluating the +inclusion-exclusion formula. We formalize this expensive closed-form solution in the following: Let +Ew denote the probability to receive a message from node w, and let P indicate all simple paths from +any perturbed w with ρv(w) = 1 to target node v. Further, let Yi denote the probability to receive a +message via path i ∈ P. Then we have: +p (E(ρv)) = p +� +� +� +ρv(w)=1 +Ew +� +� = p +� � +i∈P +Yi +� +since the probability to receive a message from any attacked node equals the probability to receive +a message from any path i from an attacked node to the target node. We now apply the inclusion- +exclusion principle: +p +� � +i∈P +Yi +� += +|P| +� +k=1 +� +� +� +�(−1)k−1 � +I⊆P +|I|=k +p +�� +i∈I +Yi +� +� +� +� +� +(2) +The remaining probability can be expressed as follows: The probability to receive messages via all +paths indicated by I is the probability that (1) all edges on those paths are not deleted, and (2) the +corresponding source nodes of the paths are not ablated. Therefore: +p +�� +i∈I +Yi +� += (1 − pd)a(1 − pa)b +(3) +where a denotes the number of (unique) edges on all paths indicated by I, and b the number of +(unique) source nodes of the paths indicated by I. Note that the above derivation assumes that the +target node v is not controlled by the adversary. In such a case (ρv(v) = 1), we have p(Ev) = 1 − pa +(since we always receive messages from non-ablated target nodes) and: +p (E(ρv)) = p +� � +i∈P +Yi +� +Ev +� +p +�� +i∈I +Yi +� +Ev +� +(1) += p +�� +i∈I +Yi +� +p(Ev) +where (1) is due to independence. +There are different ways that take additional information into account to derive faster ways of +computing p (E(ρv)), for example by exploiting that the receptive fields are trees with the target +node v as root (compare Appendix D). In general, however, computing Equation 2 is expensive since +we have to evaluate Equation 3 exactly 2|P| times. +21 + +D +Tree-shaped Receptive Fields +Given fixed ρv ∈ {0, 1}n that indicates nodes controlled by the adversary. Recall that E(ρv) +describes the event that v receives at least one messages from any attacked node indicated by the +adversarial budget vector ρv ∈ {0, 1}n. If the receptive field for target node v is a tree, we can +compute ∆ of Theorem 1 exactly. Specifically, we first provide a recursive formula to compute +p (E(ρv)) and then show that the worst-case selection of nodes by the adversary is straightforward. +We introduce the following random variables to better describe the recursion: +• Let Ri denote the event that root node i receives an adversarial message. +• Let Ai denote the event that the features of node i are ablated. +• Let Di denote the event that root i receives an adversarial message via any of its adjacent +subtrees j ∈ B (“branches”). +• Let Bj further denote the event that we receive an adversarial message via branch j. +The main idea is that branches in a tree are independent: +Theorem 4. We start the recursion with the target node v to compute p(Rv) while following edges +away from the node (j, v) (against their direction). Then the following recursive equation computes +p (E(ρv)) for tree-shaped receptive fields: +p(Ri) ≜ +�1 − pa(1 − p(Di)) +if ρv(i) = 1 +p(Di) +else +with +p(Di) ≜ 1 − +� +(j,i) +(1 − p(Bj)) +p(Bj) ≜ (1 − pd)p(Rj) +Proof. We show the three equations consecutively: +1. For p(Ri): If root i is not controlled by the adversary, then the probability to receive an +adversarial message is just the probability that we receive such a message via any of its +adjacent subtrees, that is p(Ri) = p(Di). If root i is controlled by the adversary (ρv(i) = 1), +we can exploit independence between edge deletion smoothing φ1 and node feature ablation +smoothing φ2: +p(Ri) = p( ¯Ai ∨ Di) = 1 − p(Ai ∧ ¯Di) +(1) += 1 − p(Ai)p( ¯Di) = 1 − p(Ai)(1 − p(Di)) +where (1) is due to independence. Since the probability that we do not receive any adversarial +message from root i is the probability that the features of root i are ablated: p(Ai) = pa. +We therefore have: p(Ri) = 1 − pa(1 − p(Di)). +2. For p(Di): For the probability that root i receives an adversarial message via any of its +adjacent branches j ∈ B, we exploit independence between branches (which we can do +since we have trees): +p(Di) = p +� +� � +j∈B +Bj +� +� = 1 − p +� +� � +j∈B +¯Bj +� +� (1) += 1 − +� +j∈B +p( ¯Bj) = 1 − +� +j∈B +(1 − p(Bj)) +where (1) is due to independence. +3. For p(Bj): The probability to receive a message via branch j is the probability that the +edge from branch j to root i is not deleted (1 − pd) times the probability that we receive a +message via the next root j (recursive call). +For leaves we have B = ∅ and thus the product over j ∈ B is 1, that is p(Di) = 0 for all leaves. +22 + +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 10 11 +Perturbed nodes +25 +50 +75 +Cert. ratio +(a) +distance≥2 (tight delta) +distance≥2 (m. bound) +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 10 11 +Perturbed nodes +25 +50 +75 +Cert. ratio +(b) +distance≥2 (tight delta) +distance≥2 (m. bound) +Figure 8: Comparing multiplicative bound and tight tree bound (distance at least 2). (a) Tree-certificate +only for tree-shaped receptive fields. (b) Sparsifying all receptive fields into trees. +Interestingly, we can reconstruct the following special cases: +Special case of edge deletion smoothing. Assume pa = 0. Then we directly see that p(Ri) = 1 if +root i is controlled by the adversary. This means that the adversary controls the entire sub-tree if the +root node is already attacked. Put differently, the adversary does not need to control more parts of the +tree to change the prediction if the adversary already controls the root. +Special case of node feature ablation smoothing. Assume pd = 0. Then we can directly see +that resolving the recursion just multiplies the node feature ablation probabilities pa and we get +p (E(ρv)) = 1−pρ +a for ρ = ||ρv||0. This matches the special case already discussed in Proposition 2. +Worst-case selection of nodes. Recall that our certificates are conservative and assume the additional +worst-case that the adversary attacks those nodes in the receptive field that maximize the probability +that the target node receives a message from attacked nodes (maximization in Theorem 1). This +additional assumption is required to obtain valid certificates. Notably, this worst-case adversary is +straightforward for trees: First, an adversary would always prefer closer nodes over more distant +nodes to maximize the probability that messages are getting through. Second, an adversary would +always distribute its budget over different branches to exploit independence between branches, which +also maximizes the probability that messages are getting through (also compare Appendix E). +Experiments. We find that computing ∆ tight for tree-shaped receptive fields can increase the +certifiable radius in practice (compare Figure 8). Interestingly, 25% of nodes in Cora-ML have +receptive fields that are trees (considering 2-layer GNNs). We apply our recursive scheme above +to compute tight certificates in two settings: First, we only compute tight certificates for the nodes +whose receptive fields are trees. Second, we apply sparsification that successively deletes edges in +the graph until the receptive fields of all test nodes are trees. In detail, we train GAT models on +Cora-ML and apply sparsification at test time. We use the skip-connection, train with pa = 0.68, +pd = 0.02 and compute certificates with pa = 0.79, pd = 0.36. Without sparsification we achieve +clean accuracies of 79% on average, and 77% when applying sparsification at test time. +In practice, we found that the gain in computing ∆ exactly may be rather small, as adversaries +typically distribute their budget to different branches to increase the probability that their messages +arrive. This means adversaries maximize independencies between edges. In other words, the +multiplicative bound is already quite strong in practice, and specifically tight until the degree of the +node (given that each first-hop neighbor has at least one child). +23 + +E +Proofs of Section 5 +Figure 9: Visualization of two dependent (left) and independent paths (right). When randomly +deleting edges with the same edge deletion probability pd, the probability that all messages from +both source nodes are intercepted is lower when the paths are independent (more possibilities for the +message to get through). +We first prove a more general claim that we can use to prove the multiplicative bounds of Theorem 2 +and Theorem 3. Let Xi denote the event that target node v receives a message via any path s in a set +of paths Si such that all paths start at an arbitrary source node and end at target node v. Intuitively, it +is more likely to receive at least one messages via Si and one message via Sj when there are shared +edges, compared to when we assume their paths were independent. Put differently, the probability +that all messages from all paths are intercepted is higher when paths are dependent (cf. Figure 9). +More formally: +Theorem 5. For two arbitrary sets Si and Sj of simple paths with the same target node v we have +p(Xi)p(Xj) ≤ p(Xi ∧ Xj) +under the smoothing distribution φ1 for edge deletion. +Proof. We are interested in the probability that all messages via all paths are intercepted. Consider +the following two possibilities: +1. The paths in Si and the paths in Sj are (pairwise) independent, meaning there are no edges +that appear on both - on a path si ∈ Si and on a path sj ∈ Sj. +In this case we have p(Xi ∧ Xj) = p(Xi)p(Xj) due to independence. +2. Consider the scenario where there are at least two dependent paths that share a common +edge. If we assume they were independent, there would be more possibilities how a message +can get through than there actually are. In other words, assuming independence results in +lower probability that all messages via both sets get intercepted. +Thus p(Xi)p(Xj) < p(Xi ∧ Xj). +□ +Consider the following definition of positively associated random variables (Esary et al., 1967). +Definition 1. We call a random vector x = (X1, . . . , Xn) positively associated if +Cov(φ(x), ψ(x))) ≥ 0 +for all non-decreasing, element-wise functions φ, ψ such that second moments of ψ(x) and φ(y) exist. +The concept of positively associated random variables is for example used in physical statistics +(Goldstein and Wiroonsri, 2018). We can use this concept here to prove multiplicative bounds: +Corollary 4. The random vector x = (X1, . . . , Xn) is positively associated. +Proof. Due to Theorem 5 we have p(Xi)p(Xj) ≤ p(Xi ∧ Xj) and thus +⇒E[Xi]E[Xj] ≤ E[XiXj] +⇒E[XiXj] − E[Xi]E[Xj] ≥ 0 +⇒Cov(Xi, Xj) ≥ 0 +since Xi and Xj are binary random variables. +Thus, the elements of the covariance matrix are non-negative: Cov(¯x, ¯x) ≥ 0 (variance is always +non-negative). According to Theorem 4.2 in Esary et al. (1967), ¯x is positively associated. Since ¯x is +positively associated, it follows from (BP1) in Esary et al. (1967) that x is positively associated. +24 + +Proposition 6. Given random variables Xi as defined above. Then: +1 − p +� n +� +i=1 +Xi +� +≤ 1 − +n +� +i=1 +p +� +Xi +� +Proof. Since x and ¯x are positively associated random variables, we can use Theorem 4.1 in (Esary +et al., 1967) and conclude that +p +� n +� +i=1 +Xi +� +≥ +n +� +i=1 +p +� +Xi +� +⇔ 1 − p +� n +� +i=1 +Xi +� +≤ 1 − +n +� +i=1 +p +� +Xi +� +Theorem 2 (Single Source Multiplicative Bound). Given target node v and source node w ̸= v in +the receptive field of a k-layer message-passing GNN f with respect to v. Let P k +wv denote all simple +paths from w to v of length at most k in graph G. Then ∆w ≤ ∆w for: +∆w ≜ +� +�1 − +� +q∈P k +wv +� +1 − (1 − pd)|q|� +� +� (1 − pa) +where |q| denotes the number of edges on the simple path q ∈ P k +wv from w to v. +Proof. Note in the special case of the target node v = w we just have ∆w = 1 − pa, since the +features xv of the target node v are used for the prediction independent of any edges. +For any w ̸= v in the receptive field: Let Ew denote the event that the target node v receives messages +from node w, and ∆w ≜ p(Ew). We further introduce Aw for the event that the features of node w +are ablated, and Dw for the event that v receives at least one messages from w. Then we have: +∆w = p(Ew) = p( ¯Aw ∧ Dw) +(1) += p( ¯Aw)p(Dw) = (1 − pa)p(Dw) +where (1) holds since the two smoothing distributions for node feature ablation and edge deletion are +independent. We continue with p(Dw). Therefore, recall that P ≜ Pk +wv denotes the set of simple +paths from w to v. Further, let p(q) for simple path q ∈ P denote the probability that v receives a +message via path q. Clearly, a message “arrives” only via path q if none of the edges on that path is +deleted, that is when the node is connected via path q. Since the deletion of edges is independent, +p(q) = (1 − pd)|q|, where |q| denotes the number of edges on the simple path q. We derive: +p(Di) = p +� +� � +q∈P +q +� +� = 1 − p +� +� � +q∈P +q +� +� +We can use positive association to conclude +1 − p +� +� � +q∈P +q +� +� +(1) +≤ 1 − +� +q∈P +p (q) +where (1) follows from Proposition 6. Finally, we resolve the remaining terms: +1 − +� +q∈P +p (q) = 1 − +� +q∈P +(1 − p (q)) = 1 − +� +q∈P +� +1 − (1 − pd)|q|� +Due to (1) above, we finally get ∆w ≤ ∆w, where the inequality becomes an equality if all paths are +independent (that is the paths do not share edges). +Proposition 7. We have ∆w = ∆w for ℓ-layer GNNs with ℓ ≤ 2. +Proof. For ℓ-layer GNNs with ℓ ≤ 2, all paths from a single source to the target node are independent. +25 + +Theorem 3 (Generalized multiplicative bound). Assume an adversarial budget of ρ nodes and let +∆1, . . . , ∆ρ denote the ρ largest ∆i for nodes i in the receptive field. Then we have ∆ ≤ ∆M for +∆M ≜ 1 − +ρ +� +i=1 +(1 − ∆i) +Proof. We recall from Theorem 1: +∆ = +max +||ρv||1≤ρ p (E(ρv)) +where E(ρv) describes the event that target node v receives messages from any attacked node +indicated by ρv. Recall that Ew denotes the event that the prediction for target node v is based on +information of node w in the receptive field. We further have ∆w ≜ p(Ew). Then: +p (E(ρv)) = p +� +� +� +ρv(w)=1 +Ew +� +� = 1 − p +� +� +� +ρv(w)=1 +¯Ew +� +� +where we can apply Proposition 6 and use the assumption that paths from several source nodes to the +target were independent to obtain an upper bound: +1 − p +� +� +� +ρv(w)=1 +¯Ew +� +� ≤ 1 − +� +ρv(w)=1 +p +� ¯Ew +� +Further resolving the terms yields: +1 − +� +ρv(w)=1 +p +� ¯Ew +� += 1 − +� +ρv(w)=1 +(1 − p (Ew)) = 1 − +� +ρv(w)=1 +(1 − ∆w) +Since the above equations hold for any fixed ρv: +∆ = +max +||ρv||1≤ρ p (E(ρv)) ≤ +max +||ρv||1≤ρ 1 − +� +ρv(w)=1 +(1 − ∆w) +Assume we have ordered ∆w so that ∆i ≥ ∆i+1 for all i ∈ {1, . . . , ρ}. Then: +max +||ρv||1≤ρ 1 − +� +ρv(w)=1 +(1 − ∆w) = 1 − +ρ +� +i=1 +(1 − ∆i) = ∆M +Note that instead of ∆w we can alternatively use upper bounds ∆w, which yields an even looser +upper bound on ∆ since +1 − +ρ +� +i=1 +(1 − ∆i) ≤ 1 − +ρ +� +i=1 +� +1 − ∆i +� +26 + +F +Approximation Error +Notably, the multiplicative bound derived above is tighter than the following union bound: +Proposition 8 (Union Bound). Given monotonously decreasing ∆i such that ∆i ≥ ∆i+1. Then we +have ∆ ≤ ∆U for +∆U ≜ +ρ +� +i=1 +∆i +Proof. +p (E(ρv)) = p +� +� +� +ρv(w)=1 +Ew +� +� ≤ +� +ρv(w)=1 +p (Ew) = +� +ρv(w)=1 +∆w +∆ = +max +||ρv||1≤ρ p (E(ρv)) ≤ +max +||ρv||1≤ρ +� +ρv(w)=1 +p (Ew) = +ρ +� +i=1 +∆i +The union bound is quite loose, not a probability and can even grow larger than 1. We show the +difference in practice Figure 10 (a). We also discuss the approximation error between the upper +bounds ∆U, ∆M and the tight ∆ for the following constructed example where all paths are dependent: +We assume a setting where an adversary attacks only second-hop neighbors that are connected to the +target node via the same direct neighbor of the target node. With pa = 0 we have ∆ = (1−pd)(1−pρ +d) +since we only receive a message if the bottleneck edge is not ablated, and at least one edge of the +attacked second-hop nodes is not ablated (which is the complementary probability of all second-hop +edges are ablated). In this constructed case, all paths are dependent as they share the bottleneck edge. +We show how the upper bounds compare to the tight ∆ for different edge deletion probabilities pd in +Figure 10 (b). Note that the example is constructed and worst-case adversaries aim at maximizing +independencies by choosing nodes without bottleneck edges (in which case the multiplicative bound +is a strong bound in practice). +0% +20% +40% +Perturbed nodes +25 +50 +75 +Cert. ratio (%) +(a) +multiplicative bound +union bound +0 1 2 3 4 5 6 7 8 9 10 +Radii +0 +.25 +.50 +.75 +1 +∆ +(b) +∆ (pe = 0.4) +∆M (pe = 0.4) +∆U (pe = 0.4) +∆ (pe = 0.8) +∆M (pe = 0.8) +∆U (pe = 0.8) +Figure 10: (a) Multiplicative bound is tighter than union bound and provides stronger guarantees +(Smoothed GAT model on Cora-ML with pa = 0.85, pd = 0). (b) Constructed example: All +path share the same bottleneck edge: Comparing the tight ∆ against the union bound ∆U and the +multiplicative bound ∆M for different edge deletion probabilities pd. The multiplicative bound is +tighter than the union bound, which can grow larger than 1. +27 + +G +Hyperparameters +We implement certificates for directed and undirected graphs. For our main experiments (Section 7), +however, we follow the standard procedure and prepocess all graphs into undirected graphs, only +consider the largest connected component, and binarize node features. We compute simple paths +using a modified depth first search. All datasets are included in PyTorch Geometric (Fey and Lenssen, +2019).5 We train models full-batch using Adam (learning rate = 0.001, β1 = 0.9, β2 = 0.999, +ϵ = 10−08, weight decay = 5 ∗ 10−04) for 1,000 epochs with early stopping after 50 epochs. We +use a dropout of 0.8 on the feature matrix X and on the attention coefficients. During training, we +sample a different graph from φ(G) each epoch. Each sampled graph contains nodes with features +replaced by the ablation representation t. We implement t as a parameter of our models: We initialize +t using Xavier initialization and we optimize t as we optimize the GNN weights during training. We +implement all models for two message-passing layers. We use 8 heads and 8 hidden channels for +GAT and GATv2 (Velickovic et al., 2018; Brody et al., 2022); 64 hidden channels for GCN (Kipf and +Welling, 2017); and we use k = 64 and temperature=1.0 for SMA (Geisler et al., 2021). We use the +ReLU activation function for the skip-connection. For GDC sparsification, we set the sparsification +threshold of GDC to ϵ = 0.022, and ignore edge attributes resulting from GDC preprocessing. +Training-time smoothing parameters. We also delete edges and ablate node features during training +(using different probabilities pd and pa during training and inference). Specifically, we train models +presented in Section 7 as follows: In Figure 3 (a,b) we show results for pd = 0.01, pa = 0.6 during +training (and pd = 0.31, pa = 0.794 during inference and certification). In Figure 4 (a,b) we use +pd = 0, pa = 0.59 during training (and pd = 0.31, pa = 0.71 during inference and certification). In +Figure 4 (c) we use the same probabilities pd, pa during training and inference. +In our experiments (Section 7), we also randomly sample different probabilities for training and +inference to explore the joint parameter space of the training-time and inference-time smoothing +parameters. That is, our search space is [0, 1]4 when sampling different probabilities from [0, 1] for +the Pareto-plots in Figure 6 and Appendix H (we sample separately for training and inference). +H +Detailed Results +We report certified accuracies in Figure 16 for the corresponding certified ratios in Figure 3. Moreover, +we provide detailed results for the datasets Cora-ML, Citeseer, and PubMed. We show results for +second-hop attacks against (1) smoothed GAT models in Figure 11, (2) smoothed GATv2 models +in Figure 12, (3) smoothed GCN models in Figure 13, and (4) smoothed SMA models in Figure 14. +We run 1,000 experiments for each combination, drawing random deletion and ablation probabilities +from [0, 1] for each experiment (sampling separately for training and inference). Lines connect +dominating points on the Pareto front. Comparing results with and without skip-connection we +observe that skip-connections allow higher node feature ablation probabilities while retaining high +accuracy, which can yield better robustness-accuracy tradeoffs. Moreover, as discussed in Section 7, +evaluating certificates in transductive settings comes with serious shortcomings. We nevertheless +report such results in Figure 15 for a smoothed GAT model. +Abstained predictions. Our smoothed classifier abstains from predicting if pv,y∗(G) ≤ pv,˜y(G). We +show the ratio of abstained predictions for smoothed GAT models trained on Cora-ML in Figure 17 +for different edge deletion probabilities pd and node feature ablation probabilities pa. We use the +same ablation probability during training and inference for this specific experiment. We observe that +our smoothed classifier abstains for rather large probabilities. Future work could introduce novel +architectures and training techniques to further diminish the effect of abstained predictions. +Experiments on ogbn-arxiv. We run additional experiments and compute certificates for the larger +graph ogbn-arixv with 169,343 nodes, 128 attributes and 40 classes (Hu et al., 2020). We adopt +their transductive setting, implement two-layer smoothed GCNs with skip-connection and compute +certificates for 100 randomly chosen test nodes. In Figure 18 we show results for pd = 0.1, pa = 0.4 +during training, and pd = 0.3, pa=0.8 during inference and certification. Notably, we can certify +GNNs for such large graphs. However, our approach only achieves 53% clean accuracy in this setting. +5https://pytorch-geometric.readthedocs.io +28 + +Future work could develop novel architectures and training procedures to improve clean accuracy +under our smoothing distribution. +Experiments with different confidence levels. We conduct additional experiments with varying +confidence levels α and Monte-Carlo samples. We observe strong guarantees for even smaller +confidence levels, requiring little computational efforts. The underlying reason for this is that the +theoretical largest certifiable radius of our certificates is bounded, only determined by the edge +deletion probability pd and node feature ablation probability pa, and therefore cannot increase by +changing α. Our certificates are thus less sensitive to changes in α compared to Neyman-Pearson- +based certificates (Bojchevski et al., 2020). +In fact, the difference in certifiable robustness for α = 0.05 and α = 0.0001 is already extremely +small when drawing just 2, 000 Monte-Carlo samples (Figure 19 a). We only observe differences in +robustness for considerably small amounts of Monte-Carlo samples (Figure 19 b). Drawing 2,000 +samples takes only 12 seconds on Cora-ML on average. This is significantly faster compared to +all previous probabilistic certificates for GNNs that use up to 106 Monte-Carlo samples (compare +(Bojchevski et al., 2020)). In additional experiments, we also found that the classification accuracy is +high for just a few thousand Monte-Carlo samples (Figure 20). +0% 10% 20% 30% 40% 50% +AUCRC +78 +80 +82 +Clean ACC +pd>0,pa>0 +pd=0,pa>0 +pd>0,pa=0 +0% 10% 20% 30% 40% 50% +AUCRC +74 +76 +78 +Clean ACC +pd>0,pa>0 +pd=0,pa>0 +pd>0,pa=0 +0% 10% 20% 30% 40% 50% +AUCRC +71 +73 +75 +Clean ACC +pd>0,pa>0 +pd=0,pa>0 +pd>0,pa=0 +0% 10% 20% 30% 40% 50% +AUCRC +78 +80 +82 +Clean ACC +pd>0,pa>0 +pd=0,pa>0 +pd>0,pa=0 +0% 10% 20% 30% 40% 50% +AUCRC +73 +75 +77 +Clean ACC +pd>0,pa>0 +pd=0,pa>0 +pd>0,pa=0 +0% 10% 20% 30% 40% 50% +AUCRC +71 +73 +75 +Clean ACC +pd>0,pa>0 +pd=0,pa>0 +pd>0,pa=0 +Figure 11: Robustness-accuracy tradeoffs for second-hop attacks against smoothed GAT on Cora-ML, +Citeseer and PubMed (columns). Top row without skip-connection, bottom row with skip-connection. +Lines connect dominating points on the Pareto front. +0% 10% 20% 30% 40% 50% +AUCRC +78 +80 +82 +Clean ACC +pd>0,pa>0 +pd=0,pa>0 +pd>0,pa=0 +0% 10% 20% 30% 40% 50% +AUCRC +74 +76 +78 +Clean ACC +pd>0,pa>0 +pd=0,pa>0 +pd>0,pa=0 +0% 10% 20% 30% 40% 50% +AUCRC +71 +73 +75 +Clean ACC +pd>0,pa>0 +pd=0,pa>0 +pd>0,pa=0 +0% 10% 20% 30% 40% 50% +AUCRC +79 +81 +83 +Clean ACC +pd>0,pa>0 +pd=0,pa>0 +pd>0,pa=0 +0% 10% 20% 30% 40% 50% +AUCRC +73 +75 +77 +Clean ACC +pd>0,pa>0 +pd=0,pa>0 +pd>0,pa=0 +0% 10% 20% 30% 40% 50% +AUCRC +71 +73 +75 +Clean ACC +pd>0,pa>0 +pd=0,pa>0 +pd>0,pa=0 +Figure 12: Robustness-accuracy tradeoffs for second-hop attacks against smoothed GATv2 on Cora- +ML, Citeseer and PubMed (columns). Top row without skip, bottom row with skip-connection. +29 + +0% 10% 20% 30% 40% 50% +AUCRC +78 +80 +82 +Clean ACC +pd>0,pa>0 +pd=0,pa>0 +pd>0,pa=0 +0% 10% 20% 30% 40% 50% +AUCRC +74 +76 +78 +Clean ACC +pd>0,pa>0 +pd=0,pa>0 +pd>0,pa=0 +0% 10% 20% 30% 40% 50% +AUCRC +71 +73 +75 +Clean ACC +pd>0,pa>0 +pd=0,pa>0 +pd>0,pa=0 +0% 10% 20% 30% 40% 50% +AUCRC +76 +78 +80 +Clean ACC +pd>0,pa>0 +pd=0,pa>0 +pd>0,pa=0 +0% 10% 20% 30% 40% 50% +AUCRC +71 +73 +75 +Clean ACC +pd>0,pa>0 +pd=0,pa>0 +pd>0,pa=0 +0% 10% 20% 30% 40% 50% +AUCRC +69 +71 +73 +Clean ACC +pd>0,pa>0 +pd=0,pa>0 +pd>0,pa=0 +Figure 13: Robustness-accuracy tradeoffs for second-hop attacks against smoothed GCN on Cora-ML, +Citeseer and PubMed (columns). Top row without skip-connection, bottom row with skip-connection. +0% 10% 20% 30% 40% 50% +AUCRC +75 +77 +79 +Clean ACC +pd>0,pa>0 +pd=0,pa>0 +pd>0,pa=0 +0% 10% 20% 30% 40% 50% +AUCRC +72 +74 +76 +Clean ACC +pd>0,pa>0 +pd=0,pa>0 +pd>0,pa=0 +0% 10% 20% 30% 40% 50% +AUCRC +69 +71 +73 +Clean ACC +pd>0,pa>0 +pd=0,pa>0 +pd>0,pa=0 +0% 10% 20% 30% 40% 50% +AUCRC +71 +73 +75 +Clean ACC +pd>0,pa>0 +pd=0,pa>0 +pd>0,pa=0 +0% 10% 20% 30% 40% 50% +AUCRC +68 +70 +72 +Clean ACC +pd>0,pa>0 +pd=0,pa>0 +pd>0,pa=0 +0% 10% 20% 30% 40% 50% +AUCRC +68 +70 +72 +Clean ACC +pd>0,pa>0 +pd=0,pa>0 +pd>0,pa=0 +Figure 14: Robustness-accuracy tradeoffs for second-hop attacks against smoothed SMA on Cora-ML, +Citeseer and PubMed (columns). Top row without skip-connection, bottom row with skip-connection. +0% 10% 20% 30% 40% 50% +AUCRC +77 +79 +81 +Clean ACC +pd>0,pa>0 +pd=0,pa>0 +pd>0,pa=0 +0% 10% 20% 30% 40% 50% +AUCRC +70 +72 +74 +Clean ACC +pd>0,pa>0 +pd=0,pa>0 +pd>0,pa=0 +0% 10% 20% 30% 40% 50% +AUCRC +71 +73 +75 +Clean ACC +pd>0,pa>0 +pd=0,pa>0 +pd>0,pa=0 +Figure 15: Transductive learning setting: Robustness-accuracy tradeoffs for second-hop attacks +against smoothed GAT on Cora-ML, Citeseer and PubMed. Experiments without skip-connection. +30 + +0% +20% +40% +60% +Perturbed nodes +25 +50 +75 +Cert. acc (%) +(a) +distance ≥ 2 +distance ≥ 1 +0 +1 +2 +3 +4 +5 +6 +7 +Perturbed nodes +25 +50 +75 +Certified Accuracy (%) +(b) +distance ≥ 2 +distance ≥ 1 +Figure 16: Certified accuracies for the setting of Figure 3 – Smoothed GAT on Cora-ML: (a) Robust- +ness at different distances (pd=0.31, pa=0.794, with skip-connection, ACC=0.79). +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +pd +0.9 +0.8 +0.7 +0.6 +0.5 +0.4 +0.3 +0.2 +0.1 +pa +7% +7% +11% +13% +11% +8% +6% +0% +0% +5% +4% +8% +5% +15% +10% +6% +11% +3% +4% +2% +6% +2% +9% +11% +13% +9% +11% +3% +1% +5% +3% +8% +7% +11% +22% +25% +2% +3% +3% +4% +4% +6% +8% +16% +32% +3% +3% +3% +2% +5% +3% +3% +7% +2% +0% +1% +2% +3% +2% +3% +3% +3% +3% +1% +1% +1% +3% +4% +3% +3% +3% +0% +0% +1% +1% +0% +5% +2% +4% +1% +0% +Figure 17: Abstained ratios of smoothed GAT models trained on Cora-ML for different edge deletion +probabilities pd and node feature ablation probabilities pa. +0 +1 +2 +3 +4 +5 +6 +7 +8 +Perturbed nodes +25 +50 +75 +Certified ratio +smoothed GCN +0 +1 +2 +3 +4 +5 +6 +7 +8 +Perturbed nodes +25 +50 +75 +Certified accuracy +smoothed GCN +Figure 18: Certified ratio and accuracy for smoothed two-layer GCN on ogbn-arxiv. We certify 100 +randomly selected test nodes in the graph. Certificates for nodes with distance 2 to the target node. +31 + +0 +1 +2 +3 +4 +5 +Perturbed nodes +25 +50 +75 +Cert. ratio (%) +(a) +n1 = 2, 000, α = 0.0001 +n1 = 2, 000, α = 0.05 +0 +1 +2 +3 +4 +5 +Perturbed nodes +25 +50 +75 +Cert. ratio (%) +(b) +n1 = 25, α = 0.0001 +n1 = 25, α = 0.05 +n1 = 300, α = 0.0001 +n1 = 300, α = 0.05 +Figure 19: Certified ratio of smoothed GAT on Cora-ML (pa = 0.84, pd = 0, with skip-connection) +for different confidence levels α and number of Monte-Carlo samples n1. The difference in robustness +is already considerably small for just 2,000 samples. +1e-05 0.0001 0.0005 0.001 0.005 +0.01 +0.05 +α +10,000 +3,000 +1,000 +300 +50 +20 +15 +10 +n +79% +79% +79% +79% +79% +79% +79% +78% +79% +79% +79% +79% +79% +79% +78% +78% +78% +78% +78% +78% +79% +76% +76% +77% +77% +77% +77% +78% +70% +71% +71% +72% +73% +74% +75% +58% +63% +65% +66% +68% +68% +72% +0% +59% +59% +64% +64% +67% +71% +0% +0% +0% +0% +61% +61% +67% +Figure 20: Clean accuracy of smoothed GAT on Cora-ML (pa = 0.84, pd = 0, with skip-connection). +for varying number of confidence levels α and Monte-Carlo samples n. For α = 0.05 the clean +accuracy is high for just 1, 000 samples. For smaller α, the certification accuracy decreases only +slightly. Drawing more than 3, 000 samples is not necessary except for extremely small confidence +levels such as α = 0.00001. +32 + +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +pa +0 +25 +50 +75 +100 +Radii +(a) +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +Perturbed nodes +25 +50 +75 +Certified ratio (%) +(b) +pa= 6√ +0.5 + ϵ +pa= 4√ +0.5 + ϵ +pa= 2√ +0.5 + ϵ +Figure 21: Visualizing Proposition 3. (a) Theoretically maximally certifiable radius for given node +ablation probability pa. (b) Certified ratio of smoothed GAT trained on CoraML for different node +ablation probabilities (pd = 0, ϵ = 0.01). Note: +2√ +0.5 ≈ 0.71, +4√ +0.5 ≈ 0.84 and +6√ +0.5 ≈ 0.89. +I +On Neyman-Pearson and Ablation Certificates +There are currently two types of randomized smoothing certificates for discrete data: The certificates +of Lee et al. (2019) and Bojchevski et al. (2020) are based on the Neyman-Pearson Lemma (Neyman +and Pearson, 1933), and we therefore call them Neyman-Pearson-based certificates. The other +certificates are ablation-based (Levine and Feizi, 2020b,a; Liu et al., 2021). We show that largest +certifiable radius of ablation-based certificates is bounded indepdentent of the classifier, which is not +the case for Neyman-Pearson-based certificates (see discussion in Section 6). +In ablation-based certificates, the bounding constant ∆ determines the probability mass of the +distribution pv,y(G) over labels y that the worst-case adversary controls. This probability mass ∆ +is independent of the classifier f and distribution pv,y(G) and solely determined by the smoothing +distribution. Although the final certificates still depend on the classifier f, the largest certifiable radius +of such ablation-based certificates is bounded as we show for our interception smoothing certificates: +Note again that ∆ does not depend on the base GNN f: the probability to receive at least one message +from a perturbed node is only characterized by the number of perturbed nodes ρ, and the probabilities +pd for edge deletion and pa for node ablation. Moreover, ∆ is monotonously increasing in ρ, since +the probability to receive messages from perturb nodes increases the more nodes adversaries control. +Interestingly, since ∆ is monotonously increasing in ρ, there exists a largest certifiable radius that +depends on the graph structure and changes for each target node (assuming fixed pd, pa). In the +special case of node ablation smoothing, we can directly determine the largest certifiable radius: +Proposition 3. Given fixed pa > 0 and pd = 0, it is impossible to certify a radius ρ if pa ≤ +ρ√ +0.5. +Proof. Due to Corollary 3 and Corollary 1, we only get certificates if ∆ < 1 +2, i.e. the adversary +should not control more than half of the distribution pv,y(G) over y. Thus: +∆ < 1 +2 +(1) +⇔ 1 − pρ +a < 1 +2 ⇔ pρ +a > 1 +2 ⇔ pa > +ρ√ +0.5 +since the root is monotonously increasing and pa > 0. Further, (1) stems from Proposition 2. Thus +we need an ablation probability of at least larger than +ρ√ +0.5 to certify a radius of ρ. +Proposition 3 allows us to directly determine the largest certifiable radius for given pa. We visualize +this largest radius for different ablation probabilities in Figure 21 (a). Theoretically, we can only +certify large radii for relatively large ablation probabilities: For example, to theoretically certify a +radius of 10, we already need an ablation probability of more than +10√ +0.5 ≈ 0.933. Proposition 3 +implies that we cannot certify any radius for ablation probabilities pa ≤ 0.5 (cf. Figure 2). Moreover, +we can certify a radius of only 1 for ablation probabilities between +1√ +0.5 = 0.5 and +2√ +0.5 ≈ 0.707. +Note, however, that this is only a theoretical consideration and that the certificate also depends on +the label probabilities pv,y∗(G) and pv,˜y(G) in practice (Figure 21 b), where we observe that the +certified ratio drops to zero when the largest certifiable radius is passed. +33 + +J +Message-passing-aware Derandomization +As discussed in Section 6, our certificates are probabilistic and hold with a certain confidence level α. +Here we present alternative, deterministic certificates using a simplified smoothing distribution that +just deletes nodes instead of ablating their features. We believe that future work can build upon it +towards even more efficient and scalable derandomization schemes. Specifically, our derandomized +certificates come with the following advantages: First, they are deterministic, exact certificates and +hold independent of a confidence level. Second, the smoothed classifier never abstains from making +a prediction (we resolve draws by whatever index comes first). Third, with more computation time +we obtain more derandomized certificates. This is in continuation to probabilistic certificates that can +be improved using more Monte-Carlo samples (Cohen et al., 2019). +Simplified smoothing distribution. We define a smoothed classifier that classifies node v in G +as follows: Consider a retention constant k ∈ N that represents the number of nodes not deleted +(retained) in the receptive field. Then the smoothed classifier g predicts class y with the largest +probability pv,y(G) that f classifies v as y under uniform deletion of all but k nodes: +gv(G) ≜ arg max +y +pv,y(G) +pv,y(G) ≜ pK∼U(d,k)(f(RK) = y) +where RK encodes the deletion of all nodes in the receptive field of target node v except those indexed +by K, and f(RK) denotes the predicted class of f for target node v given ablated graph RK (omitting +v for conciseness). We further denote the indexing of nodes K as follows: Define the set of all k unique +indices in [d] ≜ {1, . . . , d} including 0 as B(d, k) = {{0} ∪ M : M ∈ P([d]) ∧ |M| = k}, where P +denotes the power set (w.l.o.g. we index target nodes as 0). For example, K = {0, 1, 3, 6} ∈ B(d, k) +for retention constant k = 3 and receptive field size d = 10. Note that |K| = k + 1 for K ∈ B(d, k) +but |B(d, k)| = +�d +k +� +since we never delete the target node. Finally, let U(d, k) denote the uniform +distribution over B(d, k). +(1) +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +(2) +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +(3) +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +(4) +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +Figure 22: Given a receptive field with 10 nodes, target node 0 and k = 3. (1) If we keep nodes +K = {0, 1, 3, 6} and delete all other nodes, node 6 is disconnected. (2) If we keep nodes K = +{0, 1, 3, 7} and delete all other nodes, node 7 is disconnected. (3) In both cases, only the nodes +S(K) = {0, 1, 3} affect the prediction. (4) In the algorithm: Given S = {0, 1, 3} with neighborhood +NS = {2, 4, 5, 8, 9}. Choosing k + 1 − |S| = 1 further nodes, we find that S is a reduced +representative S(K) since there are |Vv|−|NS|−|S| = 10−5−3 = 2 nodes to choose from (6 and 7). +Computing pv,y∗(G) and pv,˜y(G) exactly is challenging. One naive approach would be to simply +iterate over the support of the smoothing distribution (all possible node deletions). For small receptive +fields, the number of possible combinations to sample k out of d nodes may be small, allowing us to +enumerate all possibilities. However, this may be infeasible for larger receptive fields. Still, similar +to how we use the message-passing structure for certification, we can also leverage it here to partition +the support of the simplified smoothing distribution into a smaller number of equivalence classes. +Specifically, we observe: First, when uniformly deleting nodes in the receptive field, some of the +remaining nodes K may be disconnected from the target node. Moreover, disconnected nodes will +not affect the prediction for the target node. Second, several possibilities for K may share the same +nodes that are still connected to v (see examples in Figure 22). This means that different possibilities +for K will lead to the same prediction by f, but the full enumeration of all possibilities is suboptimal: +We wish to avoid redundant evaluations since the evaluation of the base classifier f may be costly. +We observe that the connectivity explained above induces an equivalence relation: All sampled nodes +K that share the same nodes connected to v can be grouped into equivalence classes [K]. For any +representative K of [K] we denote the nodes still connected to v as S(K). We call S(K) a reduced +representative, since it represents a reduced form of K and only contains the nodes from which the +target node will receive messages. Note that S(K) is unique for all representatives K. +34 + +Formally, given receptive field R with d + 1 nodes and index K ∈ B(d, k) of k + 1 nodes. Consider +the subgraph RK induced by K. We observe that not necessarily all nodes in RK have to be connected +to the target node. Thus, different K ∈ B(d, k) will result in same prediction of the base classifier. +Let S(K) ⊆ K denote all nodes indexed by K without the disconnected nodes. Put differently, S(K) +stands for nodes still connected to the target node (see example in Figure 22). Then: +Proposition 9. The definition of S(K) induces an equivalence relation ∼ over B(d, k) given by +K ∼ K′ ⇔ S(K) = S(K′) and eq. classes [K] := {K′ ∈ B(d, k) : K ∼ K′} for K ∈ B(d, k). +Proof. Reflexivity, symmetry and transitivity hold by the definition of sets. +The equivalence relation ∼ partitions B(d, k) into disjoint equivalence classes, denoted by the +quotient set B(d, k)/ ∼ ≜ {[K] | K ∈ B(d, k)}. The set S(K) is uniquely defined for each +equivalence class [K] in B(d, k)/ ∼. We therefore call S(K) with 1 ≤ |S(K)| ≤ k + 1 the reduced +representative of [K]. Note that we have |S(K)| = k + 1 ⇔ S(K) = K and |[K]| = 1. We +further call S = {S(K) | K ∈ B(d, k)} the complete set of reduced representatives. Note that +S ∼= B(d, k)/ ∼ and thus |S| = |B(d, k)/ ∼ |. +To efficiently derandomize our certificates, we can leverage the fact that we only need a complete set +of reduced representatives S to compute the label probabilities pv,y(G). Given S, we only have to +evaluate f once for each reduced representative S(K) ∈ S: +Corollary 5. Given the complete set of reduced representatives S, the label probabilities are: +pv,y(G) = +�d +k +�−1 � +S∈S +I[f (RS) = y] · βS +where I[f (RS) = c] indicates whether f classifies the target node v in subgraph RS as class c, and +βS is the size of an equivalence class, βS = |[K]|. We write S ≜ S(K) and omit v for conciseness. +Proof. For all K, K′ ∈ B(d, k) with K ∼ K′ we have fv(Rv +K) = fv(Rv +K′) = fv(Rv +S(K)) as only +information from nodes of the reduced representative S(K) can be passed to the target node (other +nodes are disconnected). Thus, instead of evaluating fv(Rv +K(G)) for all K ∈ B(d, k) we only have +to evaluate fv(Rv +S(K)(G)) for each S(K) ∈ S. To do so we have to count fv(Rv +S(K)(G)) = i exactly +βS = |[K]| times. Further, as we uniformly sample K from U(d, k) over B(d, k), we have to scale +the possibilities by |B(d, k)|−1, which corresponds to the inverse binomial coefficient above. +Hence, we can compute the label probabilities pv,y(G) exactly for larger receptive fields if the number +of equivalence classes |S| is small and we have an efficient algorithm to compute S and βS. We +propose such algorithm by exploiting the sparsity of graphs as follows: +We successively enumerate all possible connected subgraphs of the receptive field R indexed by S +that contain the target node and at most k further nodes. Let S denote indices of such subgraph of R +and NS the neighborhood of S in R. If S contains k+1 nodes, then all k+1 nodes will be connected +to the target node and S is already a representative with βS = 1. If S contains less than k + 1 nodes, +then S corresponds to a reduced representative if we can choose the remaining k + 1 − |S| nodes +such that they are disconnected. Therefore, the main idea of our algorithm is that the size βS is just a +binomial coefficient: The number of disconnected nodes is given by |Vv| − |NS| − |S|, out of which +we have to choose k + 1 − |S| nodes to augment S to set of k + 1 nodes (where Vv denote nodes in +the receptive field): +βS = +�|Vv| − |NS| − |S| +k + 1 − |S| +� +If βS > 0, there must exist a representative K such that the reduced representative S(K) corresponds +to S, that is S = S(K) (compare (4) in Figure 22 for an example). Finally, our algorithm enumerates +all possible S by recursively augmenting S with nodes from the neighborhood of S (compare +algorithm 1). This way, we exploit the sparsity of graphs to find all reduced representatives S that +avoid disconnected nodes. +35 + +Algorithm 1: Compute complete set of reduced representatives S and equivalence class sizes βS +Input: Index 0 of target node v, Receptive field Rv = (Vv, Ev), Retention constant k +S ← {0} +Output: EQCGeneration(S, Vv, Ev, k) +Function EQCGeneration(S, Vv, Ev, k): +R ← {} +if |S| = k + 1 then +return {(S, 1)} +end +NS ← {w ∈ Vv \ S | ∃u ∈ S : (w, u) ∈ Ev} +// O(|Vv|) +βS ← binom(|Vv| − |NS| − |S|, k + 1 − |S|) +if βS > 0 then +R ← {(S, βS)} +end +for w ∈ NS do +// O(|Vv|) +R ← R ∪ EQCGeneration(S ∪ {w}, Vv, Ev, k) +end +return R +Note that in algorithm 1, Vv denotes nodes in the receptive field of classifier f with respect to target +node v, and Ev the edges in the receptive field. +Lemma 2 (Correctness of algorithm 1). Let S with 0 ∈ S ⊆ Vv be a set of at most k + 1 nodes +1 ≤ |S| ≤ k + 1 such that all nodes indexed by S are connected to the target node in R. We denote +the neighbors of S in R as NS ≜ {w ∈ Vv \ S | ∃u ∈ S : (w, u) ∈ Ev}. When we define the +following binomial coefficient as +βS ≜ +�|Vv| − |NS| − |S| +k + 1 − |S| +� +∈ N. +then there exists a representative K ∈ B(d, k) such that S is a reduced representative for the +equivalence class [K] if βS > 0. Then we have βS = |[K]|. +Proof. First note that for a given set S as defined above we can partition Vv into three disjoint sets +Vv = S ⊎NS ⊎Nr with S and NS defined as above, and the disconnected nodes Nr ≜ Vv \(S ∪NS). +We thus have |Nr| = |Vv| − |NS| − |S|. Now we distinguish the following cases: +Case 1: |S| = k + 1 +We have |Vv| − |NS| − |S| ∈ N0 and βS = 1 > 0. Thus for |S| = k + 1 the condition is trivially +fulfilled and we have that K ≜ S is already a representative with |[K]| = 1 as discussed before. +Note that this does not mean that all sets with k + 1 nodes are representatives, as we still have the +connectivity constraint for nodes in S. +Case 2: |S| < k + 1 +We have βS > 0 ⇔ |Vv| − |NS| − |S| ≥ k + 1 − |S| ⇔ |Nr| ≥ k + 1 − |S| where the latter means +that we can choose the remaining k + 1 − |S| nodes from Nr to augment S to representative K of +the equivalence class [K] since then |K| = |S| + k + 1 − |S| = k + 1. The corresponding size |[K]| +is given by βS. +Finally, note that the equivalence classes and the algorithm are independent of the classifier f. +36 + +Discussion. In the worst case, we have |S| = |B(d, k)| = +�d +k +� +, but we enumerate �k +i=0 +�d +i +� +≥ +�d +k +� +possibilities, as there are �k +i=0 +�d +i +� +candidates for reduced representatives in a fully connected graph. +Therefore, in the worst case of fully connected graphs, directly enumerating all +�d +k +� +possibilities +would be faster. In practice, however, we rather observe sparse graphs with |S| ≪ |B(d, k)|. The +more sparse the receptive field, the less equivalence classes exist and the larger each equivalence +class. Thus we exploit the sparsity of graphs to efficiently compute S and the corresponding sizes +|[K]| for all equivalence classes [K]. +Moreover, as our algorithm recursively enumerates all possible pairs (S, βS), we can determine a +stopping criterion at which we back off to Monte-Carlo sampling for estimating the label probabilities. +To this end, if R denotes the current set of (S, βS) pairs with βS > 0, we know that |R| is a lower +bound on the number of equivalence classes, |R| ≤ |S|. By summing up βS for all (S, βS) ∈ R we +can determine the percentage of |B(d, k)| that we already cover with R: +� +(S,βS)∈R +βS ≤ +� +S(K)∈S +|[K]| = +�d +k +� += |B(d, k)| +This allows us to use the condition � +(S,βS)∈R βS > τ ′ with threshold τ ′ ∈ N as a stopping criterion. +Using thresholds this way, our algorithm will always find more solutions in S given more time via +larger thresholds. Note that we use +�d +k +� +> τ in practice, since the binomial coefficient provides a fast +upper bound for the number of equivalence classes |S|. +J.1 +Evaluating Message-passing-aware Derandomization +Table 1: Smoothed classifier results for GCN trained on Cora-ML for different relative retention +constants. Der.: Ratio of nodes with derandomized certificates. Eq.: Mean of unique receptive fields +over all derandomized certificates. Acc.: Clean accuracy. +GCN on Cora-ML +GCN on Citeseer +GCN on PubMed +krel Der. Eq. Abstained Acc. Der. Eq. Abstained Acc. Der. Eq. Abstained Acc. +0.01 0.87 0.22 6.27e-04 0.73 1.00 0.41 0.00e+00 0.65 0.94 0.15 0.00e+00 0.73 +0.03 0.72 0.23 5.69e-04 0.73 0.94 0.42 0.00e+00 0.66 0.81 0.16 1.56e-03 0.73 +0.10 0.50 0.28 5.02e-03 0.74 0.87 0.42 1.63e-03 0.65 0.61 0.19 4.24e-03 0.74 +0.30 0.31 0.46 1.42e-02 0.80 0.73 0.53 7.61e-03 0.68 0.37 0.38 6.23e-03 0.77 +Relative retention constant. Consider a small retention constant k = 1 for a node v with deg(v) < +dv−deg(v), where dv denotes the receptive field size (excluding the target node). Then the probability +for selecting a direct neighbor of v is low and the prediction of the smoothed classifier is merely +based on the target node v itself, which amounts to traditional i.i.d. prediction. Thus, for non-trivial +robustness guarantees we use retention constants k that are relative to the receptive field size: Given a +fixed relative retention constant krel ∈ [0, 1], our smoothed classifier keeps k = ⌈dv ·krel⌉ ∈ N nodes +in the receptive field R.6 The ceiling operation ensures that we keep at least one additional node. +Derandomization results. Our certificates are deterministic for small receptive fields, and proba- +bilistic for large receptive fields: we derandomize certificates if +�d +k +� +is smaller than a threshold τ. If +the number of possibilities to choose k out of d nodes is small, we can enumerate all possibilities and +use f to predict the class of v for all possibilities. In our experiments we set τ = 100,000. There +are more possibilities to sample k out of d nodes for larger krel and thus the ratio of deterministic +certificates decreases (compare Table 1). For example, we can derandomize around 50% of the +certificates for Cora-ML given krel = 0.1. We further derandomize more certificates for Citeseer +than for Cora-ML, which can be explained by the fact that two-layer GNNs have larger receptive +fields on Cora-ML. Note that the average degree in Cora-ML is 6, in Citeseer 3 and PubMed 4. Due +to the derandomization we also hardly observe that the smoothed classifier abstains. +As discussed above, we avoid evaluating the base classifier f for equivalent receptive fields. To +represent the computations we avoid on average, we compute the mean of unique receptive fields +|S|/|B(d, k)| for all derandomized certificates. For example, out of all derandomized certificates for +krel = 0.1 on Cora-ML, we only have to evaluate 28% of all possibilities on average. +6As a disadvantage of this method, we have to process all receptive fields separately. +37 + diff --git a/ZdA0T4oBgHgl3EQfF_8g/content/tmp_files/load_file.txt b/ZdA0T4oBgHgl3EQfF_8g/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..0f6f21e29906058fb4f3f365c6ebd82140311e1c --- /dev/null +++ b/ZdA0T4oBgHgl3EQfF_8g/content/tmp_files/load_file.txt @@ -0,0 +1,1647 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf,len=1646 +page_content='Randomized Message-Interception Smoothing: Gray-box Certificates for Graph Neural Networks Yan Scholten1, Jan Schuchardt1, Simon Geisler1, Aleksandar Bojchevski2 & Stephan Günnemann1 {y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='scholten, j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='schuchardt, s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='geisler}@tum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='de bojchevski@cispa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='de, s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='guennemann@tum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='de 1Dept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' of Computer Science & Munich Data Science Institute, Technical University of Munich 2CISPA Helmholtz Center for Information Security Abstract Randomized smoothing is one of the most promising frameworks for certifying the adversarial robustness of machine learning models, including Graph Neural Networks (GNNs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Yet, existing randomized smoothing certificates for GNNs are overly pessimistic since they treat the model as a black box, ignoring the underlying architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' To remedy this, we propose novel gray-box certificates that exploit the message-passing principle of GNNs: We randomly intercept messages and carefully analyze the probability that messages from adversarially controlled nodes reach their target nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Compared to existing certificates, we certify robustness to much stronger adversaries that control entire nodes in the graph and can arbitrarily manipulate node features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Our certificates provide stronger guarantees for attacks at larger distances, as messages from farther-away nodes are more likely to get intercepted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' We demonstrate the effectiveness of our method on various models and datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Since our gray-box certificates consider the underlying graph structure, we can significantly improve certifiable robustness by applying graph sparsification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='1 1 Introduction The core principle behind the majority of Graph Neural Networks (GNNs) is message passing – the representation of a node is (recursively) computed based on the representations of its neighbors (Gilmer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' This allows for information to propagate across the graph, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' in a k-layer GNN the prediction for a node depends on the messages received from its k-hop neighborhood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' With such models, if an adversary controls a few nodes in the graph, they can manipulate node features to craft adversarial messages that in turn change the prediction for a target node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Such feature-based adversarial attacks are becoming significantly stronger in recent years and pose a realistic threat (Ma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Zou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=', 2021): Adversaries may arbitrarily manipulate features of entire nodes in their control, for example in social networks, public knowledge graphs and graphs in the financial and medical domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Detecting such adversarial perturbations is a difficult unsolved task even beyond graphs (Carlini and Wagner, 2017), meaning such attacks may go unnoticed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' How can we limit the influence of such adversarial attacks?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' We introduce a simple but powerful idea: intercept adversarial messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Specifically, we propose message-interception smoothing where we randomly delete edges and/or randomly ablate (mask) nodes, and analyze the probability that messages from adversarially controlled nodes reach the target nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' By transforming any message- passing GNN into a smoothed GNN, where the prediction is the majority vote under this randomized message interception, we can provide robustness certificates (see Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' 1Project page: https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='tum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='de/daml/interception-smoothing 36th Conference on Neural Information Processing Systems (NeurIPS 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='02039v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='LG] 5 Jan 2023 Adversarial node Ablated node Adversarial message Intercepted message Deleted edge v Receptive field (2 layers) v v v .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Adversarial messages intercepted in 2 out of 3 random samples Figure 1: Randomized message-interception smoothing: We model adversaries that can arbitrarily manipulate features of multiple nodes in their control (red) to alter the predictions for a target node v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' We intercept messages (gray) by randomly deleting edges and/or ablating (mask) all features of entire nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Our certificates are based on the majority vote under this randomized message interception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Experimentally we obtain significantly better robustness guarantees compared to previous (smoothing) certificates for GNNs (compare Section 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' This improvement stems from the fact that our certificates take the underlying architecture of the classifier into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Unlike previous randomized smoothing certificates which treat the GNN as a black-box, our certificates are gray-box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' By making the certificate message-passing aware we partially open the black-box and obtain stronger guarantees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Our approach is also in contrast to white-box certificates that apply only to very specific models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' For example, Zügner and Günnemann (2019) only certify the GCN model (Kipf and Welling, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' While newly introduced GNNs require such certificates to be derived from scratch, our approach is model-agnostic and flexible enough to accommodate the large family of message-passing GNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' We evaluate our certificates on node classification datasets and analyze the robustness of existing GNN architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' By applying simple graph sparsification we further increase the certifiable robustness while retaining high accuracy, as sparsification reduces the number of messages to intercept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' In stark contrast to previous probabilistic smoothing-based certificates for GNNs, our certificates require only a few Monte-Carlo samples and are more efficient: For example, we can compute certificates on Cora-ML in just 17 seconds and certify robustness against much stronger adversaries than previous smoothing-based certificates (Bojchevski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=', 2020) that take up to 25 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' In short, our main contributions are: The first gray-box smoothing-based certificates for GNNs that exploit the underlying message-passing principle for stronger guarantees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Novel randomized smoothing certificates for strong threat models where adversaries can arbitrarily manipulate features of multiple nodes in their control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' 2 Preliminaries and Background Threat model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' We develop certificates for feature perturbations given evasion threat models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Specifi- cally, we model adversaries that attack GNNs by entirely perturbing attributes of a few ρ nodes in the graph at inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Given an attributed graph G = (A, X) ∈ G encoded via adjacency matrix A ∈ {0, 1}n×n and feature matrix X ∈ Rn×d with n nodes and d features, we formally define the threat model of feature perturbations as a ball centered at a given graph G = (A, X): Bρ(G) ≜ {G′ = (A′, X′) | A = A′, δ(G, G′) ≤ ρ} where δ(G, G′) ≜ �n v=1 1xv̸=x′v denotes the number of nodes whose features differ in at least one dimension when comparing the clean graph G and the perturbed graph G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Intuitively, this means adversaries control up to ρ nodes in the graph and can arbitrarily manipulate node features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Graph neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' We design robustness certificates for GNNs that instantiate the so-called message-passing framework (Gilmer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' The message-passing framework describes a large family of GNN architectures that are based on the local aggregation of information from neighboring nodes in the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' To compute a new representation h(ℓ) v of node v, each message-passing layer Ψ(ℓ) transforms and aggregates the representations h(ℓ−1) v and h(ℓ−1) u of all nodes u in the local neighborhood N(v) ≜ {u | Auv = 1} of node v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' 2 We can formally describe a message-passing layer as follows: h(ℓ) v ≜ Ψ(ℓ) u∈N (v)∪{v} � h(ℓ−1) v , h(ℓ−1) u � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' For node classification, message-passing GNNs with k GNN-layers can be described as parametrized functions f : G → {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' , C}n that assign each node v in graph G class fv(G) ≜ argmaxc h(k) v,c, where h(0) v ≜ xv ∈ Rd denotes the input and h(k) v ∈ RC the final representation of node v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Randomized smoothing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Our robustness certificates for GNNs build upon the randomized smoothing framework (Cohen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Lecuyer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=', 2019): Given any base classifier f, for example a message-passing GNN, we can build a “smoothed” classifier g that classifies randomly perturbed input samples, and then takes the “majority vote” among all predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' The goal is to construct a smoothed classifier that behaves similar to f (for example in terms of accuracy) and for which we can prove (probabilistic) robustness certificates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Randomized ablation (Levine and Feizi, 2020b) is a smoothing-based certificate that “ablates” the in- put: Unlike in randomized smoothing where the input is randomly perturbed (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' by adding Gaussian noise to images), in randomized ablation the input is randomly masked, for example by replacing parts of the input with a special ablation token that “hides” the original information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' If the perturbed input is masked for the majority of predictions, we can issue certificates for the smoothed classifier g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' 3 Randomized Message-Interception Smoothing for Graph Neural Networks The main idea of our gray-box smoothing certificates is to intercept messages from perturbed nodes by (1) deleting edges to disconnect nodes, and/or (2) ablating nodes to mask their features (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' To implement this we introduce two independent smoothing distributions φ1(A) and φ2(X) that randomly apply these changes to the input graph: The first smoothing distribution φ1(A) randomly deletes edges in the adjacency matrix (1 → 0) with probability pd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' The second smoothing distribution φ2(X) randomly ablates all features of nodes with probability pa by replacing their feature represen- tations with a fixed representation token t ∈ Rd for ablated nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' The ablation representation t is a trainable parameter of our smoothed classifier and can be optimized during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Introducing two independent smoothing distributions is important since our base classifiers f are GNNs, which behave differently under structural changes in the graph than to feature ablation of nodes in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' We use this message-interception smoothing distribution φ(G) ≜ (φ1(A), φ2(X)) to randomly sample and then classify different graphs with a message-passing GNN f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Finally, our smoothed classifier g takes the majority vote among the predictions of f for the sampled graphs φ(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' We formally describe our smoothed classifier g as follows: gv(G) ≜ argmax y∈{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=',C} pv,y(G) pv,y(G) ≜ p(fv(φ(G)) = y) where pv,y(G) denotes the probability that the base GNN f classifies node v in graph G as class y under the smoothing distribution φ(G) = (φ1(A), φ2(X)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' 4 Provable Gray-box Robustness Certificates for Graph Neural Networks We derive provable certificates for the smoothed classifier g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' To this end, we develop a condition that guarantees gv(G) = gv(G′) for any graph G′ ∈ Bρ(G): We make the worst-case assumption that adversaries alter the prediction for a target node whenever it receives at least one message from perturbed nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Let E denote the event that at least one message from perturbed nodes reaches a target node v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Then the probability ∆ ≜ p(E) quantifies how much probability mass of the distribution pv,y(G) over classes y is controlled by the worst-case adversary: Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Given target node v in graph G, and adversarial budget ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Let E denote the event that the prediction fv(φ(G)) receives at least one message from perturbed nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Then the change in label probability |pv,y(G) − pv,y(G′)| is bounded by the probability ∆ = p(E) for all classes y ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' , C} and graphs G′ with G′ ∈ Bρ(G): |pv,y(G) − pv,y(G′)| ≤ ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Proof sketch (Proof in Appendix A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Whenever we intercept all adversarial messages, adversaries cannot alter the prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Thus |pv,y(G) − pv,y(G′)| is bounded by ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' □ 3 Note that we derive an upper bound on ∆ in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' We first consider the special case of node ablation smoothing, discuss its relation to randomized ablation for image classifiers (Levine and Feizi, 2020b), and then we derive our provably stronger guarantees for the general case of message-interception smoothing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Special case of node ablation smoothing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' For the special case of node feature ablation smoothing only (pd = 0), we can directly determine the probability ∆ (Proof in Appendix B): Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' For node feature ablation smoothing only (pd = 0), we have ∆ = 1 − pρ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' In this special case, our certificates for GNNs are theoretically related to the randomized ablation certificates for image classifiers (Levine and Feizi, 2020b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' We could apply their smoothing distribu- tion to GNNs by randomly ablating features of entire nodes, instead of pixels in an image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' However, their approach is specifically designed for image classifiers and comes with serious shortcomings when applied to GNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Notably, our robustness cetificates are provably tighter and experimentally stronger even in this special case without edge deletion smoothing (pd = 0): Given that ∆L denotes the bounding constant as defined by Levine and Feizi (2020b), we show ∆ < ∆L in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' We carefully discuss such differences with more technical details in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Most importantly, their certificate applied to GNNs ignores the underlying graph structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' General case of message-interception smoothing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' In contrast, our message-interception certifi- cates are specifically designed for graph-structured data, message-passing aware, and consider the interception of messages via edge deletion as follows: Consider a fixed target node v in the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' The formal condition for intercepting messages from a fixed target node v to itself is φ2(xv) = t, since we only intercept messages from the target node to the target node itself if we ablate its features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' To model the interception of messages from perturbed nodes B other than the target node, we take the graph structure A into account: We consider all simple paths P k wv = {(e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' , ei) | i ≤ k} from perturbed nodes w ∈ B to target node v of length at most k (where k is the number of GNN layers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='2 Intuitively, if any edge e on path p ∈ P k wv is deleted, or the features of w are ablated, messages via path p get intercepted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' If all messages from perturbed nodes get intercepted, adversaries cannot alter the prediction for the target node (Proof in Appendix A): Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Given a fixed target node v and perturbed nodes B in the graph with v /∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Then fv(φ(G)) = fv(φ(G′)) for any graph G′ ∈ Bρ(G) if ∀w ∈ B : � ∀p ∈ P k wv : ∃(i, j) ∈ p : φ1(A)ij = 0 � ∨ (φ2(xw) = t) Since k-layer message-passing GNNs aggregate information over local neighborhoods, only features of nodes in the receptive field affect the prediction for a target node (only via paths with a length of at most k to v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' For any perturbed node w ∈ B outside of the receptive field we have P k wv = ∅ and the message-interception condition of Lemma 1 is always fulfilled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' In practice, however, we do not know which nodes in the graph are controlled by the adversary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' To account for this, we assume adversaries control nodes indicated by ρv ∈ {0, 1}n that maximize the probability of the event E(ρv) that target node v receives perturbed messages: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' The worst-case change in label probability |pv,y(G) − pv,y(G′)| is bounded by ∆ = max ||ρv||0≤ρ p (E(ρv)) for all classes y ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' , C} and any graph G′ ∈ Bρ(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Proof in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Finally, we provide conservative robustness certificates for the smoothed classifier g by exploiting that perturbed nodes are disconnected and/or ablated and cannot send messages for the majority of predictions: Corollary 1 (Multi-class certificate).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Given ∆ as defined in Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Then we can certify the robustness gv(G) = gv(G′) for any graph G′ ∈ Bρ(G) if pv,y∗(G) − ∆ > max ˜y̸=y∗ pv,˜y(G) + ∆ where y∗ ≜ gv(G) denotes the majority class, and ˜y the follow-up (second best) class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Proof in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' We also provide a certificate for binary node classification in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' 2We consider simple paths (all nodes appear only once), since we only receive perturbed messages via more complex paths iff we receive perturbed messages via the simple part of the complex path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' 4 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='6 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='8 1 pa 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='6 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='8 1 pd (a) 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='6 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='8 1 ∆i 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='6 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='8 1 pa 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='6 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='8 1 pd (b) 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='6 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='8 1 ∆i 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='6 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='8 1 pa 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='6 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='8 1 pd (c) 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='6 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='8 1 ∆i Figure 2: Single source bounding constant ∆i for different edge deletion probabilities pd and node feature ablation probabilities pa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' White isolines indicate ∆i = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='5 and separate the theoretically certifiable region (∆i < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='5) from the uncertifiable region (∆i ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' (a) For the target node, pd does not affect ∆i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' (b) Direct neighbor of target node, single edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' (c) Second-hop neighbor, single path (two edges).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' (a-c) More distant nodes have larger theoretically certifiable regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' 5 Practical Interception Smoothing Certificates Message-interception certificates constitute two challenges in practice: (1) computing the bounding constant ∆ for arbitrary graphs, and (2) computing the label probabilities pv,y∗(G) and pv,˜y(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' We address the first problem by providing upper bounds on ∆ (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' lower bounds on the certifiable robustness).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' For the second problem we follow existing literature and estimate the smoothed classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Lower bound on certifiable robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Computing ∆ of Theorem 1 poses two problems: First, finding the worst-case nodes in arbitrary graphs involves a challenging optimization over the powerset of nodes in the receptive field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Second, computing the probability p(E(ρv)) to receive perturbed messages is challenging even for fixed ρv, since in general, it involves evaluating the inclusion- exclusion principle (Appendix C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' We can compute ∆ exactly only for special cases such as small or tree-structured receptive fields (Appendix D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Notwithstanding the challenges, we provide practical upper bounds on ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Instead of assuming a fixed ρv, we solve both problems regarding ∆ at once and directly bound the maximum over all possible ρv by assuming independence between paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Due to Corollary 1, any upper bound on ∆ result in lower bounds on the certifiable robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' We first derive an upper bound on ∆ for a single perturbed node, and then generalize to multiple nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Let Ew denote the event that the target node v receives messages from node w, and ∆w ≜ p(Ew).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Note in the special case of the target node v = w we just have ∆w = 1 − pa, since the features xv of the target node v are used for the prediction independent of any edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' For any w ̸= v in the receptive field we can derive the following upper bound for single sources (Proof in Appendix E): Theorem 2 (Single Source Multiplicative Bound).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Given target node v and source node w ̸= v in the receptive field of a k-layer message-passing GNN f with respect to v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Let P k wv denote all simple paths from w to v of length at most k in graph G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Then ∆w ≤ ∆w for: ∆w ≜ � �1 − � q∈P k wv � 1 − (1 − pd)|q|� � � (1 − pa) where |q| denotes the number of edges on the simple path q ∈ P k wv from w to v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' We visualize ∆w for different pd and pa in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' The upper bound for single sources is tight for one- and two-layer GNNs (∆ = ∆w), since then all paths from a single source to the target node are independent (Appendix E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' The single source multiplicative bound on ∆w can only be used to certify a radius of ρ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' For multiple nodes (ρ > 1), we generalize Theorem 2 as follows: Theorem 3 (Generalized multiplicative bound).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Assume an adversarial budget of ρ nodes and let ∆1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' , ∆ρ denote the ρ largest ∆i for nodes i in the receptive field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Then we have ∆ ≤ ∆M for ∆M ≜ 1 − ρ � i=1 (1 − ∆i) Proof in Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Notably, the multiplicative bound is tighter than a union bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' We specifically address the approximation error in detail in Appendix F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' 5 Estimating the smoothed classifier in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Computing the probabilities pv,y∗(G) and pv,˜y(G) exactly is challenging in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' We instead estimate them similar to previous work by drawing Monte-Carlo samples from φ (Cohen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Levine and Feizi, 2020b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Bojchevski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' We first identify the majority class y∗ and follow-up class ˜y using a few samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' We then draw more samples to estimate a lower bound pv,y∗(G) on pv,y∗(G) and an upper bound pv,˜y(G) on pv,˜y(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' We use the Clopper-Pearson Bernoulli confidence interval and apply Bonferroni correction to ensure that the bounds hold simultaneously with significance level α (with probability of at least 1 − α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Moreover, our smoothed classifier abstains from predicting if pv,y∗(G) ≤ pv,˜y(G), meaning if the estimated probabilities are too similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' We experimentally analyze abstained predic- tions in Appendix H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Practical robustness certificates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Finally, our robustness certificates also hold when bounding ∆ and the label probabilities as the following Corollary shows (Proof in Appendix A): Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' We guarantee gv(G) = gv(G′) with probability of at least 1 − α for any G′ ∈ Bρ(G) if pv,y∗(G) − ∆ > pv,˜y(G) + ∆, where y∗ denotes the majority class, and ˜y the follow-up class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' 6 Discussion Our certificates require knowledge about the graph structure A and can only account for structure perturbations if the perturbed adjacency matrix A′ is known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' While adversarial edge deletion potentially increases robustness (due to less messages to intercept), adversaries could arbitrarily increase the number of messages via edge insertion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Moreover, the number of simple paths in the graph can be huge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' We argue, however, that (1) graphs are usually sparse, (2) the number of paths can be reduced via sparsification, and (3) we have to compute paths only once for each graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Limitations of ablation certificates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Since the probability to receive messages from perturbed nodes increases the more nodes are adversarial, ∆ is monotonously increasing in ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Thus, the certifiable radius is bounded independent of the label probabilities (uncertifiable region for ∆ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='5 due to Corollary 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' This bound depends on the graph structure and changes for each target node, but in the case of node feature ablation smoothing we can directly determine the bound (Proof in Appendix I): Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Given fixed pa > 0 and pd = 0, it is impossible to certify a radius ρ if pa ≤ ρ√ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' This bound is only determined by the parameters of the smoothing distribution (pd, pa) and does not depend on the base GNN f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' The existence of an upper bound is in stark contrast to certificates whose largest certifiable radius depends on the inverse Gaussian CDF of the label probabilities (Cohen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Such certificates are theoretically tighter than ablation certificates: For example, if the base classifier f classifies all samples from φ the same (py∗ = 1), they would certify a radius of ∞, whereas the radius of ablation-based certificates is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' We leave the development of even stronger gray-box certificates for GNNs to future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Limitations of probabilistic certificates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Our certificates are probabilistic and hold with significance level α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Notably, our method still yields strong guarantees for significantly smaller confidence levels (we show additional experiments for varying α in Appendix H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' We found that α has just a minor effect on the certificate strength, since increasing it cannot increase the largest certifiable radius, which is theoretically bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Recent works also “derandomize” probabilistic certificates, that is they compute the label probabilities exactly (Levine and Feizi, 2020a, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' In Appendix J we propose the first derandomization technique that leverages message-passing structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' We believe future work can build upon it towards even more efficient derandomization schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Threat model extensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Notably, edge-deletion smoothing (pd > 0) also yields guarantees for adversarial node insertion and deletion, as disconnected nodes cannot alter the prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='3 As discussed above, we can only evaluate such certificates with structural information, that is how inserted/deleted nodes are connected to target nodes: Given clean graphs (as in our evaluation), we know which nodes adversaries could delete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Given perturbed graphs, we know which nodes could have been inserted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Note that although we can technically extend our method to certify adversarial edge deletion, we focus on the novel problem of arbitrary feature manipulations of entire nodes since there are already certificates against edge-modification attacks (Bojchevski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' 3We cannot certify node insertion/deletion with feature ablation smoothing, since e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' new nodes affect the smoothed classifier independent of whether features are ablated or not (unless we delete nodes entirely).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' 6 7 Experimental Evaluation We evaluate our certificates for different GNN architectures trained on node classification datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Our certificates work in standard transductive learning settings used throughout the literature and we report such results in Appendix H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' However, combining transductive learning with an evasion threat model comes with serious shortcomings for the evaluation of certificates, since no separate test data is available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' For example, we can usually achieve high accuracy by overfitting a Multi-Layer Perceptron (MLP) to labels predicted by GNNs during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' MLPs do not propagate information through the graph at test time and are robust to adversarial messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Instead, we evaluate our certificates in semi-supervised inductive learning settings with hold-out test nodes: Experimental setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' As labelled nodes, we draw 20 nodes per class for training and validation, and 10% of the nodes for testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' We use the labelled training nodes and all remaining unlabeled nodes as training graph, and successively insert (hold-out) validation and test nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' We train on the training graph, optimize hyperparameters against validation nodes, assume adversaries control nodes at test time, and compute certificates for all test nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' We also delete edges and ablate node features during training (Appendix G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' We use n0 = 1,000 samples for estimating the majority class, n1 = 3,000 samples for certification, and set α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' We conduct five experiments for random splits and model initializations, and report averaged results including standard deviation (shaded areas in the plots).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' When comparing settings (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' architectures), we run 1,000 experiments for each setting and draw deletion and ablation probabilities from [0, 1] for each experiment (sampling separately for training and inference).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Then, we compute dominating points on the Pareto front for each setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' For brevity, we only show points with clean accuracy of at most 5% below the maximally achieved performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Datasets and models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' We train our models on citation datasets: Cora-ML (Bojchevski and Günne- mann, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' McCallum et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=', 2000) with 2,810 nodes, 7,981 edges and 7 classes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Citeseer (Sen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=', 2008) with 2,110 nodes, 3,668 edges and 6 classes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' and PubMed (Namata et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=', 2012) with 19,717 nodes, 44,324 edges and 3 classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' We implement smoothed classifiers for four architectures with two message-passing layers: Graph convolutional networks (GCN) (Kipf and Welling, 2017), graph attention networks (GAT and GATv2) (Velickovic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Brody et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=', 2022), and soft medoid aggregation networks (SMA) (Geisler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' More details in Appendix G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' We also compute certificates for the larger graph ogbn-arxiv (Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=', 2020) in Appendix H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Evaluation metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' We report the classification accuracy of the smoothed classifier on the test set (clean accuracy), and the certified ratio, that is the number of test nodes whose predictions are certifiable robust for a given radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Since all nodes have different receptive field sizes, we also divide the certifiable radius by the receptive field size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' The resulting normalized robustness better reflects how much percentage of the “attack surface” (that is the number of nodes the adversary could attack) can be certified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Moreover, we report the area under this (normalized) certified ratio curve (AUCRC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' For completeness, we also report the certified accuracy in Appendix H, that is the number of test nodes that are correctly classified (without abstaining) and certifiable robust for a given radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' 0 1 2 3 4 5 6 7 Perturbed nodes 25 50 75 Cert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' ratio (%) (a) distance ≥ 2 distance ≥ 1 0% 20% 40% 60% Perturbed nodes 25 50 75 Cert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' ratio (%) (b) distance ≥ 2 distance ≥ 1 1 500 1,500 2,500 d 10 20 30 40 AUCRC (%) (c) ours (Bojchevski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=', 2020) Figure 3: Smoothed GAT on Cora-ML: (a) Robustness at different distances to target nodes (pd=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='31, pa=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='794, with skip, ACC=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='79).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' (b) Robustness normalized by receptive field size (“attack surface”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' (c) Naïve baseline comparison (base certificate (Bojchevski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=', 2020), 105 samples, α=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='01).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Message-interception smoothing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' In Figure 3 (a,b) we demonstrate our certificates for specific edge deletion probabilities pd and node feature ablation probabilities pa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' By making our certificates message-passing aware, we can (1) certify robustness against arbitrary feature perturbations of entire nodes, (2) analyze robustness locally in the receptive fields by incorporating the “attack surface”, and (3) provide stronger guarantees for attacks against nodes at larger distances to target nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' 7 First certificate for stronger adversaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Experimentally we obtain significantly better robustness guarantees compared to previous (smoothing-based) certificates for Graph Neural Networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Specifi- cally, existing certificates for GNNs only certify perturbations to a few attributes ˜ρ in the entire graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Our certificates are novel as they provide guarantees for much stronger adversaries that can arbitrarily manipulate features of a multiple nodes in the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' To compare these two approaches, consider a naïve baseline that certifies ρ = ˜ρ/d nodes, where d is the number of attributes per node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='4 If each node in the graph had just a single feature, the number of certifiable nodes ρ is high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' As the number of features d per node increases, however, the baseline dramatically deteriorates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' In contrast, our certificates are entirely independent of the dimension d and hold regardless of how high-dimensional the underlying node data might be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' We demonstrate this comparison in Figure 3 (c) for the first smoothing-based certificate for GNNs (Bojchevski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=', 2020), assuming attribute deletions against second-hop nodes (p+=0, p−=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' However, the superiority of our certificate regarding robustness against all features of entire nodes holds for any other GNN certificate proposed so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' 0 1 2 3 4 5 6 Perturbed nodes 25 50 75 Cert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' ratio (%) (a) sparsification w/o sparsific.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' 0% 20% 40% 60% Perturbed nodes 25 50 75 Cert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' ratio (%) (b) sparsification w/o sparsific.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' 0 1 2 3 4 5 Perturbed nodes 25 50 75 Cert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' ratio (%) (c) n1=10 n1=20 n1=50 n1=100 n1=300 n1=1,000 Figure 4: (a,b) Sparsification significantly improves certifiable robustness of our gray-box certificates to second-hop attacks since sparsification reduces (a) messages to intercept, and (b) receptive field sizes and thus the “attack surface” (Smoothed GAT, Cora-ML, pd = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='31, pa = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='71, with skip- connection, ACC = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' (c) Our certificate with largest certifiable radius of 4 with varying samples for certification (Smoothed GAT, Cora-ML, pd = 0, pa = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='85).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Our certificates are more sample efficient than existing smoothing-based certificates for GNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Stronger certificates for sparser graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Notably, our gray-box certificates incorporate graph structure and become stronger for sparser graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' This is in contrast to black-box certificates that ignore the underlying message-passing principles of GNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' We demonstrate this by applying graph sparsification, which significantly improves robustness while retaining high clean accuracy: First, sparsification reduces the number of paths in the graph and thus reduces the number of messages to intercept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Second, sparsification reduces the number of nodes in the receptive fields and thus the “attack surface”, that is the number of nodes that send messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' In Figure 4 (a,b) we apply GDC preprocessing (Gasteiger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=', 2019) to the Cora-ML graph at test time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' GDC preprocessing yields directed graphs and reduces the number of edges in the graph from 15,962 to 14,606 (we set the sparsification threshold of GDC to ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='022 and ignore resulting edge attributes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Interestingly, evaluating the model on the sparsified graph yields significantly higher certifiable robustness, although both approaches show high clean accuracy of 80%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Note that for the validity of our certificates we assume adversaries perturb nodes after sparsification and cannot attack the sparsification itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Efficient message-interception smoothing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Drawing Monte-Carlo samples from φ to estimate the smoothed classifier is usually the most costly part when computing smoothing-based certificates (Cohen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' In Figure 4 (c) we show that our certificates are much more sample efficient as we do not benefit from more than a few thousand samples from φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' This is in stark contrast to existing smoothing-based certificates for GNNs (Bojchevski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' For a fair comparison, we adopt their transductive setting and compute certificates for pd = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='3 and pa = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Bojchevski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' (2020) use 106 Monte-Carlo samples for certifying test nodes on Cora-ML, which takes up to 25 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' In contrast, our certificates saturate already for 2,000 Monte-Carlo samples in this setting, which takes only 17 seconds (preprocessing Cora-ML takes 8 additional seconds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Our gray-box certificates are significantly more sample-efficient while also providing guarantees against much stronger adversaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' We hypotheise that our certificates saturate much faster as the certifiable radius does not depend on the inverse Gaussian CDF of the label probabilities as discussed in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' 4We are the first to certify such strong adversaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Thus no baselines exist so far and we compare our method against existing certificates for GNNs using the naïve baseline we propose above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' 8 0% 10% 20% 30% AUCRC 78 80 82 Clean ACC (%) (a) GAT GATv2 GCN SMA 0% 10% 20% 30% 40% AUCRC 78 80 82 Clean ACC (%) (b) with skip-connection w/o skip-connection 0% 10% 20% AUCRC 78 80 82 Clean ACC (%) (c) pt = pe−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='1 pt = pe pt = pe+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='1 Figure 5: Second-hop attacks on Cora-ML: (a) Robustness-accuracy tradeoffs for different GNN archi- tectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' (b) Skip-connections yield improved robustness-accuracy tradeoffs for node feature ablation smoothing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' (c) Ablating less during training yields better robustness-accuracy tradeoffs (GAT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Different classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' In Figure 5 (a) we compare robustness-accuracy tradeoffs for different GNNs against second-hop attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Attention-based message-passing GNNs (Velickovic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=', 2018) are dominating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' We hypothesize that the degree-normalization of GCN (Kipf and Welling, 2017) may be problematic for the performance under randomized edge deletion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Our approach may promote novel message-passing architectures, specifically designed for smoothed classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Skip-connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' With higher node feature ablation probability, more messages from the target node itself will be intercepted, which may be detrimental for the accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Assuming adversaries do not attack target nodes, we can modify the architecture for improved robustness-accuracy tradeoffs (Figure 5b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' To this end, we forward the non-ablated input graph through the GNN without edges, and add the resulting final representation of each node to the final representation when forwarding the (ablated) graph with graph structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' We use the same weights of the base GNN, but more complex skip-connections are straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Such skip-connections yield better robustness-accuracy trade- offs against second-hop attacks, but we also loose guarantees for the target node itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' To account for that, future work could deploy existing smoothing methods for features of target nodes separately: e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=', if nodes represent images, we could deploy Gaussian smoothing (Cohen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=', 2019) on node features send through the skip-connection and still obtain robustness guarantees for target nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Training-time smoothing parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' In Figure 5 (c) we show that ablating less during training can improve the robustness-accuracy tradeoffs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Note that only inference-time smoothing parameters determine the strength of our certificates, and the probabilities pd, pa during training are just hyperpa- rameters that we can optimize to improve the robustness-accuracy tradeoffs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' In detail, we experiment with three different settings: Using the same ablation probabilities during training and inference (pt = pe), ablating 10% more during training (pt = pe+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='1), and ablating 10% less during training (pt=pe−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Note that we use max(min(pt, 1), 0) to project the training-time parameters into [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' 0% 10% 20% 30% AUCRC 78 80 82 Clean ACC (%) (a) pd>0,pa>0 pd>0,pa=0 pd=0,pa>0 0% 10% 20% 30% 40% AUCRC 74 76 78 Clean ACC (%) (b) pd>0,pa>0 pd>0,pa=0 pd=0,pa>0 0% 10% 20% 30% 40% AUCRC 72 74 76 Clean ACC (%) (c) pd>0,pa>0 pd>0,pa=0 pd=0,pa>0 Figure 6: Robustness-accuracy tradeoffs for second-hop attacks against smoothed GAT models (without skip).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Edge deletion and node ablation dominates on Cora-ML (a) and Citeseer (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' On PubMed (c), edge deletion is stronger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Lines connect dominating points on the Pareto front.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Robustness-accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' We compare robustness-accuracy tradeoffs of three different settings: (1) edge deletion and feature ablation (pd > 0, pa > 0), (2) edge deletion only (pd > 0, pa = 0), and (3) feature ablation only (pd = 0, pa > 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Our experiments show that edge deletion and feature ablation smoothing achieves significantly better robustness-accuracy tradeoffs against attribute attacks to the second-hop neighborhood and dominates on Cora-ML and Citeseer (Figure 6b,c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' On PubMed, edge deletion smoothing dominates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' More results (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' with skip-connections) in Appendix H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' 9 8 Related Work GNN robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' The vast majority of GNN robustness works focus on heuristic defenses, including adversarial graph detection (Zhang and Ma, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=', 2019a);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' architecture modifications (Brody et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=', 2019b);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' robust aggregations (Geisler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=', 2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' robust training procedures (Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Zügner and Günnemann, 2019), transfer learning (Tang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=', 2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' and graph preprocessing techniques such as edge pruning (Zhang and Zitnik, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=', 2019), low-rank approximations (Entezari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=', 2020), and graph anomaly detection (Ma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' The effectiveness of such seemingly robust defenses on the adversarial robustness of GNNs can only be assessed against existing adversarial attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Heuristic defenses do not guarantee robustness, and may even be broken by stronger attacks later on (Mujkanovic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Instead, we are interested in robustness certificates that provably guarantee the stability of predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' However, robustness certificates for GNNs are still in their infancy (Günnemann, 2022): Certificates for GNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Most certificates for GNNs are designed for specific architectures (Zügner and Günnemann, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Jin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Bojchevski and Günnemann, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Zügner and Günnemann, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Despite providing provable robustness guarantees, their applicability is limited to specific architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Bojchevski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' (2020) present the first tight and efficient smoothing-based, model- agnostic certificate for graph-structured data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' However, their method comes with crucial limitations: First, their method cannot certify robustness against arbitrary feature modifications of entire nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Second, their black-box certificate deletes edges but completely ignores the underlying message- passing principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Third, their certificate requires an expensive evaluation of the smoothed classifier, which questions the practicability of their certificate beyond theoretical robustness assessments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Randomized ablation certificates for image classifiers (Levine and Feizi, 2020b) are another approach for discrete data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Such certificates have already been applied to point cloud classifiers (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=', 2021) and even for individual attribute perturbations in GNNs (Bojchevski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' However, Bojchevski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' (2020) show that their method outperforms such ablation certificates for individual attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' In contrast, we propose to certify entire nodes, instead of only a few of their attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' As already discussed, applying their ablation certificates for image classifiers directly to GNNs comes with serious shortcomings that we overcome (Section 4 and details in Appendix B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Gray-box certificates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Exploiting model knowledge to derive tighter randomized smoothing certifi- cates constitutes a widely unexplored research problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' The first works derive tighter guarantees using information about the model’s gradients (Mohapatra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Levine et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Recently proposed collective certificates (Schuchardt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=', 2021) incorporate knowledge about the receptive fields of GNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Their certificates are orthogonal to ours, and our certificates could lead to significant improvements in such collective settings, as adversaries cannot attack first-hop neighbors of all nodes simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Schuchardt and Günnemann (2022) propose tight gray-box certificates for models that are invariant to spatial transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' 9 Conclusion We propose novel gray-box, message-passing aware robustness certificates for GNNs against strong threat models where adversaries can arbitrarily manipulate all features of multiple nodes in the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' The main idea of our certificates is to intercept adversarial messages by randomly deleting edges and/or masking features of entire nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Our certificates are significantly stronger and more sample-efficient than existing methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Future enhancements could smooth specific edges and nodes with different probabilities, for example to intercept messages from central nodes with higher probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Our gray-box certificates could lead to novel architectures, training techniques and graph preprocessing techniques to further strengthen the robustness of GNNs against adversarial examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Acknowledgments and Disclosure of Funding This work has been funded by the German Federal Ministry of Education and Research, the Bavarian State Ministry for Science and the Arts, and the German Research Foundation, grant GU 1409/4-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' The authors of this work take full responsibility for its content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} 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convolutional neural networks for semi-supervised classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' In AAAI, pages 5829–5836.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' AAAI Press, 2019b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Xu Zou, Qinkai Zheng, Yuxiao Dong, Xinyu Guan, Evgeny Kharlamov, Jialiang Lu, and Jie Tang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Tdgia: Effective injection attacks on graph neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' In Proceedings of the 27th ACM SIGKDD Conference on Knowledge Discovery & Data Mining, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Daniel Zügner and Stephan Günnemann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Certifiable robustness and robust training for graph convolutional networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' In Proceedings of the 25th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Daniel Zügner and Stephan Günnemann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Certifiable robustness of graph convolutional networks under structure perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' In Proceedings of the 26th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining, pages 1656–1665, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' 13 Checklist 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' For all authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' (a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' [Yes] All claims in abstract and introduction reflect the contributions and scope of our paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' We also provide a list of our core contributions directly in our introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' (b) Did you describe the limitations of your work?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' [Yes] We discuss the limitations of our approach in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' (c) Did you discuss any potential negative societal impacts of your work?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' [Yes] Without doubt, adversarial attacks can have negative impacts on the society.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Particularly alarming are recent attacks against GNNs for more realistic threat models (Ma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=', 2020) and attacks that scale to large graphs (Geisler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Robustness certificates are tools to assess robustness and help to (1) better understand robustness, (2) build more robust classifiers, and (3) eventually prevent adversarial attacks including their negative consequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Our certificates represent a contribution towards diminishing and preventing potential negative impacts of adversarial attacks on the society.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' (d) Have you read the ethics review guidelines and ensured that your paper conforms to them?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' [Yes] We carefully reviewed the ethics guidelines (https://neurips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' cc/public/EthicsGuidelines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Our paper conforms to all ethic guidelines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Our certificates represent a contribution to prevent negative social impacts of adversarial attacks, as discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' If you are including theoretical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' (a) Did you state the full set of assumptions of all theoretical results?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' [Yes] We state all assumptions of our theoretical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' (b) Did you include complete proofs of all theoretical results?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' [Yes] We show all statements, including additional elaborations, in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' We always link theoretical results in the paper to the corresponding proofs in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' If you ran experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' (a) Did you include the code, data, and instructions needed to reproduce the main experi- mental results (either in the supplemental material or as a URL)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' [Yes] We uploaded the code required to reproduce our main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' All required datasets are publicly available, and can be loaded for example with PyTorch Geometric (Fey and Lenssen, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' (b) Did you specify all the training details (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=', data splits, hyperparameters, how they were chosen)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' [Yes] We describe important training details directly at the beginning of our experiment section (Section 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' We further thoroughly list all training details in Appendix G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' (c) Did you report error bars (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=', with respect to the random seed after running ex- periments multiple times)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' [Yes] We repeat each experiment for five random splits and model initializations, only report averaged results, and our plots explicitly show the standard deviation over the five experiments (shaded areas in plots).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' In separate experiments, we tested our method for another set of five randomly drawn seeds, but we did not observe significant differences in the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' We also uploaded all seeds to ensure reproducibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' (d) Did you include the total amount of compute and the type of resources used (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=', type of GPUs, internal cluster, or cloud provider)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' [Yes] We discuss the runtime for each experiment in our experiment section (Section 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' We conduct all experiments in an internal cluster with the following GPU type: NVIDIA GeForce GTX 1080 Ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' If you are using existing assets (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=', code, data, models) or curating/releasing new assets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' (a) If your work uses existing assets, did you cite the creators?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' [Yes] Yes, we cite all authors of the assets we use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' (b) Did you mention the license of the assets?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' [N/A] Our datasets are well-established research datasets with MIT License or public domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' 14 (c) Did you include any new assets either in the supplemental material or as a URL?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' [Yes] We provide supplemental materials and code to reproduce our main results at https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='tum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='de/daml/interception-smoothing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' (d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' [N/A] Our assets do not require consent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' (e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' [N/A] Our assets do not contain any personal, protected or offensive data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' If you used crowdsourcing or conducted research with human subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' (a) Did you include the full text of instructions given to participants and screenshots, if applicable?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' [N/A] We do not conduct research with human subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' (b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' [N/A] We do not conduct research with human subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' (c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' [N/A] We do not conduct research with human subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' 15 A Proofs Main Certificate (Section 4) Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Given target node v in graph G, and adversarial budget ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Let E denote the event that the prediction fv(φ(G)) receives at least one message from perturbed nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Then the change in label probability |pv,y(G) − pv,y(G′)| is bounded by the probability ∆ = p(E) for all classes y ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' , C} and graphs G′ with G′ ∈ Bρ(G): |pv,y(G) − pv,y(G′)| ≤ ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' For a thorough formal proof in the context of image classifiers see (Levine and Feizi, 2020b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Here, we show the statement in the context of GNNs: Consider a fixed target node v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' We exploit that whenever we intercept all adversarial messages (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' nodes are disconnected or we mask out their features), the adversary cannot alter the prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Let ¯E denote the event that v does not receive any message from perturbed nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Then we have for any class y: p(fv(φ(G)) = y | ¯E) = p(fv(φ(G′)) = y | ¯E) since all input representations with respect to G and G′, which affect the prediction for v, are the same if all perturbed nodes are ablated or disconnected (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' their messages are intercepted).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Multiplying with p( ¯E) yields: p(fv(φ(G)) = y ∧ ¯E) = p(fv(φ(G′)) = y ∧ ¯E) (1) Following the arguments of (Levine and Feizi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' 2020b): pv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='y(G) − pv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='y(G′) (1) = p(fv(φ(G)) = y ∧ E) + p(fv(φ(G)) = y ∧ ¯E) − pv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='y(G′) (2) = p(fv(φ(G)) = y ∧ E) + p(fv(φ(G′)) = y ∧ ¯E) − pv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='y(G′) (3) = p(fv(φ(G)) = y ∧ E) − p(fv(φ(G′)) = y ∧ E) ≤ p(fv(φ(G)) = y ∧ E) (4) ≤ p(E) where (1) and (3) follow from the law of total probability,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' (2) is due to inserting Equation 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' and (4) follows from p(A ∩ B) ≤ p(B) for any events A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Analogously, pv,y(G′) − pv,y(G) ≤ p(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Thus: |pv,y(G) − pv,y(G′)| ≤ p(E) = ∆ Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Given a fixed target node v and perturbed nodes B in the graph with v /∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Then fv(φ(G)) = fv(φ(G′)) for any graph G′ ∈ Bρ(G) if ∀w ∈ B : � ∀p ∈ P k wv : ∃(i, j) ∈ p : φ1(A)ij = 0 � ∨ (φ2(xw) = t) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' The prediction fv(φ(G)) cannot differ from fv(φ(G′)) if for all perturbed nodes w ∈ B we have (1) w is disconnected from the target node v, or (2) the features of w are ablated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' If the smoothing distribution φ1 deletes an edge (i, j) (that is φ(A)ij = 0), the neighborhood N(j) changes, and thus messages from i to j get intercepted on all GNN layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' That is, the final hidden representation h(k) v of a target node v can only be changed by some non-ablated perturbed source node w if there is at least one simple path from w to v of length at most k such that no edge on this path is deleted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' The worst-case change in label probability |pv,y(G) − pv,y(G′)| is bounded by ∆ = max ||ρv||0≤ρ p (E(ρv)) for all classes y ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' , C} and any graph G′ ∈ Bρ(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Note the difference: E denotes the event that at least one message from perturbed nodes reaches a target node v E(ρv) denotes the event that at least one message from nodes indicated by ρv reaches a target node v 16 Put differently, the maximization amounts to the additional worst-case assumption that the adversary selects those nodes whose messages have the highest chance of getting to the target node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Importantly, we have to make this additional worst-case assumption to obtain valid robustness certificates for our threat model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Since the probability ∆ bounds the worst-case change |pv,y(G) − pv,y(G′)| for all classes y, we can utilize ∆ to construct robustness certificates: Intuitively, ∆ bounds how much probability mass of the distribution pv,y(G) over labels y is compromised by the worst-case adversary: If an adversary cannot shift enough probability mass to change the majority class, our smoothed classifier is robust: Corollary 3 (Binary Certificate).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Given ∆ as defined in Then we can certify the robustness gv(G) = gv(G′) for any graph G′ ∈ Bρ(G) if pv,y∗(G) − ∆ > 1 2 where y∗ ≜ gv(G) denotes the majority class predicted by smoothed classifier g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Recall that ∆ bounds how much probability mass of the distribution pv,y(G) over y is compromised by the adversary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Let y∗ ≜ g(G) denote the majority class, that is pv,y∗(G) > 1 2 in this binary classification setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Thus, to change the majority class, the adversary needs to shift enough probability mass from the majority class y∗ to the other class 1 − y∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' This is impossible if pv,y∗(G) − ∆ > 1 2, meaning the adversary cannot shift enough probability mass for a successful attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Put differently, even in the worst-case that the adversary always changes the prediction whenever adversarial messages reach the target node, the majority class cannot be altered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Corollary 1 (Multi-class certificate).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Given ∆ as defined in Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Then we can certify the robustness gv(G) = gv(G′) for any graph G′ ∈ Bρ(G) if pv,y∗(G) − ∆ > max ˜y̸=y∗ pv,˜y(G) + ∆ where y∗ ≜ gv(G) denotes the majority class, and ˜y the follow-up (second best) class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' To prove this, we utilize the same arguments as in the binary setting above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Here, given pv,y∗(G) − ∆ > max˜y̸=y∗ pv,˜y(G) + ∆, the adversary does not control enough probability mass of pv,y(G) over y to alter the second-best class ˜y into the new majority class when classifying the perturbed graph G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' We guarantee gv(G) = gv(G′) with probability of at least 1 − α for any G′ ∈ Bρ(G) if pv,y∗(G) − ∆ > pv,˜y(G) + ∆, where y∗ denotes the majority class, and ˜y the follow-up class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' We have pv,y∗(G)−∆ ≥ pv,y∗(G)−∆ > pv,˜y(G)+∆ ≥ pv,˜y(G)+∆ due to the assumption pv,y∗(G) − ∆ > pv,˜y(G) + ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' The remaining claim follows from Corollary 1 and from the fact that both bounds hold with significance level α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' 17 B Theoretical Connection to Randomized Ablation for Image Classifiers Our gray-box certificates for GNNs are theoretically related to the randomized ablation black-box certificates for image classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' In this section we thoroughly analyze the differences with more technical insights and carefully discuss how our certificates go beyond theirs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Specifically, we show that our gray-box certificates yield stronger guarantees, and are provably tighter even in the special case without additional edge deletion smoothing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' In the following we introduce their certificate again, discuss the differences to our certificate, and eventually prove that our guarantees are tighter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Randomized Ablation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Levine and Feizi (2020b) introduce randomized ablation for image clas- sifiers as follows: They define the space B(n, k) ≜ {M : M ∈ P({1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' , n}) ∧ |M| = k} of all pixel-subsets with exactly k of n total pixels (P denoting the power set here).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Then, their smoothing distribution ablates all but k pixels in a uniformly drawn subset M ∈ B(n, k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' They define ∆L as the probability to keep (not ablate) perturbed pixels in the image under this smoothing distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Assuming ρ perturbed pixels in an image: ∆L = 1 − �n−ρ k � �n k � Discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' There are various ways of applying such black-box certificates for image classifiers to certify the robustness of GNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' One way is to use them to certify threat models where adversaries control individual attributes all over the graph (Bojchevski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' We are interested in certifying robustness to adversaries that control all features of entire nodes in the graph instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' However, applying the smoothing distribution of Levine and Feizi (2020b) for certifying robustness to our threat model (that is by ablating entire node vectors) comes with several deficiencies, as their smoothing distribution is specifically designed for image classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Most importantly, applying their certificate for image classifiers to GNNs results in black-box certificates that completely ignore the underlying message-passing principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' In contrast, we propose gray-box certificates – we partially open the black-box and consider the underlying message-passing principle and paths in the graph, that is A and A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' This comes with two crucial advantages as we show experimentally in Section 7: First, additionally deleting edges leads to significantly better robustness guarantees for attacks against more distant nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Second, our certificates become increasingly stronger for sparser graphs (while their certificate applied to GNNs remains unchanged as it ignores graph structure).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='1 Special Case of Node Feature Ablation Smoothing Notably, our certificates are provably tighter even without edge deletion smoothing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Specifically, we formally show the difference between our ∆ for node feature ablation smoothing and ∆L of Levine and Feizi (2020b) when naively applying their approach to GNNs by randomly ablating features of entire nodes (instead of pixels in an image).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Specifically, while their smoothing distribution samples exactly k out of n nodes not to ablate (to keep), our smoothing distribution samples k out of n nodes in expectation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' This eventually leads to ∆ < ∆L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' We start by characterizing our certificate for node ablation smoothing: Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' For node feature ablation smoothing only (pd = 0), we have ∆ = 1 − pρ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Recall the definition of the probability ∆: E denotes the event that at least one perturbed message reaches a target node v, and ∆ ≜ p(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' When only ablating nodes (pd = 0), all nodes are equally important for the prediction fv(φ(G)), since messages are only intercepted in the input layer, not during the message-passing itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' We therefore do not have an optimization problem as in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Instead, the probability ∆ to receive perturbed messages is just the probability that at least one perturbed node is not ablated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Further, the complementary event denotes that all ρ perturbed nodes are ablated, whose probability is just pρ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Thus ∆ = 1 − pρ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' 18 Moreover, the multiplicative bound is tight in the special case of node ablation smoothing: Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' For pd = 0, the multiplicative bound is tight ∆M = ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' We have ∆i (1) = � �1 − � q∈P k wv � 1 − (1 − pd)|q|� � � (1 − pa) (2) = 1 − pa where (1) is by definition, and (2) due to our assumption pd = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Therefore: ∆M = 1 − ρ � i=1 (1 − ∆i) = 1 − ρ � i=1 pa = 1 − pρ a = ∆ where the first equality is due to definition again, and the last equality follows from Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Proposition 5 (Tighter guarantees).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Given adversarial budget ρ > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Further assume k > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Let ∆L denote the bounding constant for the smoothing distribution proposed by Levine and Feizi (2020b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Then ∆ < ∆L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Recall that due to uniform ablation we have (compare Levine and Feizi (2020b)): ∆L = 1 − �n−ρ k � �n k � To compare this to our ∆ = 1 − pρ a of Proposition 2, we first need to introduce k and n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' We note that pa is the probability to ablate a single node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' We thus have pa = 1 − k n, where k n amounts to the probability to “keep” (not ablate) a node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' In this setting, we keep n k n = k nodes in expectation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' We therefore have: ∆ = 1 − pρ a = 1 − � 1 − k n �ρ We observe: �n−ρ k � �n k � = (n − ρ)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' (n − k)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' (n − ρ − k)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' = ρ−1 � i=0 n − k − i n − i (1) < �n − k n �ρ = � 1 − k n �ρ where (1) is due to the mediant inequality (ρ > 1 and k > 0): ∀y < x ∀i > 0 : y − i x − i < y x We conclude that ∆ < ∆L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' The difference decreases for larger n, but our smoothing distribution is significantly better for small graphs/receptive fields: For example, for n = 10 and k = 1 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' pa = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='9), the largest certifiable radius with our method is 6, but only 4 using their certificate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' In detail, there are two ways of applying their method for image classifiers to certify robustness of GNNs against adversaries that control all features of entire nodes in the graph: by ablating all features of k out of n uniformly chosen nodes (1) in the entire graph, or (2) locally in each receptive field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Global randomized ablation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Assume we uniformly ablate all features of k out of n nodes in the entire graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' If the number of nodes n in the graph is large, the difference between ∆ and ∆L is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Still, the resulting black-box certificates only hold globally, not locally in the receptive fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Such certificates ignore the receptive fields, specifically that most nodes in the graph may not even be connected to the target node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' For example, in the most extreme case of A = 0 (meaning receptive fields only consist of target nodes), their certificate applied to GNNs remains entirely unchanged due to the black-box nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' In contrast, our gray-box certificates guarantee robustness for any ρ (excluding target nodes) in this case (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' normalized robustness in Section 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' 19 Local randomized ablation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' To remedy the black-box nature of their approach, one can obtain local guarantees by ablating all features of k out of the n nodes locally in the receptive field of a target node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' However, our message-interception certificates are significantly tighter even without edge deletion smoothing as receptive fields are typically small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' We demonstrate this in Figure 7 where our approach yields significantly stronger guarantees in practice (since Proposition 2 makes a significant difference).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Note that when applying their approach to GNNs by ablating nodes locally, one also needs to consider each receptive field individually and cannot use full-batch training/inference as usually implemented for GNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Our message-interception certificates are easier to implement and more efficient as we obtain local guarantees without considering and processing all receptive fields separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' 0% 20% 40% Perturbed nodes 25 50 75 Cert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' ratio (%) ours Levine’s method applied to GNNs Figure 7: Given pa = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='72, we compare our certificate against the certificate proposed by Levine and Feizi (2020b) by applying their smoothing distribution for image classifiers to GNNs (distance ≥ 1, with skip-connection).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' We locally choose k = ⌊(n − 1) ∗ pa⌋ nodes not to ablate – where n − 1 is the number of nodes in each receptive field, excluding the target node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Our certificates are experimentally stronger even without additional edge deletion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' 20 C Closed-form via Inclusion-exclusion Principle Recall that E(ρv) describes the event that v receives messages from any attacked node indicated by the adversarial budget vector ρv ∈ {0, 1}n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Computing the probability p (E(ρv)) using edge deletion probability pd and node feature ablation probability pa is challenging as it involves evaluating the inclusion-exclusion formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' We formalize this expensive closed-form solution in the following: Let Ew denote the probability to receive a message from node w, and let P indicate all simple paths from any perturbed w with ρv(w) = 1 to target node v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Further, let Yi denote the probability to receive a message via path i ∈ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Then we have: p (E(ρv)) = p � � � ρv(w)=1 Ew � � = p � � i∈P Yi � since the probability to receive a message from any attacked node equals the probability to receive a message from any path i from an attacked node to the target node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' We now apply the inclusion- exclusion principle: p � � i∈P Yi � = |P| � k=1 � � � �(−1)k−1 � I⊆P |I|=k p �� i∈I Yi � � � � � (2) The remaining probability can be expressed as follows: The probability to receive messages via all paths indicated by I is the probability that (1) all edges on those paths are not deleted, and (2) the corresponding source nodes of the paths are not ablated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Therefore: p �� i∈I Yi � = (1 − pd)a(1 − pa)b (3) where a denotes the number of (unique) edges on all paths indicated by I, and b the number of (unique) source nodes of the paths indicated by I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Note that the above derivation assumes that the target node v is not controlled by the adversary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' In such a case (ρv(v) = 1), we have p(Ev) = 1 − pa (since we always receive messages from non-ablated target nodes) and: p (E(ρv)) = p � � i∈P Yi � Ev � p �� i∈I Yi � Ev � (1) = p �� i∈I Yi � p(Ev) where (1) is due to independence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' There are different ways that take additional information into account to derive faster ways of computing p (E(ρv)), for example by exploiting that the receptive fields are trees with the target node v as root (compare Appendix D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' In general, however, computing Equation 2 is expensive since we have to evaluate Equation 3 exactly 2|P| times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' 21 D Tree-shaped Receptive Fields Given fixed ρv ∈ {0, 1}n that indicates nodes controlled by the adversary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Recall that E(ρv) describes the event that v receives at least one messages from any attacked node indicated by the adversarial budget vector ρv ∈ {0, 1}n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' If the receptive field for target node v is a tree, we can compute ∆ of Theorem 1 exactly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Specifically, we first provide a recursive formula to compute p (E(ρv)) and then show that the worst-case selection of nodes by the adversary is straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' We introduce the following random variables to better describe the recursion: Let Ri denote the event that root node i receives an adversarial message.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Let Ai denote the event that the features of node i are ablated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Let Di denote the event that root i receives an adversarial message via any of its adjacent subtrees j ∈ B (“branches”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Let Bj further denote the event that we receive an adversarial message via branch j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' The main idea is that branches in a tree are independent: Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' We start the recursion with the target node v to compute p(Rv) while following edges away from the node (j, v) (against their direction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Then the following recursive equation computes p (E(ρv)) for tree-shaped receptive fields: p(Ri) ≜ �1 − pa(1 − p(Di)) if ρv(i) = 1 p(Di) else with p(Di) ≜ 1 − � (j,i) (1 − p(Bj)) p(Bj) ≜ (1 − pd)p(Rj) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' We show the three equations consecutively: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' For p(Ri): If root i is not controlled by the adversary, then the probability to receive an adversarial message is just the probability that we receive such a message via any of its adjacent subtrees, that is p(Ri) = p(Di).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' If root i is controlled by the adversary (ρv(i) = 1), we can exploit independence between edge deletion smoothing φ1 and node feature ablation smoothing φ2: p(Ri) = p( ¯Ai ∨ Di) = 1 − p(Ai ∧ ¯Di) (1) = 1 − p(Ai)p( ¯Di) = 1 − p(Ai)(1 − p(Di)) where (1) is due to independence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Since the probability that we do not receive any adversarial message from root i is the probability that the features of root i are ablated: p(Ai) = pa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' We therefore have: p(Ri) = 1 − pa(1 − p(Di)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' For p(Di): For the probability that root i receives an adversarial message via any of its adjacent branches j ∈ B, we exploit independence between branches (which we can do since we have trees): p(Di) = p � � � j∈B Bj � � = 1 − p � � � j∈B ¯Bj � � (1) = 1 − � j∈B p( ¯Bj) = 1 − � j∈B (1 − p(Bj)) where (1) is due to independence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' For p(Bj): The probability to receive a message via branch j is the probability that the edge from branch j to root i is not deleted (1 − pd) times the probability that we receive a message via the next root j (recursive call).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' For leaves we have B = ∅ and thus the product over j ∈ B is 1, that is p(Di) = 0 for all leaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' 22 0 1 2 3 4 5 6 7 8 9 10 11 Perturbed nodes 25 50 75 Cert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' ratio (a) distance≥2 (tight delta) distance≥2 (m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' bound) 0 1 2 3 4 5 6 7 8 9 10 11 Perturbed nodes 25 50 75 Cert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' ratio (b) distance≥2 (tight delta) distance≥2 (m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' bound) Figure 8: Comparing multiplicative bound and tight tree bound (distance at least 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' (a) Tree-certificate only for tree-shaped receptive fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' (b) Sparsifying all receptive fields into trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Interestingly, we can reconstruct the following special cases: Special case of edge deletion smoothing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Assume pa = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Then we directly see that p(Ri) = 1 if root i is controlled by the adversary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' This means that the adversary controls the entire sub-tree if the root node is already attacked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Put differently, the adversary does not need to control more parts of the tree to change the prediction if the adversary already controls the root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Special case of node feature ablation smoothing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Assume pd = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Then we can directly see that resolving the recursion just multiplies the node feature ablation probabilities pa and we get p (E(ρv)) = 1−pρ a for ρ = ||ρv||0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' This matches the special case already discussed in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Worst-case selection of nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Recall that our certificates are conservative and assume the additional worst-case that the adversary attacks those nodes in the receptive field that maximize the probability that the target node receives a message from attacked nodes (maximization in Theorem 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' This additional assumption is required to obtain valid certificates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Notably, this worst-case adversary is straightforward for trees: First, an adversary would always prefer closer nodes over more distant nodes to maximize the probability that messages are getting through.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Second, an adversary would always distribute its budget over different branches to exploit independence between branches, which also maximizes the probability that messages are getting through (also compare Appendix E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' We find that computing ∆ tight for tree-shaped receptive fields can increase the certifiable radius in practice (compare Figure 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Interestingly, 25% of nodes in Cora-ML have receptive fields that are trees (considering 2-layer GNNs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' We apply our recursive scheme above to compute tight certificates in two settings: First, we only compute tight certificates for the nodes whose receptive fields are trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Second, we apply sparsification that successively deletes edges in the graph until the receptive fields of all test nodes are trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' In detail, we train GAT models on Cora-ML and apply sparsification at test time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' We use the skip-connection, train with pa = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='68, pd = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='02 and compute certificates with pa = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='79, pd = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Without sparsification we achieve clean accuracies of 79% on average, and 77% when applying sparsification at test time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' In practice, we found that the gain in computing ∆ exactly may be rather small, as adversaries typically distribute their budget to different branches to increase the probability that their messages arrive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' This means adversaries maximize independencies between edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' In other words, the multiplicative bound is already quite strong in practice, and specifically tight until the degree of the node (given that each first-hop neighbor has at least one child).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' 23 E Proofs of Section 5 Figure 9: Visualization of two dependent (left) and independent paths (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' When randomly deleting edges with the same edge deletion probability pd, the probability that all messages from both source nodes are intercepted is lower when the paths are independent (more possibilities for the message to get through).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' We first prove a more general claim that we can use to prove the multiplicative bounds of Theorem 2 and Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Let Xi denote the event that target node v receives a message via any path s in a set of paths Si such that all paths start at an arbitrary source node and end at target node v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Intuitively, it is more likely to receive at least one messages via Si and one message via Sj when there are shared edges, compared to when we assume their paths were independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Put differently, the probability that all messages from all paths are intercepted is higher when paths are dependent (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Figure 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' More formally: Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' For two arbitrary sets Si and Sj of simple paths with the same target node v we have p(Xi)p(Xj) ≤ p(Xi ∧ Xj) under the smoothing distribution φ1 for edge deletion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' We are interested in the probability that all messages via all paths are intercepted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Consider the following two possibilities: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' The paths in Si and the paths in Sj are (pairwise) independent, meaning there are no edges that appear on both - on a path si ∈ Si and on a path sj ∈ Sj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' In this case we have p(Xi ∧ Xj) = p(Xi)p(Xj) due to independence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Consider the scenario where there are at least two dependent paths that share a common edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' If we assume they were independent, there would be more possibilities how a message can get through than there actually are.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' In other words, assuming independence results in lower probability that all messages via both sets get intercepted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Thus p(Xi)p(Xj) < p(Xi ∧ Xj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' □ Consider the following definition of positively associated random variables (Esary et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=', 1967).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' We call a random vector x = (X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' , Xn) positively associated if Cov(φ(x), ψ(x))) ≥ 0 for all non-decreasing, element-wise functions φ, ψ such that second moments of ψ(x) and φ(y) exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' The concept of positively associated random variables is for example used in physical statistics (Goldstein and Wiroonsri, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' We can use this concept here to prove multiplicative bounds: Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' The random vector x = (X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' , Xn) is positively associated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Due to Theorem 5 we have p(Xi)p(Xj) ≤ p(Xi ∧ Xj) and thus ⇒E[Xi]E[Xj] ≤ E[XiXj] ⇒E[XiXj] − E[Xi]E[Xj] ≥ 0 ⇒Cov(Xi, Xj) ≥ 0 since Xi and Xj are binary random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Thus, the elements of the covariance matrix are non-negative: Cov(¯x, ¯x) ≥ 0 (variance is always non-negative).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' According to Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='2 in Esary et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' (1967), ¯x is positively associated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Since ¯x is positively associated, it follows from (BP1) in Esary et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' (1967) that x is positively associated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' 24 Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Given random variables Xi as defined above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Then: 1 − p � n � i=1 Xi � ≤ 1 − n � i=1 p � Xi � Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Since x and ¯x are positively associated random variables, we can use Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='1 in (Esary et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=', 1967) and conclude that p � n � i=1 Xi � ≥ n � i=1 p � Xi � ⇔ 1 − p � n � i=1 Xi � ≤ 1 − n � i=1 p � Xi � Theorem 2 (Single Source Multiplicative Bound).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Given target node v and source node w ̸= v in the receptive field of a k-layer message-passing GNN f with respect to v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Let P k wv denote all simple paths from w to v of length at most k in graph G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Then ∆w ≤ ∆w for: ∆w ≜ � �1 − � q∈P k wv � 1 − (1 − pd)|q|� � � (1 − pa) where |q| denotes the number of edges on the simple path q ∈ P k wv from w to v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Note in the special case of the target node v = w we just have ∆w = 1 − pa, since the features xv of the target node v are used for the prediction independent of any edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' For any w ̸= v in the receptive field: Let Ew denote the event that the target node v receives messages from node w, and ∆w ≜ p(Ew).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' We further introduce Aw for the event that the features of node w are ablated, and Dw for the event that v receives at least one messages from w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Then we have: ∆w = p(Ew) = p( ¯Aw ∧ Dw) (1) = p( ¯Aw)p(Dw) = (1 − pa)p(Dw) where (1) holds since the two smoothing distributions for node feature ablation and edge deletion are independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' We continue with p(Dw).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Therefore, recall that P ≜ Pk wv denotes the set of simple paths from w to v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Further, let p(q) for simple path q ∈ P denote the probability that v receives a message via path q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Clearly, a message “arrives” only via path q if none of the edges on that path is deleted, that is when the node is connected via path q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Since the deletion of edges is independent, p(q) = (1 − pd)|q|, where |q| denotes the number of edges on the simple path q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' We derive: p(Di) = p � � � q∈P q � � = 1 − p � � � q∈P q � � We can use positive association to conclude 1 − p � � � q∈P q � � (1) ≤ 1 − � q∈P p (q) where (1) follows from Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Finally, we resolve the remaining terms: 1 − � q∈P p (q) = 1 − � q∈P (1 − p (q)) = 1 − � q∈P � 1 − (1 − pd)|q|� Due to (1) above, we finally get ∆w ≤ ∆w, where the inequality becomes an equality if all paths are independent (that is the paths do not share edges).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' We have ∆w = ∆w for ℓ-layer GNNs with ℓ ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' For ℓ-layer GNNs with ℓ ≤ 2, all paths from a single source to the target node are independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' 25 Theorem 3 (Generalized multiplicative bound).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Assume an adversarial budget of ρ nodes and let ∆1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' , ∆ρ denote the ρ largest ∆i for nodes i in the receptive field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Then we have ∆ ≤ ∆M for ∆M ≜ 1 − ρ � i=1 (1 − ∆i) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' We recall from Theorem 1: ∆ = max ||ρv||1≤ρ p (E(ρv)) where E(ρv) describes the event that target node v receives messages from any attacked node indicated by ρv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Recall that Ew denotes the event that the prediction for target node v is based on information of node w in the receptive field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' We further have ∆w ≜ p(Ew).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Then: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='p (E(ρv)) = p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='ρv(w)=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='Ew ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='� = 1 − p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='ρv(w)=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='¯Ew ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='where we can apply Proposition 6 and use the assumption that paths from several source nodes to the ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='target were independent to obtain an upper bound: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='1 − p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='ρv(w)=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='¯Ew ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='� ≤ 1 − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='ρv(w)=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='� ¯Ew ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='Further resolving the terms yields: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='1 − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='ρv(w)=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='� ¯Ew ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='= 1 − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='ρv(w)=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='(1 − p (Ew)) = 1 − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='ρv(w)=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='(1 − ∆w) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='Since the above equations hold for any fixed ρv: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='∆ = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='max ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='||ρv||1≤ρ p (E(ρv)) ≤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='max ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='||ρv||1≤ρ 1 − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='ρv(w)=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='(1 − ∆w) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='Assume we have ordered ∆w so that ∆i ≥ ∆i+1 for all i ∈ {1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' , ρ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Then: max ||ρv||1≤ρ 1 − � ρv(w)=1 (1 − ∆w) = 1 − ρ � i=1 (1 − ∆i) = ∆M Note that instead of ∆w we can alternatively use upper bounds ∆w, which yields an even looser upper bound on ∆ since 1 − ρ � i=1 (1 − ∆i) ≤ 1 − ρ � i=1 � 1 − ∆i � 26 F Approximation Error Notably, the multiplicative bound derived above is tighter than the following union bound: Proposition 8 (Union Bound).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Given monotonously decreasing ∆i such that ∆i ≥ ∆i+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Then we have ∆ ≤ ∆U for ∆U ≜ ρ � i=1 ∆i Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' p (E(ρv)) = p � � � ρv(w)=1 Ew � � ≤ � ρv(w)=1 p (Ew) = � ρv(w)=1 ∆w ∆ = max ||ρv||1≤ρ p (E(ρv)) ≤ max ||ρv||1≤ρ � ρv(w)=1 p (Ew) = ρ � i=1 ∆i The union bound is quite loose, not a probability and can even grow larger than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' We show the difference in practice Figure 10 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' We also discuss the approximation error between the upper bounds ∆U, ∆M and the tight ∆ for the following constructed example where all paths are dependent: We assume a setting where an adversary attacks only second-hop neighbors that are connected to the target node via the same direct neighbor of the target node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' With pa = 0 we have ∆ = (1−pd)(1−pρ d) since we only receive a message if the bottleneck edge is not ablated, and at least one edge of the attacked second-hop nodes is not ablated (which is the complementary probability of all second-hop edges are ablated).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' In this constructed case, all paths are dependent as they share the bottleneck edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' We show how the upper bounds compare to the tight ∆ for different edge deletion probabilities pd in Figure 10 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Note that the example is constructed and worst-case adversaries aim at maximizing independencies by choosing nodes without bottleneck edges (in which case the multiplicative bound is a strong bound in practice).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' 0% 20% 40% Perturbed nodes 25 50 75 Cert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' ratio (%) (a) multiplicative bound union bound 0 1 2 3 4 5 6 7 8 9 10 Radii 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='25 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='50 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='75 1 ∆ (b) ∆ (pe = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='4) ∆M (pe = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='4) ∆U (pe = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='4) ∆ (pe = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='8) ∆M (pe = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='8) ∆U (pe = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='8) Figure 10: (a) Multiplicative bound is tighter than union bound and provides stronger guarantees (Smoothed GAT model on Cora-ML with pa = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='85, pd = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' (b) Constructed example: All path share the same bottleneck edge: Comparing the tight ∆ against the union bound ∆U and the multiplicative bound ∆M for different edge deletion probabilities pd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' The multiplicative bound is tighter than the union bound, which can grow larger than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' 27 G Hyperparameters We implement certificates for directed and undirected graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' For our main experiments (Section 7), however, we follow the standard procedure and prepocess all graphs into undirected graphs, only consider the largest connected component, and binarize node features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' We compute simple paths using a modified depth first search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' All datasets are included in PyTorch Geometric (Fey and Lenssen, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='5 We train models full-batch using Adam (learning rate = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='001, β1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='9, β2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='999, ϵ = 10−08, weight decay = 5 ∗ 10−04) for 1,000 epochs with early stopping after 50 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' We use a dropout of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='8 on the feature matrix X and on the attention coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' During training, we sample a different graph from φ(G) each epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Each sampled graph contains nodes with features replaced by the ablation representation t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' We implement t as a parameter of our models: We initialize t using Xavier initialization and we optimize t as we optimize the GNN weights during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' We implement all models for two message-passing layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' We use 8 heads and 8 hidden channels for GAT and GATv2 (Velickovic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Brody et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=', 2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' 64 hidden channels for GCN (Kipf and Welling, 2017);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' and we use k = 64 and temperature=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='0 for SMA (Geisler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' We use the ReLU activation function for the skip-connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' For GDC sparsification, we set the sparsification threshold of GDC to ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='022, and ignore edge attributes resulting from GDC preprocessing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Training-time smoothing parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' We also delete edges and ablate node features during training (using different probabilities pd and pa during training and inference).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Specifically, we train models presented in Section 7 as follows: In Figure 3 (a,b) we show results for pd = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='01, pa = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='6 during training (and pd = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='31, pa = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='794 during inference and certification).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' In Figure 4 (a,b) we use pd = 0, pa = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='59 during training (and pd = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='31, pa = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='71 during inference and certification).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' In Figure 4 (c) we use the same probabilities pd, pa during training and inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' In our experiments (Section 7), we also randomly sample different probabilities for training and inference to explore the joint parameter space of the training-time and inference-time smoothing parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' That is, our search space is [0, 1]4 when sampling different probabilities from [0, 1] for the Pareto-plots in Figure 6 and Appendix H (we sample separately for training and inference).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' H Detailed Results We report certified accuracies in Figure 16 for the corresponding certified ratios in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Moreover, we provide detailed results for the datasets Cora-ML, Citeseer, and PubMed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' We show results for second-hop attacks against (1) smoothed GAT models in Figure 11, (2) smoothed GATv2 models in Figure 12, (3) smoothed GCN models in Figure 13, and (4) smoothed SMA models in Figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' We run 1,000 experiments for each combination, drawing random deletion and ablation probabilities from [0, 1] for each experiment (sampling separately for training and inference).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Lines connect dominating points on the Pareto front.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Comparing results with and without skip-connection we observe that skip-connections allow higher node feature ablation probabilities while retaining high accuracy, which can yield better robustness-accuracy tradeoffs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Moreover, as discussed in Section 7, evaluating certificates in transductive settings comes with serious shortcomings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' We nevertheless report such results in Figure 15 for a smoothed GAT model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Abstained predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Our smoothed classifier abstains from predicting if pv,y∗(G) ≤ pv,˜y(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' We show the ratio of abstained predictions for smoothed GAT models trained on Cora-ML in Figure 17 for different edge deletion probabilities pd and node feature ablation probabilities pa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' We use the same ablation probability during training and inference for this specific experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' We observe that our smoothed classifier abstains for rather large probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Future work could introduce novel architectures and training techniques to further diminish the effect of abstained predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Experiments on ogbn-arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' We run additional experiments and compute certificates for the larger graph ogbn-arixv with 169,343 nodes, 128 attributes and 40 classes (Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' We adopt their transductive setting, implement two-layer smoothed GCNs with skip-connection and compute certificates for 100 randomly chosen test nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' In Figure 18 we show results for pd = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='1, pa = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='4 during training, and pd = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='3, pa=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='8 during inference and certification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Notably, we can certify GNNs for such large graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' However, our approach only achieves 53% clean accuracy in this setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' 5https://pytorch-geometric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='readthedocs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='io 28 Future work could develop novel architectures and training procedures to improve clean accuracy under our smoothing distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Experiments with different confidence levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' We conduct additional experiments with varying confidence levels α and Monte-Carlo samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' We observe strong guarantees for even smaller confidence levels, requiring little computational efforts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' The underlying reason for this is that the theoretical largest certifiable radius of our certificates is bounded, only determined by the edge deletion probability pd and node feature ablation probability pa, and therefore cannot increase by changing α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Our certificates are thus less sensitive to changes in α compared to Neyman-Pearson- based certificates (Bojchevski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' In fact, the difference in certifiable robustness for α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='05 and α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='0001 is already extremely small when drawing just 2, 000 Monte-Carlo samples (Figure 19 a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' We only observe differences in robustness for considerably small amounts of Monte-Carlo samples (Figure 19 b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Drawing 2,000 samples takes only 12 seconds on Cora-ML on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' This is significantly faster compared to all previous probabilistic certificates for GNNs that use up to 106 Monte-Carlo samples (compare (Bojchevski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=', 2020)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' In additional experiments, we also found that the classification accuracy is high for just a few thousand Monte-Carlo samples (Figure 20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' 0% 10% 20% 30% 40% 50% AUCRC 78 80 82 Clean ACC pd>0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='pa>0 pd=0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='pa>0 pd>0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='pa=0 0% 10% 20% 30% 40% 50% AUCRC 74 76 78 Clean ACC pd>0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='pa>0 pd=0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='pa>0 pd>0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='pa=0 0% 10% 20% 30% 40% 50% AUCRC 71 73 75 Clean ACC pd>0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='pa>0 pd=0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='pa>0 pd>0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='pa=0 0% 10% 20% 30% 40% 50% AUCRC 78 80 82 Clean ACC pd>0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='pa>0 pd=0,' metadata={'source': 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pd=0,pa>0 pd>0,pa=0 0% 10% 20% 30% 40% 50% AUCRC 71 73 75 Clean ACC pd>0,pa>0 pd=0,pa>0 pd>0,pa=0 Figure 15: Transductive learning setting: Robustness-accuracy tradeoffs for second-hop attacks against smoothed GAT on Cora-ML, Citeseer and PubMed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Experiments without skip-connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' 30 0% 20% 40% 60% Perturbed nodes 25 50 75 Cert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' acc (%) (a) distance ≥ 2 distance ≥ 1 0 1 2 3 4 5 6 7 Perturbed nodes 25 50 75 Certified Accuracy (%) (b) distance ≥ 2 distance ≥ 1 Figure 16: Certified accuracies for the setting of Figure 3 – Smoothed GAT on Cora-ML: (a) Robust- ness at different distances 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Cora-ML for different edge deletion ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='probabilities pd and node feature ablation probabilities pa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' 0 1 2 3 4 5 6 7 8 Perturbed nodes 25 50 75 Certified ratio smoothed GCN 0 1 2 3 4 5 6 7 8 Perturbed nodes 25 50 75 Certified accuracy smoothed GCN Figure 18: Certified ratio and accuracy for smoothed two-layer GCN on ogbn-arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' We certify 100 randomly selected test nodes in the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Certificates for nodes with distance 2 to the target node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' 31 0 1 2 3 4 5 Perturbed nodes 25 50 75 Cert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' ratio (%) (a) n1 = 2, 000, α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='0001 n1 = 2, 000, α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='05 0 1 2 3 4 5 Perturbed nodes 25 50 75 Cert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' ratio (%) (b) n1 = 25, α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='0001 n1 = 25, α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='05 n1 = 300, α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='0001 n1 = 300, α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='05 Figure 19: Certified ratio of smoothed GAT on Cora-ML (pa = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='84, pd = 0, with skip-connection) for different confidence levels α and number of Monte-Carlo samples n1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' The difference in robustness is already considerably small for just 2,000 samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' 1e-05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='0001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='0005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='05 α 10,000 3,000 1,000 300 50 20 15 10 n 79% 79% 79% 79% 79% 79% 79% 78% 79% 79% 79% 79% 79% 79% 78% 78% 78% 78% 78% 78% 79% 76% 76% 77% 77% 77% 77% 78% 70% 71% 71% 72% 73% 74% 75% 58% 63% 65% 66% 68% 68% 72% 0% 59% 59% 64% 64% 67% 71% 0% 0% 0% 0% 61% 61% 67% Figure 20: Clean accuracy of smoothed GAT on Cora-ML (pa = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='84, pd = 0, with skip-connection).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' for varying number of confidence levels α and Monte-Carlo samples n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' For α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='05 the clean accuracy is high for just 1, 000 samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' For smaller α, the certification accuracy decreases only slightly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Drawing more than 3, 000 samples is not necessary except for extremely small confidence levels such as α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='00001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' 32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='0 pa 0 25 50 75 100 Radii (a) 0 1 2 3 4 5 6 7 8 9 Perturbed nodes 25 50 75 Certified ratio (%) (b) pa= 6√ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='5 + ϵ pa= 4√ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='5 + ϵ pa= 2√ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='5 + ϵ Figure 21: Visualizing Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' (a) Theoretically maximally certifiable radius for given node ablation probability pa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' (b) Certified ratio of smoothed GAT trained on CoraML for different node ablation probabilities (pd = 0, ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='01).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Note: 2√ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='5 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='71, 4√ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='5 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='84 and 6√ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='5 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' I On Neyman-Pearson and Ablation Certificates There are currently two types of randomized smoothing certificates for discrete data: The certificates of Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' (2019) and Bojchevski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' (2020) are based on the Neyman-Pearson Lemma (Neyman and Pearson, 1933), and we therefore call them Neyman-Pearson-based certificates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' The other certificates are ablation-based (Levine and Feizi, 2020b,a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' We show that largest certifiable radius of ablation-based certificates is bounded indepdentent of the classifier, which is not the case for Neyman-Pearson-based certificates (see discussion in Section 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' In ablation-based certificates, the bounding constant ∆ determines the probability mass of the distribution pv,y(G) over labels y that the worst-case adversary controls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' This probability mass ∆ is independent of the classifier f and distribution pv,y(G) and solely determined by the smoothing distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Although the final certificates still depend on the classifier f, the largest certifiable radius of such ablation-based certificates is bounded as we show for our interception smoothing certificates: Note again that ∆ does not depend on the base GNN f: the probability to receive at least one message from a perturbed node is only characterized by the number of perturbed nodes ρ, and the probabilities pd for edge deletion and pa for node ablation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Moreover, ∆ is monotonously increasing in ρ, since the probability to receive messages from perturb nodes increases the more nodes adversaries control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Interestingly, since ∆ is monotonously increasing in ρ, there exists a largest certifiable radius that depends on the graph structure and changes for each target node (assuming fixed pd, pa).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' In the special case of node ablation smoothing, we can directly determine the largest certifiable radius: Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Given fixed pa > 0 and pd = 0, it is impossible to certify a radius ρ if pa ≤ ρ√ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Due to Corollary 3 and Corollary 1, we only get certificates if ∆ < 1 2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' the adversary should not control more than half of the distribution pv,y(G) over y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Thus: ∆ < 1 2 (1) ⇔ 1 − pρ a < 1 2 ⇔ pρ a > 1 2 ⇔ pa > ρ√ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='5 since the root is monotonously increasing and pa > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Further, (1) stems from Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Thus we need an ablation probability of at least larger than ρ√ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='5 to certify a radius of ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Proposition 3 allows us to directly determine the largest certifiable radius for given pa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' We visualize this largest radius for different ablation probabilities in Figure 21 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Theoretically, we can only certify large radii for relatively large ablation probabilities: For example, to theoretically certify a radius of 10, we already need an ablation probability of more than 10√ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='5 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='933.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Proposition 3 implies that we cannot certify any radius for ablation probabilities pa ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='5 (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Moreover, we can certify a radius of only 1 for ablation probabilities between 1√ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='5 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='5 and 2√ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='5 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='707.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Note, however, that this is only a theoretical consideration and that the certificate also depends on the label probabilities pv,y∗(G) and pv,˜y(G) in practice (Figure 21 b), where we observe that the certified ratio drops to zero when the largest certifiable radius is passed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' 33 J Message-passing-aware Derandomization As discussed in Section 6, our certificates are probabilistic and hold with a certain confidence level α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Here we present alternative, deterministic certificates using a simplified smoothing distribution that just deletes nodes instead of ablating their features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' We believe that future work can build upon it towards even more efficient and scalable derandomization schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Specifically, our derandomized certificates come with the following advantages: First, they are deterministic, exact certificates and hold independent of a confidence level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Second, the smoothed classifier never abstains from making a prediction (we resolve draws by whatever index comes first).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Third, with more computation time we obtain more derandomized certificates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' This is in continuation to probabilistic certificates that can be improved using more Monte-Carlo samples (Cohen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Simplified smoothing distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' We define a smoothed classifier that classifies node v in G as follows: Consider a retention constant k ∈ N that represents the number of nodes not deleted (retained) in the receptive field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Then the smoothed classifier g predicts class y with the largest probability pv,y(G) that f classifies v as y under uniform deletion of all but k nodes: gv(G) ≜ arg max y pv,y(G) pv,y(G) ≜ pK∼U(d,k)(f(RK) = y) where RK encodes the deletion of all nodes in the receptive field of target node v except those indexed by K, and f(RK) denotes the predicted class of f for target node v given ablated graph RK (omitting v for conciseness).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' We further denote the indexing of nodes K as follows: Define the set of all k unique indices in [d] ≜ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' , d} including 0 as B(d, k) = {{0} ∪ M : M ∈ P([d]) ∧ |M| = k}, where P denotes the power set (w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' we index target nodes as 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' For example, K = {0, 1, 3, 6} ∈ B(d, k) for retention constant k = 3 and receptive field size d = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Note that |K| = k + 1 for K ∈ B(d, k) but |B(d, k)| = �d k � since we never delete the target node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Finally, let U(d, k) denote the uniform distribution over B(d, k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' (1) 0 1 2 3 4 5 6 7 8 9 (2) 0 1 2 3 4 5 6 7 8 9 (3) 0 1 2 3 4 5 6 7 8 9 (4) 0 1 2 3 4 5 6 7 8 9 Figure 22: Given a receptive field with 10 nodes, target node 0 and k = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' (1) If we keep nodes K = {0, 1, 3, 6} and delete all other nodes, node 6 is disconnected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' (2) If we keep nodes K = {0, 1, 3, 7} and delete all other nodes, node 7 is disconnected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' (3) In both cases, only the nodes S(K) = {0, 1, 3} affect the prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' (4) In the algorithm: Given S = {0, 1, 3} with neighborhood NS = {2, 4, 5, 8, 9}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Choosing k + 1 − |S| = 1 further nodes, we find that S is a reduced representative S(K) since there are |Vv|−|NS|−|S| = 10−5−3 = 2 nodes to choose from (6 and 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Computing pv,y∗(G) and pv,˜y(G) exactly is challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' One naive approach would be to simply iterate over the support of the smoothing distribution (all possible node deletions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' For small receptive fields, the number of possible combinations to sample k out of d nodes may be small, allowing us to enumerate all possibilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' However, this may be infeasible for larger receptive fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Still, similar to how we use the message-passing structure for certification, we can also leverage it here to partition the support of the simplified smoothing distribution into a smaller number of equivalence classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Specifically, we observe: First, when uniformly deleting nodes in the receptive field, some of the remaining nodes K may be disconnected from the target node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Moreover, disconnected nodes will not affect the prediction for the target node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Second, several possibilities for K may share the same nodes that are still connected to v (see examples in Figure 22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' This means that different possibilities for K will lead to the same prediction by f, but the full enumeration of all possibilities is suboptimal: We wish to avoid redundant evaluations since the evaluation of the base classifier f may be costly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' We observe that the connectivity explained above induces an equivalence relation: All sampled nodes K that share the same nodes connected to v can be grouped into equivalence classes [K].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' For any representative K of [K] we denote the nodes still connected to v as S(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' We call S(K) a reduced representative, since it represents a reduced form of K and only contains the nodes from which the target node will receive messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Note that S(K) is unique for all representatives K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' 34 Formally, given receptive field R with d + 1 nodes and index K ∈ B(d, k) of k + 1 nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Consider the subgraph RK induced by K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' We observe that not necessarily all nodes in RK have to be connected to the target node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Thus, different K ∈ B(d, k) will result in same prediction of the base classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Let S(K) ⊆ K denote all nodes indexed by K without the disconnected nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Put differently, S(K) stands for nodes still connected to the target node (see example in Figure 22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Then: Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' The definition of S(K) induces an equivalence relation ∼ over B(d, k) given by K ∼ K′ ⇔ S(K) = S(K′) and eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' classes [K] := {K′ ∈ B(d, k) : K ∼ K′} for K ∈ B(d, k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Reflexivity, symmetry and transitivity hold by the definition of sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' The equivalence relation ∼ partitions B(d, k) into disjoint equivalence classes, denoted by the quotient set B(d, k)/ ∼ ≜ {[K] | K ∈ B(d, k)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' The set S(K) is uniquely defined for each equivalence class [K] in B(d, k)/ ∼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' We therefore call S(K) with 1 ≤ |S(K)| ≤ k + 1 the reduced representative of [K].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Note that we have |S(K)| = k + 1 ⇔ S(K) = K and |[K]| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' We further call S = {S(K) | K ∈ B(d, k)} the complete set of reduced representatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Note that S ∼= B(d, k)/ ∼ and thus |S| = |B(d, k)/ ∼ |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' To efficiently derandomize our certificates, we can leverage the fact that we only need a complete set of reduced representatives S to compute the label probabilities pv,y(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Given S, we only have to evaluate f once for each reduced representative S(K) ∈ S: Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Given the complete set of reduced representatives S, the label probabilities are: pv,y(G) = �d k �−1 � S∈S I[f (RS) = y] · βS where I[f (RS) = c] indicates whether f classifies the target node v in subgraph RS as class c, and βS is the size of an equivalence class, βS = |[K]|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' We write S ≜ S(K) and omit v for conciseness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' For all K, K′ ∈ B(d, k) with K ∼ K′ we have fv(Rv K) = fv(Rv K′) = fv(Rv S(K)) as only information from nodes of the reduced representative S(K) can be passed to the target node (other nodes are disconnected).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Thus, instead of evaluating fv(Rv K(G)) for all K ∈ B(d, k) we only have to evaluate fv(Rv S(K)(G)) for each S(K) ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' To do so we have to count fv(Rv S(K)(G)) = i exactly βS = |[K]| times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Further, as we uniformly sample K from U(d, k) over B(d, k), we have to scale the possibilities by |B(d, k)|−1, which corresponds to the inverse binomial coefficient above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Hence, we can compute the label probabilities pv,y(G) exactly for larger receptive fields if the number of equivalence classes |S| is small and we have an efficient algorithm to compute S and βS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' We propose such algorithm by exploiting the sparsity of graphs as follows: We successively enumerate all possible connected subgraphs of the receptive field R indexed by S that contain the target node and at most k further nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Let S denote indices of such subgraph of R and NS the neighborhood of S in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' If S contains k+1 nodes, then all k+1 nodes will be connected to the target node and S is already a representative with βS = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' If S contains less than k + 1 nodes, then S corresponds to a reduced representative if we can choose the remaining k + 1 − |S| nodes such that they are disconnected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Therefore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' the main idea of our algorithm is that the size βS is just a binomial coefficient: The number of disconnected nodes is given by |Vv| − |NS| − |S|,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' out of which we have to choose k + 1 − |S| nodes to augment S to set of k + 1 nodes (where Vv denote nodes in the receptive field): βS = �|Vv| − |NS| − |S| k + 1 − |S| � If βS > 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' there must exist a representative K such that the reduced representative S(K) corresponds to S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' that is S = S(K) (compare (4) in Figure 22 for an example).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Finally, our algorithm enumerates all possible S by recursively augmenting S with nodes from the neighborhood of S (compare algorithm 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' This way, we exploit the sparsity of graphs to find all reduced representatives S that avoid disconnected nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' 35 Algorithm 1: Compute complete set of reduced representatives S and equivalence class sizes βS Input: Index 0 of target node v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Receptive field Rv = (Vv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Ev),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Retention constant k S ← {0} Output: EQCGeneration(S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Vv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Ev,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' k) Function EQCGeneration(S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Vv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Ev,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' k): R ← {} if |S| = k + 1 then return {(S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' 1)} end NS ← {w ∈ Vv \\ S | ∃u ∈ S : (w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' u) ∈ Ev} // O(|Vv|) βS ← binom(|Vv| − |NS| − |S|,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' k + 1 − |S|) if βS > 0 then R ← {(S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' βS)} end for w ∈ NS do // O(|Vv|) R ← R ∪ EQCGeneration(S ∪ {w},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Vv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Ev,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' k) end return R Note that in algorithm 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Vv denotes nodes in the receptive field of classifier f with respect to target node v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' and Ev the edges in the receptive field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Lemma 2 (Correctness of algorithm 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Let S with 0 ∈ S ⊆ Vv be a set of at most k + 1 nodes 1 ≤ |S| ≤ k + 1 such that all nodes indexed by S are connected to the target node in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' We denote the neighbors of S in R as NS ≜ {w ∈ Vv \\ S | ∃u ∈ S : (w, u) ∈ Ev}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' When we define the following binomial coefficient as βS ≜ �|Vv| − |NS| − |S| k + 1 − |S| � ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' then there exists a representative K ∈ B(d, k) such that S is a reduced representative for the equivalence class [K] if βS > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Then we have βS = |[K]|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' First note that for a given set S as defined above we can partition Vv into three disjoint sets Vv = S ⊎NS ⊎Nr with S and NS defined as above, and the disconnected nodes Nr ≜ Vv \\(S ∪NS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' We thus have |Nr| = |Vv| − |NS| − |S|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Now we distinguish the following cases: Case 1: |S| = k + 1 We have |Vv| − |NS| − |S| ∈ N0 and βS = 1 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Thus for |S| = k + 1 the condition is trivially fulfilled and we have that K ≜ S is already a representative with |[K]| = 1 as discussed before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Note that this does not mean that all sets with k + 1 nodes are representatives, as we still have the connectivity constraint for nodes in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Case 2: |S| < k + 1 We have βS > 0 ⇔ |Vv| − |NS| − |S| ≥ k + 1 − |S| ⇔ |Nr| ≥ k + 1 − |S| where the latter means that we can choose the remaining k + 1 − |S| nodes from Nr to augment S to representative K of the equivalence class [K] since then |K| = |S| + k + 1 − |S| = k + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' The corresponding size |[K]| is given by βS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Finally, note that the equivalence classes and the algorithm are independent of the classifier f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' 36 Discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' In the worst case, we have |S| = |B(d, k)| = �d k � , but we enumerate �k i=0 �d i � ≥ �d k � possibilities, as there are �k i=0 �d i � candidates for reduced representatives in a fully connected graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Therefore, in the worst case of fully connected graphs, directly enumerating all �d k � possibilities would be faster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' In practice, however, we rather observe sparse graphs with |S| ≪ |B(d, k)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' The more sparse the receptive field, the less equivalence classes exist and the larger each equivalence class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Thus we exploit the sparsity of graphs to efficiently compute S and the corresponding sizes |[K]| for all equivalence classes [K].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Moreover, as our algorithm recursively enumerates all possible pairs (S, βS), we can determine a stopping criterion at which we back off to Monte-Carlo sampling for estimating the label probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' To this end, if R denotes the current set of (S, βS) pairs with βS > 0, we know that |R| is a lower bound on the number of equivalence classes, |R| ≤ |S|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' By summing up βS for all (S, βS) ∈ R we can determine the percentage of |B(d, k)| that we already cover with R: � (S,βS)∈R βS ≤ � S(K)∈S |[K]| = �d k � = |B(d, k)| This allows us to use the condition � (S,βS)∈R βS > τ ′ with threshold τ ′ ∈ N as a stopping criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Using thresholds this way, our algorithm will always find more solutions in S given more time via larger thresholds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Note that we use �d k � > τ in practice, since the binomial coefficient provides a fast upper bound for the number of equivalence classes |S|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='1 Evaluating Message-passing-aware Derandomization Table 1: Smoothed classifier results for GCN trained on Cora-ML for different relative retention constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Der.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' : Ratio of nodes with derandomized certificates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' : Mean of unique receptive fields over all derandomized certificates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' : Clean accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' GCN on Cora-ML GCN on Citeseer GCN on PubMed krel Der.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Abstained Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Der.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Abstained Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Der.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Abstained Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='01 0.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='23e-03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='77 Relative retention constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Consider a small retention constant k = 1 for a node v with deg(v) < dv−deg(v), where dv denotes the receptive field size (excluding the target node).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Then the probability for selecting a direct neighbor of v is low and the prediction of the smoothed classifier is merely based on the target node v itself, which amounts to traditional i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Thus, for non-trivial robustness guarantees we use retention constants k that are relative to the receptive field size: Given a fixed relative retention constant krel ∈ [0, 1], our smoothed classifier keeps k = ⌈dv ·krel⌉ ∈ N nodes in the receptive field R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='6 The ceiling operation ensures that we keep at least one additional node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Derandomization results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Our certificates are deterministic for small receptive fields, and proba- bilistic for large receptive fields: we derandomize certificates if �d k � is smaller than a threshold τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' If the number of possibilities to choose k out of d nodes is small, we can enumerate all possibilities and use f to predict the class of v for all possibilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' In our experiments we set τ = 100,000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' There are more possibilities to sample k out of d nodes for larger krel and thus the ratio of deterministic certificates decreases (compare Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' For example, we can derandomize around 50% of the certificates for Cora-ML given krel = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' We further derandomize more certificates for Citeseer than for Cora-ML, which can be explained by the fact that two-layer GNNs have larger receptive fields on Cora-ML.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Note that the average degree in Cora-ML is 6, in Citeseer 3 and PubMed 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' Due to the derandomization we also hardly observe that the smoothed classifier abstains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' As discussed above, we avoid evaluating the base classifier f for equivalent receptive fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' To represent the computations we avoid on average, we compute the mean of unique receptive fields |S|/|B(d, k)| for all derandomized certificates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' For example, out of all derandomized certificates for krel = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content='1 on Cora-ML, we only have to evaluate 28% of all possibilities on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' 6As a disadvantage of this method, we have to process all receptive fields separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} +page_content=' 37' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdA0T4oBgHgl3EQfF_8g/content/2301.02039v1.pdf'} diff --git a/_9AzT4oBgHgl3EQf_v5Q/content/tmp_files/2301.01952v1.pdf.txt b/_9AzT4oBgHgl3EQf_v5Q/content/tmp_files/2301.01952v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..2636c0b6856772c0ab78d6fedeb23d5881ef8e1a --- /dev/null +++ b/_9AzT4oBgHgl3EQf_v5Q/content/tmp_files/2301.01952v1.pdf.txt @@ -0,0 +1,2136 @@ +Quantum Bayesian Inference in Quasiprobability Representations +Clive Cenxin Aw,1 Kelvin Onggadinata,1 Dagomir Kaszlikowski,1, 2 and Valerio Scarani1, 2 +1Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543 +2Department of Physics, National University of Singapore, 2 Science Drive 3, Singapore 117542 +(Dated: January 6, 2023) +Bayes’ rule plays is a crucial piece of logical inference in information and physical sciences alike. +Its extension into the quantum regime has been the object of several recent works. These quantum +versions of Bayes’ rule have been expressed in the language of Hilbert spaces. In this paper, we +derive the expression of the Petz recovery map within any quasiprobability representation, with +explicit formulas for the two canonical choices of “normal quasiprobability representations” (which +include Discrete Wigner representations) and of representations based on symmetric, informationally +complete positive operator-valued measures (SIC-POVMs). By using the same mathematical syntax +of (quasi-)stochastic matrices acting on (quasi-)stochastic vectors, this construction brings to the +fore the structural similarities and the core differences in logical inference between classical and +quantum theory. +I. +INTRODUCTION +Inference is a logical necessity in every science. +In +information theory and physics, the fundamentality of +inference is particularly overt in notions of process re- +versibility and state recovery. Here, the most empirically +applied and canonical approach is Bayes’ rule: +˜Eγ(a|a′) = E(a′|a) γ(a) +˜γ(a′). +(1) +This relation gives us a recipe for obtaining various +probability-theoretic objects [1–4]. +Of particular note, +we may use it to obtain the “reverse” transition ˜Eγ for +any given (i) the forward process or transformation E, +and (ii) the reference prior γ on the input of said pro- +cess. +The posterior, ˜γ(a′) = � +a E(a′|a)γ(a), emerges +from these two objects. +This typical form of Bayes’ rule works only for classical +information theory. The extension to quantum theory re- +quires some work: as one possible reason for this, notice +that in a classical process a → a′ one can retain informa- +tion on both input and output, and thus define the joint +probability distribution P(a, a′); while nothing of the sort +can be done for the quantum process α → α′ = E(α), +where E is a completely positive trace preserving (CPTP) +map. +Various proposals have been presented over the +years, and we refer to a very recent consolidating frame- +work for all the references [5]. A special role is played by +the Petz recovery map [6–8]: +ˆEγ[•] = √γ E† +� +1 +� +E[γ] +• +1 +� +E[γ] +� +√γ, +(2) +This recovery channel is defined for any CPTP map E +and a reference density operator γ. Notably, when refer- +ence priors, input states and the channel share the same +eigenbases, the Petz map reduces to the classical Bayes +rule [8, 9]. This and other properties pertaining to what +may be called the “conservation of divergences” (which +is what led to its conception) has built up this recovery +map’s reputation as the “quantum Bayes’ rule” [10]; a +reputation recently vindicated in an axiomatic approach +[11]. The Petz map construction appears also naturally in +the definition of fluctuation theorems in thermodynamics +[12–14]. +Now, having said this, it seems that what exactly +makes the Petz map similar (or different) to the classical +Bayesian update has not been formalized as well as it can +be. From an information-theoretical perspective, there +are correspondences between the action of these recipes. +Yet, we know that there are key regime-differences in the +woodwork. This lack of formal comparison across these +regimes is at least partially because the Petz map has +thus far only been formalized in terms of CPTP maps +and density operators, living in a Hilbert space. Mean- +while, the classical Bayes rule exists as a stochastic ma- +trix mapping stochastic vectors, living in a real vector +space. +In this paper, we attempt to close this gap by inves- +tigating the Petz map in quasiprobability representation +(QPR) [15, 16]. This formalism provides a complete de- +scription of quantum theories while sharing the famil- +iar mathematical equipment found in classical probabil- +ity theory. The distinction is that quasiprobabilities (or +“negative probabilities”) are generally necessary in the +quantum case [17]. This negativity has been attributed +as a resource for advantage in quantum computation [18– +20]. As such, we seek to put the Petz map in the same +formal habitat as that of classical Bayesian inversion and +in an expression that is comparable to it. From there we +may discuss the similarities, differences and interpreta- +tions wherever appropriate. We believe this work makes +a formal step in understanding the essential distinctions +arXiv:2301.01952v1 [quant-ph] 5 Jan 2023 + +2 +between classical and quantum inference. +This paper is sectioned as follows. In Section II, we +review features of Bayesian inference for classical and +quantum transformations. In Section III, we review the +formalisms of QPR in quantum theory. Readers famil- +iar with the formal content here may skim through these +sections. In Section IV, we work towards the key expres- +sion of the Petz map in QPR, stating relevant theorems +along the way. In Section V, we discuss consequent theo- +retical observations, contrasting notable formal features +of the expression to the classical Bayesian update. We +also introduce “transition graphs” that can help visual- +ize the implications of our results. Finally in Section VI, +we summarize our findings and state some open lines of +inquiry. +II. +CLASSICAL & QUANTUM BAYESIAN +INFERENCE +In the context of classical mechanics and probabil- +ity theory, a physical transformation can be expressed +by conditional probabilities E(a′|a) mapping probability +distributions of inputs p(a) to distributions of outputs +˜p(a′) = � +a E(a′|a)p(a) residing in some given state space +A [21]. This can be captured compactly by a stochas- +tic matrix SE = {E(a′|a)}, mapping vp = {p(a)} to +v ˜p = {˜p(a′)}. +As already discussed, if we want to acquire a stochasti- +cally valid and logically sound “reverse” of this transfor- +mation E, we must invoke not only the channel in ques- +tion but also a reference prior γ on the input. This is es- +sentially a pre-existing best guess of the inputs for which +the Bayesian inverse is constructed. This process of ac- +quiring ˜Eγ from E and γ can be referred to as perform- +ing “retrodiction” (inference about the past, in contrast +to prediction, inferring about the future) on E on the +prior γ. Meanwhile, S ˜Eγv ˜p gives the “retrodicted input” +given an observation ˜p. It may also be referred to as the +“Bayesian update on γ given ˜p”. +For every each individual transition a → a′, we may +consult (1) for the corresponding retrodiction a′ → a. +For the mapping of distributions, it is more instructive +to write the retrodiction map as a stochastic matrix: +S +˜Eγ +CL = Dγ(SE)TD−1 +E[γ] +(3) +Here Dp is a diagonal matrix with entries corresponding +to some distribution p. +As introduced in Section I, the counterpart to Bayes +rule in quantum theory, is the Petz map (2). It is well- +defined and CPTP for any full-rank E[γ].[22] It may also +be expressed as +ˆEγ = Mγ1/2 ◦ E† ◦ ME[γ]−1/2, +(4) +where Mαr[•] = αr • αr for any density operator α and +r ∈ R, and E† is the adjoint of E. This is the unique map +for which +Tr(E[ρ]σ) = Tr +� +E†[σ]ρ +� +(5) +for all ρ, σ. +Before continuing, +it is important to stress that +Bayesian inference is generically not inversion. +Infer- +ence is possible for any map, while inversion is only +possible for invertible maps (information-preserving) – +and even then, the two operations are generally not the +same, since the inverse of a map is generically not a valid +map. In fact, it can be proved that inference and inver- +sion coincide if and only if SE is a permutation (for the +classical case), or E is a unitary channel (in the quan- +tum case) [14]. In general therefore, S +˜Eγ +CLSEvρ ̸= vρ and +ˆEγ ◦ E[ρ] ̸= ρ; although the reference state is recovered: +S +˜Eγ +CLSEvγ = vγ and ˆEγ ◦ E[γ] = γ for all γ. +III. +QUASIPROBABILITY REPRESENTATIONS +A. +Generalities +We now move on to provide a brief review of the es- +sential elements of QPRs for quantum theory. To bridge +quantum theoretic objects in a d-dimensional Hilbert +space to a QPR, the architectural core is given by the +so-called frame {Fj}j∈Λ, which is a set of Hermitian op- +erators spanning the Hermitian space equipped with in- +ner product. +We denote Λ as the discrete state space +with a minimal cardinality of d2 [23]. Those saturating +the lower bound referred as minimal bases, which we will +assume for the remainder of this paper. A counterpart +to the frame is known as the dual frame {Gj}j∈Λ, which +is defined such that: +∀ A, B : +� +j +Tr[FjA] Tr[GjB] = Tr[AB] . +(6) +In general, the dual is not unique given a frame. However, +for a minimal basis, the frame and dual always enjoy an +orthogonality relation Tr[FjGk] = δjk. +As long as these objects are known to the user, we can +describe all Hibert space objects in terms of QPR. The +morphisms are summarized in Table I. By requiring that +our state quasiprobability to be normalized, � +a vρ +a = 1, +this immediately implies a constraint on the frame op- +erators: � +a Fa = 1. Moreover, with each POVM {Em} + +3 +Object +Hilbert space formalism +Quasiprobability formalism +State +ρ = � +i λi |λi⟩ ⟨λi| +vρ : vρ +a = Tr[ρ Fa] +POVM +{Em | Em ≥ 0 , � +m Em = 1} +¯vm : ¯vm +a′ = Tr[Em Ga′] +Unitary +U[•] = U • U †, UU † = 1 +SU : SU +a′a = Tr +� +Fa′UGaU †� +Channel +E[•] = � +l κl • κ† +l , � +l κ† +l κl = 1 +SE : SE +a′a = Tr[Fa′E[Ga]] +Born Rule +Tr[ρEm] +vρ · ¯vm ∈ [0, 1] +Dimensionality +dim[Cd] = d +dim[Zd ⊗ Zd] = d2 +TABLE I: Morphisms between the Hilbert space formalism and quantum theory. vp +a = p(a) indicates the a-th entry +in a p-distribution. SE +a′a = E(a′|a) indicates the entry on the a′-column and a-row of a matrix SE. +satisfying a unity sum � +m Em = 1, we also have a con- +straint for the dual frame operators: Tr[Gj] = 1 for all +j ∈ Λ. Likewise, it is the case that Tr[Fj] = 1/d for all +j ∈ Λ. As such, the QPR of any CPTP map E is a quasi- +stochastic matrix SE. With a slight abuse of notation, +for ease of correspondence with the classical formalism, +we shall also denote the elements of the quasi-stochastic +matrix as SE +a′a ≡ E(a′|a). +Now despite the vast plurality of valid representations +that adhere to these rules, there are two canonical choices +of QPR used in the relevant literature. To these, we turn. +B. +Normal quasiprobability representation +The first class of representations are those for which +the frame and dual frame operators are proportional to +each other up to some scaling factor c, i.e., Gj = cFj +for all j. For minimal bases, the constant c is equal to +the Hilbert space dimension d. The class of representa- +tions satisfying this is known as normal quasiprobability +representation (NQPR) [24]. +An example of NQPR, and perhaps the most widely +used representation, is the discrete Wigner (DW) rep- +resentation [25–27], which is well-defined for prime di- +mension d and composites of them. For a qubit system +(d = 2), the frame has a simple expression given by +Fk = Fr,s = 1 +4 +� +1+(−1)rσx+(−1)sσz+(−1)r+sσy +� +, (7) +where k = (r, s) ∈ Z2 × Z2. For composite d = d1 × +d2 × · · · × dL, where d1, d2, . . . , dL are primes, a tensor +structure applies for the total frame. That is, the frame +operators decompose as +Fk = Fk1 ⊗ Fk2 ⊗ · · · ⊗ FkL, +where k → (k1, k2, . . . , kL) with each kl = (rl, sl) ∈ Zdl × +Zdl. This tensor structure is enjoyed by any NQPR and +thus affords them an aesthetic benefit when dealing with +composite states and purifications +C. +SIC-POVM representation +Under NQPR, negativity can be found in states, +POVM elements, and transformations alike. Symmetric, +informationally complete, positive operator-valued mea- +sure (SIC-POVM) representations seek to avoid this by +ensuring that all state vectors are positive [28, 29]. Neg- +ativity features are thus consolidated into the transfor- +mations and POVMs +For d-dimensional Hilbert space, a SIC-POVM is +defined as a set of sub-normalized rank-1 projectors +{ 1 +dΠj}d2 +j=1, Πj = |ψj⟩⟨ψj|, such that the elements have +equal pairwise Hilbert-Schmidt inner product: +Tr +� +Π† +jΠk +� += |⟨ψj|ψk⟩|2 = dδjk + 1 +d + 1 . +The solution to the vectors of SIC-POVM have been +found for vast number of dimensions (see [30] for the +list), and is believed to exist for all [31]. Since the set +is informationally complete (i.e. it forms a basis) we can +use it as the definition of the SIC-POVM representation’s +frame {Fj = 1 +dΠj}. From the orthogonality relation, it +can be easily deduced that the dual frame is given by +Gj = d(d + 1)Fj − 1 = (d + 1)Πj − 1. +(8) +As an example, the canonical choice for the one-qubit + +4 +scenario is the tetrahedron +F0 = 1 +4 +� +1 + 1 +√ +3(1, −1, 1) · ⃗σ +� +, +(9) +F1 = 1 +4 +� +1 + 1 +√ +3(1, 1, −1) · ⃗σ +� +, +(10) +F2 = 1 +4 +� +1 + 1 +√ +3(−1, 1, 1) · ⃗σ +� +, +(11) +F3 = 1 +4 +� +1 + 1 +√ +3(−1, −1, −1) · ⃗σ +� +, +(12) +where ⃗σ = (σx, σy, σz) is the vector of Pauli matrices. In +our calculations, the choice of representation, when rele- +vant, will be stated in context and distinguished. If not, +the derivation will apply generally to all representations. +IV. +THE PETZ MAP IN QUASIPROBABILITY +FORMALISMS +Now, our task is to express the Petz recovery map in +its QPR, which we denote as S ˆEγ. This obviously can be +done by invoking the morphism for channels in Table I +and then connecting it with (2). This gives: +S +ˆEγ +aa′ = Tr +� +Fa +√γE† +� +1 +� +E[γ] +Ga′ +1 +� +E[γ] +� +√γ +� +(13) +But, of course, this affords us no new insight. +We +are still relying entirely on the Hilbert space formal- +ism. Nothing novel can be said in comparison to clas- +sical Bayesian inference as found in (3). +Our specific +task is as illustrated in FIG. 1: +write the Petz in a +way that only quasiprobability-theoretic objects (quasi- +stochastic vectors, matrices and frames) are required. +E, γ +SE, vγ +ˆEγ +S +ˆ +Eγ +QPR +PETZ +? +QPR +FIG. 1: The task, illustrated commutatively. +The naive guess that S ˆEγ could be obtained by graft- +ing the quasiprobabilistic formalism onto the classical +Bayesian inverse (3) is easily dismissed: the S +˜Eγ +CL obtained +by such a recipe is in general not a valid map (see Ap- +pendix E for explicit counterexamples). Rather, taking a +hint from (4), we note that channel isomorphism works +also when a map is not CPTP. Hence it is the case that +S +ˆEγ = Mγ1/2 +� +SE†� +ME[γ]−1/2 . +(14) +with +SMαr +a′a +:= (Mαr)a′a = Tr[Fa′αrGaαr] . +(15) +Now, it is crucial for our goals that all objects entering +(14) can be constructed within the quasiprobability for- +malism: so we have to prove that this holds for Mαr. +As a first check, we notice that all the entries of these +matrices are real. Indeed, one can rewrite (Mαr)a′a = +Tr[Fr +a′Gr +a] with Fr +a = αr/2Faαr/2 and Gr +a = αr/2Gaαr/2. +These are Hermitian operators, and so Tr[Fr +a′Gr +a] = +1 +2Tr[{Fr +a′, Gr +a}] is real. Next, we provide a recipe to ex- +plicitly compute the Mαr (recalling that we shall need it +for r = ± 1 +2). For r = 1, it is relatively straightforward +that +(Mα)a′a = Tr[Fa′αGaα] +(16) += +� +xy +vα +x vα +y Tr[Fa′GxGaGy] +(17) +:= +� +xy +vα +x vα +y ξa′xay +(18) +where the ξpqrs = Tr[FpGqGrGs] are referred to as struc- +ture coefficients. Here we have invoked the fact that ev- +ery density operator α can be reconstructed from vα as +α = � +x vα +x Gx. While such a closed expression cannot +be found for r = ± 1 +2, fortunately for any r ∈ R one can +prove (see Appendix A) that +Mαr = M r +α . +(19) +Thus, to compute the Mαr for r = ± 1 +2, one first writes +down Mα and then takes the suitable roots. The resulting +matrices are guaranteed to contain only positive entries +by the remark above, which was valid for every r. +In summary, we have obtained our main result: +Result. The Petz map in any QPR reads +S +ˆEγ +QM = M 1/2 +γ +� +SE†� +M −1/2 +E[γ] +(20) +where +(Mγ)a′a = +� +xy +vγ +xvγ +y ξa′xay +� +ME[γ] +� +a′a = +� +xy +(SEvγ)x(SEvγ)y ξa′xay +and ξpqrs = Tr[FpGqGrGs] are structure coefficients de- +termined by the specific QPR. Everything is expressed +exclusively in the quasiprobabilistic formalism: no knowl- +edge of Hilbert space renditions of the quantum channel +or reference state is required. +For the two canonical choices of QPR introduced +above, we prove in Appendix B that +NQPR : SE† +NQ =(SE)T +(21) + +5 +SIC-POVM : SE† +SP =(SE)T + KE +(22) +where (KE)ij = 1 +d(� +a E(j|a) − 1); whence explicitly +S +ˆEγ +NQ = M 1/2 +γ +(SE)TM −1/2 +E[γ] +(23) +S +ˆEγ +SP = M 1/2 +γ +� +(SE)T + KE +� +M −1/2 +E[γ] +(24) +Since the QPR of unital maps (i.e. E[1] = 1) are quasi- +bistochastic matrices (that is, � +a E(j|a) = 1 for all j), +for such maps KE vanishes and the expressions for NQPR +and SIC-POVM representations are formally identical. +V. +DISCUSSION +A. +Formal Comparisons Across Regimes +Here we discuss and compare formal features across +classical and quantum Bayesian inference, as expressed +in (3) and (20). The key points of comparison are sum- +marized in Table II. +We first express (3) in the following form: +S +˜Eγ +CL = W 1/2 +γ +� +SE†� +W −1/2 +E[γ] +(25) +Here, we have highlighted two things about the classical +retrodiction map. Firstly, (25) highlights the fact that +one can always write Dγ as the square root of its own +square Wγ = D2 +γ. This cosmetic change has advantages +for comparing with (20) later. We leave also a reminder +that Dγ is a diagonal matrix with entries corresponding +to the distribution of γ (i.e. (Dγ)ij = vγ +i δij). +Secondly, (25) highlights the fact that (in parallel with +the opposite relation found in NQPR) for classical chan- +nels the transpose of the channel corresponds to the ad- +joint (SE)T = SE†. That is, (SE)T satisfies the relation +(5) by morphism (see Appendix C). With this, there are +a few similarities and differences worth noting. +Firstly, the retrodiction maps, across both regimes, +feature the same structure: a central “adjoint” matrix +SE†, a prior-dependent matrix (i.e. M 1/2 +γ +, W 1/2 +γ += Dγ) +acting on its left, and a posterior-dependent matrix (i.e. +M −1/2 +E[γ] , W −1/2 +E[γ] = D−1 +E[γ]) acting on its right. +Secondly, this central adjoint object in S +ˆEγ +NQ and S +˜Eγ +CL +are both the transpose of the channel matrix itself. For +S +ˆEγ +SP, the additional KE term may be thought of as cor- +recting for the positivity of the states. +Thirdly, the prior (and posterior) dependent matrices +Xγ differ structurally between classical and quantum in- +ference. Having expressed Dγ as a function of Wγ we see +how these matrices can be generally defined as +(Xγ)ij = +� +xy +vγ +xvγ +yξixjy. +(26) +With this, it becomes clear from that the key difference +between these two domains of inference is the nature of +the “structure coeffecients” ξpqrs. +While in the quan- +tum scenario ξpqrs = Tr[FpGqGrGs], classical Bayesian +inference calls us to something much more reductive: +ξpqrs = δpqδrsδpr. This singular difference in the classical +expression casts out many structural features necessary +in the general quantum case. These may be enumerated: +General Retrodictive Expression +S +¯ +Eγ +RT = X1/2 +γ +� +SE†� +X−1/2 +E[γ] +(Xγ)ij = � +xy vγ +xvγ +yξixjy +Object +Quantum +Classical +SE† +NQ : (SE)T +SP : (SE)T + KE +(SE)T +ξixjy +Tr[FiGxGjGy] +δixδjyδij +TABLE II: Retrodiction maps for classical probabilities [S +¯Eγ +RT → S +˜Eγ +CL, Eq. (3)] and quantum quasiprobabilities +[S +¯Eγ +RT → S +ˆEγ +QM, Eq. (20)]. + +6 +• Firstly, the classical matrix neglects the depen- +dence (present in the quantum matrix) of every en- +try on the aggregation of every parameter in the +prior distribution. +• Secondly and relatedly, the classical case only has +diagonal entries, and only depends on the cor- +responding parameter in the prior distribution. +Meanwhile, the quantum matrix has non-diagonal +“coherences”. +• Thirdly, while the entries of the classical matrix +correspond trivially to values in the prior distribu- +tion, entries in the quantum matrix are weighted +depending on the representation via the trace of +four frame and dual operators. +• Finally, the presence of coherences make it such +that quantum retrodiction finding the root of Mγ. +In the classical scenario, Wγ is already diagonal. +All these features emerge simply because of the differ- +ences between the structure coefficients present in these +scenarios. We elaborate on the significance of these dif- +ferences in Section VI. +There are other resultant properties of Mα, on the ma- +trix level, that may be worth noting. Generally, it is a +real, semi-definite matrix with a unit trace. That is, so +defined, Mα ≥ 0 and Tr[Mα] = 1. For SIC-POVM, it +is not generally symmetric and thus not Hermitian. For +NQPR, however, it does have symmetry and is thus a +density operator under such a representation. That said, +Tr[M r +α] ̸= 1 when rank(Mα) ̸= 1. +Finally, since the square roots of Mγ and ME[γ] are cer- +tainly functions of the E(a′|a) and the γ(a), the quantum +Bayes rule can in principle be written as +ˆEγ(a|a′) = f +� +{E(a′|a)}, {γ(a)} +� +(27) +in full analogy to Eq. (1). But writing down this expres- +sion in practice requires the explicit expressions. For the +simplest quantum case (the qubit) we would be working +to solve for the roots of a quartic characteristic equation, +for which no general analytical solution exists at present. +B. +Visualizing Quantum Inference via QPR +1. +Introducing Transition Graphs +A notable advantage of stochastic maps is their ease +of visualization. +One can draw what might be called +“transition graphs”, where transition between ai to a′ +j +are depicted by arrows going from the former to the lat- +ter. +The probabiltiy weights on these transitions may +be then depicted by a number or by a colour function. +These kinds of graphs are not straightforward to write +for the standard Hilbert space formalism. This is sim- +ply due to the use of complex terms, probability ampli- +tudes and the plurality of possible basis choices. With +QPR, we can illustrate transformations and their quan- +tum Bayesian inverses with transition graphs just as we +would for classical stochastic channels, albeit with the +added task of depicting negativity in these transitions. +In Appendix F and this section, we consider some +choices of E that give rise to SE and their retrodictions +S +ˆEγ +DW and S +ˆEγ +SP. +These are then depicted as transition +graphs. +We have chosen to include, in particular, a +Half-SWAP channel with a |1⟩⟨1| ancilla to visually il- +lustrate and explore the properties of quantum retrod- +iction. Other transformations are also noted in passing +with their graphs and expressions consolidated in Ap- +pendix F. Before these, we note some illustrative ele- +ments of these figures. +Firstly, with transition arrows we depict negative (pos- +itive) quasiprobabilities with cooler (warmer) shades. +Furthermore, these negative (positive) arrows will be +drawn with dashed (solid) lines. A colour legend is in- +cluded in FIG. 2a. +Secondly, in order to get a sensing of how irreversible +a forward map is and which states it tends to erase to- +ward, we add coloured “bubbles” around the output side +(denoted {a′ +j}) of every graph for a given SE. The in- +tensity and colour of the bubbles are weighted according +quasiprobability distribution of the state E[1/d]. Hence, +one should expect that these bubbles are coloured uni- +formly for all unital maps. +Thirdly, a similar feature is added for the retrodictive +transition graphs, drawn for S ˆEγ matrices. Crucial for +understanding the Bayesian inverse is the reference prior. +Hence, for Bayesian inverting transition graphs we add +coloured bubbles on the input (denoted {aj}, that is, the +input of the forward map) side of the graph, weighted +according to the distribution of γ. Finally, for simplic- +ity, we stick to channels acting on qubits. We also use +the most canonical choices of frames for both DW (r, s +starting from 0) and SIC-POVM representations (consol- +idated in (9)). +2. +Fully Reversible & Fully Irreversible +As depicted in Figures 5 and 6 (found in Appendix +F), we observe the provable property that S ˆUγ = S ˆU = +(SU)T, for unitary channels U. +The Bayesian inverses +simply reflect the transition trajectories back, doing so +with equal probability and negativity and regardless of +what reference prior is chosen. More interesting features + +7 +occur for non-unitary channels. We may write any CPTP +map as a dilation defined by a global unitary U acting +on an extended state space HA ⊗HB for which the input +system •A and an environment or ancilla βB is defined: +E[•] = TrB[U • ⊗β U †] +(28) +We stick to the case where both the target and the ancilla +are qubits. Arbitrary qubits may be written as: +β(ω, θ, φ) = sin2(ω) |ψ⟩⟨ψ| + cos2(ω) +��ψ⊥�� +ψ⊥�� +(29) +Where |ψ⟩ = cos(θ/2) |0⟩ + eiφ sin(θ/2) |1⟩ and |ψ⊥⟩ = +e−iφ sin(θ/2) |0⟩ + cos(θ/2) |1⟩. In maximal contrast to +unitary channels, one may consider a quantum total era- +sure channel. This is simply a kind of replacement map +where a Full-SWAP (F1) acts on a qubit and an ancilla +and we trace out the environment. The Bayesian inverse +of such quantum channels follow their classical counter- +parts: they erase back to reference prior [14]. Since the +channel is totally irreversible, the quantum Bayes rule +simply reverts our inference to our best guess about the +initial state (illustrated by FIG. 4). +3. +Liminally (Ir)reversible +For a more conceptually involved and instructive sce- +nario, we consider the Half-SWAP U�, which may be rep- +resented in the computational basis as: +U� ˆ= +1 +√ +2 +� +� +� +� +� +� +� +� +� +√ +2 0 +0 +0 +0 +1 +1 +0 +0 +1 −1 +0 +0 +0 +0 +√ +2 +� +� +� +� +� +� +� +� +� +(30) +As depicted in 2, we have the forward and retrodictive +transition graphs for a channel given by E[•] = TrB[U� • +⊗ |1⟩⟨1| U † +�]. To understand the retrodictive action given +by the Petz, we can gain some intuitions from by writing +out these mappings: +|01⟩ +U� +−−→ +1 +√ +2 +� +|01⟩ + |10⟩ +� TrB +−−→ 1 +21 +|+1⟩ +U� +−−→ 1 +2 |11⟩ + 1 +√ +2 +� +|01⟩ + |10⟩ +� +TrB +−−→ +1 +4 |0⟩⟨0| + 3 +4 |1⟩⟨1| + +1 +2 +√ +2 +� +|1⟩⟨0| + |0⟩⟨1| +� +|11⟩ +U� +−−→ |11⟩ +TrB +−−→ |1⟩⟨1| +(a) Colour Legend for Transition Graphs +(b) SE +DW for E[•] = TrB[U� • ⊗ |1⟩⟨1| U † +�] +(c) S +ˆ +Eγ +DW for U�, β = |1⟩⟨1| , γ = |0⟩⟨0| +(d) S +ˆ +Eγ +DW for U�, β = |1⟩⟨1| , γ = |+⟩⟨+| +(e) S +ˆ +Eγ +DW for U�, β = |1⟩⟨1| , γ = |1⟩⟨1| +(f) S +ˆ +Eγ +DW for U�, β = |1⟩⟨1| , γ( π +16, π +5 , π +3 ) +as per (29) +FIG. 2: Transition Graphs for a “Half-SWAP” with +|1⟩⟨1|, and various retrodictions with a range of +reference priors. + +8 +We see that if the reference state is γ = |0⟩⟨0| or |+⟩⟨+|, +then any state is compatible to its output (they are un- +ambiguously full rank in C2). Hence, the Petz Map erases +all (output) states back to the reference, in full consis- +tency with the earlier comments about the quantum total +erasure channel. This is depicted in Figures 2c and 2d. +A very different situation occurs for γ = |1⟩⟨1|. In this +case only |1⟩⟨1| is allowed as an output. Thus, the Petz +sends |1⟩⟨1| to itself while all other states are retrodicted +in (complicated but logically consistent) ways dependent +on channel’s forward transitions, reflected in FIG. 2e. +To explain this more symmetrically: in the former two +scenarios, all outputs are compatible with the absolute +conviction (as enforced by state purity) given to the refer- +ence state, hence all outputs are retrodicted to it. Mean- +while, in this latter case, only one pure output (which +just so happens to be the same as the reference) is com- +patible with the pure reference state. Hence, all other +states (beside the expected output) are retrodicted in +accordance to the channel without any regard the refer- +ence, since the reference already excludes the possibility +of such states. These more complicated Bayesian inver- +sions come together and cumulate into a vertical reflec- +tion of the forward channel, as FIG. 2e depicts. For an +arbitrary γ, we get a classical mixture of all these key ef- +fects together. We depict the case where γ = γ( π +16, π +5 , π +3 ) +in FIG. 2f. +It should be said the interplay of reference and chan- +nel dependencies we have reviewed here is fundamental in +classical retrodiction scenarios as well. The Half-SWAP +illustrates that these same fundamental Bayesian princi- +ples hold in the quantum regime via the inferential struc- +ture of the Petz Map, even when complementarity and +entanglement is introduced. +VI. +CONCLUSIONS +By expressing the Petz Recovery map as a decomposi- +tion of matrices given by (20) we have situated quantum +Bayesian inference in the same formal language as that +of its classical counterpart given by (26). We have also +highlighted what we have found to be the most note- +worthy (and interpretation-neutral) differences between +these two levels of inference. +It should be clear to the reader that, in keeping with +the Bayesian character of the Petz, the crucial formal dif- +ference is found in the prior-dependent (and in turn, pos- +terior dependent) matrices Xγ. The properties of these +objects encode the most significant differences between +classical and quantum inferences. +Particularly, it for- +malizes how prior and posterior variables are taken into +account to the central adjoint map, structurally speak- +ing. In the classical case we neglect the total aggrega- +tion (all parameters are involved in every entry), “coher- +ences” (non-diagonal terms with sums of product pairs +of the parameters, weighted depending on the choice of +representation) and “eigenstructure” (finding the matrix +root of prior-dependent matrix is not generally circum- +vented) found in the quantum case. This is all emergent +from the simple differences between the “structure coef- +fecients” found across these regimes. Adding to all this +the characteristic notion of negativity embedded in all +three component matrices, we have a sense of just how +wide the conceptual gap we have on our hands. +The reduction in structure that we find in the classical +regime may be understood as a kind of classical epis- +temic prejudice, which leads to absurdities if applied to +general quantum scenarios. This prejudice (or reduction) +is, of course, unsurprising. Classical Bayesian thinking is +deeply intuitive and subconscious for us, regardless of +being mathematically initiated or not. Nevertheless, ar- +ticulating or formalizing such strong intuitions already +has its complications. +Given the orders of differences +between this kind of inference and that of the quantum +regime, we see why the classical prejudice presides our +everyday experience. +That said, this work also illustrates noteworthy sim- +ilarities. Despite profound structural differences, many +Bayesian intuitions seem to nevertheless come together +as illustrated in transition graphs. The prior and pos- +terior dependent matrices and the role of channel’s ad- +joint presides in both regimes. Under this exploration, we +take a step closer to what a regime-independent Bayesian +foundation could be. +Many open questions remain. +For one, we can look +into the overlap between classical and quantum inference: +what (quantum) processes does classical and quantum +retrodiction become equivalent (that is, for every choice +of reference prior)? It is easily checked that this holds +for channels that give permutative SE and quantum to- +tal erasure channels. Aside from these two extreme cases, +are there other channels where this retrodiction property +holds? Related to this, it may be worth investigating if +for some frames or under some gauge transformations the +current decomposition (in (20)) may be simplified into a +more classically intuitive form. No such simpler alterna- +tive has been found for every channel and every refer- +ence. Finally, throughout this paper we have identified +the Petz with quantum Bayesian inference. Nevertheless, +other transpose maps exist that have been seen as retro- +diction channels. It could be noteworthy to perform a +similar decomposition on those maps under quasiproba- +bility schemes. + +9 +ACKNOWLEDGMENTS +This research was supported by the National Research +Foundation and the Ministry of Education, Singapore, +under the Research Centres of Excellence programme (till +6 December 2022); and by the National Research Foun- +dation, Singapore, and A∗Star under the CQT Bridging +Grant (from 7 December 2022 onwards). We also thank +Zaw Lin Htoo and Eugene Koh for helpful discussions. +Appendix A: Mαr = M r +α for all r ∈ R +We know that in general, +(Aij(z) → A) ̸⇒ (Aij(zr) → Ar) . +(A1) +Informally speaking, powers on the level of entry param- +eters do not necessarily translate to powers on the level of +matrices. Thankfully, this does obtain in our case. The +derivation is as follows. We first note that +(Mαr)ij = Tr[FiαrGjαr] +(A2) +It may be tempting to invoke that since +SESF = SE◦F +(A3) +we already have desired theorem proven. However, this +will only do for r ∈ Z+. If we wish to do so for values +of r ∈ R, we must be careful not to already assume the +conclusion in our derivation. Our proof comes in three +main steps. Firstly, +(Mα1/2Mα1/2)ij = +� +k +(Mα1/2)ik(Mα1/2)kj += +� +k +Tr +� +Fi +√αGk +√α +� +Tr +� +Fk +√αGj +√α +� += +� +k +Tr +� +Gk +√αFi +√α +� +Tr +� +Fk +√αGj +√α +� += Tr +�√αFi +√α√αGj +√α +� += Tr[FiαGjα] = (Mα)ij +Crucially, in the fourth equality we use the property (6) +in QPR. Hence, +Mα1/2 = M 1/2 +α +By reiterating this (i.e. sending α → √α), we obtain the +more general relation that +∀x ∈ Z+ : Mα1/2x = M +1 +2x +α +Which is really just to say: +Mαϵ = M ϵ +α, +for an arbitrarily small and positive real number ϵ. In- +voking (A3) and the definition of Mαr, for N ∈ Z+, we +have, +MαNϵ = SMαNϵ += S +N times +� +�� +� +Mαϵ ◦ Mαϵ ◦ · · · ◦ Mαϵ += +� +N +Mαϵ = +� +N +M ϵ +α. +Together, we may thus conclude that for N ∈ Z+: +MαNϵ = M Nϵ +α . +(A4) +Secondly, we note that +(MαMα−1)ij = +� +k +Tr[FiαGkα] Tr +� +Fkα−1Gjα−1� += +� +k +Tr[GkαFiα] Tr +� +Fkα−1Gjα−1� += Tr +� +αFiαα−1Gjα−1� += Tr[FiGj] = δij = 1ij +Hence, Mα−1 = M −1 +α . Repeating this, we can easily see +that for any N ′ ∈ Z+: +Mα−N′ = M −N ′ +α +(A5) +Finally, taking from (A3), (A4) and (A5), we find that: +MαNϵ−N′ = SMαNϵ−N′ = SMαN ϵ◦Mα−N ′ += MαNϵMα−N′ = M Nϵ +α M −N ′ +α += M Nϵ−N ′ +α +Since it holds for any arbitrarily small, positive ϵ and for +any arbitrarily large positive integers N and N ′, we write +Nϵ − N ′ ∈ R and denote this Nϵ − N ′ → r. Thus we +obtain our desired result: Mαr = M r +α for any r ∈ R. +Appendix B: SE† in terms of (SE)T for QPRs +We derive the QPR expressions for SE† for some CPTP +map E[•] = � +l κl • κ† +l . For NQPRs, we find easily that: +(SE† +NQ)ij = Tr +� +FiE†[Gj] +� += Tr +� +Fi +� +l +κ† +l Gjκl +� += +� +l +Tr +� +GjκlFiκ† +l +� += +� +l +Tr +� +FjκlGiκ† +l +� += Tr +� +Fj +� +l +κlGiκ† +l +� += Tr[FjE[Gi]] = SE +ji +Thus, for NQPRs +SE† +NQ = (SE)T +(B1) + +10 +For SIC-POVM representations, we have a more compli- +cated expression. We first use (8), +(SE† +SP)ij = Tr +� +FiE†[Gj] +� += Tr +� Gi + 1 +d(d + 1) E† [d(d + 1)Fj − 1] +� +By expanding the terms and noting the unitality of every +adjoint map (i.e. E†[1] = 1), we arrive at the expression: +(SE† +SP)ij = Tr +� +Fi +� +l +κ† +l Gjκl +� ++ Tr +� +E†[Fj] +� +− Tr[Fi] += SE +ji + Tr +� +E†[Fj] +� +− 1 +d +By taking note of the isomorphisms found in Ta- +ble I, we may write Tr +� +E†[Fj] +� += � +l Tr +� +κ† +l Fjκl +� += +� +l Tr +� +Fjκl1κ† +l +� += +Tr[FjE[1]] += +1 +d(SEv1)j += +1 +d +� +a E(j|a). Hence, we can write the total expression +of each entry for SIC-POVM representation as: +(SE† +SP)ij = SE +ji + 1 +d +�� +a +E(j|a) − 1 +� +(B2) +This can be written, on the matrix level, as (22). +Appendix C: (SE)T as SE† for Classical +In the previous section we proved that for quantum +channels (expressed in QPRs), we can express the adjoint +channel in terms of the tranpose of the channel. Here, +we prove the opposite relation for classical channels: that +the transpose of a classical channel is the adjoint of that +channel. Namely, the transpose map is the map for which +(5) is fulfilled in the case of classical scenarios. Noting +first the commutative diagram found in FIG. 3 (which +invokes the morphisms found in Table I), we see how (5) +is fulfilled by a map for which +(SEvρ) · ¯vσ = (SE†vσ) · ¯vρ, +(C1) +for all ρ and σ. With this, we first expand the LHS of +(C1): +(SEvρ) · ¯vσ = +� +y +(SEvρ)y¯vσ +y +(C2) += +� +xy +SE +yxvρ +x¯vσ +y +(C3) +Next we expand the following, in order to check if the +transpose qualifies as the adjoint: +((SE)Tvσ) · ¯vρ = +� +xy +(SE)T +yxvσ +x ¯vρ +y +(C4) +vρ · ¯vσ +Tr[ρ σ] +(SEvρ) · ¯vσ +Tr [E[ρ]σ] +(SE†vσ) · ¯vρ +Tr +� +E†[σ]ρ +� +∀ρσ +∀ρσ +FIG. 3: Relations between formalisms pertaining the +adjoint map, illustrated commutatively. += +� +xy +SE +xyvσ +x ¯vρ +y +(C5) += +� +xy +SE +yx¯vρ +xvσ +y +(C6) +Now for classical scenarios the trace of two states, if +treated like quantum states in Hilbert space, would sim- +ply be the inner product of its density spectra: vρ · vσ = +Tr[ρ σ]. This is because the states, being classical distri- +butions, would be diagonalized in the same way. Thus +we could have replaced ¯vρ with vρ in all the above calcu- +lations and in FIG. 3. The reason why we have written +¯vρ as opposed to vρ is to simply highlight that while +indeed (C3) is identical to (C6) for classical scenarios be- +cause ¯vρ = vρ there (and NQPR for that matter since +¯vρ = c vρ), the same does not hold for SIC-POVM. The +transpose qualifies as an adjoint for both NQPR and clas- +sical channels, but not for SIC-POVM. Hence, the rela- +tion proved for classical states and channels does not con- +tradict the ones proved in the previous section for QPRs. +Appendix D: Properties of Mα +Here we note some interesting properties of Mα. +Namely that it is a matrix with all real entries and non- +negative eigenvalues that sum to 1. +1. +Real Entries +It can be shown that all the entries of Mα are real: +(Mα)ij = Tr[FiαGjα] ∈ R. +A proof was given in the +main text, valid for any Mαr; we repeat it here for com- +pleteness. +We first note that the anticommutator for any two +Hermitian operators A and B is always also Hermitian: +{A, B}† = {A, B}; while the trace of the commutator of +any two operators is always zero (in finite dimension) due + +11 +to cyclicity: Tr +� +[A, B] +� += Tr[AB] − Tr[BA] = 0. Hence +Tr[AB] = Tr +�{A, B} +2 ++ [A, B] +2 +� += 1 +2Tr[{A, B}] ∈ R +(D1) +Noting that √αFi +√α and √αGj +√α are both Hermitian +(frame and dual operators are always Hermitian, and α +is a density operator in Hilbert Space), we apply (D1) to +(Mα)ij. The entries of Mα are thus proven to be always +real. +2. +Positive Semi-Definitiveness +For NQPR, we can always write +(Mα)ij = c Tr +�√αFi +√α +�√αFj +√α +�†� +Hence, Mα is a Gram matrix with some positive factor +c. Thus it is positive semi-definite. For SIC-POVM, we +expand (Mα)ij via (8), arriving at: +(Mα)ij = +1 +d(d + 1) +� +Tr +�√αGi +√α +�√αGj +√α +�†� ++Tr +� +Gjα2� � +The first term, as with the NQPR case, corresponds to +a Gram Matrix, which is positive semi-definite. One can +then note that the second term corresponds to a matrix +Jα (i.e. (Jα)ij = Tr +� +Gjα2� +) with duplicate rows (every j- +th column with filled with identical entries. This simply +implies that the only non-zero eigenvalue would be the +sum of the entries of any given row. Which just means: +eig[Jα] = {� +j Tr +� +Gjα2� +, 0} = {Tr +� +α2� +, 0} ≥ 0. So Mα +is the sum of two positive semi-definitive matrices and +thus we may conclude Mα ≥ 0 for SIC-POVM as well. +3. +Unit Trace +The trace of Mα is given by +Tr[Mα] = +� +i +(Mα)ii = Tr +� � +i +FiαGi +� +�� +� +1 +α +� += 1 +To prove the relation invoked for the final equality we +will use the previously found result in [32]. Consider the +superoperator +Λ[•] = +d2 +� +i=1 +Πi • Πi , +(D2) +it can be shown that +Λ[Πi] = +d +d + 1(Πi + 1) +(D3) +Since the set {Πi} forms a basis, we can express the su- +peroperator as +Λ = +d +d + 1(I + 1) +(D4) +where I[A] = Tr[A] 1. Using this we can easily show that +for SIC-POVM representation we have +� +i +FiαGi = d + 1 +d +� +i +ΠiαΠi − 1 +dα +� +i +Πi += 1 . +(D5) +For discrete Wigner representation, Zhu [24] showed that +the dual frame can always be expressed as such: +Gi = − +√ +d + 1Πi + +�1 + +√ +d + 1 +d +� +1 +(D6) +Thus, it can also be easily shown that � +i FiαGi = 1 in +this representation. +Appendix E: Examples for S +ˆ +Eγ ̸= S +˜ +Eγ +CL +As discussed in Section V B 3, it is the case that ˆEγ[ρ] = +ˆE+[ρ] = |+⟩⟨+| for all ρ when E[•] = TrB[U� •⊗ |1⟩⟨1| U † +�] +and γ = |+⟩⟨+|. +Yet we can easily find that, for the +canonical state representations for DW and SIC-POVM, +we have: +S +˜E+ +DW = +� +� +� +� +� +� +� +� +� +1 +1 +7 +� +3 − +√ +2 +� +1 +1 +7 +�√ +2 + 3 +� +0 +1 +7 +�√ +2 + 4 +� +0 +1 +7 +� +4 − +√ +2 +� +0 +0 +0 +0 +0 +0 +0 +0 +� +� +� +� +� +� +� +� +� +̸= 1 +2 +� +� +� +� +� +� +� +� +� +1 1 1 1 +1 1 1 1 +0 0 0 0 +0 0 0 0 +� +� +� +� +� +� +� +� +� += S +ˆE+ +DW +Likewise, +S +˜E+ +SP = +� +� +� +� +� +� +� +� +� +0.925 +0.183 +−0.264 +0.353 +0.0744 +0.744 +0.275 +0.168 +−0.0191 0.0491 +0.915 +0.0947 +0.0199 +0.0233 0.0737 +0.384 +� +� +� +� +� +� +� +� +� + +12 +̸= 1 +12 +� +� +� +� +� +� +� +� +� +√ +3 + 3 +√ +3 + 3 +√ +3 + 3 +√ +3 + 3 +√ +3 + 3 +√ +3 + 3 +√ +3 + 3 +√ +3 + 3 +3 − +√ +3 3 − +√ +3 3 − +√ +3 3 − +√ +3 +3 − +√ +3 3 − +√ +3 3 − +√ +3 3 − +√ +3 +� +� +� +� +� +� +� +� +� += S +ˆE+ +SP +Indeed, for some channels one can find states for which +the post-measurement probabilities violate acceptable +bounds. This means S +˜Eγ +CL fails to represent a generally +valid quantum transformation. For instance, for a uni- +tary transformation U[•] = U •U † where U = i +2 +� √ +3 −1 +1 +√ +3 +� +We find that +(S +˜U+ +DWv+) · ¯v0 = 1 +2(1 + +√ +3) > 1 +(S +˜U+ +SPv0) · ¯v+ = 1 +13(2 − 5 +√ +3) < 0. +S ˆEγ ̸= S +˜Eγ +CL is thus easily shown. +Appendix F: Other Transition Graphs +In this appendix, we include illustrative cases of SE, +some respective retrodictions and their transition graphs. +In FIG. 5, the transition graphs are depicted for very +familiar Pauli rotations. It so happens that these uni- +taries translate to SE that give permutations. +This is +seen in the bold bijective transition arrows. Like other +unitary channels, all retrodictions are reference-prior in- +dependent. Transition graphs of such retrodictions are +thus always mirror images of the corresponding forward +transition graph. That said, most unitaries do not en- +joy a permutative structure that exists for these SU(2) +rotations. The Hadamard gate for instance defined by +the following computationally represented operator and +gives the respective quasi-stochastic matrix: +UH ˆ= 1 +√ +2 +� +�1 +1 +1 −1 +� +� , +SUH = 1 +2 +� 1 +1 +1 +−1 +1 +−1 +1 +1 +1 +1 +−1 +1 +−1 +1 +1 +1 +� +, +which is consistent across the canonical choices of the +DW and SIC-POVM representations. +Likewise, an arbitrarily chosen unitary Ueg: +Ueg ˆ= 1 +4 +� +� +i( +√ +3+2i) +0 +3i +0 +0 +i( +√ +3+2i) +0 +3i +−3i +0 +2+i +√ +3 +0 +0 +−3i +0 +2+i +√ +3 +� +� +has the following quasiprobability objects: +SUeg +DW = 1 +16 +� +� +9 +√ +3−6 4−3 +√ +3 2 +√ +3+9 +− +√ +3−6 +9 +9−2 +√ +3 3 +√ +3+4 +3 +√ +3+4 2 +√ +3+9 +−3 +6−5 +√ +3 +9−2 +√ +3 4−3 +√ +3 5 +√ +3+6 +−3 +� +� +SUeg +SP = 1 +16 +� +� +−3 +5 +√ +3+6 4−3 +√ +3 9−2 +√ +3 +6−5 +√ +3 +−3 +2 +√ +3+9 3 +√ +3+4 +3 +√ +3+4 9−2 +√ +3 +9 +− +√ +3−6 +2 +√ +3+9 4−3 +√ +3 +√ +3−6 +9 +� +� +It is clear that these forward channels do not give per- +mutative QPRs. Nevertheless the property that S ˆUγ = +S ˆU = (SU)T is still reflected clearly in FIG. 6. In contrast +to these reversible maps, we can speak of the quantum +total erasure channel mentioned in section V B 2. The +full swap is expressed as such: +U� ˆ= +� +� +� +� +� +� +� +� +� +1 0 0 0 +0 0 1 0 +0 1 0 0 +0 0 0 1 +� +� +� +� +� +� +� +� +� +(F1) +As depicted clearly in FIG. 4, both the forward channel +and its retrodiction are erase perfectly to the relevant +state (the ancilla for the forward map and the reference +state for the retrodiction). +(a) SE +DW for E[•] = TrB[U� • ⊗β( 7π +16 , 3π +5 , π +6 ) U † +�] +(b) S +ˆ +Eγ +DW for U�, β( 7π +16 , 3π +5 , π +6 ), γ( π +16, π +5 , π +8 ) +FIG. 4: Transition Graphs for a Quantum Total +Erasure channel with arbitrary ancilla β and a +corresponding retrodiction with reference prior γ. + +13 +(a) SE for E[•] = σz • σz +(b) S +ˆ +Eγ +SP and S +ˆ +Eγ +NQ for E[•] = σz • σz +(c) SE for E[•] = σy • σy +(d) S +ˆ +Eγ +SP and S +ˆ +Eγ +NQ for E[•] = σy • σy +(e) SE for E[•] = σx • σx +(f) S +ˆ +Eγ +SP and S +ˆ +Eγ +NQ for E[•] = σx • σx +FIG. 5: Transition Graphs for σz, σx and σy and their +respective retrodictions. +(a) SUH for UH[•] = UH • U † +H +(b) S +ˆ +UH for UH[•] = UH • U † +H +(c) S +Ueg +DW for Ueg[•] = Ueg • U † +eg +(d) S +ˆ +Ueg +DW for Ueg[•] = Ueg • U † +eg +(e) S +Ueg +SP for Ueg[•] = Ueg • U † +eg +(f) S +ˆ +Ueg +SP for Ueg[•] = Ueg • U † +eg +FIG. 6: Transition Graphs for the Hadamard gate and +an arbitrarily chosen qubit unitary and their respective +retrodictions. + +14 +[1] S. 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Leifer and R. W. Spekkens, Towards a formula- +tion of quantum theory as a causally neutral theory of +bayesian inference, Phys. Rev. A 88, 052130 (2013). +[11] A. J. Parzygnat and F. Buscemi, Axioms for retrod- +iction: achieving time-reversal symmetry with a prior, +arXiv preprint arXiv:2210.13531 (2022). +[12] H. Kwon and M. S. Kim, Fluctuation theorems for a +quantum channel, Phys. Rev. X 9, 031029 (2019). +[13] F. Buscemi and V. Scarani, Fluctuation theorems from +bayesian retrodiction, Phys. Rev. E 103, 052111 (2021). +[14] C. C. Aw, +F. Buscemi, and V. Scarani, Fluctua- +tion theorems with retrodiction rather than reverse +processes, AVS Quantum Science 3, 045601 (2021), +https://doi.org/10.1116/5.0060893. +[15] C. Ferrie and J. Emerson, Framed hilbert space: hanging +the quasi-probability pictures of quantum theory, New +Journal of Physics 11, 063040 (2009). +[16] C. 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Lett. 115, 070501 (2015). +[21] Of course, one can have it that a′ and a are defined in +different state spaces A and A′, but we can always take +A′′ = A ∪ A′ and characterize the channel in this larger +alphabet. +[22] This constraint also exists in the classical Bayes update +and is likewise of no practical concern as one can always +ensure that that γ is full-rank by adding some arbitrar- +ily small weights into its spectrum and adding some ar- +bitrarily small mapping probability in E as well. These +contributions can then be sent to zero on the recovered +state. +[23] The state space Λ may be continuous as well. +[24] H. Zhu, Quasiprobability representations of quantum me- +chanics with minimal negativity, Phys. Rev. Lett. 117, +120404 (2016). +[25] W. K. Wootters, A wigner-function formulation of finite- +state quantum mechanics, Annals of Physics 176, 1 +(1987). +[26] K. S. Gibbons, M. J. Hoffman, and W. K. Wootters, Dis- +crete phase space based on finite fields, Phys. Rev. A 70, +062101 (2004). +[27] D. Gross, Hudson’s theorem for finite-dimensional quan- +tum systems, Journal of Mathematical Physics 47, +122107 (2006), https://doi.org/10.1063/1.2393152. +[28] M. Appleby, C. A. Fuchs, B. C. Stacey, and H. Zhu, In- +troducing the qplex: a novel arena for quantum theory, +The European Physical Journal D 71, 1 (2017). +[29] E. O. Kiktenko, A. O. Malyshev, A. S. Mastiukova, V. I. +Man’ko, A. K. Fedorov, and D. Chru´sci´nski, Probabil- +ity representation of quantum dynamics using pseudos- +tochastic maps, Phys. Rev. A 101, 052320 (2020). +[30] J. +DeBrota, +C. +Fuchs, +and +B. +Stacey, +Qbism +re- +search group, http://www.physics.umb.edu/Research/ +QBism/. +[31] D. M. Appleby, H. Yadsan-Appleby, and G. Zauner, Ga- +lois automorphisms of a symmetric measurement, Quan- +tum Info. Comput. 13, 672–720 (2013). +[32] J. M. Renes, R. Blume-Kohout, A. J. Scott, and C. M. +Caves, Symmetric informationally complete quantum +measurements, Journal of Mathematical Physics 45, 2171 +(2004). + diff --git a/_9AzT4oBgHgl3EQf_v5Q/content/tmp_files/load_file.txt b/_9AzT4oBgHgl3EQf_v5Q/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d6a22d8a7f016f27d294ab8536315fde20957765 --- /dev/null +++ b/_9AzT4oBgHgl3EQf_v5Q/content/tmp_files/load_file.txt @@ -0,0 +1,674 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf,len=673 +page_content='Quantum Bayesian Inference in Quasiprobability Representations Clive Cenxin Aw,1 Kelvin Onggadinata,1 Dagomir Kaszlikowski,1, 2 and Valerio Scarani1, 2 1Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543 2Department of Physics, National University of Singapore, 2 Science Drive 3, Singapore 117542 (Dated: January 6, 2023) Bayes’ rule plays is a crucial piece of logical inference in information and physical sciences alike.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' Its extension into the quantum regime has been the object of several recent works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' These quantum versions of Bayes’ rule have been expressed in the language of Hilbert spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' In this paper, we derive the expression of the Petz recovery map within any quasiprobability representation, with explicit formulas for the two canonical choices of “normal quasiprobability representations” (which include Discrete Wigner representations) and of representations based on symmetric, informationally complete positive operator-valued measures (SIC-POVMs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' By using the same mathematical syntax of (quasi-)stochastic matrices acting on (quasi-)stochastic vectors, this construction brings to the fore the structural similarities and the core differences in logical inference between classical and quantum theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' INTRODUCTION Inference is a logical necessity in every science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' In information theory and physics, the fundamentality of inference is particularly overt in notions of process re- versibility and state recovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' Here, the most empirically applied and canonical approach is Bayes’ rule: ˜Eγ(a|a′) = E(a′|a) γ(a) ˜γ(a′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' (1) This relation gives us a recipe for obtaining various probability-theoretic objects [1–4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' Of particular note, we may use it to obtain the “reverse” transition ˜Eγ for any given (i) the forward process or transformation E, and (ii) the reference prior γ on the input of said pro- cess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' The posterior, ˜γ(a′) = � a E(a′|a)γ(a), emerges from these two objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' This typical form of Bayes’ rule works only for classical information theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' The extension to quantum theory re- quires some work: as one possible reason for this, notice that in a classical process a → a′ one can retain informa- tion on both input and output, and thus define the joint probability distribution P(a, a′);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' while nothing of the sort can be done for the quantum process α → α′ = E(α), where E is a completely positive trace preserving (CPTP) map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' Various proposals have been presented over the years, and we refer to a very recent consolidating frame- work for all the references [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' A special role is played by the Petz recovery map [6–8]: ˆEγ[•] = √γ E† � 1 � E[γ] 1 � E[γ] � √γ, (2) This recovery channel is defined for any CPTP map E and a reference density operator γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' Notably, when refer- ence priors, input states and the channel share the same eigenbases, the Petz map reduces to the classical Bayes rule [8, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' This and other properties pertaining to what may be called the “conservation of divergences” (which is what led to its conception) has built up this recovery map’s reputation as the “quantum Bayes’ rule” [10];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' a reputation recently vindicated in an axiomatic approach [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' The Petz map construction appears also naturally in the definition of fluctuation theorems in thermodynamics [12–14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' Now, having said this, it seems that what exactly makes the Petz map similar (or different) to the classical Bayesian update has not been formalized as well as it can be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' From an information-theoretical perspective, there are correspondences between the action of these recipes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' Yet, we know that there are key regime-differences in the woodwork.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' This lack of formal comparison across these regimes is at least partially because the Petz map has thus far only been formalized in terms of CPTP maps and density operators, living in a Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' Mean- while, the classical Bayes rule exists as a stochastic ma- trix mapping stochastic vectors, living in a real vector space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' In this paper, we attempt to close this gap by inves- tigating the Petz map in quasiprobability representation (QPR) [15, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' This formalism provides a complete de- scription of quantum theories while sharing the famil- iar mathematical equipment found in classical probabil- ity theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' The distinction is that quasiprobabilities (or “negative probabilities”) are generally necessary in the quantum case [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' This negativity has been attributed as a resource for advantage in quantum computation [18– 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' As such, we seek to put the Petz map in the same formal habitat as that of classical Bayesian inversion and in an expression that is comparable to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' From there we may discuss the similarities, differences and interpreta- tions wherever appropriate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' We believe this work makes a formal step in understanding the essential distinctions arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content='01952v1 [quant-ph] 5 Jan 2023 2 between classical and quantum inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' This paper is sectioned as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' In Section II, we review features of Bayesian inference for classical and quantum transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' In Section III, we review the formalisms of QPR in quantum theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' Readers famil- iar with the formal content here may skim through these sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' In Section IV, we work towards the key expres- sion of the Petz map in QPR, stating relevant theorems along the way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' In Section V, we discuss consequent theo- retical observations, contrasting notable formal features of the expression to the classical Bayesian update.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' We also introduce “transition graphs” that can help visual- ize the implications of our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' Finally in Section VI, we summarize our findings and state some open lines of inquiry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' CLASSICAL & QUANTUM BAYESIAN INFERENCE In the context of classical mechanics and probabil- ity theory, a physical transformation can be expressed by conditional probabilities E(a′|a) mapping probability distributions of inputs p(a) to distributions of outputs ˜p(a′) = � a E(a′|a)p(a) residing in some given state space A [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' This can be captured compactly by a stochas- tic matrix SE = {E(a′|a)}, mapping vp = {p(a)} to v ˜p = {˜p(a′)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' As already discussed, if we want to acquire a stochasti- cally valid and logically sound “reverse” of this transfor- mation E, we must invoke not only the channel in ques- tion but also a reference prior γ on the input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' This is es- sentially a pre-existing best guess of the inputs for which the Bayesian inverse is constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' This process of ac- quiring ˜Eγ from E and γ can be referred to as perform- ing “retrodiction” (inference about the past, in contrast to prediction, inferring about the future) on E on the prior γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' Meanwhile, S ˜Eγv ˜p gives the “retrodicted input” given an observation ˜p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' It may also be referred to as the “Bayesian update on γ given ˜p”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' For every each individual transition a → a′, we may consult (1) for the corresponding retrodiction a′ → a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' For the mapping of distributions, it is more instructive to write the retrodiction map as a stochastic matrix: S ˜Eγ CL = Dγ(SE)TD−1 E[γ] (3) Here Dp is a diagonal matrix with entries corresponding to some distribution p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' As introduced in Section I, the counterpart to Bayes rule in quantum theory, is the Petz map (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' It is well- defined and CPTP for any full-rank E[γ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' [22] It may also be expressed as ˆEγ = Mγ1/2 ◦ E† ◦ ME[γ]−1/2, (4) where Mαr[•] = αr • αr for any density operator α and r ∈ R, and E† is the adjoint of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' This is the unique map for which Tr(E[ρ]σ) = Tr � E†[σ]ρ � (5) for all ρ, σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' Before continuing, it is important to stress that Bayesian inference is generically not inversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' Infer- ence is possible for any map, while inversion is only possible for invertible maps (information-preserving) – and even then, the two operations are generally not the same, since the inverse of a map is generically not a valid map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' In fact, it can be proved that inference and inver- sion coincide if and only if SE is a permutation (for the classical case), or E is a unitary channel (in the quan- tum case) [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' In general therefore, S ˜Eγ CLSEvρ ̸= vρ and ˆEγ ◦ E[ρ] ̸= ρ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' although the reference state is recovered: S ˜Eγ CLSEvγ = vγ and ˆEγ ◦ E[γ] = γ for all γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' QUASIPROBABILITY REPRESENTATIONS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' Generalities We now move on to provide a brief review of the es- sential elements of QPRs for quantum theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' To bridge quantum theoretic objects in a d-dimensional Hilbert space to a QPR, the architectural core is given by the so-called frame {Fj}j∈Λ, which is a set of Hermitian op- erators spanning the Hermitian space equipped with in- ner product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' We denote Λ as the discrete state space with a minimal cardinality of d2 [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' Those saturating the lower bound referred as minimal bases, which we will assume for the remainder of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' A counterpart to the frame is known as the dual frame {Gj}j∈Λ, which is defined such that: ∀ A, B : � j Tr[FjA] Tr[GjB] = Tr[AB] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' (6) In general, the dual is not unique given a frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' However, for a minimal basis, the frame and dual always enjoy an orthogonality relation Tr[FjGk] = δjk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' As long as these objects are known to the user, we can describe all Hibert space objects in terms of QPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' The morphisms are summarized in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' By requiring that our state quasiprobability to be normalized, � a vρ a = 1, this immediately implies a constraint on the frame op- erators: � a Fa = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' Moreover,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' with each POVM {Em} 3 Object Hilbert space formalism Quasiprobability formalism State ρ = � i λi |λi⟩ ⟨λi| vρ : vρ a = Tr[ρ Fa] POVM {Em | Em ≥ 0 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' � m Em = 1} ¯vm : ¯vm a′ = Tr[Em Ga′] Unitary U[•] = U • U †,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' UU † = 1 SU : SU a′a = Tr � Fa′UGaU †� Channel E[•] = � l κl • κ† l ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' � l κ† l κl = 1 SE : SE a′a = Tr[Fa′E[Ga]] Born Rule Tr[ρEm] vρ · ¯vm ∈ [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' 1] Dimensionality dim[Cd] = d dim[Zd ⊗ Zd] = d2 TABLE I: Morphisms between the Hilbert space formalism and quantum theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' vp a = p(a) indicates the a-th entry in a p-distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' SE a′a = E(a′|a) indicates the entry on the a′-column and a-row of a matrix SE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' satisfying a unity sum � m Em = 1, we also have a con- straint for the dual frame operators: Tr[Gj] = 1 for all j ∈ Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' Likewise, it is the case that Tr[Fj] = 1/d for all j ∈ Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' As such, the QPR of any CPTP map E is a quasi- stochastic matrix SE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' With a slight abuse of notation, for ease of correspondence with the classical formalism, we shall also denote the elements of the quasi-stochastic matrix as SE a′a ≡ E(a′|a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' Now despite the vast plurality of valid representations that adhere to these rules, there are two canonical choices of QPR used in the relevant literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' To these, we turn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' Normal quasiprobability representation The first class of representations are those for which the frame and dual frame operators are proportional to each other up to some scaling factor c, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=', Gj = cFj for all j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' For minimal bases, the constant c is equal to the Hilbert space dimension d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' The class of representa- tions satisfying this is known as normal quasiprobability representation (NQPR) [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' An example of NQPR, and perhaps the most widely used representation, is the discrete Wigner (DW) rep- resentation [25–27], which is well-defined for prime di- mension d and composites of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' For a qubit system (d = 2), the frame has a simple expression given by Fk = Fr,s = 1 4 � 1+(−1)rσx+(−1)sσz+(−1)r+sσy � , (7) where k = (r, s) ∈ Z2 × Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' For composite d = d1 × d2 × · · · × dL, where d1, d2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' , dL are primes, a tensor structure applies for the total frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' That is, the frame operators decompose as Fk = Fk1 ⊗ Fk2 ⊗ · · · ⊗ FkL, where k → (k1, k2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' , kL) with each kl = (rl, sl) ∈ Zdl × Zdl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' This tensor structure is enjoyed by any NQPR and thus affords them an aesthetic benefit when dealing with composite states and purifications C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' SIC-POVM representation Under NQPR, negativity can be found in states, POVM elements, and transformations alike.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' Symmetric, informationally complete, positive operator-valued mea- sure (SIC-POVM) representations seek to avoid this by ensuring that all state vectors are positive [28, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' Neg- ativity features are thus consolidated into the transfor- mations and POVMs For d-dimensional Hilbert space, a SIC-POVM is defined as a set of sub-normalized rank-1 projectors { 1 dΠj}d2 j=1, Πj = |ψj⟩⟨ψj|, such that the elements have equal pairwise Hilbert-Schmidt inner product: Tr � Π† jΠk � = |⟨ψj|ψk⟩|2 = dδjk + 1 d + 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' The solution to the vectors of SIC-POVM have been found for vast number of dimensions (see [30] for the list), and is believed to exist for all [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' Since the set is informationally complete (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' it forms a basis) we can use it as the definition of the SIC-POVM representation’s frame {Fj = 1 dΠj}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' From the orthogonality relation, it can be easily deduced that the dual frame is given by Gj = d(d + 1)Fj − 1 = (d + 1)Πj − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' (8) As an example, the canonical choice for the one-qubit 4 scenario is the tetrahedron F0 = 1 4 � 1 + 1 √ 3(1, −1, 1) · ⃗σ � , (9) F1 = 1 4 � 1 + 1 √ 3(1, 1, −1) · ⃗σ � , (10) F2 = 1 4 � 1 + 1 √ 3(−1, 1, 1) · ⃗σ � , (11) F3 = 1 4 � 1 + 1 √ 3(−1, −1, −1) · ⃗σ � , (12) where ⃗σ = (σx, σy, σz) is the vector of Pauli matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' In our calculations, the choice of representation, when rele- vant, will be stated in context and distinguished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' If not, the derivation will apply generally to all representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' THE PETZ MAP IN QUASIPROBABILITY FORMALISMS Now, our task is to express the Petz recovery map in its QPR, which we denote as S ˆEγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' This obviously can be done by invoking the morphism for channels in Table I and then connecting it with (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' This gives: S ˆEγ aa′ = Tr � Fa √γE† � 1 � E[γ] Ga′ 1 � E[γ] � √γ � (13) But, of course, this affords us no new insight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' We are still relying entirely on the Hilbert space formal- ism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' Nothing novel can be said in comparison to clas- sical Bayesian inference as found in (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' Our specific task is as illustrated in FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' 1: write the Petz in a way that only quasiprobability-theoretic objects (quasi- stochastic vectors, matrices and frames) are required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' E, γ SE, vγ ˆEγ S ˆ Eγ QPR PETZ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' QPR FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' 1: The task, illustrated commutatively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' The naive guess that S ˆEγ could be obtained by graft- ing the quasiprobabilistic formalism onto the classical Bayesian inverse (3) is easily dismissed: the S ˜Eγ CL obtained by such a recipe is in general not a valid map (see Ap- pendix E for explicit counterexamples).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' Rather, taking a hint from (4), we note that channel isomorphism works also when a map is not CPTP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' Hence it is the case that S ˆEγ = Mγ1/2 � SE†� ME[γ]−1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' (14) with SMαr a′a := (Mαr)a′a = Tr[Fa′αrGaαr] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' (15) Now, it is crucial for our goals that all objects entering (14) can be constructed within the quasiprobability for- malism: so we have to prove that this holds for Mαr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' As a first check, we notice that all the entries of these matrices are real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' Indeed, one can rewrite (Mαr)a′a = Tr[Fr a′Gr a] with Fr a = αr/2Faαr/2 and Gr a = αr/2Gaαr/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' These are Hermitian operators, and so Tr[Fr a′Gr a] = 1 2Tr[{Fr a′, Gr a}] is real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' Next, we provide a recipe to ex- plicitly compute the Mαr (recalling that we shall need it for r = ± 1 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' For r = 1, it is relatively straightforward that (Mα)a′a = Tr[Fa′αGaα] (16) = � xy vα x vα y Tr[Fa′GxGaGy] (17) := � xy vα x vα y ξa′xay (18) where the ξpqrs = Tr[FpGqGrGs] are referred to as struc- ture coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' Here we have invoked the fact that ev- ery density operator α can be reconstructed from vα as α = � x vα x Gx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' While such a closed expression cannot be found for r = ± 1 2, fortunately for any r ∈ R one can prove (see Appendix A) that Mαr = M r α .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' (19) Thus, to compute the Mαr for r = ± 1 2, one first writes down Mα and then takes the suitable roots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' The resulting matrices are guaranteed to contain only positive entries by the remark above, which was valid for every r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' In summary, we have obtained our main result: Result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' The Petz map in any QPR reads S ˆEγ QM = M 1/2 γ � SE†� M −1/2 E[γ] (20) where (Mγ)a′a = � xy vγ xvγ y ξa′xay � ME[γ] � a′a = � xy (SEvγ)x(SEvγ)y ξa′xay and ξpqrs = Tr[FpGqGrGs] are structure coefficients de- termined by the specific QPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' Everything is expressed exclusively in the quasiprobabilistic formalism: no knowl- edge of Hilbert space renditions of the quantum channel or reference state is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' For the two canonical choices of QPR introduced above, we prove in Appendix B that NQPR : SE† NQ =(SE)T (21) 5 SIC-POVM : SE† SP =(SE)T + KE (22) where (KE)ij = 1 d(� a E(j|a) − 1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' whence explicitly S ˆEγ NQ = M 1/2 γ (SE)TM −1/2 E[γ] (23) S ˆEγ SP = M 1/2 γ � (SE)T + KE � M −1/2 E[γ] (24) Since the QPR of unital maps (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' E[1] = 1) are quasi- bistochastic matrices (that is, � a E(j|a) = 1 for all j), for such maps KE vanishes and the expressions for NQPR and SIC-POVM representations are formally identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' DISCUSSION A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' Formal Comparisons Across Regimes Here we discuss and compare formal features across classical and quantum Bayesian inference, as expressed in (3) and (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' The key points of comparison are sum- marized in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' We first express (3) in the following form: S ˜Eγ CL = W 1/2 γ � SE†� W −1/2 E[γ] (25) Here, we have highlighted two things about the classical retrodiction map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' Firstly, (25) highlights the fact that one can always write Dγ as the square root of its own square Wγ = D2 γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' This cosmetic change has advantages for comparing with (20) later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' We leave also a reminder that Dγ is a diagonal matrix with entries corresponding to the distribution of γ (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' (Dγ)ij = vγ i δij).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' Secondly, (25) highlights the fact that (in parallel with the opposite relation found in NQPR) for classical chan- nels the transpose of the channel corresponds to the ad- joint (SE)T = SE†.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' That is, (SE)T satisfies the relation (5) by morphism (see Appendix C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' With this, there are a few similarities and differences worth noting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' Firstly, the retrodiction maps, across both regimes, feature the same structure: a central “adjoint” matrix SE†, a prior-dependent matrix (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' M 1/2 γ , W 1/2 γ = Dγ) acting on its left, and a posterior-dependent matrix (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' M −1/2 E[γ] , W −1/2 E[γ] = D−1 E[γ]) acting on its right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' Secondly, this central adjoint object in S ˆEγ NQ and S ˜Eγ CL are both the transpose of the channel matrix itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' For S ˆEγ SP, the additional KE term may be thought of as cor- recting for the positivity of the states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' Thirdly, the prior (and posterior) dependent matrices Xγ differ structurally between classical and quantum in- ference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' Having expressed Dγ as a function of Wγ we see how these matrices can be generally defined as (Xγ)ij = � xy vγ xvγ yξixjy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' (26) With this, it becomes clear from that the key difference between these two domains of inference is the nature of the “structure coeffecients” ξpqrs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' While in the quan- tum scenario ξpqrs = Tr[FpGqGrGs], classical Bayesian inference calls us to something much more reductive: ξpqrs = δpqδrsδpr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' This singular difference in the classical expression casts out many structural features necessary in the general quantum case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' These may be enumerated: General Retrodictive Expression S ¯ Eγ RT = X1/2 γ � SE†� X−1/2 E[γ] (Xγ)ij = � xy vγ xvγ yξixjy Object Quantum Classical SE† NQ : (SE)T SP : (SE)T + KE (SE)T ξixjy Tr[FiGxGjGy] δixδjyδij TABLE II: Retrodiction maps for classical probabilities [S ¯Eγ RT → S ˜Eγ CL, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' (3)] and quantum quasiprobabilities [S ¯Eγ RT → S ˆEγ QM, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' (20)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' 6 Firstly, the classical matrix neglects the depen- dence (present in the quantum matrix) of every en- try on the aggregation of every parameter in the prior distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' Secondly and relatedly, the classical case only has diagonal entries, and only depends on the cor- responding parameter in the prior distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' Meanwhile, the quantum matrix has non-diagonal “coherences”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' Thirdly, while the entries of the classical matrix correspond trivially to values in the prior distribu- tion, entries in the quantum matrix are weighted depending on the representation via the trace of four frame and dual operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' Finally, the presence of coherences make it such that quantum retrodiction finding the root of Mγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' In the classical scenario, Wγ is already diagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' All these features emerge simply because of the differ- ences between the structure coefficients present in these scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' We elaborate on the significance of these dif- ferences in Section VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' There are other resultant properties of Mα, on the ma- trix level, that may be worth noting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' Generally, it is a real, semi-definite matrix with a unit trace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' That is, so defined, Mα ≥ 0 and Tr[Mα] = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' For SIC-POVM, it is not generally symmetric and thus not Hermitian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' For NQPR, however, it does have symmetry and is thus a density operator under such a representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' That said, Tr[M r α] ̸= 1 when rank(Mα) ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' Finally, since the square roots of Mγ and ME[γ] are cer- tainly functions of the E(a′|a) and the γ(a), the quantum Bayes rule can in principle be written as ˆEγ(a|a′) = f � {E(a′|a)}, {γ(a)} � (27) in full analogy to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' But writing down this expres- sion in practice requires the explicit expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' For the simplest quantum case (the qubit) we would be working to solve for the roots of a quartic characteristic equation, for which no general analytical solution exists at present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' Visualizing Quantum Inference via QPR 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' Introducing Transition Graphs A notable advantage of stochastic maps is their ease of visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' One can draw what might be called “transition graphs”, where transition between ai to a′ j are depicted by arrows going from the former to the lat- ter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' The probabiltiy weights on these transitions may be then depicted by a number or by a colour function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' These kinds of graphs are not straightforward to write for the standard Hilbert space formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' This is sim- ply due to the use of complex terms, probability ampli- tudes and the plurality of possible basis choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' With QPR, we can illustrate transformations and their quan- tum Bayesian inverses with transition graphs just as we would for classical stochastic channels, albeit with the added task of depicting negativity in these transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' In Appendix F and this section, we consider some choices of E that give rise to SE and their retrodictions S ˆEγ DW and S ˆEγ SP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' These are then depicted as transition graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' We have chosen to include, in particular, a Half-SWAP channel with a |1⟩⟨1| ancilla to visually il- lustrate and explore the properties of quantum retrod- iction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' Other transformations are also noted in passing with their graphs and expressions consolidated in Ap- pendix F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' Before these, we note some illustrative ele- ments of these figures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' Firstly, with transition arrows we depict negative (pos- itive) quasiprobabilities with cooler (warmer) shades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' Furthermore, these negative (positive) arrows will be drawn with dashed (solid) lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' A colour legend is in- cluded in FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' 2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' Secondly, in order to get a sensing of how irreversible a forward map is and which states it tends to erase to- ward, we add coloured “bubbles” around the output side (denoted {a′ j}) of every graph for a given SE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' The in- tensity and colour of the bubbles are weighted according quasiprobability distribution of the state E[1/d].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' Hence, one should expect that these bubbles are coloured uni- formly for all unital maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' Thirdly, a similar feature is added for the retrodictive transition graphs, drawn for S ˆEγ matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' Crucial for understanding the Bayesian inverse is the reference prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' Hence, for Bayesian inverting transition graphs we add coloured bubbles on the input (denoted {aj}, that is, the input of the forward map) side of the graph, weighted according to the distribution of γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' Finally, for simplic- ity, we stick to channels acting on qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' We also use the most canonical choices of frames for both DW (r, s starting from 0) and SIC-POVM representations (consol- idated in (9)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' Fully Reversible & Fully Irreversible As depicted in Figures 5 and 6 (found in Appendix F), we observe the provable property that S ˆUγ = S ˆU = (SU)T, for unitary channels U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' The Bayesian inverses simply reflect the transition trajectories back, doing so with equal probability and negativity and regardless of what reference prior is chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' More interesting features 7 occur for non-unitary channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' We may write any CPTP map as a dilation defined by a global unitary U acting on an extended state space HA ⊗HB for which the input system •A and an environment or ancilla βB is defined: E[•] = TrB[U • ⊗β U †] (28) We stick to the case where both the target and the ancilla are qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' Arbitrary qubits may be written as: β(ω, θ, φ) = sin2(ω) |ψ⟩⟨ψ| + cos2(ω) ��ψ⊥�� ψ⊥�� (29) Where |ψ⟩ = cos(θ/2) |0⟩ + eiφ sin(θ/2) |1⟩ and |ψ⊥⟩ = e−iφ sin(θ/2) |0⟩ + cos(θ/2) |1⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' In maximal contrast to unitary channels, one may consider a quantum total era- sure channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' This is simply a kind of replacement map where a Full-SWAP (F1) acts on a qubit and an ancilla and we trace out the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' The Bayesian inverse of such quantum channels follow their classical counter- parts: they erase back to reference prior [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' Since the channel is totally irreversible, the quantum Bayes rule simply reverts our inference to our best guess about the initial state (illustrated by FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' Liminally (Ir)reversible For a more conceptually involved and instructive sce- nario, we consider the Half-SWAP U�, which may be rep- resented in the computational basis as: U� ˆ= 1 √ 2 � � � � � � � � � √ 2 0 0 0 0 1 1 0 0 1 −1 0 0 0 0 √ 2 � � � � � � � � � (30) As depicted in 2, we have the forward and retrodictive transition graphs for a channel given by E[•] = TrB[U� • ⊗ |1⟩⟨1| U † �].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' To understand the retrodictive action given by the Petz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' we can gain some intuitions from by writing out these mappings: |01⟩ U� −−→ 1 √ 2 � |01⟩ + |10⟩ � TrB −−→ 1 21 |+1⟩ U� −−→ 1 2 |11⟩ + 1 √ 2 � |01⟩ + |10⟩ � TrB −−→ 1 4 |0⟩⟨0| + 3 4 |1⟩⟨1| + 1 2 √ 2 � |1⟩⟨0| + |0⟩⟨1| � |11⟩ U� −−→ |11⟩ TrB −−→ |1⟩⟨1| (a) Colour Legend for Transition Graphs (b) SE DW for E[•] = TrB[U� • ⊗ |1⟩⟨1| U † �] (c) S ˆ Eγ DW for U�,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' β = |1⟩⟨1| ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' γ = |0⟩⟨0| (d) S ˆ Eγ DW for U�,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' β = |1⟩⟨1| ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' γ = |+⟩⟨+| (e) S ˆ Eγ DW for U�,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' β = |1⟩⟨1| ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' γ = |1⟩⟨1| (f) S ˆ Eγ DW for U�,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' β = |1⟩⟨1| ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' γ( π 16,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' π 5 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' π 3 ) as per (29) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' 2: Transition Graphs for a “Half-SWAP” with |1⟩⟨1|, and various retrodictions with a range of reference priors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' 8 We see that if the reference state is γ = |0⟩⟨0| or |+⟩⟨+|, then any state is compatible to its output (they are un- ambiguously full rank in C2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' Hence, the Petz Map erases all (output) states back to the reference, in full consis- tency with the earlier comments about the quantum total erasure channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' This is depicted in Figures 2c and 2d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' A very different situation occurs for γ = |1⟩⟨1|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' In this case only |1⟩⟨1| is allowed as an output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' Thus, the Petz sends |1⟩⟨1| to itself while all other states are retrodicted in (complicated but logically consistent) ways dependent on channel’s forward transitions, reflected in FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' 2e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' To explain this more symmetrically: in the former two scenarios, all outputs are compatible with the absolute conviction (as enforced by state purity) given to the refer- ence state, hence all outputs are retrodicted to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' Mean- while, in this latter case, only one pure output (which just so happens to be the same as the reference) is com- patible with the pure reference state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' Hence, all other states (beside the expected output) are retrodicted in accordance to the channel without any regard the refer- ence, since the reference already excludes the possibility of such states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' These more complicated Bayesian inver- sions come together and cumulate into a vertical reflec- tion of the forward channel, as FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' 2e depicts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' For an arbitrary γ, we get a classical mixture of all these key ef- fects together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' We depict the case where γ = γ( π 16, π 5 , π 3 ) in FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' 2f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' It should be said the interplay of reference and chan- nel dependencies we have reviewed here is fundamental in classical retrodiction scenarios as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' The Half-SWAP illustrates that these same fundamental Bayesian princi- ples hold in the quantum regime via the inferential struc- ture of the Petz Map, even when complementarity and entanglement is introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' CONCLUSIONS By expressing the Petz Recovery map as a decomposi- tion of matrices given by (20) we have situated quantum Bayesian inference in the same formal language as that of its classical counterpart given by (26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' We have also highlighted what we have found to be the most note- worthy (and interpretation-neutral) differences between these two levels of inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' It should be clear to the reader that, in keeping with the Bayesian character of the Petz, the crucial formal dif- ference is found in the prior-dependent (and in turn, pos- terior dependent) matrices Xγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' The properties of these objects encode the most significant differences between classical and quantum inferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' Particularly, it for- malizes how prior and posterior variables are taken into account to the central adjoint map, structurally speak- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' In the classical case we neglect the total aggrega- tion (all parameters are involved in every entry), “coher- ences” (non-diagonal terms with sums of product pairs of the parameters, weighted depending on the choice of representation) and “eigenstructure” (finding the matrix root of prior-dependent matrix is not generally circum- vented) found in the quantum case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' This is all emergent from the simple differences between the “structure coef- fecients” found across these regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' Adding to all this the characteristic notion of negativity embedded in all three component matrices, we have a sense of just how wide the conceptual gap we have on our hands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' The reduction in structure that we find in the classical regime may be understood as a kind of classical epis- temic prejudice, which leads to absurdities if applied to general quantum scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' This prejudice (or reduction) is, of course, unsurprising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' Classical Bayesian thinking is deeply intuitive and subconscious for us, regardless of being mathematically initiated or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' Nevertheless, ar- ticulating or formalizing such strong intuitions already has its complications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' Given the orders of differences between this kind of inference and that of the quantum regime, we see why the classical prejudice presides our everyday experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' That said, this work also illustrates noteworthy sim- ilarities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' Despite profound structural differences, many Bayesian intuitions seem to nevertheless come together as illustrated in transition graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' The prior and pos- terior dependent matrices and the role of channel’s ad- joint presides in both regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' Under this exploration, we take a step closer to what a regime-independent Bayesian foundation could be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' Many open questions remain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' For one, we can look into the overlap between classical and quantum inference: what (quantum) processes does classical and quantum retrodiction become equivalent (that is, for every choice of reference prior)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' It is easily checked that this holds for channels that give permutative SE and quantum to- tal erasure channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' Aside from these two extreme cases, are there other channels where this retrodiction property holds?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' Related to this, it may be worth investigating if for some frames or under some gauge transformations the current decomposition (in (20)) may be simplified into a more classically intuitive form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' No such simpler alterna- tive has been found for every channel and every refer- ence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' Finally, throughout this paper we have identified the Petz with quantum Bayesian inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' Nevertheless, other transpose maps exist that have been seen as retro- diction channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' It could be noteworthy to perform a similar decomposition on those maps under quasiproba- bility schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' 9 ACKNOWLEDGMENTS This research was supported by the National Research Foundation and the Ministry of Education, Singapore, under the Research Centres of Excellence programme (till 6 December 2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' and by the National Research Foun- dation, Singapore, and A∗Star under the CQT Bridging Grant (from 7 December 2022 onwards).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' We also thank Zaw Lin Htoo and Eugene Koh for helpful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' Appendix A: Mαr = M r α for all r ∈ R We know that in general, (Aij(z) → A) ̸⇒ (Aij(zr) → Ar) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' (A1) Informally speaking, powers on the level of entry param- eters do not necessarily translate to powers on the level of matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' Thankfully, this does obtain in our case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' The derivation is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' We first note that (Mαr)ij = Tr[FiαrGjαr] (A2) It may be tempting to invoke that since SESF = SE◦F (A3) we already have desired theorem proven.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' However, this will only do for r ∈ Z+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' If we wish to do so for values of r ∈ R, we must be careful not to already assume the conclusion in our derivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' Our proof comes in three main steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' Firstly, (Mα1/2Mα1/2)ij = � k (Mα1/2)ik(Mα1/2)kj = � k Tr � Fi √αGk √α � Tr � Fk √αGj √α � = � k Tr � Gk √αFi √α � Tr � Fk √αGj √α � = Tr �√αFi √α√αGj √α � = Tr[FiαGjα] = (Mα)ij Crucially, in the fourth equality we use the property (6) in QPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' Hence, Mα1/2 = M 1/2 α By reiterating this (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' sending α → √α), we obtain the more general relation that ∀x ∈ Z+ : Mα1/2x = M 1 2x α Which is really just to say: Mαϵ = M ϵ α, for an arbitrarily small and positive real number ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' In- voking (A3) and the definition of Mαr, for N ∈ Z+, we have, MαNϵ = SMαNϵ = S N times � �� � Mαϵ ◦ Mαϵ ◦ · · · ◦ Mαϵ = � N Mαϵ = � N M ϵ α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' Together, we may thus conclude that for N ∈ Z+: MαNϵ = M Nϵ α .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' (A4) Secondly, we note that (MαMα−1)ij = � k Tr[FiαGkα] Tr � Fkα−1Gjα−1� = � k Tr[GkαFiα] Tr � Fkα−1Gjα−1� = Tr � αFiαα−1Gjα−1� = Tr[FiGj] = δij = 1ij Hence, Mα−1 = M −1 α .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' Repeating this, we can easily see that for any N ′ ∈ Z+: Mα−N′ = M −N ′ α (A5) Finally, taking from (A3), (A4) and (A5), we find that: MαNϵ−N′ = SMαNϵ−N′ = SMαN ϵ◦Mα−N ′ = MαNϵMα−N′ = M Nϵ α M −N ′ α = M Nϵ−N ′ α Since it holds for any arbitrarily small, positive ϵ and for any arbitrarily large positive integers N and N ′, we write Nϵ − N ′ ∈ R and denote this Nϵ − N ′ → r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' Thus we obtain our desired result: Mαr = M r α for any r ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' Appendix B: SE† in terms of (SE)T for QPRs We derive the QPR expressions for SE† for some CPTP map E[•] = � l κl • κ† l .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' For NQPRs, we find easily that: (SE† NQ)ij = Tr � FiE†[Gj] � = Tr � Fi � l κ† l Gjκl � = � l Tr � GjκlFiκ† l � = � l Tr � FjκlGiκ† l � = Tr � Fj � l κlGiκ† l � = Tr[FjE[Gi]] = SE ji Thus, for NQPRs SE† NQ = (SE)T (B1) 10 For SIC-POVM representations, we have a more compli- cated expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' We first use (8), (SE† SP)ij = Tr � FiE†[Gj] � = Tr � Gi + 1 d(d + 1) E† [d(d + 1)Fj − 1] � By expanding the terms and noting the unitality of every adjoint map (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' E†[1] = 1), we arrive at the expression: (SE† SP)ij = Tr � Fi � l κ† l Gjκl � + Tr � E†[Fj] � − Tr[Fi] = SE ji + Tr � E†[Fj] � − 1 d By taking note of the isomorphisms found in Ta- ble I, we may write Tr � E†[Fj] � = � l Tr � κ† l Fjκl � = � l Tr � Fjκl1κ† l � = Tr[FjE[1]] = 1 d(SEv1)j = 1 d � a E(j|a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' Hence, we can write the total expression of each entry for SIC-POVM representation as: (SE† SP)ij = SE ji + 1 d �� a E(j|a) − 1 � (B2) This can be written, on the matrix level, as (22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' Appendix C: (SE)T as SE† for Classical In the previous section we proved that for quantum channels (expressed in QPRs), we can express the adjoint channel in terms of the tranpose of the channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' Here, we prove the opposite relation for classical channels: that the transpose of a classical channel is the adjoint of that channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' Namely, the transpose map is the map for which (5) is fulfilled in the case of classical scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' Noting first the commutative diagram found in FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' 3 (which invokes the morphisms found in Table I), we see how (5) is fulfilled by a map for which (SEvρ) · ¯vσ = (SE†vσ) · ¯vρ, (C1) for all ρ and σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' With this, we first expand the LHS of (C1): (SEvρ) · ¯vσ = � y (SEvρ)y¯vσ y (C2) = � xy SE yxvρ x¯vσ y (C3) Next we expand the following, in order to check if the transpose qualifies as the adjoint: ((SE)Tvσ) · ¯vρ = � xy (SE)T yxvσ x ¯vρ y (C4) vρ · ¯vσ Tr[ρ σ] (SEvρ) · ¯vσ Tr [E[ρ]σ] (SE†vσ) · ¯vρ Tr � E†[σ]ρ � ∀ρσ ∀ρσ FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' 3: Relations between formalisms pertaining the adjoint map, illustrated commutatively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' = � xy SE xyvσ x ¯vρ y (C5) = � xy SE yx¯vρ xvσ y (C6) Now for classical scenarios the trace of two states, if treated like quantum states in Hilbert space, would sim- ply be the inner product of its density spectra: vρ · vσ = Tr[ρ σ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' This is because the states, being classical distri- butions, would be diagonalized in the same way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' Thus we could have replaced ¯vρ with vρ in all the above calcu- lations and in FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' The reason why we have written ¯vρ as opposed to vρ is to simply highlight that while indeed (C3) is identical to (C6) for classical scenarios be- cause ¯vρ = vρ there (and NQPR for that matter since ¯vρ = c vρ), the same does not hold for SIC-POVM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' The transpose qualifies as an adjoint for both NQPR and clas- sical channels, but not for SIC-POVM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' Hence, the rela- tion proved for classical states and channels does not con- tradict the ones proved in the previous section for QPRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' Appendix D: Properties of Mα Here we note some interesting properties of Mα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' Namely that it is a matrix with all real entries and non- negative eigenvalues that sum to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' Real Entries It can be shown that all the entries of Mα are real: (Mα)ij = Tr[FiαGjα] ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' A proof was given in the main text, valid for any Mαr;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' we repeat it here for com- pleteness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' We first note that the anticommutator for any two Hermitian operators A and B is always also Hermitian: {A, B}† = {A, B};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' while the trace of the commutator of any two operators is always zero (in finite dimension) due 11 to cyclicity: Tr � [A, B] � = Tr[AB] − Tr[BA] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' Hence Tr[AB] = Tr �{A, B} 2 + [A, B] 2 � = 1 2Tr[{A, B}] ∈ R (D1) Noting that √αFi √α and √αGj √α are both Hermitian (frame and dual operators are always Hermitian, and α is a density operator in Hilbert Space), we apply (D1) to (Mα)ij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' The entries of Mα are thus proven to be always real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' Positive Semi-Definitiveness For NQPR, we can always write (Mα)ij = c Tr �√αFi √α �√αFj √α �†� Hence, Mα is a Gram matrix with some positive factor c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' Thus it is positive semi-definite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' For SIC-POVM, we expand (Mα)ij via (8), arriving at: (Mα)ij = 1 d(d + 1) � Tr �√αGi √α �√αGj √α �†� +Tr � Gjα2� � The first term, as with the NQPR case, corresponds to a Gram Matrix, which is positive semi-definite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' One can then note that the second term corresponds to a matrix Jα (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' (Jα)ij = Tr � Gjα2� ) with duplicate rows (every j- th column with filled with identical entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' This simply implies that the only non-zero eigenvalue would be the sum of the entries of any given row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' Which just means: eig[Jα] = {� j Tr � Gjα2� , 0} = {Tr � α2� , 0} ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' So Mα is the sum of two positive semi-definitive matrices and thus we may conclude Mα ≥ 0 for SIC-POVM as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' Unit Trace The trace of Mα is given by Tr[Mα] = � i (Mα)ii = Tr � � i FiαGi � �� � 1 α � = 1 To prove the relation invoked for the final equality we will use the previously found result in [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' Consider the superoperator Λ[•] = d2 � i=1 Πi • Πi , (D2) it can be shown that Λ[Πi] = d d + 1(Πi + 1) (D3) Since the set {Πi} forms a basis, we can express the su- peroperator as Λ = d d + 1(I + 1) (D4) where I[A] = Tr[A] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' Using this we can easily show that for SIC-POVM representation we have � i FiαGi = d + 1 d � i ΠiαΠi − 1 dα � i Πi = 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' (D5) For discrete Wigner representation, Zhu [24] showed that the dual frame can always be expressed as such: Gi = − √ d + 1Πi + �1 + √ d + 1 d � 1 (D6) Thus, it can also be easily shown that � i FiαGi = 1 in this representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' Appendix E: Examples for S ˆ Eγ ̸= S ˜ Eγ CL As discussed in Section V B 3, it is the case that ˆEγ[ρ] = ˆE+[ρ] = |+⟩⟨+| for all ρ when E[•] = TrB[U� •⊗ |1⟩⟨1| U † �] and γ = |+⟩⟨+|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' Yet we can easily find that, for the canonical state representations for DW and SIC-POVM, we have: S ˜E+ DW = � � � � � � � � � 1 1 7 � 3 − √ 2 � 1 1 7 �√ 2 + 3 � 0 1 7 �√ 2 + 4 � 0 1 7 � 4 − √ 2 � 0 0 0 0 0 0 0 0 � � � � � � � � � ̸= 1 2 � � � � � � � � � 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 � � � � � � � � � = S ˆE+ DW Likewise, S ˜E+ SP = � � � � � � � � � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content='925 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content='183 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content='264 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content='353 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content='0744 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content='744 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content='275 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content='168 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content='0191 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content='0491 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content='915 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content='0947 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content='0199 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content='0233 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content='0737 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content='384 � � � � � � � � � 12 ̸= 1 12 � � � � � � � � � √ 3 + 3 √ 3 + 3 √ 3 + 3 √ 3 + 3 √ 3 + 3 √ 3 + 3 √ 3 + 3 √ 3 + 3 3 − √ 3 3 − √ 3 3 − √ 3 3 − √ 3 3 − √ 3 3 − √ 3 3 − √ 3 3 − √ 3 � � � � � � � � � = S ˆE+ SP Indeed, for some channels one can find states for which the post-measurement probabilities violate acceptable bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' This means S ˜Eγ CL fails to represent a generally valid quantum transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' For instance, for a uni- tary transformation U[•] = U •U † where U = i 2 � √ 3 −1 1 √ 3 � We find that (S ˜U+ DWv+) · ¯v0 = 1 2(1 + √ 3) > 1 (S ˜U+ SPv0) · ¯v+ = 1 13(2 − 5 √ 3) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' S ˆEγ ̸= S ˜Eγ CL is thus easily shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' Appendix F: Other Transition Graphs In this appendix, we include illustrative cases of SE, some respective retrodictions and their transition graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' In FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' 5, the transition graphs are depicted for very familiar Pauli rotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' It so happens that these uni- taries translate to SE that give permutations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' This is seen in the bold bijective transition arrows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' Like other unitary channels, all retrodictions are reference-prior in- dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' Transition graphs of such retrodictions are thus always mirror images of the corresponding forward transition graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' That said, most unitaries do not en- joy a permutative structure that exists for these SU(2) rotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' The Hadamard gate for instance defined by the following computationally represented operator and gives the respective quasi-stochastic matrix: UH ˆ= 1 √ 2 � �1 1 1 −1 � � , SUH = 1 2 � 1 1 1 −1 1 −1 1 1 1 1 −1 1 −1 1 1 1 � , which is consistent across the canonical choices of the DW and SIC-POVM representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' Likewise,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' an arbitrarily chosen unitary Ueg: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content='Ueg ˆ= 1 ' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content='has the following quasiprobability objects: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content='SUeg ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content='DW = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content='� ' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content='SUeg ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content='SP = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content='−3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content='5 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content='3−6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content='3+9 4−3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content='3−6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content='9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content='It is clear that these forward channels do not give per- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content='mutative QPRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' Nevertheless the property that S ˆUγ = S ˆU = (SU)T is still reflected clearly in FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' In contrast to these reversible maps, we can speak of the quantum total erasure channel mentioned in section V B 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' The full swap is expressed as such: U� ˆ= � � � � � � � � � 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 � � � � � � � � � (F1) As depicted clearly in FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' 4, both the forward channel and its retrodiction are erase perfectly to the relevant state (the ancilla for the forward map and the reference state for the retrodiction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' (a) SE DW for E[•] = TrB[U� • ⊗β( 7π 16 , 3π 5 , π 6 ) U † �] (b) S ˆ Eγ DW for U�, β( 7π 16 , 3π 5 , π 6 ), γ( π 16, π 5 , π 8 ) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' 4: Transition Graphs for a Quantum Total Erasure channel with arbitrary ancilla β and a corresponding retrodiction with reference prior γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' 13 (a) SE for E[•] = σz • σz (b) S ˆ Eγ SP and S ˆ Eγ NQ for E[•] = σz • σz (c) SE for E[•] = σy • σy (d) S ˆ Eγ SP and S ˆ Eγ NQ for E[•] = σy • σy (e) SE for E[•] = σx • σx (f) S ˆ Eγ SP and S ˆ Eγ NQ for E[•] = σx • σx FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' 5: Transition Graphs for σz, σx and σy and their respective retrodictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' (a) SUH for UH[•] = UH • U † H (b) S ˆ UH for UH[•] = UH • U † H (c) S Ueg DW for Ueg[•] = Ueg • U † eg (d) S ˆ Ueg DW for Ueg[•] = Ueg • U † eg (e) S Ueg SP for Ueg[•] = Ueg • U † eg (f) S ˆ Ueg SP for Ueg[•] = Ueg • U † eg FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' 6: Transition Graphs for the Hadamard gate and an arbitrarily chosen qubit unitary and their respective retrodictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' 14 [1] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' Watanabe, Conditional probabilities in physics, Progr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' Suppl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' E65, 135 (1965).' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} +page_content=' Caves, Symmetric informationally complete quantum measurements, Journal of Mathematical Physics 45, 2171 (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQf_v5Q/content/2301.01952v1.pdf'} diff --git a/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf b/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..0a0bdf79ff55aa7bdf485d08cfabfdc6c1e8b1a3 --- /dev/null +++ b/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:62f6971eeaead6ab3694fb262999b47c011d4588651eb781c6fbb2d664230238 +size 390843 diff --git a/aNFST4oBgHgl3EQfBTj3/vector_store/index.faiss 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20 Jan 2023 +LIOUVILLE’S THEOREMS FOR LÉVY OPERATORS +TOMASZ GRZYWNY, MATEUSZ KWA´SNICKI +Abstract. Let L be a Lévy operator. A function h is said to be harmonic with respect +to L if Lh “ 0 in an appropriate sense. We prove Liouville’s theorem for positive +functions harmonic with respect to a general Lévy operator L: such functions are +necessarily mixtures of exponentials. For signed harmonic functions we provide a +fairly general result, which encompasses and extends all Liouville-type theorems +previously known in this context, and which allows to trade regularity assumptions +on L for growth restrictions on h. Finally, we construct an explicit counterexample +which shows that Liouville’s theorem for signed functions harmonic with respect to +a general Lévy operator L does not hold. +1. Introduction +1.1. Liouville’s theorem and Lévy operators. A classical result due to Liouville +and Cauchy, traditionally called Liouville’s theorem, states that every bounded har- +monic function in Rd is constant. This was extended by Bôcher and Picard, who +proved that a one-sided bound is sufficient: every positive harmonic function in +Rd is constant. A yet another variant of Liouville’s theorem asserts that every har- +monic function bounded by a polynomial (and again a one-sided bound is sufficient) +is in fact a harmonic polynomial. +Liouville’s theorem has been extended in various directions. Here we study vari- +ants of Liouville’s theorem for Lévy operators, that is, operators L of the form +Lfpxq “ +dÿ +j,k“1 +ajkBjkfpxq ` +dÿ +j“1 +bjBjfpxq +` +ż +Rdzt0u +ˆ +fpx ` yq ´ fpxq ´ 1Bpyq +dÿ +j“1 +yjBjfpxq +˙ +νpdyq, +(1.1) +acting on an appropriate class of functions on Rd. Here B is the unit ball, paijq P +Rdˆd is a nonnegative definite symmetric matrix, pbjq P Rd is a vector, and ν is a +nonnegative measure on Rdzt0u such that +ş +Rdzt0u mint1, |y|2uνpdyq ă 8, the so-called +Lévy measure. +Various equivalent descriptions of the class of Lévy operators are available. In +particular, the following conditions are equivalent: L is a Lévy operator; L is the +generator of a Lévy process Xt; L generates a strongly continuous semigroup of +translation invariant Markov operators; ´L is a Fourier multiplier whose symbol +Date: January 23, 2023. +2010 Mathematics Subject Classification. 35B08, 35B09, 35B10, 35B53, 35R09, 58J65, 60G51, +60J35, 60J45. +Key words and phrases. Harmonic function, Lévy process, Liouville’s theorem. +Work +supported +by +the +Polish +National +Science +Centre +(NCN) +grants +no. +2017/27/B/ST1/01339 (TG) and 2019/33/B/ST1/03098 (MK). +1 + +2 +TOMASZ GRZYWNY, MATEUSZ KWA´SNICKI +Ψ is a continuous negative definite function vanishing at the origin; L is a trans- +lation invariant integro-differential operator vanishing on constants and satisfying +the maximum principle. We remark that the function Ψ mentioned above is the +characteristic exponent of the Lévy process Xt, given by the Lévy–Khintchine for- +mula +Ψpξq “ apξ, ξq ´ ibξ ` +ż +Rdzt0u +` +1 ´ eiξy ` iξy 1Bpyq +˘ +νpdyq, +(1.2) +and for every t P r0, 8q, the characteristic function of the random variable Xt is +equal to e´tΨ. Here apξ, ξq “ řd +j,k“1 ajkξjξk, and bξ “ řk +j“1 bjξj. +The Laplace operator ∆ is a prime example of a Lévy operator, and a smooth +function f is harmonic in Rd if and only if ∆f “ 0 in Rd. In a similar way, given a +general Lévy operator L, a smooth function f is said to be harmonic with respect +to L, or L-harmonic, if Lf “ 0 in Rd. In probabilistic terms, the Lévy operator L +is the generator of a Lévy process Xt, and f is L-harmonic if and only if fpXtq is a +local martingale. +The question that we address in this article is: which variants of Liouville’s the- +orem remain true for L-harmonic functions, where L is a given Lévy operator? +This problem was tackled by several authors over the last few years, and in fact +its history can be traced back to the seminal work of M. Riesz [19] on functions +harmonic with respect to the fractional Laplace operator ´p´∆qs, another widely +studied Lévy operator. Indeed: Liouville’s theorem for bounded harmonic functions +follows immediately from Harnack’s inequality, and an appropriate variant of the +latter result for the fractional Laplace operator is given in [19]. +The goal of this paper is three-fold. First, we prove a general Liouville’s the- +orem for nonnegative L-harmonic functions: every such function is a mixture of +L-harmonic exponentials; see Section 1.2. Next, we provide a general framework +for proving Liouville’s theorems for signed polynomially bounded L-harmonic func- +tions by means of Fourier transformation. Under appropriate additional assump- +tions, we prove that all signed L-harmonic functions which correspond to tempered +distributions have their spectrum (that is, the support of the Fourier transform) +contained in the zero set of the characteristic exponent of L; see Section 1.3. Fi- +nally, we construct a surprising example which proves that one cannot completely +drop the additional assumptions mentioned above: Liouville’s theorem for signed, +polynomially bounded L-harmonic functions does not hold in full generality; see +Section 1.4. +In the introduction, we state simplified variants of our main results, and we pro- +vide references to the corresponding full statements given later in the text. We +commonly impose the following standard assumption. +Definition 1.1. A Lévy operator L is said not to be concentrated on a proper closed +subgroup of Rd if the Fourier symbol Ψ of ´L (that is, the characteristic exponent +of the corresponding Lévy process Xt) is equal to zero only at the origin. +The above condition holds if and only if the closed subgroup G of Rd gener- +ated by the union of the supports of the distributions of Xt ´ X0 is equal to Rd. +The subgroup G can be described as the smallest closed subgroup which contains: +(a) eigenspaces of the matrix paijq which correspond to nonzero eigenvalues; (b) the +support of ν; and (c) an appropriately defined drift vector. This result is due to + +LIOUVILLE’S THEOREMS FOR LÉVY OPERATORS +3 +Tortrat [24]; we refer to Théorème 3 in [23] and to [1] for details, as well as to +Section 24 in [21] (in particular, Definitions 24.13 and 24.21 and Proposition 24.14 +therein) for further discussion. We also note that other names, such as nonlattice +condition, are commonly used for the condition described in Definition 1.1. +We remark that if a Lévy operator L is concentrated on a proper closed subgroup +G of Rd, then L acts on each coset x ` G separately. In particular, every G-periodic +function on Rd (that is, a function f such that fpx ` zq “ fpxq for every x P Rd +and z P G) is L-harmonic. The proper way to state Liouville’s theorem for a Lévy +operator L concentrated on G is to consider functions defined only on G. Although +we do not discuss this extension in the introduction, our main results carry over to +this situation: for positive L-harmonic functions we state this explicitly in Section 3, +while for signed L-harmonic function this extension corresponds to Fourier symbols +Ψ with zero sets other than t0u, which are allowed in Section 5. +1.2. Positive L-harmonic functions. In order to state Liouville’s theorem for +positive L-harmonic functions, we need an auxiliary definition. +If L is a Lévy operator, we denote by L: the Lévy operator dual to L, obtained by +conjugation of L with the reflection x ÞÑ ´x. More precisely, if L is given by (1.1), +then L: has a similar representation with the same matrix pa: +ijq “ paijq, with vector +pb: +jq “ p´bjq, and with Lévy measure ν:pAq “ νp´Aq. We remark that L: is an op- +erator formally adjoint to the operator L, and if both are considered as unbounded +operators on L2pRdq, then L and L: are indeed adjoint operators. +Definition 1.2. A locally integrable function h is said to be L-harmonic (in the +weak sense) if +ż +Rd hpxqL:ϕpxqdx “ 0 +for every ϕ P C8 +c pRdq. +Here we implicitly assume absolute convergence of the +integral in the left-hand side. +In fact, we will allow for a slightly more general definition; see Lemma 2.1 in Sec- +tion 2.3 for a detailed discussion. Of course, whenever we say that an L-harmonic +function h in the above sense is constant or equal to a polynomial, in fact we mean +equality almost everywhere with respect to the Lebesgue measure. +Recall that B is the unit ball in Rd. It is relatively simple to see that if h has +continuous second-order partial derivatives, +ż +RdzB +|hpx ` yq|νpdyq +is a locally integrable function of x P Rd, and Lh “ 0 in Rd in the pointwise sense, +then h is L-harmonic in the weak sense. +The following statement, which builds upon and extends Theorem 5.7 in [3], is +our first main result. +Theorem 1.3 (see Theorem 3.2). Let L be a Lévy operator which is not concen- +trated on a proper closed subgroup of Rd, and let f be a nonnegative function on +Rd which satisfies Lf “ 0 in the weak sense. Then +fpxq “ +ż +Λ +eλxµpdλq + +4 +TOMASZ GRZYWNY, MATEUSZ KWA´SNICKI +for a unique nonnegative measure µ on the set Λ of those vectors λ P Rd for which +the function eλpxq “ eλx satisfies Leλpxq “ 0. +The above result follows directly from the more general Theorem 3.2, which +allows h to be an arbitrary Schwartz distribution such that Definition 1.2 makes +sense, and L to be concentrated on a proper closed subgroup G of Rd. +Theorem 1.3 for Lévy operators which generate random walks was proved by +Deny in [9] (see Théorème 3 therein). In terms of the representation (1.1), this +result corresponds to aij “ 0, bj “ 0 and ν a finite measure. We stress that this spe- +cial case is a crucial ingredient of our proof. A variant of Theorem 1.3 was proved +by Berger and Schilling in [3] (see Theorem 5.7 therein) with the additional as- +sumption that h is bounded by a submultiplicative function integrable with respect +to the Lévy measure ν over the complement of the unit ball. An alternative proof +was given recently by Berger, Schilling and Shargorodsky in [4] (see Theorem 13 +therein). +Since every bounded mixture of exponentials is constant, Theorem 1.3 in partic- +ular implies that bounded L-harmonic functions are necessarily constant, provided +that L is not concentrated on a proper closed subgroup of Rd. This variant of The- +orem 1.3 has some history. For the fractional Laplace operator, this result follows +directly from Harnack’s inequality proved by M. Riesz in [19] (see formula (5) in +Chapter V therein); an essentially equivalent argument was given by Bogdan, Kul- +czycki and Nowak in [6] (see Lemma 3.2 therein). Similar Liouville’s theorem for +Lévy operators with measure νpdyq decaying exponentially fast at infinity and com- +parable with |y|´d´1dy near the origin was given by Barlow, Bass and Gui in [2] (see +Theorem 1.17 therein), while the result for rather general Lévy operators with non- +degenerate second order local term was proved by Priola and Zabczyk in [18] (see +Theorem 3.8 therein). The case of general Lévy operators L was solved completely +by Alibaud, del Teso, Endal and Jakobsen in [1] (see Theorems 1.1 and 1.2 therein) +using analytical methods. We remark that the probabilistic argument applied by +Berger and Schilling in [3] (see Theorem 4.4 therein for the statement for bounded +L-harmonic functions) is quite different and more general. +1.3. Signed L-harmonic functions. In order to study signed L-harmonic func- +tions, we apply Fourier transform methods. These are limited to the class of tem- +pered distributions, and for this reason we need to modify slightly the definition of +an L-harmonic function. We denote by S the Schwartz class of rapidly decreasing +smooth functions on Rd, and by S 1 the class of tempered distributions. +Definition 1.4. A locally integrable function h with at most polynomial growth at +infinity, or, more generally, a tempered distribution h, is said to be L-harmonic (in +the sense of tempered distributions) if +ż +Rd h ˚ ψpxqL:ϕpxqdx “ 0 +for every ϕ, ψ P S . Here we implicitly assume absolute convergence of the integral +in the left-hand side. + +LIOUVILLE’S THEOREMS FOR LÉVY OPERATORS +5 +We remark that if h is an L-harmonic function in the weak sense (according to +Definition 1.2, and if additionally +|hpxq| ` +ż +RdzB +|hpx ` yq|νpdyq +is bounded by a polynomial (as a function of x P Rd), then h is L-harmonic in the +sense of tempered distributions (according to Definition 1.4); see Lemma 2.2 in +Section 2.3. +A Lévy operator L is a convolution operator. +The corresponding convolution +kernel is an appropriate tempered distribution, which we denote by ˇL. The dis- +tributional Fourier transform F ˇL of the tempered distribution ˇL is a continuous +function: it is equal to ´Ψ, the Fourier symbol of L. Recall that, in probabilistic +terms, Ψ is the characteristic exponent of the corresponding Lévy process Xt. +For simplicity, below we assume that L is not concentrated on a proper closed +subgroup of Rd, that is, Ψpξq ‰ 0 for ξ ‰ 0. We stress, however, that the most +general version of our result stated in Theorem 5.1 also covers general Lévy oper- +ators, as well as other convolution operators, as long as their Fourier symbols are +continuous functions in an appropriate class. +Before we state our main result in this section, Theorem 1.7, it is instructive +to describe briefly our argument. +Suppose that h is an L-harmonic function in +the sense of tempered distributions. Our proof of Liouville’s theorem for signed +L-harmonic functions consists of the following steps. +(I) Observe that ˇL and h are convolvable (as tempered distributions), and +ˇL ˚ h “ Lh “ 0. +(II) Use the Fourier exchange formula to find out that the muliplicative product +of Fourier transforms of ˇL and h is well-defined (in the sense of tempered +distributions), and +F ˇL ¨ Fh “ FpˇL ˚ hq “ 0. +(III) Use an appropriate variant of Wiener’s 1{f theorem to deduce that Fh “ 0 +on Rdzt0u. +(IV) Conclude that h is a polynomial +Steps (I) and (IV) present no difficulties, while step (II) is a known result in the +theory of tempered distributions; we refer to Sections 2.2 and 2.3 for a detailed +discussion. The only problematic part in the above argument is step (III). +We remark that if Ψ “ ´F ˇL is a smooth function in Rdzt0u, then also step (III) is +completely standard, and in this way we recover the variant of Liouville’s theorem +given in Theorem 3.2 in [3]. +If we assume that h is a bounded function, then the usual Wiener’s 1{f theorem +can be easily adapted to make step (III) rigorous. +This leads to a significantly +shorter proof of Liouville’s theorem originally given in Theorem 1.1 in [1] and, +independently, in Theorem 4.4 in [3]. +The two examples discussed above are in some sense extreme cases. We also +provide intermediate variants. However, we need to keep balance between smooth- +ness conditions on Ψ and growth restrictions of h. Detailed statements are given +in Corollary 1.8 below. + +6 +TOMASZ GRZYWNY, MATEUSZ KWA´SNICKI +A rigorous statement of our general result requires the following two auxiliary +definitions. +Definition 1.5. We say that an algebra W of continuous functions on Rd is a +Wiener-type algebra if: +(a) every element of W is a tempered distribution; +(b) if Φ P W and ϕ P S , then ϕΦ P W ; +(c) if K Ď Rd is a compact set, Φ P W and Φpξq ‰ 0 for every ξ P K, then there +is ˜Φ P W such that Φpξq˜Φpξq “ 1 for every ξ P K. +We say that a tempered distribution Ψ belongs to W locally on an open set U if +for every compact set K Ď U there is a distribution Φ P W such that Ψ “ Φ in a +neighbourhood of K. +We note that the Schwartz class S , or the class of Fourier transforms of inte- +grable functions (that is, the usual Wiener algebra), are Wiener-type algebras. +Clearly, conditions (a) and (b) are rather natural, and typically they are easy to +check. +The essential property of Wiener-type algebras is given in condition (c), +which can be thought of as a variant of Wiener’s 1{f theorem. +Definition 1.6. We say that a tempered distribution H acts on a Wiener-type al- +gebra W if for every Φ, Ψ P W we have the following identity of multiplicative +products of tempered distributions: +pH ¨ Φq ¨ Ψ “ H ¨ pΦ ¨ Ψq. +In particular, we require that all products of tempered distributions in the above +identity are well-defined. +The following result apparently covers all known Liouville-type theorems on +signed L-harmonic functions for Lévy operators L, and, together with Corollary 1.8 +below, it is our second main result. +Theorem 1.7 (see Theorem 5.1). Let W be a Wiener-type algebra of continuous +functions on Rd, and let L be a Lévy operator which is not concentrated on a proper +closed subgroup of Rd. Suppose that the Fourier symbol ´Ψ of L belongs to W +locally on Rdzt0u. Let h be an L-harmonic function in the sense of tempered distri- +butions, and suppose that Fh acts on W . Then f is a polynomial. +We remark that if f is a polynomial and Lf is well-defined, then Lf is again a +polynomial, and the degree of Lf is less than the degree of f. Thus, in order to find +L-harmonic polynomials of degree n it is sufficient to evaluate Lf for every mono- +mial f of degree at most n, and solve a system of linear equations. We also remark +that if L is isotropic (invariant under rotations), then L-harmonic polynomials h are +harmonic in the classical sense, that is, they satisfy ∆h “ 0. +The proof of Theorem 1.7 is similar to the argument applied independently in [4]. +By specifying the Wiener-type algebra W , we obtain the following family of less ab- +stract results. More variants of Liouville’s theorem which follow from Theorem 1.7 +can be found in Section 5. +Corollary 1.8. Let L be a Lévy operator which is not concentrated on a proper +closed subgroup of Rd, and let h be an L-harmonic function in the sense of tempered +distributions. Then h is necessarily a polynomial if any of the following conditions +holds: + +LIOUVILLE’S THEOREMS FOR LÉVY OPERATORS +7 +(a) the Fourier symbol ´Ψ of L is smooth on Rdzt0u (Corollary 5.2); +(b) h is a bounded function (Corollary 5.4); +(c) for some α ě 0, the function |x|α is integrable with respect to the Lévy mea- +sure ν on RdzB, while p1 ` |x|q´αhpxq is a bounded function (Example 5.10); +(d) for some β ě 0, the function plog |x|qβ is integrable with respect to the Lévy +measure ν on RdzB, while plogpe ` |x|qq´βhpxq is a bounded function (Exam- +ple 5.11); +(e) for some α ą 0, there is a constant c such that the Lévy measure of the ball +with centre x and radius 1 is bounded by c|x|´d´α for every x P Rd such that +|x| ě 2, while p1 ` |x|q´d´αhpxq is an integrable function (Example 5.18); +(f) for some α ą 0 and β P R, or for α “ 0 and some β ą 1, there is a con- +stant c such that the Lévy measure of the ball with centre x and radius 1 +is bounded by c|x|´d´αplog |x|q´β for every x P Rd such that |x| ě 2, while +p1 ` |x|q´d´αplogpe ` |x|qq´βhpxq is an integrable function (Example 5.19). +As discussed after the statement of Theorem 1.3, Liouville’s theorem for bounded +L-harmonic functions has been studied previously, and the general result stated in +Corollary 1.8(a) is due to Alibaud, del Teso, Endal and Jakobsen [1] (see Theo- +rems 1.1 and 1.2 therein). An independent proof was given by Berger and Schilling +in [3] (see Theorem 4.4 therein). +The variant for smooth symbols, given in Corollary 1.8(b), follows easily from +standard properties of tempered distributions and their Fourier transforms. Essen- +tially the same statement is given by Berger and Schilling in [3] (see Theorem 3.2 +therein). +Liouville’s theorem for L-harmonic functions under a moment condition on the +Lévy measure similar to Corollary 1.8(c) was proved by Ros-Oton and Serra in [20] +(see Theorem 2.1 therein) for homogeneous Lévy-type operators (that is, genera- +tors of stable Lévy processes). General Lévy operators were considered by Kühn +in [16] (see Theorem 1 therein), and the present statement of Corollary 1.8(c) was +independently found by Berger, Schilling and Shargorodsky in [4] (see Theorem 8 +therein). The same work contains a more general result (see Theorem 11 therein), +equivalent to our Corollary 5.9, which encompasses Corollary 1.8(c) and (d). +Corollary 1.8(e) for the fractional Laplace operator was proved by Fall in [12] +(see Theorem 1.1 therein) and Chen, D’Ambrosio and Li in [7] (see Theorem 1.3 +therein). Extension similar to Corollary 1.8(e) was given by Fall and Weth in [13] +(see Theorem 1.4 therein). +Corollary 1.8(f), as well as the more general Corol- +lary 5.16, seem to be new. +We remark that if α ą 0 and the Lévy measure ν of the Lévy operator L is com- +parable with |y|´d´αdy when |y| is large enough, then Corollary 1.8(e) provides a +complete Liouville’s theorem, with no restrictions on the L-harmonic function h. +Indeed: in this case p1 ` |x|q´d´αhpxq is automatically integrable whenever h is +L-harmonic, and additionally the two notions of L-harmonicity (Definitions 1.2 +and 1.4) are equivalent. +The same remark applies to Lévy operators with Lévy +measure comparable with |y|´d´αplog |y|q´βdy when |y| is large enough, where α +and β are as in Corollary 1.8(f). +1.4. A counterexample. Although rather general, Theorem 1.7 does not apply to +arbitrary L-harmonic functions (in the sense of tempered distributions), and there +is a reason for that: it turns out that it is not possible to make step (III) of our + +8 +TOMASZ GRZYWNY, MATEUSZ KWA´SNICKI +proof of Theorem 1.7 rigorous without additional assumptions. +In other words, +even though Ψ “ ´F ˇL is a positive, continuous function in Rdzt0u, there may exist +a tempered distribution Fh such that the multiplicative product of Ψ ¨ Fh is well- +defined in the sense of tempered distribution and Ψ ¨ Fh “ 0, but nevertheless Fh +is nonzero in Rdzt0u. That is, in general, division by Ψ turns out to be impossible. +A rigorous statement is given in the following theorem, which is our third main +result. +Theorem 1.9. +(a) There is a bounded, positive, continuous function f and a +tempered distribution g such that the S 1-product f ¨ g is well-defined and +equal to zero even though g is not identically zero (see Corollary 4.8). +(b) There is a nontrivial probability measure µ on Z and a function h on Z such +that lim|x|Ñ8p|x|´εhpxqq “ 0 for every ε ą 0 and h˚µ “ µ, but h is not constant, +and hence not a polynomial (see Theorem 4.2). +(c) There is a Lévy operator L and a smooth function h such that L is not +concentrated on a proper closed subgroup of Rd, for every ε ą 0 we have +lim|x|Ñ8p|x|´εhpxqq “ 0, and Lh “ 0 (both pointwise and in the weak sense), +but h is not constant, and hence not a polynomial (see Theorem 4.1). +1.5. Structure of the paper. The remaining part of the article is divided into four +sections. +In Preliminaries we introduce the notation used in the remaining part of the +paper (Section 2.1), we discuss the notions of convolution and multiplication of +distributions (Section 2.2), and connect them with Lévy operators (Section 2.3). +Theorem 1.3, describing positive L-harmonic functions, is proved in Section 3. +In Section 4, we construct counterexamples to Liouville’s theorem which prove +Theorem 1.9. We begin with operators on Z (Section 4.1), then we reinterpret this +example in terms of multiplication of distributions (Section 4.2), and finally we deal +with the case of operators on Rd (Section 4.3). +The final part of the paper contains the proof of Theorem 1.7 and Corollary 1.8, +which provides various variants of Liouville’s theorem for signed L-harmonic func- +tions. We begin with the abstract result given in Theorem 1.7 (Section 5.1). Then, +by choosing appropriate Wiener-type algebras, we show how this result leads to Li- +ouville’s theorems for operators with smooth symbols (Section 5.2) and for bounded +L-harmonic functions (Section 5.3), which correspond to Corollary 1.8(a) and (b), +respectively. +In order to prove Corollary 1.8(c) and (d), we construct another +Wiener-type algebra in Proposition 5.8 (Section 5.4). +Similarly, Corollary 1.8(e) +and (f) is a consequence of Proposition 5.15 and auxiliary Lemma 5.17 (Section 5.5). +Acknowledgements. We thank Moritz Kaßmann for stimulating discussions about +Liouville theorems. +2. Preliminaries +2.1. General notation. We use x, y, z P Rd for spatial variables, and ξ, η P Rd for +Fourier variables. If ξ, x P Rd, by ξx we denote the dot product of ξ and x. We use +the symbol δxpdyq for the Dirac delta measure at x. +We write D for the class of smooth, compactly supported functions on Rd, and D1 +for the dual space, the class of Schwartz distributions. As it is customary, if f P D1 +and ϕ P D, we write xf, ϕy for the value of f at ϕ. + +LIOUVILLE’S THEOREMS FOR LÉVY OPERATORS +9 +We denote by S the Schwartz class of rapidly decreasing smooth functions on +Rd, and by S 1 the dual space, the class of tempered distributions. Again, if f P S 1 +and ϕ P S , we write xf, ϕy for the value of f at ϕ. Note that D Ă S and S 1 Ă D1. +The support of a distribution f is the smallest closed set K such that xf, ϕy “ 0 +whenever ϕ P D and ϕpxq “ 0 for every x in some neighbourhood of K. +We say that a distribution f corresponds to a (locally integrable) function ˜f in +an open set U if xf, ϕy “ +ş +Rd ˜fpxqϕpxqdx whenever ϕ P D has a compact support +contained in U. In this case we do not distinguish between f and ˜f, and we use a +single symbol for both the distribution f and the function ˜f. In a similar way, we +identify (locally finite) measures with the corresponding distributions. +The Fourier transform of ϕ P S is defined by Fϕpξq “ +ş +Rd e´iξxfpxqdx. The distri- +butional Fourier transform of a tempered distribution f is a tempered distribution +Ff, defined by the exchange formula xFf, ϕy “ xf, Fϕy for every ϕ P S . The +inverse Fourier transform F ´1 is defined in a similar way, with kernel p2πq´deiξx +rather than e´iξx. +The spectrum of a tempered distribution is the support of its distributional +Fourier transform. +A (tempered) distribution f is bounded if the convolution f ˚ ϕ (defined later in +this section) is bounded for every ϕ P S . A bounded distribution extends to a con- +tinuous functional on the class of smooth functions with all derivatives integrable. +In a similar way, a (tempered) distribution f is integrable if f ˚ϕ is integrable for +every ϕ P S . An integrable distribution extends to a continuous functional on the +class of smooth functions with all derivatives bounded. +The Fourier transform of an integrable distribution f coincides with a continuous +function, defined by Ffpξq “ xf, eξy, where eξpxq “ e´iξx. +These and many other properties of Schwartz distributions and tempered distri- +butions can be found in [25]. +2.2. Crash course in convolution and multiplication of distributions. The +convolution f ˚ ϕ of f P D1 and ϕ P D is a smooth function defined by f ˚ ϕpxq “ +xf, ϕxy, where ϕxpyq “ ϕpx ´ yq. The convolution of two Schwartz distributions f +and g exists if there is a Schwartz distribution f f g such that +pf f gq ˚ pϕ ˚ ψq “ pf ˚ ϕq ˚ pg ˚ ψq +(2.1) +for every ϕ, ψ P D; in particular, we assume that the convolution of functions f ˚ ϕ +and g˚ψ in the right-hand side of (2.1) exists in the usual sense. It is straightforward +to check that if ϕ P D, then pf fgq˚ϕ “ f fpg˚ϕq “ pf ˚ϕqfg. Similarly, if f, g, h P D1, +f fg is well-defined and h is compactly supported, then pf fgqfh “ f fpgfhq (and +in particular all convolutions are well-defined). However, convolution of Schwartz +distributions is not associative in general. +For further information, we refer to +Section 1 in [10]. +In a similar way, the convolution f ˚ ϕ of f P S 1 and ϕ P S is a smooth function +determined by f ˚ ϕpxq “ xf, ϕxy, where again ϕxpyq “ ϕpx ´ yq. The convolution of +two tempered distributions f and g exists if there is a tempered distribution f f g +such that +pf f gq ˚ pϕ ˚ ψq “ pf ˚ ϕq ˚ pg ˚ ψq +(2.2) + +10 +TOMASZ GRZYWNY, MATEUSZ KWA´SNICKI +for every ϕ, ψ P S ; here, too, we assume that the middle convolution in the right- +hand side of (2.2) exists in the usual sense. The convolution f f g is well-defined if, +for example, f, g P S 1 and f has compact support. It is again a simple exercise to +check that if ϕ P S , then pf f gq ˚ ϕ “ f f pg ˚ ϕq “ pf ˚ ϕq f g, but the convolution of +tempered distributions is not associative in general. We refer to Section 2 in [10] +for further details. +The two notions of the convolution of distributions: D1-convolution f f g and S 1- +convolution f fg, are clearly closely related to each other. However, we stress that +there are tempered distributions f, g P S 1 such that f f g is defined, while f f g is +not; see Section 3 in [10] for a detailed discussion. +The S 1-convolution of a bounded distribution f and an integrable distribution g +is a bounded distribution, and the S 1-convolution of two integrable distributions +is again an integrable distribution. In fact, S 1-convolution of any number of in- +tegrable distributions and a bounded distribution is commutative and associative. +Obviously, every Schwartz class function corresponds to a distribution which is +integrable and bounded. We refer to [10] for further discussion. +The product ψ ¨ f of f P S 1 and ψ P S is a tempered distribution defined by +xψ ¨ f, ϕy “ xf, ϕψy for every ϕ P S . The product of two tempered distributions f +and g exists if there is a tempered distribution f d g such that +xf d g, ϕy “ lim +nÑ8 +ż +Rd f ˚ αnpxqg ˚ βnpxqϕpxqdx +for every ϕ P S and every sequences αn and βn of nonnegative functions in D such +that the integrals of αn and βn are equal to 1 and the supports of αn and βn shrink to +t0u as n Ñ 8. Multiplication of distributions extends multiplication of continuous +functions, or a continuous function and a locally finite measure. +Furthermore, +multiplication of distributions is a local operation: if f1 “ f2 and g1 “ g2 in an open +set U, then f1dg1 “ f2dg2 in U (provided that both products exist). Finally, if ψ P S , +then ψ ¨pf dgq “ pψ ¨fqdg “ f dpψ ¨gq, but multiplication of tempered distributions +is not associative in general. For a detailed discussion, we refer to [22]. +For every f P S 1 and ϕ P S , we have the exchange formula Fpf ˚ ϕq “ pFϕq ¨ +pFfq. +As it was proved in [14], this result extends to the general case: if the +convolution of tempered distributions f and g exists, then the product of their +Fourier transforms is well-defined, and we have +Fpf f gq “ pFfq d pFgq. +2.3. Lévy operators and distributions. Let L be a Lévy operator, and let L: be +the dual operator, as in Definition 1.2. The map which assigns to ϕ P S the value +L:ϕp0q is easily checked to be continuous, and hence it corresponds to some dis- +tribution, that we denote by ˇL: L:ϕp0q “ xˇL, ϕy. Furthermore, if ϕxpyq “ ϕpx ´ yq, +then Lϕpxq “ L:ϕxp0q “ xˇL, ϕxy, that is, Lϕpxq “ ˇL ˚ ϕpxq. Thus, L is a convolution +operator, with convolution kernel ˇL P S 1. It is straightforward to see that Lϕ is +integrable for every ϕ P S , and therefore ˇL is in fact an integrable distribution. +By a classical result in the theory of Lévy processes, the Fourier transform of ˇL +is the characteristic exponent Ψ defined in (1.2). In fact, this will serve us as the +definition of the continuous function Ψ, and we will never need (1.2). +In Section 3 we work with a general, distributional definition of an L-harmonic +function. Here we prove that this definition indeed extends Definition 1.2. + +LIOUVILLE’S THEOREMS FOR LÉVY OPERATORS +11 +Lemma 2.1. Let L be a Lévy operator. If a function h is L-harmonic in the weak +sense (according to Definition 1.2), then h is L-harmonic in the sense of Schwartz +distributions: h corresponds to a Schwartz distribution which is convolvable with +ˇL, the convolution kernel of L, and we have ˇL f h “ 0. +Proof. By Definition 1.2, h is a locally integrable function such that for every ϕ P D +we have +ż +Rd hpyqL:ϕpyqdy “ 0. +Recall that xˇL, ϕy “ L:ϕp0q. Thus, if ˇϕpxq “ ϕp´xq, then L:ϕpxq “ ˇL ˚ ˇϕp´xq. It +follows that +ż +Rd hpyqˇL ˚ ˇϕp´yqdy “ 0, +and in particular the integral is absolutely convergent. Replacing ϕ with ϕxpyq “ +ϕpx ´ yq, we find that +ż +Rd hpyqˇL ˚ ϕpx ´ yqdx “ 0 +(2.3) +for every ϕ P D and x P Rd. We will momentarily show that we can use Fubini’s +theorem to find that for every ϕ, ψ P D and x P Rd, +ph ˚ ψq ˚ pˇL ˚ ϕqpxq “ +ż +Rd h ˚ ψpx ´ yqˇL ˚ ϕpyqdy +“ +ż +Rd +ˆż +Rd hpx ´ y ´ zqψpzqdz +˙ +ˇL ˚ ϕpyqdy +“ +ż +Rd +ˆż +Rd hpx ´ y ´ zqˇL ˚ ϕpyqdy +˙ +ψpzqdz +“ +ż +Rd +ˆż +Rd hpyqˇL ˚ ϕpx ´ z ´ yqdy +˙ +ψpzqdz “ 0. +(2.4) +Of course, this means that ˇL f h is well-defined and equal to zero, as desired. +In order to show absolute integrability of the double integral above, we denote by +C the supremum of |ϕ|`|ψ|, and we choose R large enough, so that ϕpxq “ ψpxq “ 0 +when |x| ě R. Let Brpxq denote the ball of radius r centred at x. By (1.1), we have +|ˇL ˚ ϕpxq| ď CνpBRp´xqq +whenever |x| ą R. If |x ´ y| ą 2R and |z| ă R, then |x ´ y ´ z| ą R. Thus, +ż +Rd |ˇL ˚ ϕpx ´ z ´ yqψpzq|dz ď C +ż +BRp0q +|ˇL ˚ ϕpx ´ z ´ yq|dz +ď C2 +ż +BRp0q +νpBRpy ` z ´ xqqdz ď C2|BRp0q|νpB2Rpy ´ xqq. +On the other hand, if ˜ϕ P D is chosen in such a way that 0 ď ˜ϕpxq ď 1 for all x P Rd, +˜ϕpxq “ 1 when |x| ď 2R and ˜ϕpxq “ 0 when |x| ě 3R, then, again by (1.1), we have +|ˇL ˚ ˜ϕpxq| ě νpB2Rp´xqq +whenever |x| ą 3R. It follows that if |x ´ y| ą 3R, then +ż +Rd |ˇL ˚ ϕpx ´ z ´ yqψpzq|dz ď C2|BRp0q||ˇL ˚ ˜ϕpx ´ yq|. + +12 +TOMASZ GRZYWNY, MATEUSZ KWA´SNICKI +Thus, +ż +RdzB3Rpxq +ż +Rd |hpyqˇL ˚ ϕpx ´ z ´ yqψpzq|dzdy +ď C2|BRp0q| +ż +RdzB3Rpxq +|hpyqˇL ˚ ˜ϕpx ´ yq|dy, +and the right-hand side is finite by (2.3). Additionally, h is integrable over B3Rpxq, +ψ is integrable over Rd, and ˇL ˚ ϕ is bounded, and hence +ż +B3Rpxq +ż +Rd |hpyqˇL ˚ ϕpx ´ z ´ yqψpzq|dzdy ă 8. +Absolute integrability of the double integral in (2.4) follows, and the proof is com- +plete. +□ +Finally, we prove that under a natural growth condition, L-harmonic functions +in the weak sense coincide with L-harmonic functions in the sense of tempered +distributions. +Note that with the notation introduced above, a function h is L- +harmonic in the sense of tempered distributions (according to Definition 1.4) if and +only if ˇL f h is well-defined and equal to 0. +Lemma 2.2. Let L be a Lévy operator, let ν be the corresponding Lévy measure, +and let B denote the unit ball in Rd. Let h be a function which is L-harmonic in the +weak sense (according to Definition 1.2), and such that +|hpxq| ` +ż +RdzB +|hpx ` yq|νpdyq, +as a function of x P Rd, is bounded by a polynomial. Then h is L-harmonic in the +sense of tempered distributions (according to Definition 1.4). +Proof. By Lemma 2.1, we know that ˇL f h is well-defined and equal to zero, that +is, for every ϕ, ψ P D the convolution of ˇL ˚ ϕ and h ˚ ψ is well-defined and equal to +zero. Our goal is to prove a similar result for ϕ, ψ P S . +We choose a function ϕ0 P D which takes values in r0, 1s, and such that ϕ0pxq “ 1 +when |x| ă 1. Let ˇL0 “ ϕ0 ˇL and ˇL1 “ ˇL ´ ˇL0. Then ˇL0 is a Schwartz distribution +with compact support, and ˇL1 “ p1 ´ ϕ0qν is a nonnegative finite measure. +Since the support of ˇL0 is compact and h defines a tempered distribution, the +convolution ˇL0 fh is well-defined. Furthermore, ˇL1 is a finite nonnegative measure +and, by assumption, ˇL1 ˚ |h| is bounded by a polynomial. This implies that ˇL1 f h is +well-defined, too. Thus, ˇL f h is well-defined. +Clearly, ˇL f h “ ˇL f h. However, by Lemma 2.1, ˇL f h “ 0, and the proof is +complete. +□ +3. Positive harmonic functions +In this section we prove Theorem 1.3: we show that nonnegative L-harmonic +functions are essentially mixtures of L-harmonic exponentials. +The idea of the +proof is very similar to the one used in [3]: we reduce the problem to a convo- +lution equation studied by Deny in [9]. However, instead of the heat kernel (the +distribution of the corresponding Lévy process at a fixed time) as in [3], we use the +harmonic measure (the distribution of the Lévy process at the first exit time). This + +LIOUVILLE’S THEOREMS FOR LÉVY OPERATORS +13 +allows us to avoid integrability problems and thus remove unnecessary restrictions +on the class of L-harmonic functions h. +For simplicity, Theorem 1.3 is stated for Lévy operators on Rd which are not +concentrated on a proper closed subgroup of Rd. In the general case, the smallest +closed subgroup G of Rd on which L is concentrated is isomorphic to Rd´k ˆ Zk for +some k P t0, 1, . . . , du. In order to cover this case, here we consider Lévy operators +acting on G “ Rd ˆ Zk for arbitrary d and k, and we continue to assume that L is +not concentrated on a proper subgroup of G. We omit the obvious extensions of the +notions discussed above for Rd to this more general case. +As mentioned above, the key tool in our proof is the main result (Théorème 3) +of [9]. The original statement allows G to be an arbitrary separable locally compact +abelian group. We restrict our attention to G “ Rd ˆ Zk. In this case the Haar +measure on G is the product of the Lebesgue measure on Rd and the counting +measure on Zk, and for simplicity we call it simply the Lebesgue measure on G. +Theorem 3.1 (Deny’s theorem). Let ν be a probability measure on G “ Rd ˆ Zk +such that the closed group generated by the support of ν is equal to G, and let h be +a locally finite nonnegative measure on G such that h ˚ ν “ h. Then h is absolutely +continuous with respect to the Lebesgue measure on G, and the density function is +equal to +hpxq “ +ż +Λ +eλxµpdλq +for a unique nonnegative measure µ on the set Λ of those vectors λ P Rd ˆ Rk for +which the function eλpxq “ eλx satisfies eλ ˚ ν “ eλ. +Although we avoid as much as possible probabilistic tools, in this section some +well-known results from the theory of Lévy processes play an important role. +Namely, in the next paragraph we introduce two standard potential-theoretic ob- +jects: the Green kernel and the harmonic measure, and in the proof of our Li- +ouville’s theorem we use the probabilistic definition of the notion of L not being +concentrated on a proper closed subgroup of G. +For every bounded open set D Ă G there are associated Green kernel GDpx, dyq +and the harmonic measure HDpx, dzq, with the following properties. If x P D, then +GDpx, dyq is a finite, nonnegative measure with respect to y, concentrated on D, +while HDpx, dzq is a probability measure with respect to z, concentrated on GzD. If +x R D, then GDpx, dyq is a zero measure, while HDpx, dzq “ δxpdzq. Finally, for every +ϕ P D and x P G we have +ϕpxq “ +ż +G +ϕpzqHDpx, dzq ´ +ż +G +LϕpyqGDpx, dyq. +(3.1) +While the above objects can be constructed analytically, their probabilistic descrip- +tion is much simpler and far more intuitive. Let Xt be the Lévy process with gen- +erator L, and let Px and Ex denote the probability and expectation corresponding +to the process Xt started at x P G. Let τD denote the first exit time from D: +τD “ inftt P r0, 8q : Xt R Du. + +14 +TOMASZ GRZYWNY, MATEUSZ KWA´SNICKI +Then HDpx, dzq is the distribution of Xt at the first exit time τD, and GDpx, dyq is the +mean occupation measure up to time τD: +HDpx, Aq “ PxpXτD P Aq, +GDpx, Aq “ Ex +ż τD +0 +1ApXtqdt. +Formula (3.1) is known as Dynkin’s formula, and it is one of the fundamental tools +in the study of Markov processes; see Theorem 5.1 in [11]. +For a bounded open set D which contains the origin, we denote +ˇGDpAq “ GDp0, ´Aq +and +ˇHDpAq “ HDp0, ´Aq. +Recall that by ˇL we denote the convolution kernel of the Lévy operator L. +Then (3.1) implies that for every ϕ P D we have +ϕp0q “ +ż +G +ϕp´zq ˇHDpdzq ´ +ż +G +Lϕp´yq ˇGDpdyq +“ ˇHD ˚ ϕp0q ´ pˇL ˚ ϕq ˚ ˇGDp0q. +Since ˇGD is a finite measure with compact support, the convolution ˇL f ˇGD is well- +defined, and pˇL ˚ ϕq ˚ ˇGD “ pˇL ˚ ϕq f ˇGD “ ϕ ˚ pˇL f ˇGDq. Thus, +ϕp0q “ ϕ ˚ +` ˇHD ´ ˇL f ˇGD +˘ +p0q +for every ϕ P D. But this is another way to say that +δ0 “ ˇHD ´ ˇL f ˇGD. +(3.2) +We will use the above identity in the proof of the following theorem, which is the +main result of this section. +Theorem 3.2 (Liouville’s theorem for positive solutions). Let L be a Lévy operator +on G “ Rd ˆ Zk, which is not concentrated on a proper closed subgroup of G, and +let h be a locally finite nonnegative measure on G which is L-harmonic in the weak +sense, or, more generally, in the sense of Schwartz distributions (see Lemma 2.1). +Then h is absolutely continuous with respect to the Lebesgue measure on G, and +the density function is equal to +hpxq “ +ż +Λ +eλxµpdλq +(3.3) +for a unique nonnegative measure µ on the set Λ of those vectors λ P Rd ˆ Rk for +which the function eλpxq “ eλx satisfies Leλpxq “ 0. +We remark that when we write Leλpxq “ 0 above, we mean that Leλpxq is well +defined pointwise, according to the definition (1.1). +However, as we will see in +the proof of Theorem 3.2, this is equivalent to eλ being L-harmonic in the weak +sense. Additionally, since Leλpxq “ eλpxqLeλp0q, we have Leλpxq “ 0 for every x P G +whenever Leλpxq “ 0 for a single x P G. +Proof. Step 1. Suppose that the convolution ˇLfh is well-defined and equal to 0. We +consider a bounded open set D such that 0 P D, and we use the notation introduced +above. Recall that the convolution of distributions is associative when one of the +factors is compactly supported. Thus, +0 “ ph f ˇLq f ˇGD “ h f pˇL f ˇGDq. + +LIOUVILLE’S THEOREMS FOR LÉVY OPERATORS +15 +By (3.2), we have ˇL f ˇGD “ ˇHD ´ δ0, and so h f p ˇHD ´ δ0q is well-defined and equal +to zero. Clearly, h f δ0 “ h is well-defined, and so +h “ 0 ` h f δ0 “ h f p ˇHD ´ δ0q ` h f δ0 “ h f ˇHD. +Since h and ˇHD are nonnegative measures, we have h f ˇHD “ h ˚ ˇHD. Thus, +h ˚ ˇHD “ h. +Step 2. The desired result essentially follows now from Deny’s theorem (Theo- +rem 3.1): if the support of ˇHD generates a dense subgroup of G, then the equality +h ˚ ˇHD “ h implies the desired representation (3.3) of h, with Λ replaced by the set +ΛD of those vectors λ P Rd for which the function eλpxq “ eλx satisfies eλ ˚ ˇHD “ eλ. +A detailed argument, however, requires some care: we need to show that ΛD “ Λ, +and we need to handle the case when the support of ˇHD is contained in a proper +closed subgroup of G. +Step 3. We first show that ΛD “ Λ. In fact, we prove a stronger statement: if +eλ ˚ ˇHD is not everywhere infinite, then eλ f ˇL is well-defined, and the sign of eλ f ˇL +is the same as the sign of eλ ˚ ˇHD ´ eλ. +Denote αλ,D “ eλ ˚ ˇHDp0q. Observe that +eλ ˚ ˇHDpxq “ +ż +G +eλpx ´ yq ˇHDpdyq “ eλpxq +ż +G +eλp´yq ˇHDpdyq “ αλ,Deλpxq. +In particular, since eλ ˚ ˇHD is not everywhere infinite, αλ,D is finite. We find that +pαλ,D ´ 1qeλ “ eλ ˚ p ˇHD ´ δ0q “ eλ f p ˇHD ´ δ0q “ eλ f pˇL f ˇGDq. +Suppose that ϕ P D is a nonnegative function which is not identically equal to zero. +We have +pαλ,D ´ 1qeλ ˚ ϕ “ peλ f pˇL f ˇGDqq ˚ ϕ “ eλ f ppˇL f ˇGDq ˚ ϕq “ eλ f pˇL f p ˇGD ˚ ϕqq. +Observe that also ψ “ ˇGD ˚ ϕ is a nonnegative function in D, not identically equal +to zero (because ˇGD is compactly supported, nonnegative and not identically equal +to zero). Hence, +pαλ,D ´ 1qeλ ˚ ϕ “ eλ f pˇL f ψq “ eλ f pˇL ˚ ψq “ peλ f ˇLq ˚ ψ “ peλ ˚ ψq f ˇL. +However, if βλ,D “ eλ ˚ ϕp0q and γλ,D “ eλ ˚ ψp0q, then, by the argument already +applied above, we have eλ ˚ ϕ “ βλ,Deλ and eλ ˚ ψ “ γλ,Deλ. And since ϕ and ψ are +nonnegative and not identically equal to zero, we have βλ,D ą 0 and γλ,D ą 0. We +conclude that +pαλ,D ´ 1qβλ,Deλ “ γλ,Deλ f ˇL. +It follows that the sign of αλ,D ´ 1 is the same as the sign of eλ f ˇL, and our claim is +proved. Additionally, λ P ΛD if and only if αλ,D “ 1, which is equivalent to eλf ˇL “ 0, +that is, λ P Λ. In other words, ΛD “ Λ. +Step 4. We claim that there is no proper closed subgroup of G which contains the +support of ˇHD for every bounded open set D such that 0 P D. We use the following +interpretation of our assumption that L is not concentrated on a proper closed +subgroup of G: if Xt is the Lévy process generated by L, then the union of supports +of all random variables Xt ´ X0 is not contained in a proper closed subgroup of G. + +16 +TOMASZ GRZYWNY, MATEUSZ KWA´SNICKI +Suppose that A is a compact set, 0 R A, and ˇHDp´Aq “ 0 for every bounded open +set D such that 0 P D. Then +P0pXτD P Aq “ 0 +for every bounded open set D such that 0 P D. By considering D “ tx P GzA : |x| ă +ru and passing to the limit as r Ñ 8, we find that with probability P0 one, Xt R A +for all t ą 0. Here we use the fact that with probability one, τD is equal to τGzA +for r large enough (and this, in turn, is a consequence of quasi-left continuity of +Lévy processes). In particular, A is disjoint from the support of Xt ´ X0 for every +t ą 0. It follows that the union of supports of measures ˇHD (where D is allowed to +be an arbitrary bounded open set such that 0 P D) contains the union of supports of +random variables X0´Xt (where t ą 0). The latter one generates a dense subgroup +of G, and hence the same is true for the former one. Our claim is proved. +Step 5. +Let FD denote the support of ˇHD. +We choose a countable family of +bounded open sets Dn such that 0 P Dn, with the following property: the closure +of the union of the supports FDn is equal to the closure of the union of FD over +all bounded open sets D such that 0 P D. In order to do that, we may apply the +following procedure: choose a countable base of open sets in G, and for every +basic open set G which intersects the union of all FD, choose a set Dn so that FDn +intersects G. +Let ˇH0 be an arbitrary convex combination of the measures ˇHDn with positive +coefficients. +Then the support of ˇH0 is equal to the closure of the union of the +supports FDn. But this set contains the union of all supports FD, and by the result +of the previous step, the latter is contained in no proper closed subgroup of G. +Hence, the support of ˇH0 is contained in no proper closed subgroup of G. Since +h ˚ ˇHDn “ h for every n, by Fubini’s theorem we find that h ˚ ˇH0 “ h. Thus, we may +apply Deny’s theorem to conclude that h is absolutely continuous with respect to +the Lebesgue measure on G, and the density function is given by +hpxq “ +ż +Λ0 +eλxµpdλq +for a unique nonnegative measure µ on the set Λ0 of those vectors λ P Rd for which +the function eλpxq “ eλx satisfies eλ ˚ ˇH0 “ eλ. It remains to show that Λ0 “ Λ. +Suppose that λ P Λ0. Then for every n the convolution eλ ˚ ˇHDn is not everywhere +infinite. By the result of step 3, eλ f ˇL is well-defined, and the sign of eλ f ˇL is the +same as the sign of eλ ˚ ˇHDn ´ eλ, regardless of n. Applying again Fubini’s theorem, +we find that the sign of eλ f ˇL is the same as the sign of eλ ˚ ˇH0 ´ eλ, and since +λ P Λ0, the latter is zero by assumption. Thus, eλ f ˇL “ 0, that is, λ P Λ. Conversely, +if λ P Λ, then eλ f ˇHD “ eλ for every bounded open set D such that 0 P D, and thus +eλ f ˇH0 “ eλ, that is, λ P Λ0. This completes the proof. +□ +We remark that if L is the one-dimensional Laplace operator and D “ p´r, rq, +then ˇHD “ +1 +2δ´r ` 1 +2δr has support contained in a proper closed subgroup rZ of +G “ R. However, we conjecture that for an arbitrary Lévy operator L satisfying +the assumption of Theorem 3.2 it is always possible find a single bounded open +set D such that the support of ˇHD is not contained in a proper closed subgroup +of G. For example, if L is the one-dimensional Laplace operator and D “ p´a, bq +with incommensurable a, b ą 0, then ˇHD “ pa ` bq´1pbδ´a ` aδbq has support t´a, bu, +which is not contained in a proper closed subgroup of G “ R. + +LIOUVILLE’S THEOREMS FOR LÉVY OPERATORS +17 +Remark 3.3. Let L be an arbitrary Lévy operator on Rd, and let G be the smallest +closed subgroup of Rd such that L is concentrated on G. Then L can be viewed +as an operator which acts independently on each coset x ` G of G in Rd. Thus, +L-harmonic functions or measures can be constructed independently on each coset +x ` G (as long as the resulting function is locally integrable on G, or the resulting +measure is locally finite on G). On each coset, nonnegative L-harmonic measures +are described by Theorem 3.2. +4. An unusual L-harmonic function +In this section we prove the following unexpected result. +Theorem 4.1 (counterexample to the general Liouville’s theorem). There is a one- +dimensional Lévy operator L, and a smooth function h, with the following proper- +ties: +(a) L is not concentrated on a proper subgroup of R; +(b) for every ε ą 0 we have lim|x|Ñ8 |x|´εhpxq “ 0; +(c) Lhpxq “ 0 for every x P R (so that, in particular, the integral in the definition +of Lhpxq is absolutely convergent); +(d) h is L-harmonic in the sense of tempered distributions: ˇLf h is well-defined +and equal to zero, where ˇL is the convolution kernel of L; +but h is not a polynomial. +More precisely, we consider a one-dimensional symmetric Lévy operator L of the +form +Lfpxq “ f 2pxq ` +8 +ÿ +k“0 +pk +` +fpx ` xkq ` fpx ´ xkq ´ 2fpxq +˘ +, +where pk “ 2´k´2 and xk is a rapidly increasing sequence. +Theorem 4.1 extends trivially to Rd by considering the Lévy operator which is +the sum of operators L defined above acting on each coordinate xj, and the cor- +responding harmonic function which is the product of hpxjq for each coordinate +xj. +The construction of h is somewhat technical. For this reason, we begin with a +simpler, discrete variant of Theorem 4.1. +4.1. Discrete case. In this section, we prove the following result, which will pre- +pare us for the proof of Theorem 4.1. +Theorem 4.2 (counterexample to Liouville’s theorem for lattice random walks). +Let +pk “ 2´k´2 +xk “ 22k2. +There is a doubly infinite sequence hpnq which satisfies +hpnq “ +8 +ÿ +k“0 +pk +` +hpn ` xkq ` hpn ´ xkq +˘ +for every n P Z, and such that for every ε ą 0, +lim +|n|Ñ8 +hpnq +|n|ε “ 0, + +18 +TOMASZ GRZYWNY, MATEUSZ KWA´SNICKI +but hpnq is not a polynomial sequence. Furthermore, for every ε ą 0, we have +lim +|n|Ñ8 +1 +|n|ε +8 +ÿ +k“0 +pkp|hpn ` xkq| ` |hpn ´ xkq|q “ 0. +The sequence hpnq constructed in the proof is extremely sparse: we have hpnq “ 0 +unless n “ 0 or |n| “ k ` xk for some k “ 0, 1, 2, . . . More precisely, we set hp0q “ 1, +hpnq “ ak if |n| “ k ` xk, with appropriately chosen ak, and hpnq “ 0 for all other +indices n. +Before we proceed with the construction of the sequence hpnq, we make the +following observation. +Lemma 4.3. If xk`1 ě 2xk and xk ě 2k for k “ 0, 1, 2, . . ., then +|n ˘ xk| “ m ` xm +implies that +|n| “ m +or +|n| ě 1 +2xm. +In particular, the above property holds when xk “ 22k2. +Proof. Suppose that |n ˘ xk| “ m ` xm. If k ă m, then +|n| ě m ` xm ´ xk ě xm ´ xm´1 ě xm ´ 1 +2xm “ 1 +2xm. +If k “ m, then either |n| “ m or |n| “ m ` 2xm ě 1 +2xm. Finally, if k ą m, then +|n| ě xk ´ m ´ xm ě xm`1 ´ m ´ xm ě 2xm ´ m ´ xm “ xm ´ m ě 1 +2xm. +□ +From now on we let pk “ 2´k´2 and xk “ 22k2, as in Theorem 4.2. For convenience, +let us write +Lfpnq “ +8 +ÿ +k“0 +pk +` +fpn ` xkq ` fpn ´ xkq ´ 2fpnq +˘ +“ +8 +ÿ +k“0 +pk +` +fpn ` xkq ` fpn ´ xkq +˘ +´ fpnq +(4.1) +whenever fpnq is a doubly infinite sequence such that the above series converges +absolutely (so that L is a Lévy operator acting on Z). +The construction of the +sequence hpnq is an iterative procedure, which can be summarised as follows. In +the initial step, we let h´1pnq “ 1t0upnq. Next, for m “ 0, 1, 2, . . . we define hmpnq “ +hm´1pnq except at two values of n, namely, n “ ˘pm ` xmq. +At these values we +modify hmpnq in such a way that Lhmpmq “ Lhmp´mq “ 0. The key observation is +that, by Lemma 4.3, we also have Lhmpnq “ Lhm´1pnq “ 0 if |n| ă m. Finally, we +define hpnq to be the limit of hmpnq as m Ñ 8. +We proceed with the detailed construction of hpnq. We let +h´1pnq “ 1t0upnq, +and +hmpnq “ hm´1pnq ` am 1t´m´xm,m`xmupnq +“ 1t0upnq ` +m +ÿ +j“0 +aj 1t´j´xj,j`xjupnq +for m “ 0, 1, 2, . . ., where +am “ ´Lhm´1pmq +pm +“ ´ 1 +pm +ˆ 8 +ÿ +k“0 +pk +` +hm´1pm ` xkq ` hm´1pm ´ xkq +˘ +´ hm´1pmq +˙ + +LIOUVILLE’S THEOREMS FOR LÉVY OPERATORS +19 +if m “ 1, 2, . . ., and +a0 “ ´Lh´1p0q +2p0 +“ 2. +Finally, we define +hpnq “ lim +mÑ8 hmpnq +“ 1t0upnq ` +8 +ÿ +m“0 +am 1t´m´xm,m`xmupnq. +Below we prove Theorem 4.2 by showing that the sequence hpnq constructed above +has all the desired properties. We break the proof into three lemmas. First, we +prove that hpnq is L-harmonic. +Lemma 4.4. With the above definitions, we have +Lhpnq “ 0 +for every n P Z. +Proof. By Lemma 4.3, for a fixed n P Z and all k “ 0, 1, 2, . . . we have +|n ˘ xk| ‰ m ` xm +if m ą |n|. +It follows that hpn ˘ xkq “ hmpn ˘ xkq, where m “ |n|. Therefore, +Lhpnq “ +8 +ÿ +k“0 +pk +` +hpn ` xkq ` hpn ´ xkq +˘ +´ hpnq +“ +8 +ÿ +k“0 +pk +` +hmpn ` xkq ` hmpn ´ xkq +˘ +´ hmpnq +(4.2) +“ Lhmpnq. +We claim that Lhmpnq “ 0. If n “ m “ 0, then we simply have +Lh0p0q “ p0 +` +h0px0q ` h0p´x0q +˘ +´ h0p0q “ 2p0a0 ´ 1 “ 0. +Suppose now that n “ m ą 0. By the definition of hm, we have +Lhmpmq “ Lhm´1pmq ` amL 1t´m´xm,m`xmupmq. +Since |m ˘ xk| ‰ m ` xm if k ‰ m, and also |m ´ xm| ‰ m ` xm, we find that +L 1t´m´xm,m`xmupmq “ pm 1t´m´xm,m`xmupm ` xmq ´ 1t´m´xm,m`xmupmq “ pm. +Combining the above two identities and the definition am “ ´p´1 +m Lhm´1pmq of am, +we conclude that +Lhmpmq “ Lhm´1pmq ` ampm “ 0, +(4.3) +as desired. By a similar argument (or by symmetry), we also have Lhmp´mq “ 0, +and our claim is thus proved. +□ +In order to prove Theorem 4.2, it remains to show appropriate estimates of hpnq. +Recall that for m “ 1, 2, . . ., +pmam “ ´Lhm´1pmq “ ´ +8 +ÿ +k“0 +pk +` +hm´1pm ` xk ` hm´1pm ´ xkq +˘ +` hm´1pmq. + +20 +TOMASZ GRZYWNY, MATEUSZ KWA´SNICKI +Lemma 4.5. With the above definitions, for every ε ą 0 there is Cε such that +pm|am| ď +8 +ÿ +k“0 +pk +` +|hm´1pm ` xkq| ` |hm´1pm ´ xkq| +˘ +` |hm´1pmq| ď Cεp1 ` mqε +(4.4) +for every m “ 0, 1, 2, . . . +Proof. Fix ε ą 0. Recall that pk “ 2´k´2 and xk “ 22k2. For j large enough, say, +j ą j0, we have +2j`3p1 ` jqε ď p1 ` 1 +2xjqε. +We choose m0 so that +2j0`3p1 ` j0qε ď p1 ` m0qε. +With this choice, we have the following property: +2j`3p1 ` jqε ď p1 ` mqε +when m ě m0 and m ě 1 +2xj. +(4.5) +We choose Cε large enough, so that the desired bound (4.4) holds for m “ +1, 2, . . . , m0, and we prove by induction that (4.4) also holds for m ą m0. +Suppose that for some m ą m0 formula (4.4) holds with m replaced by any +smaller number. The first inequality in (4.4) follows from the definition of am, so in +order to complete the proof we only need to show the other inequality. +If hm´1pmq ‰ 0, then m “ j ` xj and hm´1pmq “ aj for some j ă m. Since (4.4) +holds with m replaced by j, we obtain +|hm´1pmq| “ |aj| ď Cεp1 ` jqε +pj +“ Cε2j`2p1 ` jqε. +Since m ą xj, we may apply (4.5) to find that +|hm´1pmq| ď Cε +2 2j`3p1 ` jqε ď Cε +2 p1 ` mqε. +Similarly, if hm´1pm ˘ xkq ‰ 0, then |m ˘ xk| “ j ` xj and hm´1pm ˘ xkq “ aj for some +j ă m, so that again +|hm´1pm ˘ xkq| “ |aj| ď Cεp1 ` jqε +pj +“ Cε2j`2p1 ` jqε. +By Lemma 4.3 we find that m ě 1 +2xj, and hence (4.5) again leads to +|hm´1pm ˘ xkq| ď Cε +2 2j`3p1 ` jqε ď Cε +2 p1 ` mqε. +It follows that +8 +ÿ +k“0 +pk +` +|hm´1pm ` xkq| ` |hm´1pm ´ xkq| +˘ +` |hm´1pmq| +ď +8 +ÿ +k“0 +2pk ˆ Cε +2 p1 ` mqε ` Cε +2 p1 ` mqε “ Cεp1 ` mqε. +The proof is complete. +□ +Lemma 4.6. With the above definitions, for every ε ą 0 we have +lim +|n|Ñ8 +hpnq +|n|ε “ 0 + +LIOUVILLE’S THEOREMS FOR LÉVY OPERATORS +21 +and +lim +|n|Ñ8 +1 +|n|ε +8 +ÿ +k“0 +pk +` +|hpn ` xkq| ` |hpn ´ xkq| +˘ +“ 0. +Proof. If n ‰ 0 and |n| ‰ m ` xm for every m “ 0, 1, 2, . . ., then hpnq “ 0. +If +|n| “ m ` xm for some m “ 0, 1, 2, . . ., then hpnq “ am, and, by Lemma 4.5, +|hpnq| +|n|ε +“ +|am| +pm ` xmqε ď Cεp1 ` mqε +pmpm ` xmqε “ Cε2m`2p1 ` mqε +pm ` 22m2qε +. +The right-hand side clearly converges to zero as m Ñ 8, and the first part of the +lemma follows. +To prove the other one, we consider m “ 0, 1, 2, . . . and we recall that, as in (4.2) +and (4.3), we have +8 +ÿ +k“0 +pk +` +|hpm ` xkq| ` |hpm ´ xkq| +˘ +“ +8 +ÿ +k“0 +pk +` +|hmpm ` xkq| ` |hmpm ´ xkq| +˘ +“ pm|am| ` +8 +ÿ +k“0 +pkp|hm´1pm ` xkq| ` |hm´1pm ´ xkq| +˘ +. +By Lemma 4.5, the right-hand side does not exceed 2Cε{2p1`mqε{2, and consequently +lim +mÑ8 +1 +mε +8 +ÿ +k“0 +pk +` +|hpm ` xkq| ` |hpm ´ xkq| +˘ +“ 0, +as desired. In a similar way (or by symmetry), +lim +mÑ8 +1 +mε +8 +ÿ +k“0 +pk +` +|hp´m ` xkq| ` |hp´m ´ xkq| +˘ +“ 0, +and the proof is complete. +□ +Theorem 4.2 is an immediate corollary of the above series of lemmas. We prove +one additional property of the sequence hpnq, which in fact proves Theorem 4.1 +without the assumption that L is not concentrated on a proper closed subgroup of +R. +Lemma 4.7. With the above definitions, the convolution kernel ˇL of the Lévy op- +erator L is S 1-convolvable with the measure +H “ +8 +ÿ +n“´8 +hpnqδn, +and +ˇL f H “ 0. +Proof. By the last assertion of Theorem 4.2, for a given ε ą 0, there is a constant +C1 such that +8 +ÿ +k“0 +pk +` +|hpn ` xkq| ` |hpn ´ xkq| +˘ +` |hpnq| ď C1p1 ` |n|qε. + +22 +TOMASZ GRZYWNY, MATEUSZ KWA´SNICKI +Furthermore, if ϕ, ψ P S and x P R is fixed, then there is a constant C2 such that +|ϕ| ˚ |ψ|px ´ nq ď C2p1 ` |n|q´2´ε. +It follows that +ż +R +8 +ÿ +n“´8 +ˆ 8 +ÿ +k“0 +pk +` +|hpn ` xkq| ` |hpn ´ xkq| +˘ +` |ϕpnq| +˙ +|ϕpy ´ nq||ψpx ´ yq|dy ă 8. +Thus, we may apply the result of Theorem 4.2 and Fubini’s theorem to find that +0 “ +ż +R +8 +ÿ +n“´8 +ˆ 8 +ÿ +k“0 +pk +` +hpn ` xkq ` hpn ´ xkq +˘ +´ hpnq +˙ +ϕpy ´ nqψpx ´ yqdy +“ +ż +R +8 +ÿ +n“´8 +8 +ÿ +k“0 +pkhpn ` xkqϕpy ´ nqψpx ´ yqdy +` +ż +R +8 +ÿ +n“´8 +8 +ÿ +k“0 +pkhpn ´ xkqϕpy ´ nqψpx ´ yqdy +´ +ż +R +8 +ÿ +n“´8 +hpnqϕpy ´ nqψpx ´ yqdy +“ +ż +R +8 +ÿ +n“´8 +8 +ÿ +k“0 +pkhpnqϕpy ´ n ` xkqψpx ´ yqdy +` +ż +R +8 +ÿ +n“´8 +8 +ÿ +k“0 +pkhpnqϕpy ´ n ´ xkqψpx ´ yqdy +´ +ż +R +8 +ÿ +n“´8 +hpnqϕpy ´ nqψpx ´ yqdy +“ +ż +R +8 +ÿ +n“´8 +8 +ÿ +k“0 +pkhpnqϕpy ` xkqψpx ´ y ´ nqdy +` +ż +R +8 +ÿ +n“´8 +8 +ÿ +k“0 +pkhpnqϕpy ´ xkqψpx ´ y ´ nqdy +´ +ż +R +8 +ÿ +n“´8 +hpnqϕpyqψpx ´ y ´ nqdy +“ +ż +R +ˆ 8 +ÿ +k“0 +pk +` +ϕpy ` xkq ` ϕpy ` xkq +˘ +´ ϕpyq +˙ˆ +8 +ÿ +n“´8 +hpnqψpx ´ y ´ nq +˙ +dy +“ +ż +R +pˇL ˚ ϕqpyqpH ˚ ψqpx ´ yqdy +“ pˇL ˚ ϕq ˚ pH ˚ ψqpxq, +as desired. +□ +4.2. Multiplication of distributions. Before dealing with the case of Lévy op- +erators on R, we state the following counterintuitive result about S 1-product of +distributions. +Corollary 4.8. There is a strictly positive continuous function f on R, and a +nonzero tempered distribution g on R, such that f d g “ 0. + +LIOUVILLE’S THEOREMS FOR LÉVY OPERATORS +23 +Proof. Consider the one-dimensional Lévy operator L defined in (4.1), its convolu- +tion kernel ˇL, and the corresponding characteristic exponent Ψ. Thus, +Ψpξq “ 2 +8 +ÿ +k“0 +pk +` +1 ´ cospxkξq +˘ +“ +8 +ÿ +k“0 +2´k´1` +1 ´ cosp22k2ξq +˘ +is a Weierstrass-type nowhere differentiable function. Furthermore, let hpnq be the +doubly-infinite sequence from Theorem 4.2, and let +H “ +8 +ÿ +n“´8 +hpnqδn. +By Lemma 4.7, ˇL and H are S 1-convolvable, and ˇL f H “ 0. The desired result +essentially follows by the exchange formula: we have +Ψ d FH “ ´F ˇL d FH “ FpˇL f Hq “ 0. +Clearly, Ψ is a nonnegative continuous function, and by (1.2), Ψ is strictly positive +everywhere except 2πZ (recall that ˇL is concentrated on Z, and it has an atom at +1). On the other hand, FH is a periodic tempered distribution with period 2π, and +since hpnq is not a polynomial sequence, the support of FH is not contained in 2πZ. +Thus, to get the desired result, we only need to correct Ψ and ˇL so that Ψ is strictly +positive everywhere. +One way to do this would be to replace ˇL by ˇL ´ δ0 and repeat the construction +of hpnq. However, there is a simpler solution: it is sufficient to define +f “ Ψ ` ϕ, +g “ ψ ¨ FH, +where ψ P S is chosen in such a way that ψ “ 0 on 2πZ and ψ¨FH is not identically +zero, while ϕ P S is a nonnegative function such that ϕ ą 0 on 2πZ, but ϕ¨ψ “ 0 on +R. Indeed: f is then a strictly positive continuous function, g is a nonzero tempered +distribution, and +f d g “ Ψ d pψ ¨ FHq ` ϕ ¨ pψ ¨ FHq +“ ψ ¨ pΨ d FHq ` pϕ ¨ ψq ¨ FH “ 0, +as desired. +□ +4.3. Continuous case: proof of Theorem 4.1. The proof of Theorem 4.1 is very +similar to the argument used in the discrete case, in the proof of Theorem 4.2. +Thus, we omit some details and leave them to the interested reader. +We consider a Lévy operator similar to the one given in (4.1), but with an addi- +tional one-dimensional Laplace operator. That is, we consider +Lfpxq “ f 2pxq ` +8 +ÿ +k“0 +pk +` +fpx ` xkq ` fpx ´ xkq ´ 2fpxq +˘ +“ f 2pxq ` +8 +ÿ +k“0 +pk +` +fpx ` xkq ` fpx ´ xkq +˘ +´ fpxq, +where again pk “ 2´k´2, and xk is a rapidly increasing sequence to be spec- +ified later. +The construction of an L-harmonic function hpxq is very similar to +the construction of the sequence hpnq in Section 4.1, but for each n P Z we re- +place the single number hpnq by an appropriate compactly supported function hpxq, + +24 +TOMASZ GRZYWNY, MATEUSZ KWA´SNICKI +x P pn ´ 1 +2, n ` 1 +2q. Additionally, we specify the value of xk on the fly, but in any case +we will have +x0 “ 1, +xk`1 ě 2xk and xk ě 4k for k “ 0, 1, 2, . . . , +(4.6) +so that, in particular, Lemma 4.3 applies. +We consider a nonzero smooth even function h´1 with support contained in +p´ 1 +2, 1 +2q. +Then, in step m “ 0, 1, 2, . . ., we define hm by appropriately modifying +hm´1 on the intervals |x| P pm ` xm ´ 1 +2, m ` xm ` 1 +2q in such a way that hm is smooth, +even, and Lhmpxq “ 0 when |x| P pm ´ 1 +2, m ` 1 +2q. +Let us describe more precisely step m “ 0, 1, 2, . . . of the construction. Suppose +that hm´1 and x0, x1, . . . , xm´1 have already been defined. For z P p´ 1 +2, 1 +2q we define +hmpxq “ hm´1pxq ` ϕmpx ´ m ´ xmq ` ϕmp´x ´ m ´ xmq +“ h´1pxq ` +m +ÿ +j“0 +` +ϕjpx ´ j ´ xjq ` ϕjp´x ´ j ´ xjq +˘ +, +where xm is specified below, +ϕmpzq “ ´Lhm´1pz ` mq +pm +“ ´Lhm´1p´z ´ mq +pm +“ h2 +m´1pz ` mq ` +m´1 +ÿ +k“0 +pk +` +hm´1pz ` m ` xkq ` hm´1pz ` m ´ xkq +˘ +´ hm´1pz ` mq +if m “ 1, 2, . . ., and +ϕ0pzq “ ´Lh´1pzq +2p0 +“ ´2h2 +´1pzq ` 2h´1pzq +if m “ 0. For convenience, we set ϕmpzq “ 0 when |z| ě 1 +2. Then ϕm is a smooth +function with compact support in p´ 1 +2, 1 +2q. +Note that in the above calculation of +Lhm´1pz ` mq we truncated the series at k “ m ´ 1. This is because, as we now +prove, all terms corresponding to k ě m are zero. Indeed: by construction, we have +hm´1pxq “ 0 when |x| ě pm ´ 1q ` xm´1 ` 1 +2, and if k ě m, then, by (4.6), +|m ˘ xk| ě xk ´ m ě xm ´ m ě 1 +2xm ` xm´1 ´ m ě m ` xm´1. +Thus, hm´1pz ` m ˘ xkq “ 0, as claimed. In other words, the values of xk for k ě m +are not needed in order to evaluate ϕmpzq “ ´p´1 +m Lhm´1pz ` mq, as long as condi- +tion (4.6) is satisfied, and this allows us to specify the value of xm only in step m. +If m ą 0, then we choose xm to be an integer large enough, so that xm ě 2xm´1, +xm ě 4m, xm ě 22m2, and +log xm ě supt|ϕmpzq| ` |ϕ2 +mpzq| : z P p´ 1 +2, 1 +2qu. +(4.7) +If m “ 0, we simply let x0 “ 1. +We now define +hpxq “ lim +mÑ8 hmpxq +“ h´1pxq ` +8 +ÿ +m“0 +` +ϕmpx ´ m ´ xmq ` ϕmp´x ´ m ´ xmq +˘ +. + +LIOUVILLE’S THEOREMS FOR LÉVY OPERATORS +25 +We stress that h is a smooth function with a very sparse support; namely, we have +hpxq “ +$ +’ +’ +’ +’ +& +’ +’ +’ +’ +% +ϕ0pxq +if x P p´ 1 +2, 1 +2q, +ϕmpx ´ m ´ xmq +if x P pm ` xm ´ 1 +2, m ` xm ` 1 +2q with m “ 0, 1, . . . , +ϕmp´x ´ m ´ xmq +if x P p´m ´ xm ´ 1 +2, ´m ´ xm ` 1 +2q with m “ 0, 1, . . ., +0 +otherwise. +We now follow closely the arguments used in the discrete case in Sections 4.1 +and 4.2. +Lemma 4.9. With the above definitions, we have +Lhpxq “ 0 +for every x P R. +Proof. The argument is almost exactly the same as in the proof of Lemma 4.4, +except that we replace n by n ` z, with z P p´ 1 +2, 1 +2q, and we need to consider z “ 1 +2 +separately. +By construction, hpn` 1 +2q “ h2pn` 1 +2q “ 0 for every n P Z. Thus, Lhpn` 1 +2q “ 0, and +we only need to show that Lhpz ` nq “ 0 when n P Z and z P p´ 1 +2, 1 +2q. By Lemma 4.3, +for every k “ 0, 1, 2, . . . we have +|n ˘ xk| ‰ m ` xm +if m ą |n|, +and thus hpz ` n ˘ xkq “ hmpz ` n ˘ xkq, where m “ |n|. Therefore, +Lhpz ` nq “ h2pz ` nq ` +8 +ÿ +k“0 +pk +` +hpz ` n ` xkq ` hpz ` n ´ xkq +˘ +´ hpz ` nq +“ h2 +mpz ` nq ` +8 +ÿ +k“0 +pk +` +hmpz ` n ` xkq ` hmpz ` n ´ xkq +˘ +´ hmpz ` nq +“ Lhmpz ` nq. +We claim that Lhmpz ` nq “ 0. If n “ m “ 0, then +Lh0pzq “ h2 +0p0q ` p0 +` +h0pz ` x0q ` h0pz ´ x0q +˘ +´ h0pzq +“ h2 +´1pzq ` 2p0ϕ0pzq ´ h´1pzq “ 0 +by the definitions of h0 and ϕ0. Suppose now that n “ m ą 0. By the definition of +hm, we have +Lhmpz ` mq “ Lhm´1pz ` mq ` Lϕmpz ´ xmq ` Lϕmpz ` 2m ` xmq. +Note that |z ´ xm| ą 1 +2, |z ´ xm ˘ xk| ą 1 +2 if k ‰ m, and also |z ´ 2xm| ą 1 +2. Thus, +Lϕmpz ´ xmq “ ϕ2 +mpz ´ xmq ´ ϕmpz ´ xmq +` +8 +ÿ +k“0 +pk +` +ϕmpz ´ xm ` xkq ` ϕmpz ´ xm ´ xkq +˘ +“ pmϕmpzq “ ´Lhm´1pm ` zq. + +26 +TOMASZ GRZYWNY, MATEUSZ KWA´SNICKI +Similarly, |z ` 2m ` xm| ą 1 +2 and |z ` 2m ` xm ˘ xk| ą 1 +2 for every k, and hence +Lϕmpz ` 2m ` xmq “ ϕ2 +mpz ` 2m ` xmq ´ ϕmpz ` 2m ` xmq +` +8 +ÿ +k“0 +pk +` +ϕmpz ` 2m ` xm ` xkq ` ϕmpz ` 2m ` xm ´ xkq +˘ +“ 0. +Therefore, +Lhmpz ` mq “ Lhm´1pz ` mq ´ Lhm´1pz ` mq “ 0. +By a similar argument (or by symmetry), we also have Lhmp´z ´ mq “ 0, and our +claim is thus proved. +□ +Lemma 4.10. If xk grows sufficiently fast, then, with the above definitions, for +every ε ą 0 there is a constant Cε such that +pm|ϕmpzq| ď |h2 +m´1pz ` mq| ` +8 +ÿ +k“0 +pk +` +|hm´1pz ` m ` xkq| ` |hm´1px ´ xkq| +˘ +` |hm´1pz ` mq| ď Cεp1 ` mqε +(4.8) +for m “ 0, 1, 2, . . . and z P p´ 1 +2, 1 +2q. +Proof. The proof is actually simpler than the proof of Lemma 4.5, due to flexibility +in the choice of xm. The first inequality in (4.8) follows by the definition of ϕm, and +so we are left with the proof of the other inequality. +Let ε ą 0. By (4.7), there is a constant Cε such that +supt|ϕmpzq| ` |ϕ2 +mpzq| : z P p´ 1 +2, 1 +2qu ď Cεp1 ` xmqε +(4.9) +for m “ 0, 1, 2, . . . +Fix m “ 0, 1, 2, . . . If hm´1pz ` mq ‰ 0 for some z P p´ 1 +2, 1 +2q, then m “ j ` xj and +hm´1pz ` mq “ ϕjpzq for some j ă m. By (4.9), we obtain +|hm´1pz ` mq| ` |h2 +m´1pz ` mq| “ |ϕjpzq| ` |ϕ2 +jpzq| ď Cεp1 ` xjqε ď Cεp1 ` mqε. +Similarly, if hm´1pz`m˘xkq ‰ 0, then |m˘xk| “ j`xj and hm´1pz`m˘xkq “ ϕjp˘zq +for some j ă m, so that again +|hm´1pz ` m ˘ xkq| “ |ϕjp˘zq| ď Cεp1 ` xjqε. +By Lemma 4.3 we find that m ě 1 +2xj, and hence +|hm´1pz ` m ˘ xkq| ď Cεp1 ` 2mqε ď 2εCεp1 ` mqε. +It follows that +|h2 +m´1pz ` mq| ` +8 +ÿ +k“0 +pk +` +|hm´1pz ` m ` xkq| ` |hm´1pz ` m ´ xkq| +˘ +` |hm´1pz ` mq| +ď Cεp1 ` mqε ` 2εCεp1 ` mqε ď 21`εCεp1 ` mqε. +The proof is complete. +□ +Lemma 4.11. With the above definitions, for every ε ą 0 we have +lim +|x|Ñ8 +|hpxq| ` |h2pxq| +|x|ε +“ 0, + +LIOUVILLE’S THEOREMS FOR LÉVY OPERATORS +27 +and +lim +|x|Ñ8 +1 +|x|ε +8 +ÿ +k“0 +pk +` +|hpx ` xkq| ` |hpx ´ xkq| +˘ +“ 0. +Proof. The proof is very similar to the proof of Lemma 4.6. By definition, if n P Z, +z P p´ 1 +2, 1 +2q, |n| ě 2 and hpz ` nq ‰ 0 or h2pz ` nq ‰ 0, then |n| “ m ` xm for some +m “ 1, 2, . . ., and thus, by (4.7), +|hpz ` nq| ` |h2pz ` nq| +|z ` n|ε +“ |ϕmpzq| ` |ϕ2 +mpzq| +|z ` n|ε +ď log xm +|n|ε +ď log xm +xεm +. +The right-hand side clearly converges to zero as m Ñ 8, and the first part of the +lemma follows. +To prove the other one, we consider m “ 0, 1, 2, . . . and z P p´ 1 +2, 1 +2q, and we recall +that, as in the proof of Lemma 4.9, we have +8 +ÿ +k“0 +pk +` +|hpz ` m ` xkq| ` |hpz ` m ´ xkq| +˘ +“ +8 +ÿ +k“0 +pk +` +|hmpz ` m ` xkq| ` |hmpz ` m ´ xkq| +˘ +“ pm|ϕmpzq| ` +8 +ÿ +k“0 +pkp|hm´1pz ` m ` xkq| ` |hm´1pz ` m ´ xkq| +˘ +. +By Lemma 4.10, the right-hand side does not exceed Cε{2p1 ` mqε{2, and thus +lim +mÑ8 +1 +mε +8 +ÿ +k“0 +pk +` +|hpz ` m ` xkq| ` |hpz ` m ´ xkq| +˘ +“ 0, +uniformly with respect to z P p´ 1 +2, 1 +2q. By a similar argument (or by symmetry), +lim +mÑ8 +1 +mε +8 +ÿ +k“0 +pk +` +|hp´z ´ m ` xkq| ` |hp´z ´ m ´ xkq| +˘ +“ 0 +uniformly with respect to z P p´ 1 +2, 1 +2q, and the proof is complete. +□ +Lemma 4.12. With the above definitions, the function h corresponds to a tempered +distribution, which is S 1-convolvable with the convolution kernel ˇL of the Lévy +operator L, and we have ˇL f h “ 0. +Proof. The argument is virtually the same as in the proof of Lemma 4.7. +By +Lemma 4.11, for a given ε ą 0, there is a constant C1 such that +|h2pzq| ` +8 +ÿ +k“0 +pk +` +|hpz ` xkq| ` |hpz ´ xkq| +˘ +` |hpzq| ď C1p1 ` |z|qε. +Furthermore, if ϕ, ψ P S and x P R is fixed, then there is a constant C2 such that +|ϕ| ˚ |ψ|px ´ zq ď C2p1 ` |z|q´2´ε. +It follows that +ż +R +ż +R +ˆ +|h2pzq| ` +8 +ÿ +k“0 +pk +` +|hpz ` xkq| ` |hpz ´ xkq| +˘ +´ hpzq +˙ +|ϕpy ´ zq||ψpx ´ yq|dzdy ă 8. + +28 +TOMASZ GRZYWNY, MATEUSZ KWA´SNICKI +Thus, we may apply Lemma 4.9, Fubini’s theorem and integration by parts to find +that +0 “ +ż +R +ż +R +ˆ +h2pzq ` +8 +ÿ +k“0 +pk +` +hpz ` xkq ` hpz ´ xkq +˘ +´ hpzq +˙ +ϕpy ´ zqψpx ´ yqdzdy +“ +ż +R +ż +R +h2pzqϕpy ´ zqψpx ´ yqdzdy +` +ż +R +ż +R +8 +ÿ +k“0 +pkhpz ` xkqϕpy ´ zqψpx ´ yqdzdy +` +ż +R +ż +R +8 +ÿ +k“0 +pkhpz ´ xkqϕpy ´ zqψpx ´ yqdzdy +´ +ż +R +ż +R +hpzqϕpy ´ zqψpx ´ yqdzdy +“ +ż +R +ż +R +hpzqϕ2py ´ zqψpx ´ yqdzdy +` +ż +R +ż +R +8 +ÿ +k“0 +pkhpzqϕpy ´ z ` xkqψpx ´ yqdzdy +` +ż +R +ż +R +8 +ÿ +k“0 +pkhpzqϕpy ´ z ´ xkqψpx ´ yqdzdy +´ +ż +R +ż +R +hpzqϕpy ´ zqψpx ´ yqdzdy +“ +ż +R +ż +R +hpzqϕ2pyqψpx ´ y ´ zqdzdy +` +ż +R +ż +R +8 +ÿ +k“0 +pkhpzqϕpy ` xkqψpx ´ y ´ zqdzdy +` +ż +R +ż +R +8 +ÿ +k“0 +pkhpzqϕpy ´ xkqψpx ´ y ´ zqdzdy +´ +ż +R +ż +R +hpzqϕpyqψpx ´ y ´ zqdzdy +“ +ż +R +ˆ +ϕ2pyq ` +8 +ÿ +k“0 +pk +` +ϕpy ` xkq ` ϕpy ` xkq +˘ +´ ϕpyq +˙ˆż +R +hpzqψpx ´ y ´ zqdz +˙ +dy +“ +ż +R +pˇL ˚ ϕqpyqph ˚ ψqpx ´ yqdy +“ pˇL ˚ ϕq ˚ ph ˚ ψqpxq, +and the proof is complete. +□ +Theorem 4.1 follows directly from the above series of lemmas. +5. Signed harmonic functions +In this final section of the article we prove various variants of Liouville’s theorem +for signed polynomially bounded functions using Fourier transform approach. We + +LIOUVILLE’S THEOREMS FOR LÉVY OPERATORS +29 +begin with an abstract result in Theorem 5.1, and then, by choosing an appropriate +Wiener-type algebra W , we obtain specific Liouville’s theorems as corollaries. +5.1. General result. Recall that according to Definition 1.5, an algebra W of con- +tinuous functions on Rd is a Wiener-type algebra if every Φ P W corresponds to +a tempered distribution, ϕΦ P W whenever ϕ P S and Φ P W , and the following +variant of Wiener’s 1{f theorem holds: +if K Ď Rd is a compact set, Φ P W and Φpξq ‰ 0 for every ξ P K, +then there is ˜Φ P W such that Φpξq˜Φpξq “ 1 for every ξ P K. +A tempered distribution Ψ is said to belong to W locally on an open set U if for +every compact set K Ď U there is a distribution Φ P W such that Ψ “ Φ in a +neighbourhood of K. In particular, in this case the restriction of Ψ to U is given by +a continuous function, and if ϕ P S has a compact support contained in U, then ϕΨ +is an element of W . +Using the notation introduced in Section 2.2, Definition 1.6 reads as follows. A +tempered distribution H is said to act on W if for every Φ, Ψ P W we have: +pH d Φq d Ψ “ H d pΦΨq. +The main result of this section is the following extension of Theorem 1.7. Note that +we do not require ˇL to be the convolution kernel of a Lévy operator. +Theorem 5.1 (Liouville’s theorem factory). Let W be an Wiener-type algebra of +continuous functions on Rd, and let ˇL be a tempered distribution. Suppose that +F ˇL belongs to W locally on an open set U, and assume that F ˇLpξq ‰ 0 for every +ξ P U. Let h be a tempered distribution such that Fh acts on W , and such that +ˇL f h “ 0. +Then the spectrum of h is contained in RdzU. +In other words: the +restriction of Fh to U is zero. +In particular, if U “ Rdzt0u, then h is a polynomial. +Proof. Fix a compact subset K of U. +Suppose that the spectrum of ϕ P S is a +compact subset of U and Fϕpξq ‰ 0 for every ξ P K. Define f “ ´ˇL ˚ ϕ, Ψ “ ´F ˇL, +Φ “ Ff and H “ Fh. Observe that +Φ “ Ff “ ´FpˇL ˚ ϕq “ ´Fϕ ¨ F ˇL “ Fϕ ¨ Ψ. +By assumption, Fϕ P S , the support of Fϕ is a compact subset of U, and Ψ +belongs to W locally on U. Thus, Φ P W . Furthermore, +0 “ pˇL f hq ˚ ϕ “ h f pˇL ˚ ϕq “ h f f, +and hence, by the Fourier exchange formula, +0 “ Fph f fq “ H d Φ. +On the other hand, Φpξq “ FϕpξqΨpξq ‰ 0 for every ξ P K, and therefore there is +˜Φ P W such that Φpξq˜Φpξq “ 1 for every ξ P K. Since H acts on W , we conclude that +0 “ pH d Φq d ˜Φ “ H d pΦ˜Φq. +Recall that Φpξq˜Φpξq “ 1 for every ξ P K. By definition, multiplication of distribu- +tions is a local operation. Hence, in the interior of K, we have +H “ H d 1 “ H d pΦ˜Φq “ 0. +Since K is an arbitrary subset of U, we conclude that H “ 0 on U, as desired. + +30 +TOMASZ GRZYWNY, MATEUSZ KWA´SNICKI +If U “ Rdzt0u, then H “ Fh is supported in RdzU “ t0u, and hence h is necessar- +ily a polynomial (see Sections 2.6 and 6.3.2 in [25]). +□ +5.2. Operators with smooth symbols. It is straightforward to verify that W “ S , +the Schwartz class of rapidly decaying functions, is a Wiener-type algebra. Clearly, +Ψ belongs to W locally on U if and only if Ψ is smooth on U. Finally, if Φ, Ψ P W and +H is an arbitrary tempered distribution, we have pΦΨq¨H “ Φ¨pΨ¨Hq, and so every +tempered distribution acts on W . This leads to the following statement, which is a +minor extension of Theorem 3.2 in [3]. +Corollary 5.2 (Liouville’s theorem for operators with smooth symbols). Let ˇL be a +tempered distribution. Suppose that on an open set U, F ˇL corresponds to a smooth +function without zeroes. Let h be a tempered distribution such that ˇLfh “ 0. Then +the spectrum of h is contained in RdzU. +In particular, if U “ Rdzt0u, then h is a polynomial. +Example 5.3. Let α P p0, 2q and let L “ ´p´∆qα{2 be the fractional Laplace opera- +tor. If ˇL denotes the corresponding convolution kernel, then F ˇLpξq “ ´|ξ|α. Since +F ˇL is smooth in Rdzt0u, we find that the only polynomially bounded L-harmonic +functions (in the sense of tempered distributions) are polynomials. +However, Lh is not well-defined if h is a polynomial of degree rαs or higher, and so +all L-harmonic functions are constant when α P p0, 1s, and all L-harmonic functions +are affine when α P p1, 2q. +This is exactly the main result of [12] (Theorem 1.1 therein) and of [7] (Theo- +rem 1.3 therein). +5.3. Bounded L-harmonic functions. Let W be the Wiener algebra: the class +of Fourier transforms of integrable functions. It is straightforward to see that W +satisfies the first two conditions of Definition 1.5, and the last one is a variant of +the classical Wiener’s 1{f theorem. We refer to the Division Lemma in Section 3 +in [8] for the proof in dimension one, and to Section 5.4 below for a more general +statement. +If ˇL is an integrable distribution, r ą 0 and ϕ P S satisfies Fϕpξq “ 1 when +|ξ| ă r, then ˇL ˚ ϕ is an integrable function with Fourier transform Fϕ ¨ F ˇL in +W . Since r is arbitrary, we see that F ˇL belongs to W locally on Rd. In particular, +Fourier symbols of Lévy operators belong to W locally on Rd. +Observe that if h is a bounded distribution and Φ, Ψ P W , then there are inte- +grable functions f, g such that Φ “ Ff and Ψ “ Fg. Since hfpf ˚gq “ hfpf fgq “ +ph f fq f g, by the exchange formula we find that Fh d pΦΨq “ pFh d Φq d Ψ. Thus, +Fourier transforms of bounded distributions act on W . +The above observations immediately lead to the following minor extension of +the general Liouville’s theorem for bounded L-harmonic functions given in Theo- +rem 1.1 in [1] and in Theorem 4.4 in [3]. +Corollary 5.4 (Liouville’s theorem for bounded functions). Let ˇL be an integrable +distribution. Suppose that F ˇL, which is necessarily a continuous function, has no +zeroes in an open set U. Let h be a bounded distribution such that ˇL f h “ 0. Then +the spectrum of h is contained in RdzU. +In particular, if U “ Rdzt0u, then h is a constant. + +LIOUVILLE’S THEOREMS FOR LÉVY OPERATORS +31 +5.4. Operators with finite generalised moments. We define the following class +of Wiener-type algebras, parameterised by auxiliary functions Y . Examples of ad- +missible functions Y are discussed at the end of this section. Here we remark that +a typical choice is Y pxq « |x|α, where α ą 0 is a parameter. +Definition 5.5. Suppose that Y is a nonnegative function on Rd with the following +properties: +(a) we have 1 ` Y px ` yq ď p1 ` Y pxqqp1 ` Y pyqq for every x, y P Rd; +(b) for some positive constants c, q we have Y pxq ď cp1 ` |x|qq. +Define W 8 +Y to be the class of Fourier transforms of integrable functions f such that +also Y f is integrable. +Our main goal in this section is to prove that W 8 +Y +is a Wiener-type algebra ac- +cording to Definition 1.5. +Lemma 5.6. The set W 8 +Y +defined in Definition 5.5 is an algebra of continuous +functions which satisfies conditions (a) and (b) of Definition 1.5. +Proof. Clearly, W 8 +Y is a linear space. If Φ P W 8 +Y , then Φ is the Fourier transform of +an integrable function f, and hence Φ is a bounded continuous function. Thus, W 8 +Y +is a class of continuous functions, and every Φ P W 8 +Y +corresponds to a tempered +distribution. +We claim that W 8 +Y is indeed an algebra of functions. If Φ, Ψ P W 8 +Y , then Φ “ Ff +and Ψ “ Fg for some integrable functions f, g such that also Y f, Y g are inte- +grable. It follows that h “ f ˚ g is an integrable function, and by condition (a) in +Definition 5.5, +ż +Rd |p1 ` Y pxqqhpxq|dx “ +ż +Rd +ˇˇˇˇp1 ` Y pxqq +ż +Rd fpyqgpx ´ yqdy +ˇˇˇˇdx +ď +ż +Rd +ż +Rdp1 ` Y pxqq|fpyq|gpx ´ yq|dydx +“ +ż +Rd +ż +Rdp1 ` Y py ` zqq|fpyq||gpzq|dydz +ď +ż +Rdp1 ` Y pyqq|fpyq|dy ¨ +ż +Rdp1 ` Y pzqq|gpzq|dz ă 8, +(5.1) +that is, also Y h is integrable. Since ΦΨ “ Ff ¨ Fg “ Fpf ˚ gq “ Fh, we conclude +that ΦΨ P W 8 +Y , as desired. +We now show that W 8 +Y +contains S . +Indeed: if ϕ P S and ψ “ F ´1ϕ, then +ψ P S , and so, in particular, ψ is integrable. Furthermore, by condition (b) in Defi- +nition 5.5, p1 ` |x|qd`1Y pxqψpxq is a bounded function of x P Rd, and so in particular +Y ψ is integrable. Thus, ϕ “ Fψ indeed belongs to W 8 +Y . +It remains to observe that if ϕ P S and Φ P W 8 +Y , then ϕ P W 8 +Y , and therefore +ϕΦ P W 8 +Y . +□ +In order to prove that W 8 +Y is a Wiener-type algebra, we only need to verify that +W 8 +Y satisfies condition (c) of Definition 1.5, that is, a variant of Wiener’s 1{f theo- +rem holds in W 8 +Y . While this result is known (see [4] for further discussion and +references), we provide a complete proof in order to prepare the reader for a +similar, but slightly more involved argument in the next section, in the proof of +Lemma 5.14. + +32 +TOMASZ GRZYWNY, MATEUSZ KWA´SNICKI +Lemma 5.7. Let W 8 +Y be the set defined in Definition 5.5. Suppose that K Ď Rd is +a compact set, Φ P W 8 +Y +and Φpξq ‰ 0 for every ξ P K. Then there is ˜Φ P W 8 +Y +such +that Φpξq˜Φpξq “ 1 for every ξ P K. +Proof. We follow the proof of the classical Wiener’s 1{f lemma given in [17]. Let +us denote by } ¨ }p the usual norm in LppRdq, where p P r1, 8s. We begin with the +following elementary observation. +Clearly, if Φ “ Ff and f is integrable, then +}Φ}8 ď }f}1. +Conversely, for every k “ 1, 2, . . . there is a constant cd,k such that if Φ is smooth, +both Φ and ∆kΦ are integrable, and f “ F ´1Φ, then +suptp1 ` |x|2kq|fpxq| : x P Rdu ď cd,k}Φ ` p´∆qkΦ}1. +In particular, if 2k ą d, we find that f is integrable, while if 2k ą d ` q with q as in +condition (b) in Definition 5.5, then Y f is integrable. It follows that if 2k ą d ` q, +then there is a constant cd,k such that whenever Φ is smooth, and Φ and ∆kΦ are +integrable, then Φ P W 8 +Y and +}p1 ` Y qf}1 ď cd,k}Φ ` p´∆qkΦ}1, +(5.2) +where f “ F ´1Φ. +We return to the proof of the lemma. Suppose that Φ P W 8 +Y , that is, Φ “ Ff for +an integrable function f such that also Y f is integrable. Suppose furthermore that +K is a compact set and Φpξq ‰ 0 for every ξ P K. Let ε ą 0 be small enough, so +that |Φpξq| ą 3ε for every ξ in some bounded neighbourhood U of K. With no loss +of generality we assume that ε ă 1 +2. Choose ϕ P S so that the support of ϕ is a +compact subset of U and ϕpξq “ 1 for ξ P K. Our goal is to prove that +˜Φpξq “ ϕpξq +Φpξq +is an element of W 8 +Y ; here, of course, ˜Φpξq “ 0 for ξ P RdzU. Once this is proved, +we have Φpξq˜Φpξq “ ϕpξq “ 1 for every ξ P K, and so ˜Φ has the desired property. +Define gpxq “ fpxq when |x| ă r and gpxq “ 0 otherwise, where r is large enough, +so that +}f ´ g}1 ` }Y f ´ Y g}1 ă ε. +Clearly, g and Y g are integrable, and therefore Ψ “ Fg is an element of W 8 +Y . +Furthermore, +}Φ ´ Ψ}8 ď }f ´ g}1 ă ε. +In particular, for ξ P U we have +|Ψpξq| ě |Φpξq| ´ |Φpξq ´ Ψpξq| ą 3ε ´ ε “ 2ε, +and +ˇˇˇˇ +Ψpξq ´ Φpξq +Ψpξq +ˇˇˇˇ ă ε +2ε “ 1 +2 . +It follows that for ξ P U, +1 +Φpξq “ +8 +ÿ +n“0 +pΨpξq ´ Φpξqqn +pΨpξqqn`1 +, + +LIOUVILLE’S THEOREMS FOR LÉVY OPERATORS +33 +and therefore for every ξ P Rd, +˜Φpξq “ ϕpξq +Φpξq “ +8 +ÿ +n“0 +pΨpξq ´ Φpξqqn +ϕpξq +pΨpξqqn`1 . +(5.3) +We study the terms pΨ ´ Φqn and ϕ{Ψn`1 separately. +Recall that Ψ ´ Φ “ Fpg ´ fq and +}p1 ` Y qpg ´ fq}1 “ }g ´ f}1 ` }Y g ´ Y f}1 ă ε. +If pg ´ fq˚n denotes the n-fold convolution of g ´ f, then Fpg ´ fq˚n “ pΨ ´ Φqn. +Furthermore, using condition (a) in Definition 5.5 as in (5.1), we find that +››p1 ` Y qpg ´ fq˚n›› +1 ď +` +}p1 ` Y qpg ´ fq}1 +˘n ă εn. +(5.4) +This is the desired bound for pΨ ´ Φqn, and we turn to the estimate of ϕ{Ψn`1. +Since Ψ “ Fg and g has compact support, Ψ is smooth in Rd. +Additionally, +|Ψpξq| ě 2ε ą 0 for ξ P U, and ϕpξq “ 0 for ξ outside a compact subset of U. It +follows that ϕ{Ψn`1 is smooth on Rd, and equal to zero in RdzU. +In particular, +by (5.2), we find that ϕ{Ψn`1 P W 8 +Y , and if hn “ F ´1pϕ{Ψn`1q, then +››p1 ` Y qhn +›› +1 ď cd,k +››ϕ{Ψn`1 ` p´∆qkpϕ{Ψn`1q +›› +1, +where k is a fixed sufficiently large positive integer and cd,k is an appropriate con- +stant. By applying the product rule to p´∆qkpϕ{Ψn`1q, we obtain a fixed number of +terms, each of which is a product of: the derivative of ϕ of some order j, where +0 ď j ď 2k; a finite number of derivatives of Ψ of total order 2k ´ j; Ψ´n´1´i, +where 0 ď i ď 2k ´ j; and a coefficient, which is an appropriate polynomial of +n of degree at most 2k. Furthermore, |Ψpξq| ě 2ε for ξ P U, and 2ε ă 1. Thus, +Ψ´n´1´ipξq ď p2εq´n´1´2k when ξ P U and 0 ď i ď 2k. +It follows that there is a +constant cd,k,ϕ,Ψ such that +››p1 ` Y qhn +›› +1 ď cd,k,ϕ,Ψ|U|p1 ` n2kqp2εq´n´1´2k. +(5.5) +This is the desired estimate for ϕ{Ψn`1. +We combine (5.4) and (5.5) as in (5.1): +››p1 ` Y qpg ´ fq˚n ˚ hn +›› +1 ď +››p1 ` Y qpg ´ fq˚n›› +1 ¨ +››p1 ` Y qhn +›› +1 +ď cd,k,ϕ,Ψ|U|p1 ` nkqp2εq´1´2k ¨ 2´n. +By Fubini’s theorem, we obtain +››››p1 ` Y q +8 +ÿ +n“0 +pg ´ fq˚n ˚ hn +›››› +1 +ă 8. +In particular, the series +˜f “ +8 +ÿ +n“0 +pg ´ fq˚n ˚ hn + +34 +TOMASZ GRZYWNY, MATEUSZ KWA´SNICKI +defines an integrable function such that also Y ˜f is integrable, and again by Fubini’s +theorem we find that +F ˜f “ +8 +ÿ +n“0 +F +` +pg ´ fq˚n ˚ hn +˘ +“ +8 +ÿ +n“0 +F +` +pg ´ fq˚n˘ +Fhn +“ +8 +ÿ +n“0 +pΨ ´ Φqnpϕ{Ψn`1q. +Thus, by (5.3), F ˜f “ ˜Φ, and therefore ˜Φ P W 8 +Y , as desired. +□ +Lemmas 5.6 and 5.7 immediately lead to the following result. +Proposition 5.8. The set W 8 +Y defined in Definition 5.5 is a Wiener-type algebra. +Suppose that Y satisfies the conditions of Definition 5.5, ˇY pxq “ Y p´xq, and ˇL is +an integrable distribution such that ˇLf ˇY is well-defined. Then for every ϕ P S the +integrable function ˇL˚ϕ is convolvable with ˇY , and so, in particular, }pˇL˚ϕq¨Y }1 ă 8. +Thus, Fϕ ¨ F ˇL “ FpˇL ˚ ϕq P W 8 +Y . By choosing ϕ such that Fϕpξq “ 1 for ξ in a +given compact set, we find that F ˇL belongs to W 8 +Y locally on Rd. +We claim that if h is a function on Rd such that h{p1 ` ˇY q is bounded, then h cor- +responds to a tempered distribution and Fh acts on W 8 +Y . Indeed: by condition (b) +in Definition 5.5, h is bounded by some polynomial and hence it corresponds to a +tempered distribution. Furthermore, if Φ, Ψ P W 8 +Y , then Φ “ Ff and Ψ “ Fg for +some integrable functions f, g such that also Y f and Y g are integrable. Thus, by +condition (a) in Definition 5.5 and Fubini’s theorem, we find that +|h| ˚ |f| ˚ |g|pxq “ +ż +Rd +ż +Rd hpx ´ y ´ zqfpyqgpzqdydz +ď p1 ` Y p´xqq +ż +Rd +ż +Rd +hpx ´ y ´ zq +1 ` Y p´x ` y ` zq p1 ` Y pyqqfpyqp1 ` Y pzqqgpzqdydz +ď p1 ` ˇY pxqq}h{p1 ` ˇY q}8}p1 ` Y qg}1}p1 ` Y qh}1. +Therefore, by Fubini’s theorem, we have +h ˚ pf ˚ gq “ ph ˚ fq ˚ g, +and in fact, due to condition (b) in Definition 5.5, +h f pf ˚ gq “ ph f fq f g. +Applying the exchange formula, we conclude that +FH d pΦ ¨ Ψq “ pFH d Φq d Ψ, +which completes the proof of our claim. +As an immediate corollary of Theorem 5.1 and Proposition 5.8, as well as the two +properties discussed above, and after exchanging the roles of Y and ˇY , we obtain +the following variant of Liouville’s theorem. +Corollary 5.9 (Liouville’s theorem under generalised moment condition). Suppose +that Y satisfies the conditions of Definition 5.5. Let ˇL be an integrable distribution +which is convolvable with Y . Suppose that F ˇL, which is necessarily a continuous + +LIOUVILLE’S THEOREMS FOR LÉVY OPERATORS +35 +function, has no zeroes in an open set U. Let h be a function such that h{p1 ` Y q is +bounded on Rd and ˇL f h “ 0. Then the spectrum of h is contained in RdzU. +In particular, if U “ Rdzt0u, then h is a polynomial. +We conclude this section with examples of admissible functions Y . +Example 5.10. If α ě 0, then Y pxq “ p1 ` |x|qα ´ 1 satisfies conditions (a) and (b) +in Definition 5.5. Indeed, +p1 ` Y pxqqp1 ` Y pyqq “ p1 ` |x|qαp1 ` |y|qα +ě p1 ` |x| ` |y|qα ě p1 ` |x ` y|qα “ 1 ` Y px ` yq. +Thus, if L is a Lévy operator which is not concentrated on a proper closed subgroup +of Rd, and the Lévy measure ν of L satisfies +ż +RdzB +|y|ανpdyq ă 8, +then every L-harmonic function h (in the sense of tempered distributions) such that +p1 ` |x|q´αhpxq is a bounded function of x P Rd is necessarily a polynomial. When +α ă 1, then it follows that h is in fact constant. For α “ 0 we thus recover Liouville’s +theorem for bounded L-harmonic functions given in Corollary 5.4. +Example 5.11. If β ě 0, then it is easy to see that Y pxq “ plogpe2 ` |x|qqβ satisfies +conditions (a) and (b) in Definition 5.5. Indeed: we have +Y pxqY pyq “ +` +logpe2 ` |x|q logpe2 ` |y|q +˘β +ě +` +logpe2q logpe2 ` maxt|x|, |y|uq +˘β +ě +` +2 logpe2 ` 1 +2p|x| ` |y|qq +˘β +“ +` +logpe4 ` e2p|x| ` |y|q ` 1 +4p|x| ` |y|q2q +˘β +ě +` +logpe2 ` |x ` y|q +˘β “ Y px ` yq, +and hence 1 ` Y px ` yq ď 1 ` Y pxqY pyq ď p1 ` Y pxqqp1 ` Y pyqq. +It follows that if L is a Lévy operator which is not concentrated on a proper closed +subgroup of Rd, and the Lévy measure ν of L satisfies +ż +RdzB +plog |y|qβνpdyq ă 8, +then every L-harmonic function h (in the sense of tempered distributions) such that +plogpe ` |x|qq´βhpxq is a bounded function of x P Rd is necessarily constant. +5.5. Harmonic functions with finite generalised negative moments. As in the +previous section, we define a class of Wiener-type algebras, again parameterised +by auxiliary functions Y . Once again we discuss examples of admissible functions +Y at the end of this section, and here we remark that a typical example is Y pxq “ +cd,αp1`|x|q´d´α, where α ą 0 is a parameter and cd,α is a sufficiently small constant. +Definition 5.12. Suppose that Y is an integrable function on Rd with the following +properties: + +36 +TOMASZ GRZYWNY, MATEUSZ KWA´SNICKI +(a) we have Y ˚ Y pxq ď Y pxq for every x P Rd, and if Yrpxq “ Y pxq when |x| ě r +and Yrpxq “ 0 otherwise, then +lim +rÑ8 sup +"Yr ˚ Yrpxq +Y pxq +: x P Rd +* +“ 0; +(b) for some positive constants c, q we have Y pxq ě cp1 ` |x|q´q for every x P Rd. +Define W 1 +Y to be the class of Fourier transforms of integrable functions f such that +f{Y is bounded. +We remark that if Y ˚ Y ď cY for some constant c, then we may replace Y by +c´1Y to get a function which satisfies Y ˚ Y ď Y . +We also note that condition +Y ˚ Y pxq ď cY pxq is called direct jump property in [15], and we refer to that paper +for further discussion and references. +Below we prove that W 1 +Y is a Wiener-type algebra according to Definition 1.5. +The argument is similar to the one applied in the previous section. Nevertheless, +since there are essential differences between the two, we provide all details. +Lemma 5.13. The set W 1 +Y defined in Definition 5.12 is an algebra of continuous +functions which satisfies conditions (a) and (b) of Definition 1.5. +Proof. Clearly, W 1 +Y is a linear space. If Φ P W 1 +Y , then Φ is the Fourier transform of +an integrable function f, and hence Φ is a bounded continuous function. Thus, W 1 +Y +is a class of continuous functions, and every Φ P W 1 +Y corresponds to a tempered +distribution. +We claim that W 1 +Y is indeed an algebra of functions. If Φ, Ψ P W 1 +Y , then Φ “ Ff +and Ψ “ Fg for some integrable functions f, g, and for some constants c1, c2 we +have |fpxq| ď c1Y pxq and |gpxq| ď c2Y pxq for every x P Rd. It follows that h “ f ˚ g is +an integrable function, and +|hpxq| “ |f ˚ gpxq| ď |f| ˚ |g|pxq ď pc1Y q ˚ pc2Y qpxq “ c1c2Y ˚ Y pxq ď c1c2Y pxq, +(5.6) +that is, h{Y is bounded. Since ΦΨ “ Ff ¨ Fg “ Fpf ˚ gq “ Fh, we conclude that +ΦΨ P W 1 +Y , as desired. +We now show that W 1 +Y contains S . +Indeed: if ϕ P S and ψ “ F ´1ϕ, then +ψ P S , and so, in particular, ψ is integrable. +Furthermore, by condition (b) in +Definition 5.12, ψ{Y is bounded. Thus, ϕ “ Fψ indeed belongs to W 1 +Y . +It remains to observe that if ϕ P S and Φ P W 1 +Y , then ϕ P W 1 +Y , and therefore +ϕΦ P W 1 +Y . +□ +As before, in order to prove that W 1 +Y is a Wiener-type algebra, it remains to verify +that W 1 +Y satisfies condition (c) of Definition 1.5, that is, a variant of Wiener’s 1{f +theorem holds in W 1 +Y . +Lemma 5.14. Let W 1 +Y be the set defined in Definition 5.12. Suppose that K Ď Rd +is a compact set, Φ P W 1 +Y and Φpξq ‰ 0 for every ξ P K. Then there is ˜Φ P W 1 +Y such +that Φpξq˜Φpξq “ 1 for every ξ P K. +Proof. Once again we follow the proof of the classical Wiener’s 1{f lemma given +in [17]. We denote by } ¨ }p the usual norm in LppRdq, where p P r1, 8s. As in the +proof of Lemma 5.7, we have the following two observations. +If Φ “ Ff and f is integrable, then +}Φ}8 ď }f}1. + +LIOUVILLE’S THEOREMS FOR LÉVY OPERATORS +37 +Conversely, if k “ 1, 2, . . ., Φ is smooth, both Φ and ∆kΦ are integrable, and f “ +F ´1Φ, then +suptp1 ` |x|2kq|fpxq| : x P Rdu ď cd,k}Φ ` p´∆qkΦ}1, +where cd,k is an appropriate constant. In particular, if 2k ą d, then f is integrable, +while if 2k ě q with q as in condition (b) in Definition 5.12, then f{Y is bounded. It +follows that if 2k ą d and 2k ě q, then there is a constant cd,k such that whenever Φ +is smooth, and Φ and ∆kΦ are integrable, then Φ P W 1 +Y and +}f}1 ` }f{Y }8 ď cd,k}Φ ` p´∆qkΦ}1, +(5.7) +where f “ F ´1Φ. +We return to the proof of the lemma. Suppose that Φ P W 1 +Y , that is, Φ “ Ff for +an integrable function f such that f{Y is bounded. Suppose furthermore that K +is a compact set and Φpξq ‰ 0 for every ξ P K. Let ε ą 0 be small enough, so that +|Φpξq| ą 3ε for every ξ in some bounded neighbourhood U of K. Choose ϕ P S so +that the support of ϕ is a compact subset of U and ϕpξq “ 1 for ξ P K. As in the +proof of Lemma 5.7, our goal is to prove that +˜Φpξq “ ϕpξq +Φpξq +is an element of W 1 +Y ; here, of course, ˜Φpξq “ 0 for ξ P RdzU. Once this is shown, we +have Φpξq˜Φpξq “ ϕpξq “ 1 for every ξ P K, and so ˜Φ has the desired property. +For r ą 0, let Br denote the ball of radius r centred at the origin, and denote +Yrpxq “ Y pxq 1Brpxq, as in condition (a) in Definition 5.12. Choose r large enough, +so that if gpxq “ fpxq 1Brpxq, then +}f ´ g}1 ă ε +and +Λr ˚ Λrpxq ď +ε2 +}f{Λ}2 +8 +Λprq +(5.8) +for every x P Rd (see condition (a) in Definition 5.12). Clearly, g is integrable and +g{Y is bounded, and therefore Ψ “ Fg is an element of W 1 +Y . Furthermore, +}Φ ´ Ψ}8 ď }f ´ g}1 ă ε. +In particular, for ξ P U we have +|Ψpξq| ě |Φpξq| ´ |Φpξq ´ Ψpξq| ą 3ε ´ ε “ 2ε, +and +ˇˇˇˇ +Ψpξq ´ Φpξq +Ψpξq +ˇˇˇˇ ă ε +2ε “ 1 +2 , +so that for ξ P U we have +1 +Φpξq “ +1 +Ψpξq +8 +ÿ +n“0 +ˆΨpξq ´ Φpξq +Ψpξq +˙n +. +We conclude that for every ξ P Rd, +˜Φpξq “ ϕpξq +Φpξq “ +8 +ÿ +n“0 +ϕpξq +pΨpξqqn`1 pΨpξq ´ Φpξqqn, +(5.9) + +38 +TOMASZ GRZYWNY, MATEUSZ KWA´SNICKI +and we study the terms ϕ{Ψn`1 and pΨ ´ Φqn separately. +Recall that Ψ ´ Φ “ Fpg ´ fq and }g ´ f}1 ă ε. Furthermore, if λ “ }f{Y }8, then +|gpxq ´ fpxq| ď λYrpxq +for every x P Rd. If pg´fq˚n denotes the n-fold convolution of g´f, then Fpg´fq˚n “ +pΨ ´ Φqn, and +}pg ´ fq˚n}1 ď }g ´ f}n +1 ă εn. +(5.10) +The estimate of pg ´ fq˚n{Y is more involved. If n is even, then, by (5.8), +|pg ´ fq˚npxq| ď pλYrq˚npxq +ď λnpYr ˚ Yrq˚n{2pxq +ď λnpλ´2ε2Y q˚n{2pxq +“ εnY ˚n{2pxq ď εnY pxq, +and if n is odd, then, in a similar manner, +|pg ´ fq˚npxq| ď pλYrq˚npxq +ď λnpYr ˚ Yrq˚tn{2u ˚ Yrpxq +ď λnpλ´2ε2Y q˚tn{2u ˚ Y pxq +“ λεn´1Y ˚tn{2u`1pxq ď λεn´1Y pxq. +Thus, for an arbitrary n, +|pg ´ fq˚npxq| ď pλ ` εqεn´1Y pxq. +(5.11) +These are the desired bounds for pΨ ´ Φqn, and we now study ϕ{Ψn`1. +Since Ψ “ Fg and g has compact support, Ψ is smooth in Rd. +Additionally, +|Ψpξq| ě 2ε ą 0 for ξ P U, and ϕpξq “ 0 for ξ outside a compact subset of U. It +follows that ϕ{Ψn`1 is smooth on Rd, and equal to zero in RdzU. +In particular, +by (5.7), we find that ϕ{Ψn`1 P W 1 +Y , and if hn “ F ´1pϕ{Ψn`1q, then +}hn}1 ` }hn{Y }8 ď cd,k +››ϕ{Ψn`1 ` p´∆qkpϕ{Ψn`1q +›› +1, +where k is a fixed sufficiently large positive integer and cd,k is an appropriate con- +stant. By applying the product rule to p´∆qkpϕ{Ψn`1q, as in the proof of Lemma 5.7 +we find that there is a constant cd,k,ϕ,Ψ such that +}hn}1 ` }hn{Y }8 ď cd,k,ϕ,Ψ|U|p1 ` nkqp2εq´n´1´2k, +(5.12) +and this is the desired estimate for ϕ{Ψn`1. +We combine (5.10) and (5.12) to find that +}pg ´ fq˚n ˚ hn}1 ď }pg ´ fq˚n}1}hn}1 ď ¨cd,k,ϕ,Ψ|U|p1 ` nkqp2εq´1´2k ¨ 2´n, +and similarly we combine combine (5.11) and (5.12) as in (5.6) to find that +|pg ´ fq˚n ˚ hnpxq| ď cd,k,ϕ,Ψ|U|p1 ` nkqp2εq´2´2kpλ ` εq ¨ 21´nY pxq. +By Fubini’s theorem, we obtain +›››› +8 +ÿ +n“0 +pg ´ fq˚n ˚ hn +›››› +1 +ă 8, + +LIOUVILLE’S THEOREMS FOR LÉVY OPERATORS +39 +and +›››› +1 +Y +8 +ÿ +n“0 +pg ´ fq˚n ˚ hn +›››› +8 +ă 8. +In particular, the series +˜f “ +8 +ÿ +n“0 +pg ´ fq˚n ˚ hn +defines an integrable function such that ˜f{Y is bounded, and as in the proof of +Lemma 5.7, by Fubini’s theorem we find that +F ˜f “ +8 +ÿ +n“0 +pΨ ´ Φqnpϕ{Ψn`1q. +Thus, by (5.9), F ˜f “ ˜Φ, and therefore ˜Φ P W 1 +Y , as desired. +□ +As an immediate corollary of Lemmas 5.13 and 5.14, we obtain the following +result. +Proposition 5.15. The set W 1 +Y defined in Definition 5.12 is a Wiener-type algebra. +Suppose that Y satisfies the conditions of Definition 5.12, and ˇL is an integrable +distribution such that for every ϕ P S the Fourier transform of pˇL ˚ ϕq{Y is a +bounded function. Then, by definition, Fϕ ¨ F ˇL “ FpˇL ˚ ϕq P W 1 +Y . By choosing ϕ +such that Fϕpξq “ 1 for ξ in a given compact set, we find that F ˇL belongs to W 1 +Y +locally on Rd. +Let ˇY pxq “ Y p´xq. We claim that if h is a function on Rd such that ˇY h is inte- +grable, then h corresponds to a tempered distribution and Fh acts on W 1 +Y . Indeed: +by condition (b) in Definition 5.12, h is bounded by the product of an integrable +function ˇY h and a polynomial, and hence it corresponds to a tempered distribu- +tion. Using both conditions in Definition 5.12, we find that for some constants c, q +we have Y px ´ yq ď cp1 ` |x|qqY p´yq for every x, y P Rd. It follows that h and Y are +convolvable, and for some constants c, q we have +|h| ˚ Y pxq ď c|h| ˚ Y p0qp1 ` |x|qq. +Finally, if Φ, Ψ P W 1 +Y , then Φ “ Ff and Ψ “ Fg for some integrable functions f, g +such that f{Y and g{Y are bounded. Thus, by condition (a) in Definition 5.12 and +Fubini’s theorem, we find that +|h| ˚ |f| ˚ |g|pxq ď }f{Y }8}g{Y }8|h| ˚ Y ˚ Y pxq +ď }f{Y }8}g{Y }8|h| ˚ Y pxq +ď c}f{Y }8}g{Y }8|h| ˚ Y p0qp1 ` |x|qq +(5.13) +for every x P Rd. Therefore, by Fubini’s theorem, we have +h ˚ pf ˚ gq “ ph ˚ fq ˚ g, +and in fact, by (5.13), +h f pf ˚ gq “ ph f fq f g. +Applying the exchange formula, we conclude that +FH d pΦΨq “ pFH d Φq d Ψ, + +40 +TOMASZ GRZYWNY, MATEUSZ KWA´SNICKI +which completes the proof of our claim. +After exchanging the roles of Y and ˇY , Theorem 5.1 and Proposition 5.15, as well +as the two properties discussed above, immediately lead to the following variant of +Liouville’s theorem. +Corollary 5.16 (Liouville’s theorem under generalised negative moment condi- +tion). Suppose that Y satisfies the conditions of Definition 5.12. Let ˇL be an in- +tegrable distribution such that for every ϕ P S there is a constant c such that +|ˇL ˚ ϕpxq| ď cY p´xq for every x P Rd. Suppose that F ˇL, which is necessarily a +continuous function, has no zeroes in an open set U. Let h be a function such that +Y h is integrable on Rd and ˇL f h “ 0. Then the spectrum of h is contained in RdzU. +In particular, if U “ Rdzt0u, then h is a polynomial. +We conclude this section with examples of admissible functions Y . For this, we +need the following auxiliary result, which resembles Theorem 1.1 in [5]. +Lemma 5.17. Suppose that ϕ is a positive, decreasing, continuous function on +r0, 8q such that rd´1ϕprq is integrable with respect to r P r0, 8q and +lim inf +rÑ8 +ϕp2rq +ϕprq ą 0. +(5.14) +Then, for ε ą 0 small enough, the function Y pxq “ εϕp|x|q satisfies all conditions of +Definition 5.12. +Proof. Clearly, Y is integrable over Rd. We need to verify conditions (a) and (b) in +Definition 5.12. +Since ϕ is decreasing, we have +min +␣ +ϕp|x ´ y|q, ϕp|y|q +( +“ ϕ +` +maxt|x ´ y|, |y|u +˘ +ď ϕp 1 +2|x|q. +Hence, +ϕp|x ´ y|qϕp|y|q “ min +␣ +ϕp|x ´ y|q, ϕp|y|q +( +max +␣ +ϕp|x ´ y|q, ϕp|y|q +( +ď ϕp 1 +2|x|q +` +ϕp|x ´ y|q ` ϕp|y|q +˘ +. +It follows that +Y ˚ Y pxq “ ε2 +ż +Rd ϕp|x ´ y|qϕp|y|qdy +ď ε2ϕp 1 +2|x|q +ż +Rd +` +ϕp|x ´ y|q ` ϕp|y|q +˘ +dy “ 2ε2c1ϕp 1 +2|x|q, +where c1 is the integral of ϕp|x|q over x P Rd. +By assumption (5.14), there is a +constant c2 such that ϕp 1 +2|x|q ď c2ϕp|x|q for every x P Rd. It follows that +Y ˚ Y pxq ď 2c1c2εY pxq. +Hence, if ε is small enough, we have Y ˚ Y pxq ď Y pxq, as desired. + +LIOUVILLE’S THEOREMS FOR LÉVY OPERATORS +41 +Let Yr be defined as in condition (a) in Definition 5.12, and let Br denote the +centred ball of radius r. By the same argument as above, we have +Yr ˚ Yrpxq “ ε2 +ż +Rd 1RdzBrpx ´ yqϕp|x ´ y|q 1RdzBrpyqϕp|y|qdy +ď ε2ϕp 1 +2|x|q +ż +Rd 1RdzBrpx ´ yq 1RdzBrpyq +` +ϕp|x ´ y|q ` ϕp|y|q +˘ +dy +ď c2ε2ϕp|x|q +ż +Rd +` +1RdzBrpx ´ yqϕp|x ´ y|q ` 1RdzBrpyqϕp|y|q +˘ +dy +ď 2c2εY pxq +ż +Rd 1RdzBrpyqϕp|y|qdy. +Thus, +lim +rÑ8 sup +"Yr ˚ Yrpxq +Y pxq +: x P Rd +* +ď lim +rÑ8 +ˆ +2c2ε +ż +Rd 1RdzBrpyqϕp|y|qdy +˙ +“ 0. +We have thus proved that Y indeed satisfies condition (a) in Definition 5.12. +Condition (b) in Definition 5.12 is related to general theory of regular variation, +but in fact it is an elementary result: if, as above, ϕp 1 +2|x|q ď c2ϕp|x|q for every x P Rd +and c3 is the infimum of ϕ over r0, 1s, then for x P Rdzt0u such that 2n´1 ă |x| ď 2n, +where n “ 0, 1, 2, . . ., we have +c3 ď ϕp2´n|x|q ď cn +2ϕp|x|q “ c2p2n´1qlog c2{ log 2ϕp|x|q ď c2|x|log c2{ log 2ϕp|x|q, +and hence ϕp|x|q ě pc3{c2qp1 ` |x|q´ log c2{ log 2 for every x P Rd, as desired. +□ +Example 5.18. Using Lemma 5.17 it is straightforward to verify that if α ą 0, then +Y pxq “ cd,αp1 ` |x|q´d´α satisfies the conditions of Definition 5.12. Thus, we get the +following result. +Let B denote the unit ball in Rd. Suppose that L is a Lévy operator which is not +concentrated on a proper closed subgroup of Rd, and for some constant c the Lévy +measure ν of L satisfies +νpx ` Bq ď c|x|´d´α +for every x P Rd such that |x| ě 2. Then every L-harmonic function h such that +p1 ` |x|q´d´αhpxq is an integrable function of x P Rd is necessarily a polynomial. +Suppose now that ν has a density function which is comparable with |x|´d´α +for |x| large enough. +In this case our integrability condition on h, namely, that +p1 ` |x|q´d´αhpxq is an integrable function, is automatically satisfied by every L- +harmonic function in the weak sense (as defined in Definition 1.2). +Thus, by +Lemma 2.2, the result given above characterises all functions h which are L- +harmonic in the weak sense, with no additional conditions on h. +The above result is a variant of Theorem 1.4 in [13], while the more general result +given in Corollary 5.16 with this choice of Y is a minor extension of Theorem 1.1 +in [13]. +Example 5.19. More generally, by Lemma 5.17, we find that if Y pxq “ εϕp|x|q for a +positive, decreasing, continuous function ϕ on r0, 8q such that ϕp|x|q is integrable +with respect to x P Rd, and such that ϕprq is regularly varying as r Ñ 8, then Y +satisfies the conditions of Definition 5.12 if ε is small enough. In particular, this +implies the following result. + +42 +TOMASZ GRZYWNY, MATEUSZ KWA´SNICKI +Let α ą 0 and β P R, or α ą 0 and β ą 1. Let L be a Lévy operator which is not +concentrated on a proper closed subgroup of Rd, and such that for some constant +c the Lévy measure ν of L satisfies +νpx ` Bq ď c|x|´d´αplog |x|q´β +for every x P Rd such that |x| ě 2 (where B is the unit ball). Then every L-harmonic +function h such that p1 ` |x|q´d´αplogpe ` |x|qq´βhpxq is an integrable function of +x P Rd is necessarily a polynomial. +As in the previous example, if ν has a density function which is comparable with +|x|´d´αplog |x|q´β for |x| large enough, then L-harmonic functions in the weak sense +(see Definition 1.2) automatically satisfy the integrability condition, and hence, by +Lemma 2.2, the above result describes all L-harmonic functions in the weak sense. +Example 5.20. Suppose that for j “ 1, 2, . . . , k a function Yj on Rdj satisfies the +conditions of Definition 5.12, and define d “ d1 ` d2 ` . . . ` dk and +Y pxq “ +k +ź +j“1 +Yjpxjq, +where x “ px1, x2, . . . , xkq P Rd with xj P Rdj. It is then immediate to check that Y +satisfies the conditions of Definition 5.12. Combining this with Example 5.18, we +arrive at the following result. +Let B denote the unit ball in Rd. Suppose that L “ L1 ` L2 ` . . . ` Ld, where for +every j “ 1, 2, . . . , d the operator Lj is a dj-dimensional Lévy operator which is not +concentrated on a proper closed subgroup of Rdj, and for some constants cj and +αj ą 0 the Lévy measure νj of L satisfies +νjpx ` Bjq ď c|x|´1´αj +for every x P Rdj such that |x| ě 2; here Bj is the unit ball in Rdj. Then every +L-harmonic function h such that +ˆ d +ź +j“1 +1 +p1 ` |xj|qd`αj +˙ +hpxq +is an integrable function of x P Rd is necessarily a polynomial. +We remark that, unlike in the previous examples, the integrability condition on h +does not seem to follow automatically from the definition of an L-harmonic function +in the weak sense. +References +[1] N. Alibaud, F. del Teso, J. Endal, E.R. Jakobsen, The Liouville theorem and linear operators +satisfying the maximum principle. J. Math. Pures Appl. 142(2020): 229–242. +[2] M.T. Barlow, R.F. Bass, C. Gui, The Liouville property and a conjecture of De Giorgi. Commun. +Pure Appl. 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Deny, Sur l’equation de convolution µ “ µ ˚ σ. Séminaire Brelot-Choquet-Deny. Théorie du +potentiel, tome 4 (1959–1960), exp. no 5: 1–11. +[10] P. Dierolf, J. Voigt, Convolution and S1-convolution of distributions. Collect. Math. 29(3) (1978): +185–196. +[11] E.B. Dynkin, Markov processes, Vols. I and II. Springer-Verlag, Berlin-Götingen-Heidelberg, +1965. +[12] M.M. Fall, Entire s-harmonic functions are affine. Proc. Amer. Math. Soc. 144 (2016): 2587– +2592. +[13] M.M. Fall, T. Weth. Liouville Theorems for a General Class of Nonlocal Operators. Potential +Anal. 45(1) (2016): 187–200. +[14] Y. Hirata, H. Ogata, On the exchange formula for distributions. J. Sci. Hiroshima Univ. Ser. A +22 (1958): 147–152. +[15] K. Kaleta, R.L. Schilling, Progressive intrinsic ultracontractivity and heat kernel estimates for +non-local Schrödinger operators. J. Funct. Anal. 279(6) (2020), no. 108606: 1–69. +[16] F. Kühn, A Liouville theorem for Lévy generators. Positivity 25 (2021): 997–1012. +[17] D.J. Newman, A simple proof of Wiener’s 1{f theorem. Proc. Amer. Math. Soc. 48(1) (1975): +264–265. +[18] E. Priola, J. Zabczyk, Liouville theorems for non-local operators. J. Funct. Anal. 216(2) (2004): +455–490. +[19] M. Riesz, Intégrales de Riemann–Liouville et potentiels. Acta Sci. Math. Szeged 9 (1938), 1–42. +[20] X. Ros-Oton, J. Serra, Regularity theory for general stable operators. J. Diff. Equations 260 +(2016): 8675–8715. +[21] K. Sato, Lévy Processes and Infinitely Divisible Distributions. Cambridge Univ. Press, Cam- +bridge, 1999. +[22] R. Shiraishi, M. Itano, On the multiplicative products of distributions. J. Sci. Hiroshima Univ. +Ser. A-I Math. 28(2) (1964): 223–235. +[23] T. Simon, Petites déviations et support d’un processus de Lévy (Small deviations and support +of a Lévy process). C. R. Acad. Sci. Paris 329 (1999): 331–334. +[24] A. Tortrat, Le support des lois indéfiniment divisibles dans un groupe abélien localement com- +pact. Math. Z. 197 (1988): 231–250. +[25] V.S. Vladimirov, Methods of the theory of generalized functions. Taylor and Francis, New York, +2002. +Tomasz Grzywny, Mateusz Kwa´snicki, Faculty of Pure and Applied Mathematics, Wrocław Uni- +versity of Science and Technology, ul. Wybrze˙ze Wyspia´nskiego 27, 50-370 Wrocław, Poland +Email address: tomasz.grzywny@pwr.edu.pl,mateusz.kwasnicki@pwr.edu.pl + diff --git a/c9FAT4oBgHgl3EQfYh2h/content/tmp_files/load_file.txt b/c9FAT4oBgHgl3EQfYh2h/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..9b15998d2a69e51ac4d3cfc7b14d693a4ab9487a --- /dev/null +++ b/c9FAT4oBgHgl3EQfYh2h/content/tmp_files/load_file.txt @@ -0,0 +1,1814 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf,len=1813 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='08540v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='AP] 20 Jan 2023 LIOUVILLE’S THEOREMS FOR LÉVY OPERATORS TOMASZ GRZYWNY, MATEUSZ KWA´SNICKI Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Let L be a Lévy operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' A function h is said to be harmonic with respect to L if Lh “ 0 in an appropriate sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' We prove Liouville’s theorem for positive functions harmonic with respect to a general Lévy operator L: such functions are necessarily mixtures of exponentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' For signed harmonic functions we provide a fairly general result, which encompasses and extends all Liouville-type theorems previously known in this context, and which allows to trade regularity assumptions on L for growth restrictions on h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Finally, we construct an explicit counterexample which shows that Liouville’s theorem for signed functions harmonic with respect to a general Lévy operator L does not hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Introduction 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Liouville’s theorem and Lévy operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' A classical result due to Liouville and Cauchy, traditionally called Liouville’s theorem, states that every bounded har- monic function in Rd is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' This was extended by Bôcher and Picard, who proved that a one-sided bound is sufficient: every positive harmonic function in Rd is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' A yet another variant of Liouville’s theorem asserts that every har- monic function bounded by a polynomial (and again a one-sided bound is sufficient) is in fact a harmonic polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Liouville’s theorem has been extended in various directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Here we study vari- ants of Liouville’s theorem for Lévy operators, that is, operators L of the form Lfpxq “ dÿ j,k“1 ajkBjkfpxq ` dÿ j“1 bjBjfpxq ` ż Rdzt0u ˆ fpx ` yq ´ fpxq ´ 1Bpyq dÿ j“1 yjBjfpxq ˙ νpdyq, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='1) acting on an appropriate class of functions on Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Here B is the unit ball, paijq P Rdˆd is a nonnegative definite symmetric matrix, pbjq P Rd is a vector, and ν is a nonnegative measure on Rdzt0u such that ş Rdzt0u mint1, |y|2uνpdyq ă 8, the so-called Lévy measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Various equivalent descriptions of the class of Lévy operators are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' In particular, the following conditions are equivalent: L is a Lévy operator;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' L is the generator of a Lévy process Xt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' L generates a strongly continuous semigroup of translation invariant Markov operators;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' ´L is a Fourier multiplier whose symbol Date: January 23, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' 2010 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' 35B08, 35B09, 35B10, 35B53, 35R09, 58J65, 60G51, 60J35, 60J45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Harmonic function, Lévy process, Liouville’s theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Work supported by the Polish National Science Centre (NCN) grants no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' 2017/27/B/ST1/01339 (TG) and 2019/33/B/ST1/03098 (MK).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' 1 2 TOMASZ GRZYWNY, MATEUSZ KWA´SNICKI Ψ is a continuous negative definite function vanishing at the origin;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' L is a trans- lation invariant integro-differential operator vanishing on constants and satisfying the maximum principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' We remark that the function Ψ mentioned above is the characteristic exponent of the Lévy process Xt, given by the Lévy–Khintchine for- mula Ψpξq “ apξ, ξq ´ ibξ ` ż Rdzt0u ` 1 ´ eiξy ` iξy 1Bpyq ˘ νpdyq, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='2) and for every t P r0, 8q, the characteristic function of the random variable Xt is equal to e´tΨ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Here apξ, ξq “ řd j,k“1 ajkξjξk, and bξ “ řk j“1 bjξj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' The Laplace operator ∆ is a prime example of a Lévy operator, and a smooth function f is harmonic in Rd if and only if ∆f “ 0 in Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' In a similar way, given a general Lévy operator L, a smooth function f is said to be harmonic with respect to L, or L-harmonic, if Lf “ 0 in Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' In probabilistic terms, the Lévy operator L is the generator of a Lévy process Xt, and f is L-harmonic if and only if fpXtq is a local martingale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' The question that we address in this article is: which variants of Liouville’s the- orem remain true for L-harmonic functions, where L is a given Lévy operator?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' This problem was tackled by several authors over the last few years, and in fact its history can be traced back to the seminal work of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Riesz [19] on functions harmonic with respect to the fractional Laplace operator ´p´∆qs, another widely studied Lévy operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Indeed: Liouville’s theorem for bounded harmonic functions follows immediately from Harnack’s inequality, and an appropriate variant of the latter result for the fractional Laplace operator is given in [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' The goal of this paper is three-fold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' First, we prove a general Liouville’s the- orem for nonnegative L-harmonic functions: every such function is a mixture of L-harmonic exponentials;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' see Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Next, we provide a general framework for proving Liouville’s theorems for signed polynomially bounded L-harmonic func- tions by means of Fourier transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Under appropriate additional assump- tions, we prove that all signed L-harmonic functions which correspond to tempered distributions have their spectrum (that is, the support of the Fourier transform) contained in the zero set of the characteristic exponent of L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' see Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Fi- nally, we construct a surprising example which proves that one cannot completely drop the additional assumptions mentioned above: Liouville’s theorem for signed, polynomially bounded L-harmonic functions does not hold in full generality;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' see Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' In the introduction, we state simplified variants of our main results, and we pro- vide references to the corresponding full statements given later in the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' We commonly impose the following standard assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' A Lévy operator L is said not to be concentrated on a proper closed subgroup of Rd if the Fourier symbol Ψ of ´L (that is, the characteristic exponent of the corresponding Lévy process Xt) is equal to zero only at the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' The above condition holds if and only if the closed subgroup G of Rd gener- ated by the union of the supports of the distributions of Xt ´ X0 is equal to Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' The subgroup G can be described as the smallest closed subgroup which contains: (a) eigenspaces of the matrix paijq which correspond to nonzero eigenvalues;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' (b) the support of ν;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' and (c) an appropriately defined drift vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' This result is due to LIOUVILLE’S THEOREMS FOR LÉVY OPERATORS 3 Tortrat [24];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' we refer to Théorème 3 in [23] and to [1] for details, as well as to Section 24 in [21] (in particular, Definitions 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='13 and 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='21 and Proposition 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='14 therein) for further discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' We also note that other names, such as nonlattice condition, are commonly used for the condition described in Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' We remark that if a Lévy operator L is concentrated on a proper closed subgroup G of Rd, then L acts on each coset x ` G separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' In particular, every G-periodic function on Rd (that is, a function f such that fpx ` zq “ fpxq for every x P Rd and z P G) is L-harmonic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' The proper way to state Liouville’s theorem for a Lévy operator L concentrated on G is to consider functions defined only on G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Although we do not discuss this extension in the introduction, our main results carry over to this situation: for positive L-harmonic functions we state this explicitly in Section 3, while for signed L-harmonic function this extension corresponds to Fourier symbols Ψ with zero sets other than t0u, which are allowed in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Positive L-harmonic functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' In order to state Liouville’s theorem for positive L-harmonic functions, we need an auxiliary definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' If L is a Lévy operator, we denote by L: the Lévy operator dual to L, obtained by conjugation of L with the reflection x ÞÑ ´x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' More precisely, if L is given by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='1), then L: has a similar representation with the same matrix pa: ijq “ paijq, with vector pb: jq “ p´bjq, and with Lévy measure ν:pAq “ νp´Aq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' We remark that L: is an op- erator formally adjoint to the operator L, and if both are considered as unbounded operators on L2pRdq, then L and L: are indeed adjoint operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' A locally integrable function h is said to be L-harmonic (in the weak sense) if ż Rd hpxqL:ϕpxqdx “ 0 for every ϕ P C8 c pRdq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Here we implicitly assume absolute convergence of the integral in the left-hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' In fact, we will allow for a slightly more general definition;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' see Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='1 in Sec- tion 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='3 for a detailed discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Of course, whenever we say that an L-harmonic function h in the above sense is constant or equal to a polynomial, in fact we mean equality almost everywhere with respect to the Lebesgue measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Recall that B is the unit ball in Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' It is relatively simple to see that if h has continuous second-order partial derivatives, ż RdzB |hpx ` yq|νpdyq is a locally integrable function of x P Rd, and Lh “ 0 in Rd in the pointwise sense, then h is L-harmonic in the weak sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' The following statement, which builds upon and extends Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='7 in [3], is our first main result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='3 (see Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Let L be a Lévy operator which is not concen- trated on a proper closed subgroup of Rd, and let f be a nonnegative function on Rd which satisfies Lf “ 0 in the weak sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Then fpxq “ ż Λ eλxµpdλq 4 TOMASZ GRZYWNY, MATEUSZ KWA´SNICKI for a unique nonnegative measure µ on the set Λ of those vectors λ P Rd for which the function eλpxq “ eλx satisfies Leλpxq “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' The above result follows directly from the more general Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='2, which allows h to be an arbitrary Schwartz distribution such that Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='2 makes sense, and L to be concentrated on a proper closed subgroup G of Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='3 for Lévy operators which generate random walks was proved by Deny in [9] (see Théorème 3 therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' In terms of the representation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='1), this result corresponds to aij “ 0, bj “ 0 and ν a finite measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' We stress that this spe- cial case is a crucial ingredient of our proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' A variant of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='3 was proved by Berger and Schilling in [3] (see Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='7 therein) with the additional as- sumption that h is bounded by a submultiplicative function integrable with respect to the Lévy measure ν over the complement of the unit ball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' An alternative proof was given recently by Berger, Schilling and Shargorodsky in [4] (see Theorem 13 therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Since every bounded mixture of exponentials is constant, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='3 in partic- ular implies that bounded L-harmonic functions are necessarily constant, provided that L is not concentrated on a proper closed subgroup of Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' This variant of The- orem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='3 has some history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' For the fractional Laplace operator, this result follows directly from Harnack’s inequality proved by M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Riesz in [19] (see formula (5) in Chapter V therein);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' an essentially equivalent argument was given by Bogdan, Kul- czycki and Nowak in [6] (see Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='2 therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Similar Liouville’s theorem for Lévy operators with measure νpdyq decaying exponentially fast at infinity and com- parable with |y|´d´1dy near the origin was given by Barlow, Bass and Gui in [2] (see Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='17 therein), while the result for rather general Lévy operators with non- degenerate second order local term was proved by Priola and Zabczyk in [18] (see Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='8 therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' The case of general Lévy operators L was solved completely by Alibaud, del Teso, Endal and Jakobsen in [1] (see Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='1 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='2 therein) using analytical methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' We remark that the probabilistic argument applied by Berger and Schilling in [3] (see Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='4 therein for the statement for bounded L-harmonic functions) is quite different and more general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Signed L-harmonic functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' In order to study signed L-harmonic func- tions, we apply Fourier transform methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' These are limited to the class of tem- pered distributions, and for this reason we need to modify slightly the definition of an L-harmonic function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' We denote by S the Schwartz class of rapidly decreasing smooth functions on Rd, and by S 1 the class of tempered distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' A locally integrable function h with at most polynomial growth at infinity, or, more generally, a tempered distribution h, is said to be L-harmonic (in the sense of tempered distributions) if ż Rd h ˚ ψpxqL:ϕpxqdx “ 0 for every ϕ, ψ P S .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Here we implicitly assume absolute convergence of the integral in the left-hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' LIOUVILLE’S THEOREMS FOR LÉVY OPERATORS 5 We remark that if h is an L-harmonic function in the weak sense (according to Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='2, and if additionally |hpxq| ` ż RdzB |hpx ` yq|νpdyq is bounded by a polynomial (as a function of x P Rd), then h is L-harmonic in the sense of tempered distributions (according to Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='4);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' see Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='2 in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' A Lévy operator L is a convolution operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' The corresponding convolution kernel is an appropriate tempered distribution, which we denote by ˇL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' The dis- tributional Fourier transform F ˇL of the tempered distribution ˇL is a continuous function: it is equal to ´Ψ, the Fourier symbol of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Recall that, in probabilistic terms, Ψ is the characteristic exponent of the corresponding Lévy process Xt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' For simplicity, below we assume that L is not concentrated on a proper closed subgroup of Rd, that is, Ψpξq ‰ 0 for ξ ‰ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' We stress, however, that the most general version of our result stated in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='1 also covers general Lévy oper- ators, as well as other convolution operators, as long as their Fourier symbols are continuous functions in an appropriate class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Before we state our main result in this section, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='7, it is instructive to describe briefly our argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Suppose that h is an L-harmonic function in the sense of tempered distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Our proof of Liouville’s theorem for signed L-harmonic functions consists of the following steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' (I) Observe that ˇL and h are convolvable (as tempered distributions), and ˇL ˚ h “ Lh “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' (II) Use the Fourier exchange formula to find out that the muliplicative product of Fourier transforms of ˇL and h is well-defined (in the sense of tempered distributions), and F ˇL ¨ Fh “ FpˇL ˚ hq “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' (III) Use an appropriate variant of Wiener’s 1{f theorem to deduce that Fh “ 0 on Rdzt0u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' (IV) Conclude that h is a polynomial Steps (I) and (IV) present no difficulties, while step (II) is a known result in the theory of tempered distributions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' we refer to Sections 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='2 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='3 for a detailed discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' The only problematic part in the above argument is step (III).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' We remark that if Ψ “ ´F ˇL is a smooth function in Rdzt0u, then also step (III) is completely standard, and in this way we recover the variant of Liouville’s theorem given in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='2 in [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' If we assume that h is a bounded function, then the usual Wiener’s 1{f theorem can be easily adapted to make step (III) rigorous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' This leads to a significantly shorter proof of Liouville’s theorem originally given in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='1 in [1] and, independently, in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='4 in [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' The two examples discussed above are in some sense extreme cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' We also provide intermediate variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' However, we need to keep balance between smooth- ness conditions on Ψ and growth restrictions of h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Detailed statements are given in Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='8 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' 6 TOMASZ GRZYWNY, MATEUSZ KWA´SNICKI A rigorous statement of our general result requires the following two auxiliary definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' We say that an algebra W of continuous functions on Rd is a Wiener-type algebra if: (a) every element of W is a tempered distribution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' (b) if Φ P W and ϕ P S , then ϕΦ P W ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' (c) if K Ď Rd is a compact set, Φ P W and Φpξq ‰ 0 for every ξ P K, then there is ˜Φ P W such that Φpξq˜Φpξq “ 1 for every ξ P K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' We say that a tempered distribution Ψ belongs to W locally on an open set U if for every compact set K Ď U there is a distribution Φ P W such that Ψ “ Φ in a neighbourhood of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' We note that the Schwartz class S , or the class of Fourier transforms of inte- grable functions (that is, the usual Wiener algebra), are Wiener-type algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Clearly, conditions (a) and (b) are rather natural, and typically they are easy to check.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' The essential property of Wiener-type algebras is given in condition (c), which can be thought of as a variant of Wiener’s 1{f theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' We say that a tempered distribution H acts on a Wiener-type al- gebra W if for every Φ, Ψ P W we have the following identity of multiplicative products of tempered distributions: pH ¨ Φq ¨ Ψ “ H ¨ pΦ ¨ Ψq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' In particular, we require that all products of tempered distributions in the above identity are well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' The following result apparently covers all known Liouville-type theorems on signed L-harmonic functions for Lévy operators L, and, together with Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='8 below, it is our second main result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='7 (see Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Let W be a Wiener-type algebra of continuous functions on Rd, and let L be a Lévy operator which is not concentrated on a proper closed subgroup of Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Suppose that the Fourier symbol ´Ψ of L belongs to W locally on Rdzt0u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Let h be an L-harmonic function in the sense of tempered distri- butions, and suppose that Fh acts on W .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Then f is a polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' We remark that if f is a polynomial and Lf is well-defined, then Lf is again a polynomial, and the degree of Lf is less than the degree of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Thus, in order to find L-harmonic polynomials of degree n it is sufficient to evaluate Lf for every mono- mial f of degree at most n, and solve a system of linear equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' We also remark that if L is isotropic (invariant under rotations), then L-harmonic polynomials h are harmonic in the classical sense, that is, they satisfy ∆h “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' The proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='7 is similar to the argument applied independently in [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' By specifying the Wiener-type algebra W , we obtain the following family of less ab- stract results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' More variants of Liouville’s theorem which follow from Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='7 can be found in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Let L be a Lévy operator which is not concentrated on a proper closed subgroup of Rd, and let h be an L-harmonic function in the sense of tempered distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Then h is necessarily a polynomial if any of the following conditions holds: LIOUVILLE’S THEOREMS FOR LÉVY OPERATORS 7 (a) the Fourier symbol ´Ψ of L is smooth on Rdzt0u (Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' (b) h is a bounded function (Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='4);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' (c) for some α ě 0, the function |x|α is integrable with respect to the Lévy mea- sure ν on RdzB, while p1 ` |x|q´αhpxq is a bounded function (Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='10);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' (d) for some β ě 0, the function plog |x|qβ is integrable with respect to the Lévy measure ν on RdzB, while plogpe ` |x|qq´βhpxq is a bounded function (Exam- ple 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='11);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' (e) for some α ą 0, there is a constant c such that the Lévy measure of the ball with centre x and radius 1 is bounded by c|x|´d´α for every x P Rd such that |x| ě 2, while p1 ` |x|q´d´αhpxq is an integrable function (Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='18);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' (f) for some α ą 0 and β P R, or for α “ 0 and some β ą 1, there is a con- stant c such that the Lévy measure of the ball with centre x and radius 1 is bounded by c|x|´d´αplog |x|q´β for every x P Rd such that |x| ě 2, while p1 ` |x|q´d´αplogpe ` |x|qq´βhpxq is an integrable function (Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' As discussed after the statement of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='3, Liouville’s theorem for bounded L-harmonic functions has been studied previously, and the general result stated in Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='8(a) is due to Alibaud, del Teso, Endal and Jakobsen [1] (see Theo- rems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='1 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='2 therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' An independent proof was given by Berger and Schilling in [3] (see Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='4 therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' The variant for smooth symbols, given in Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='8(b), follows easily from standard properties of tempered distributions and their Fourier transforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Essen- tially the same statement is given by Berger and Schilling in [3] (see Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='2 therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Liouville’s theorem for L-harmonic functions under a moment condition on the Lévy measure similar to Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='8(c) was proved by Ros-Oton and Serra in [20] (see Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='1 therein) for homogeneous Lévy-type operators (that is, genera- tors of stable Lévy processes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' General Lévy operators were considered by Kühn in [16] (see Theorem 1 therein), and the present statement of Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='8(c) was independently found by Berger, Schilling and Shargorodsky in [4] (see Theorem 8 therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' The same work contains a more general result (see Theorem 11 therein), equivalent to our Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='9, which encompasses Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='8(c) and (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='8(e) for the fractional Laplace operator was proved by Fall in [12] (see Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='1 therein) and Chen, D’Ambrosio and Li in [7] (see Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='3 therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Extension similar to Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='8(e) was given by Fall and Weth in [13] (see Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='4 therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='8(f), as well as the more general Corol- lary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='16, seem to be new.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' We remark that if α ą 0 and the Lévy measure ν of the Lévy operator L is com- parable with |y|´d´αdy when |y| is large enough, then Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='8(e) provides a complete Liouville’s theorem, with no restrictions on the L-harmonic function h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Indeed: in this case p1 ` |x|q´d´αhpxq is automatically integrable whenever h is L-harmonic, and additionally the two notions of L-harmonicity (Definitions 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='2 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='4) are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' The same remark applies to Lévy operators with Lévy measure comparable with |y|´d´αplog |y|q´βdy when |y| is large enough, where α and β are as in Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='8(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' A counterexample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Although rather general, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='7 does not apply to arbitrary L-harmonic functions (in the sense of tempered distributions), and there is a reason for that: it turns out that it is not possible to make step (III) of our 8 TOMASZ GRZYWNY, MATEUSZ KWA´SNICKI proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='7 rigorous without additional assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' In other words, even though Ψ “ ´F ˇL is a positive, continuous function in Rdzt0u, there may exist a tempered distribution Fh such that the multiplicative product of Ψ ¨ Fh is well- defined in the sense of tempered distribution and Ψ ¨ Fh “ 0, but nevertheless Fh is nonzero in Rdzt0u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' That is, in general, division by Ψ turns out to be impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' A rigorous statement is given in the following theorem, which is our third main result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' (a) There is a bounded, positive, continuous function f and a tempered distribution g such that the S 1-product f ¨ g is well-defined and equal to zero even though g is not identically zero (see Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' (b) There is a nontrivial probability measure µ on Z and a function h on Z such that lim|x|Ñ8p|x|´εhpxqq “ 0 for every ε ą 0 and h˚µ “ µ, but h is not constant, and hence not a polynomial (see Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' (c) There is a Lévy operator L and a smooth function h such that L is not concentrated on a proper closed subgroup of Rd, for every ε ą 0 we have lim|x|Ñ8p|x|´εhpxqq “ 0, and Lh “ 0 (both pointwise and in the weak sense), but h is not constant, and hence not a polynomial (see Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Structure of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' The remaining part of the article is divided into four sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' In Preliminaries we introduce the notation used in the remaining part of the paper (Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='1), we discuss the notions of convolution and multiplication of distributions (Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='2), and connect them with Lévy operators (Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='3, describing positive L-harmonic functions, is proved in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' In Section 4, we construct counterexamples to Liouville’s theorem which prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' We begin with operators on Z (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='1), then we reinterpret this example in terms of multiplication of distributions (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='2), and finally we deal with the case of operators on Rd (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' The final part of the paper contains the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='7 and Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='8, which provides various variants of Liouville’s theorem for signed L-harmonic func- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' We begin with the abstract result given in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='7 (Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Then, by choosing appropriate Wiener-type algebras, we show how this result leads to Li- ouville’s theorems for operators with smooth symbols (Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='2) and for bounded L-harmonic functions (Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='3), which correspond to Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='8(a) and (b), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' In order to prove Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='8(c) and (d), we construct another Wiener-type algebra in Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='8 (Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Similarly, Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='8(e) and (f) is a consequence of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='15 and auxiliary Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='17 (Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' We thank Moritz Kaßmann for stimulating discussions about Liouville theorems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Preliminaries 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' General notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' We use x, y, z P Rd for spatial variables, and ξ, η P Rd for Fourier variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' If ξ, x P Rd, by ξx we denote the dot product of ξ and x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' We use the symbol δxpdyq for the Dirac delta measure at x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' We write D for the class of smooth, compactly supported functions on Rd, and D1 for the dual space, the class of Schwartz distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' As it is customary, if f P D1 and ϕ P D, we write xf, ϕy for the value of f at ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' LIOUVILLE’S THEOREMS FOR LÉVY OPERATORS 9 We denote by S the Schwartz class of rapidly decreasing smooth functions on Rd, and by S 1 the dual space, the class of tempered distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Again, if f P S 1 and ϕ P S , we write xf, ϕy for the value of f at ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Note that D Ă S and S 1 Ă D1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' The support of a distribution f is the smallest closed set K such that xf, ϕy “ 0 whenever ϕ P D and ϕpxq “ 0 for every x in some neighbourhood of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' We say that a distribution f corresponds to a (locally integrable) function ˜f in an open set U if xf, ϕy “ ş Rd ˜fpxqϕpxqdx whenever ϕ P D has a compact support contained in U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' In this case we do not distinguish between f and ˜f, and we use a single symbol for both the distribution f and the function ˜f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' In a similar way, we identify (locally finite) measures with the corresponding distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' The Fourier transform of ϕ P S is defined by Fϕpξq “ ş Rd e´iξxfpxqdx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' The distri- butional Fourier transform of a tempered distribution f is a tempered distribution Ff, defined by the exchange formula xFf, ϕy “ xf, Fϕy for every ϕ P S .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' The inverse Fourier transform F ´1 is defined in a similar way, with kernel p2πq´deiξx rather than e´iξx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' The spectrum of a tempered distribution is the support of its distributional Fourier transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' A (tempered) distribution f is bounded if the convolution f ˚ ϕ (defined later in this section) is bounded for every ϕ P S .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' A bounded distribution extends to a con- tinuous functional on the class of smooth functions with all derivatives integrable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' In a similar way, a (tempered) distribution f is integrable if f ˚ϕ is integrable for every ϕ P S .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' An integrable distribution extends to a continuous functional on the class of smooth functions with all derivatives bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' The Fourier transform of an integrable distribution f coincides with a continuous function, defined by Ffpξq “ xf, eξy, where eξpxq “ e´iξx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' These and many other properties of Schwartz distributions and tempered distri- butions can be found in [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Crash course in convolution and multiplication of distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' The convolution f ˚ ϕ of f P D1 and ϕ P D is a smooth function defined by f ˚ ϕpxq “ xf, ϕxy, where ϕxpyq “ ϕpx ´ yq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' The convolution of two Schwartz distributions f and g exists if there is a Schwartz distribution f f g such that pf f gq ˚ pϕ ˚ ψq “ pf ˚ ϕq ˚ pg ˚ ψq (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='1) for every ϕ, ψ P D;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' in particular, we assume that the convolution of functions f ˚ ϕ and g˚ψ in the right-hand side of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='1) exists in the usual sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' It is straightforward to check that if ϕ P D, then pf fgq˚ϕ “ f fpg˚ϕq “ pf ˚ϕqfg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Similarly, if f, g, h P D1, f fg is well-defined and h is compactly supported, then pf fgqfh “ f fpgfhq (and in particular all convolutions are well-defined).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' However, convolution of Schwartz distributions is not associative in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' For further information, we refer to Section 1 in [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' In a similar way, the convolution f ˚ ϕ of f P S 1 and ϕ P S is a smooth function determined by f ˚ ϕpxq “ xf, ϕxy, where again ϕxpyq “ ϕpx ´ yq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' The convolution of two tempered distributions f and g exists if there is a tempered distribution f f g such that pf f gq ˚ pϕ ˚ ψq “ pf ˚ ϕq ˚ pg ˚ ψq (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='2) 10 TOMASZ GRZYWNY, MATEUSZ KWA´SNICKI for every ϕ, ψ P S ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' here, too, we assume that the middle convolution in the right- hand side of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='2) exists in the usual sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' The convolution f f g is well-defined if, for example, f, g P S 1 and f has compact support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' It is again a simple exercise to check that if ϕ P S , then pf f gq ˚ ϕ “ f f pg ˚ ϕq “ pf ˚ ϕq f g, but the convolution of tempered distributions is not associative in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' We refer to Section 2 in [10] for further details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' The two notions of the convolution of distributions: D1-convolution f f g and S 1- convolution f fg, are clearly closely related to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' However, we stress that there are tempered distributions f, g P S 1 such that f f g is defined, while f f g is not;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' see Section 3 in [10] for a detailed discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' The S 1-convolution of a bounded distribution f and an integrable distribution g is a bounded distribution, and the S 1-convolution of two integrable distributions is again an integrable distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' In fact, S 1-convolution of any number of in- tegrable distributions and a bounded distribution is commutative and associative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Obviously, every Schwartz class function corresponds to a distribution which is integrable and bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' We refer to [10] for further discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' The product ψ ¨ f of f P S 1 and ψ P S is a tempered distribution defined by xψ ¨ f, ϕy “ xf, ϕψy for every ϕ P S .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' The product of two tempered distributions f and g exists if there is a tempered distribution f d g such that xf d g, ϕy “ lim nÑ8 ż Rd f ˚ αnpxqg ˚ βnpxqϕpxqdx for every ϕ P S and every sequences αn and βn of nonnegative functions in D such that the integrals of αn and βn are equal to 1 and the supports of αn and βn shrink to t0u as n Ñ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Multiplication of distributions extends multiplication of continuous functions, or a continuous function and a locally finite measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Furthermore, multiplication of distributions is a local operation: if f1 “ f2 and g1 “ g2 in an open set U, then f1dg1 “ f2dg2 in U (provided that both products exist).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Finally, if ψ P S , then ψ ¨pf dgq “ pψ ¨fqdg “ f dpψ ¨gq, but multiplication of tempered distributions is not associative in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' For a detailed discussion, we refer to [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' For every f P S 1 and ϕ P S , we have the exchange formula Fpf ˚ ϕq “ pFϕq ¨ pFfq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' As it was proved in [14], this result extends to the general case: if the convolution of tempered distributions f and g exists, then the product of their Fourier transforms is well-defined, and we have Fpf f gq “ pFfq d pFgq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Lévy operators and distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Let L be a Lévy operator, and let L: be the dual operator, as in Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' The map which assigns to ϕ P S the value L:ϕp0q is easily checked to be continuous, and hence it corresponds to some dis- tribution, that we denote by ˇL: L:ϕp0q “ xˇL, ϕy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Furthermore, if ϕxpyq “ ϕpx ´ yq, then Lϕpxq “ L:ϕxp0q “ xˇL, ϕxy, that is, Lϕpxq “ ˇL ˚ ϕpxq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Thus, L is a convolution operator, with convolution kernel ˇL P S 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' It is straightforward to see that Lϕ is integrable for every ϕ P S , and therefore ˇL is in fact an integrable distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' By a classical result in the theory of Lévy processes, the Fourier transform of ˇL is the characteristic exponent Ψ defined in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' In fact, this will serve us as the definition of the continuous function Ψ, and we will never need (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' In Section 3 we work with a general, distributional definition of an L-harmonic function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Here we prove that this definition indeed extends Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' LIOUVILLE’S THEOREMS FOR LÉVY OPERATORS 11 Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Let L be a Lévy operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' If a function h is L-harmonic in the weak sense (according to Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='2), then h is L-harmonic in the sense of Schwartz distributions: h corresponds to a Schwartz distribution which is convolvable with ˇL, the convolution kernel of L, and we have ˇL f h “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' By Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='2, h is a locally integrable function such that for every ϕ P D we have ż Rd hpyqL:ϕpyqdy “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Recall that xˇL, ϕy “ L:ϕp0q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Thus, if ˇϕpxq “ ϕp´xq, then L:ϕpxq “ ˇL ˚ ˇϕp´xq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' It follows that ż Rd hpyqˇL ˚ ˇϕp´yqdy “ 0, and in particular the integral is absolutely convergent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Replacing ϕ with ϕxpyq “ ϕpx ´ yq, we find that ż Rd hpyqˇL ˚ ϕpx ´ yqdx “ 0 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='3) for every ϕ P D and x P Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' We will momentarily show that we can use Fubini’s theorem to find that for every ϕ, ψ P D and x P Rd, ph ˚ ψq ˚ pˇL ˚ ϕqpxq “ ż Rd h ˚ ψpx ´ yqˇL ˚ ϕpyqdy “ ż Rd ˆż Rd hpx ´ y ´ zqψpzqdz ˙ ˇL ˚ ϕpyqdy “ ż Rd ˆż Rd hpx ´ y ´ zqˇL ˚ ϕpyqdy ˙ ψpzqdz “ ż Rd ˆż Rd hpyqˇL ˚ ϕpx ´ z ´ yqdy ˙ ψpzqdz “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='4) Of course, this means that ˇL f h is well-defined and equal to zero, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' In order to show absolute integrability of the double integral above, we denote by C the supremum of |ϕ|`|ψ|, and we choose R large enough, so that ϕpxq “ ψpxq “ 0 when |x| ě R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Let Brpxq denote the ball of radius r centred at x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' By (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='1), we have |ˇL ˚ ϕpxq| ď CνpBRp´xqq whenever |x| ą R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' If |x ´ y| ą 2R and |z| ă R, then |x ´ y ´ z| ą R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Thus, ż Rd |ˇL ˚ ϕpx ´ z ´ yqψpzq|dz ď C ż BRp0q |ˇL ˚ ϕpx ´ z ´ yq|dz ď C2 ż BRp0q νpBRpy ` z ´ xqqdz ď C2|BRp0q|νpB2Rpy ´ xqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' On the other hand, if ˜ϕ P D is chosen in such a way that 0 ď ˜ϕpxq ď 1 for all x P Rd, ˜ϕpxq “ 1 when |x| ď 2R and ˜ϕpxq “ 0 when |x| ě 3R, then, again by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='1), we have |ˇL ˚ ˜ϕpxq| ě νpB2Rp´xqq whenever |x| ą 3R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' It follows that if |x ´ y| ą 3R, then ż Rd |ˇL ˚ ϕpx ´ z ´ yqψpzq|dz ď C2|BRp0q||ˇL ˚ ˜ϕpx ´ yq|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' 12 TOMASZ GRZYWNY, MATEUSZ KWA´SNICKI Thus, ż RdzB3Rpxq ż Rd |hpyqˇL ˚ ϕpx ´ z ´ yqψpzq|dzdy ď C2|BRp0q| ż RdzB3Rpxq |hpyqˇL ˚ ˜ϕpx ´ yq|dy, and the right-hand side is finite by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Additionally, h is integrable over B3Rpxq, ψ is integrable over Rd, and ˇL ˚ ϕ is bounded, and hence ż B3Rpxq ż Rd |hpyqˇL ˚ ϕpx ´ z ´ yqψpzq|dzdy ă 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Absolute integrability of the double integral in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='4) follows, and the proof is com- plete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' □ Finally, we prove that under a natural growth condition, L-harmonic functions in the weak sense coincide with L-harmonic functions in the sense of tempered distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Note that with the notation introduced above, a function h is L- harmonic in the sense of tempered distributions (according to Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='4) if and only if ˇL f h is well-defined and equal to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Let L be a Lévy operator, let ν be the corresponding Lévy measure, and let B denote the unit ball in Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Let h be a function which is L-harmonic in the weak sense (according to Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='2), and such that |hpxq| ` ż RdzB |hpx ` yq|νpdyq, as a function of x P Rd, is bounded by a polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Then h is L-harmonic in the sense of tempered distributions (according to Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='1, we know that ˇL f h is well-defined and equal to zero, that is, for every ϕ, ψ P D the convolution of ˇL ˚ ϕ and h ˚ ψ is well-defined and equal to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Our goal is to prove a similar result for ϕ, ψ P S .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' We choose a function ϕ0 P D which takes values in r0, 1s, and such that ϕ0pxq “ 1 when |x| ă 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Let ˇL0 “ ϕ0 ˇL and ˇL1 “ ˇL ´ ˇL0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Then ˇL0 is a Schwartz distribution with compact support, and ˇL1 “ p1 ´ ϕ0qν is a nonnegative finite measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Since the support of ˇL0 is compact and h defines a tempered distribution, the convolution ˇL0 fh is well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Furthermore, ˇL1 is a finite nonnegative measure and, by assumption, ˇL1 ˚ |h| is bounded by a polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' This implies that ˇL1 f h is well-defined, too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Thus, ˇL f h is well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Clearly, ˇL f h “ ˇL f h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' However, by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='1, ˇL f h “ 0, and the proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Positive harmonic functions In this section we prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='3: we show that nonnegative L-harmonic functions are essentially mixtures of L-harmonic exponentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' The idea of the proof is very similar to the one used in [3]: we reduce the problem to a convo- lution equation studied by Deny in [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' However, instead of the heat kernel (the distribution of the corresponding Lévy process at a fixed time) as in [3], we use the harmonic measure (the distribution of the Lévy process at the first exit time).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' This LIOUVILLE’S THEOREMS FOR LÉVY OPERATORS 13 allows us to avoid integrability problems and thus remove unnecessary restrictions on the class of L-harmonic functions h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' For simplicity, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='3 is stated for Lévy operators on Rd which are not concentrated on a proper closed subgroup of Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' In the general case, the smallest closed subgroup G of Rd on which L is concentrated is isomorphic to Rd´k ˆ Zk for some k P t0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' , du.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' In order to cover this case, here we consider Lévy operators acting on G “ Rd ˆ Zk for arbitrary d and k, and we continue to assume that L is not concentrated on a proper subgroup of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' We omit the obvious extensions of the notions discussed above for Rd to this more general case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' As mentioned above, the key tool in our proof is the main result (Théorème 3) of [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' The original statement allows G to be an arbitrary separable locally compact abelian group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' We restrict our attention to G “ Rd ˆ Zk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' In this case the Haar measure on G is the product of the Lebesgue measure on Rd and the counting measure on Zk, and for simplicity we call it simply the Lebesgue measure on G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='1 (Deny’s theorem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Let ν be a probability measure on G “ Rd ˆ Zk such that the closed group generated by the support of ν is equal to G, and let h be a locally finite nonnegative measure on G such that h ˚ ν “ h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Then h is absolutely continuous with respect to the Lebesgue measure on G, and the density function is equal to hpxq “ ż Λ eλxµpdλq for a unique nonnegative measure µ on the set Λ of those vectors λ P Rd ˆ Rk for which the function eλpxq “ eλx satisfies eλ ˚ ν “ eλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Although we avoid as much as possible probabilistic tools, in this section some well-known results from the theory of Lévy processes play an important role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Namely, in the next paragraph we introduce two standard potential-theoretic ob- jects: the Green kernel and the harmonic measure, and in the proof of our Li- ouville’s theorem we use the probabilistic definition of the notion of L not being concentrated on a proper closed subgroup of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' For every bounded open set D Ă G there are associated Green kernel GDpx, dyq and the harmonic measure HDpx, dzq, with the following properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' If x P D, then GDpx, dyq is a finite, nonnegative measure with respect to y, concentrated on D, while HDpx, dzq is a probability measure with respect to z, concentrated on GzD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' If x R D, then GDpx, dyq is a zero measure, while HDpx, dzq “ δxpdzq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Finally, for every ϕ P D and x P G we have ϕpxq “ ż G ϕpzqHDpx, dzq ´ ż G LϕpyqGDpx, dyq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='1) While the above objects can be constructed analytically, their probabilistic descrip- tion is much simpler and far more intuitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Let Xt be the Lévy process with gen- erator L, and let Px and Ex denote the probability and expectation corresponding to the process Xt started at x P G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Let τD denote the first exit time from D: τD “ inftt P r0, 8q : Xt R Du.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' 14 TOMASZ GRZYWNY, MATEUSZ KWA´SNICKI Then HDpx, dzq is the distribution of Xt at the first exit time τD, and GDpx, dyq is the mean occupation measure up to time τD: HDpx, Aq “ PxpXτD P Aq, GDpx, Aq “ Ex ż τD 0 1ApXtqdt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Formula (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='1) is known as Dynkin’s formula, and it is one of the fundamental tools in the study of Markov processes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' see Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='1 in [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' For a bounded open set D which contains the origin, we denote ˇGDpAq “ GDp0, ´Aq and ˇHDpAq “ HDp0, ´Aq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Recall that by ˇL we denote the convolution kernel of the Lévy operator L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Then (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='1) implies that for every ϕ P D we have ϕp0q “ ż G ϕp´zq ˇHDpdzq ´ ż G Lϕp´yq ˇGDpdyq “ ˇHD ˚ ϕp0q ´ pˇL ˚ ϕq ˚ ˇGDp0q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Since ˇGD is a finite measure with compact support, the convolution ˇL f ˇGD is well- defined, and pˇL ˚ ϕq ˚ ˇGD “ pˇL ˚ ϕq f ˇGD “ ϕ ˚ pˇL f ˇGDq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Thus, ϕp0q “ ϕ ˚ ` ˇHD ´ ˇL f ˇGD ˘ p0q for every ϕ P D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' But this is another way to say that δ0 “ ˇHD ´ ˇL f ˇGD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='2) We will use the above identity in the proof of the following theorem, which is the main result of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='2 (Liouville’s theorem for positive solutions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Let L be a Lévy operator on G “ Rd ˆ Zk, which is not concentrated on a proper closed subgroup of G, and let h be a locally finite nonnegative measure on G which is L-harmonic in the weak sense, or, more generally, in the sense of Schwartz distributions (see Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Then h is absolutely continuous with respect to the Lebesgue measure on G, and the density function is equal to hpxq “ ż Λ eλxµpdλq (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='3) for a unique nonnegative measure µ on the set Λ of those vectors λ P Rd ˆ Rk for which the function eλpxq “ eλx satisfies Leλpxq “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' We remark that when we write Leλpxq “ 0 above, we mean that Leλpxq is well defined pointwise, according to the definition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' However, as we will see in the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='2, this is equivalent to eλ being L-harmonic in the weak sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Additionally, since Leλpxq “ eλpxqLeλp0q, we have Leλpxq “ 0 for every x P G whenever Leλpxq “ 0 for a single x P G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Suppose that the convolution ˇLfh is well-defined and equal to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' We consider a bounded open set D such that 0 P D, and we use the notation introduced above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Recall that the convolution of distributions is associative when one of the factors is compactly supported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Thus, 0 “ ph f ˇLq f ˇGD “ h f pˇL f ˇGDq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' LIOUVILLE’S THEOREMS FOR LÉVY OPERATORS 15 By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='2), we have ˇL f ˇGD “ ˇHD ´ δ0, and so h f p ˇHD ´ δ0q is well-defined and equal to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Clearly, h f δ0 “ h is well-defined, and so h “ 0 ` h f δ0 “ h f p ˇHD ´ δ0q ` h f δ0 “ h f ˇHD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Since h and ˇHD are nonnegative measures, we have h f ˇHD “ h ˚ ˇHD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Thus, h ˚ ˇHD “ h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' The desired result essentially follows now from Deny’s theorem (Theo- rem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='1): if the support of ˇHD generates a dense subgroup of G, then the equality h ˚ ˇHD “ h implies the desired representation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='3) of h, with Λ replaced by the set ΛD of those vectors λ P Rd for which the function eλpxq “ eλx satisfies eλ ˚ ˇHD “ eλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' A detailed argument, however, requires some care: we need to show that ΛD “ Λ, and we need to handle the case when the support of ˇHD is contained in a proper closed subgroup of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' We first show that ΛD “ Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' In fact, we prove a stronger statement: if eλ ˚ ˇHD is not everywhere infinite, then eλ f ˇL is well-defined, and the sign of eλ f ˇL is the same as the sign of eλ ˚ ˇHD ´ eλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Denote αλ,D “ eλ ˚ ˇHDp0q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Observe that eλ ˚ ˇHDpxq “ ż G eλpx ´ yq ˇHDpdyq “ eλpxq ż G eλp´yq ˇHDpdyq “ αλ,Deλpxq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' In particular, since eλ ˚ ˇHD is not everywhere infinite, αλ,D is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' We find that pαλ,D ´ 1qeλ “ eλ ˚ p ˇHD ´ δ0q “ eλ f p ˇHD ´ δ0q “ eλ f pˇL f ˇGDq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Suppose that ϕ P D is a nonnegative function which is not identically equal to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' We have pαλ,D ´ 1qeλ ˚ ϕ “ peλ f pˇL f ˇGDqq ˚ ϕ “ eλ f ppˇL f ˇGDq ˚ ϕq “ eλ f pˇL f p ˇGD ˚ ϕqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Observe that also ψ “ ˇGD ˚ ϕ is a nonnegative function in D, not identically equal to zero (because ˇGD is compactly supported, nonnegative and not identically equal to zero).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Hence, pαλ,D ´ 1qeλ ˚ ϕ “ eλ f pˇL f ψq “ eλ f pˇL ˚ ψq “ peλ f ˇLq ˚ ψ “ peλ ˚ ψq f ˇL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' However, if βλ,D “ eλ ˚ ϕp0q and γλ,D “ eλ ˚ ψp0q, then, by the argument already applied above, we have eλ ˚ ϕ “ βλ,Deλ and eλ ˚ ψ “ γλ,Deλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' And since ϕ and ψ are nonnegative and not identically equal to zero, we have βλ,D ą 0 and γλ,D ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' We conclude that pαλ,D ´ 1qβλ,Deλ “ γλ,Deλ f ˇL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' It follows that the sign of αλ,D ´ 1 is the same as the sign of eλ f ˇL, and our claim is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Additionally, λ P ΛD if and only if αλ,D “ 1, which is equivalent to eλf ˇL “ 0, that is, λ P Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' In other words, ΛD “ Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Step 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' We claim that there is no proper closed subgroup of G which contains the support of ˇHD for every bounded open set D such that 0 P D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' We use the following interpretation of our assumption that L is not concentrated on a proper closed subgroup of G: if Xt is the Lévy process generated by L, then the union of supports of all random variables Xt ´ X0 is not contained in a proper closed subgroup of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' 16 TOMASZ GRZYWNY, MATEUSZ KWA´SNICKI Suppose that A is a compact set, 0 R A, and ˇHDp´Aq “ 0 for every bounded open set D such that 0 P D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Then P0pXτD P Aq “ 0 for every bounded open set D such that 0 P D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' By considering D “ tx P GzA : |x| ă ru and passing to the limit as r Ñ 8, we find that with probability P0 one, Xt R A for all t ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Here we use the fact that with probability one, τD is equal to τGzA for r large enough (and this, in turn, is a consequence of quasi-left continuity of Lévy processes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' In particular, A is disjoint from the support of Xt ´ X0 for every t ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' It follows that the union of supports of measures ˇHD (where D is allowed to be an arbitrary bounded open set such that 0 P D) contains the union of supports of random variables X0´Xt (where t ą 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' The latter one generates a dense subgroup of G, and hence the same is true for the former one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Our claim is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Step 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Let FD denote the support of ˇHD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' We choose a countable family of bounded open sets Dn such that 0 P Dn, with the following property: the closure of the union of the supports FDn is equal to the closure of the union of FD over all bounded open sets D such that 0 P D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' In order to do that, we may apply the following procedure: choose a countable base of open sets in G, and for every basic open set G which intersects the union of all FD, choose a set Dn so that FDn intersects G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Let ˇH0 be an arbitrary convex combination of the measures ˇHDn with positive coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Then the support of ˇH0 is equal to the closure of the union of the supports FDn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' But this set contains the union of all supports FD, and by the result of the previous step, the latter is contained in no proper closed subgroup of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Hence, the support of ˇH0 is contained in no proper closed subgroup of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Since h ˚ ˇHDn “ h for every n, by Fubini’s theorem we find that h ˚ ˇH0 “ h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Thus, we may apply Deny’s theorem to conclude that h is absolutely continuous with respect to the Lebesgue measure on G, and the density function is given by hpxq “ ż Λ0 eλxµpdλq for a unique nonnegative measure µ on the set Λ0 of those vectors λ P Rd for which the function eλpxq “ eλx satisfies eλ ˚ ˇH0 “ eλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' It remains to show that Λ0 “ Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Suppose that λ P Λ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Then for every n the convolution eλ ˚ ˇHDn is not everywhere infinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' By the result of step 3, eλ f ˇL is well-defined, and the sign of eλ f ˇL is the same as the sign of eλ ˚ ˇHDn ´ eλ, regardless of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Applying again Fubini’s theorem, we find that the sign of eλ f ˇL is the same as the sign of eλ ˚ ˇH0 ´ eλ, and since λ P Λ0, the latter is zero by assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Thus, eλ f ˇL “ 0, that is, λ P Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Conversely, if λ P Λ, then eλ f ˇHD “ eλ for every bounded open set D such that 0 P D, and thus eλ f ˇH0 “ eλ, that is, λ P Λ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' □ We remark that if L is the one-dimensional Laplace operator and D “ p´r, rq, then ˇHD “ 1 2δ´r ` 1 2δr has support contained in a proper closed subgroup rZ of G “ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' However, we conjecture that for an arbitrary Lévy operator L satisfying the assumption of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='2 it is always possible find a single bounded open set D such that the support of ˇHD is not contained in a proper closed subgroup of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' For example, if L is the one-dimensional Laplace operator and D “ p´a, bq with incommensurable a, b ą 0, then ˇHD “ pa ` bq´1pbδ´a ` aδbq has support t´a, bu, which is not contained in a proper closed subgroup of G “ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' LIOUVILLE’S THEOREMS FOR LÉVY OPERATORS 17 Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Let L be an arbitrary Lévy operator on Rd, and let G be the smallest closed subgroup of Rd such that L is concentrated on G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Then L can be viewed as an operator which acts independently on each coset x ` G of G in Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Thus, L-harmonic functions or measures can be constructed independently on each coset x ` G (as long as the resulting function is locally integrable on G, or the resulting measure is locally finite on G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' On each coset, nonnegative L-harmonic measures are described by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' An unusual L-harmonic function In this section we prove the following unexpected result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='1 (counterexample to the general Liouville’s theorem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' There is a one- dimensional Lévy operator L, and a smooth function h, with the following proper- ties: (a) L is not concentrated on a proper subgroup of R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' (b) for every ε ą 0 we have lim|x|Ñ8 |x|´εhpxq “ 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' (c) Lhpxq “ 0 for every x P R (so that, in particular, the integral in the definition of Lhpxq is absolutely convergent);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' (d) h is L-harmonic in the sense of tempered distributions: ˇLf h is well-defined and equal to zero, where ˇL is the convolution kernel of L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' but h is not a polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' More precisely, we consider a one-dimensional symmetric Lévy operator L of the form Lfpxq “ f 2pxq ` 8 ÿ k“0 pk ` fpx ` xkq ` fpx ´ xkq ´ 2fpxq ˘ , where pk “ 2´k´2 and xk is a rapidly increasing sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='1 extends trivially to Rd by considering the Lévy operator which is the sum of operators L defined above acting on each coordinate xj, and the cor- responding harmonic function which is the product of hpxjq for each coordinate xj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' The construction of h is somewhat technical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' For this reason, we begin with a simpler, discrete variant of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Discrete case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' In this section, we prove the following result, which will pre- pare us for the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='2 (counterexample to Liouville’s theorem for lattice random walks).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Let pk “ 2´k´2 xk “ 22k2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' There is a doubly infinite sequence hpnq which satisfies hpnq “ 8 ÿ k“0 pk ` hpn ` xkq ` hpn ´ xkq ˘ for every n P Z, and such that for every ε ą 0, lim |n|Ñ8 hpnq |n|ε “ 0, 18 TOMASZ GRZYWNY, MATEUSZ KWA´SNICKI but hpnq is not a polynomial sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Furthermore, for every ε ą 0, we have lim |n|Ñ8 1 |n|ε 8 ÿ k“0 pkp|hpn ` xkq| ` |hpn ´ xkq|q “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' The sequence hpnq constructed in the proof is extremely sparse: we have hpnq “ 0 unless n “ 0 or |n| “ k ` xk for some k “ 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' More precisely, we set hp0q “ 1, hpnq “ ak if |n| “ k ` xk, with appropriately chosen ak, and hpnq “ 0 for all other indices n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Before we proceed with the construction of the sequence hpnq, we make the following observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' If xk`1 ě 2xk and xk ě 2k for k “ 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=', then |n ˘ xk| “ m ` xm implies that |n| “ m or |n| ě 1 2xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' In particular, the above property holds when xk “ 22k2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Suppose that |n ˘ xk| “ m ` xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' If k ă m, then |n| ě m ` xm ´ xk ě xm ´ xm´1 ě xm ´ 1 2xm “ 1 2xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' If k “ m, then either |n| “ m or |n| “ m ` 2xm ě 1 2xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Finally, if k ą m, then |n| ě xk ´ m ´ xm ě xm`1 ´ m ´ xm ě 2xm ´ m ´ xm “ xm ´ m ě 1 2xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' □ From now on we let pk “ 2´k´2 and xk “ 22k2, as in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' For convenience, let us write Lfpnq “ 8 ÿ k“0 pk ` fpn ` xkq ` fpn ´ xkq ´ 2fpnq ˘ “ 8 ÿ k“0 pk ` fpn ` xkq ` fpn ´ xkq ˘ ´ fpnq (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='1) whenever fpnq is a doubly infinite sequence such that the above series converges absolutely (so that L is a Lévy operator acting on Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' The construction of the sequence hpnq is an iterative procedure, which can be summarised as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' In the initial step, we let h´1pnq “ 1t0upnq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Next, for m “ 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' we define hmpnq “ hm´1pnq except at two values of n, namely, n “ ˘pm ` xmq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' At these values we modify hmpnq in such a way that Lhmpmq “ Lhmp´mq “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' The key observation is that, by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='3, we also have Lhmpnq “ Lhm´1pnq “ 0 if |n| ă m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Finally, we define hpnq to be the limit of hmpnq as m Ñ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' We proceed with the detailed construction of hpnq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' We let h´1pnq “ 1t0upnq, and hmpnq “ hm´1pnq ` am 1t´m´xm,m`xmupnq “ 1t0upnq ` m ÿ j“0 aj 1t´j´xj,j`xjupnq for m “ 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=', where am “ ´Lhm´1pmq pm “ ´ 1 pm ˆ 8 ÿ k“0 pk ` hm´1pm ` xkq ` hm´1pm ´ xkq ˘ ´ hm´1pmq ˙ LIOUVILLE’S THEOREMS FOR LÉVY OPERATORS 19 if m “ 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=', and a0 “ ´Lh´1p0q 2p0 “ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Finally, we define hpnq “ lim mÑ8 hmpnq “ 1t0upnq ` 8 ÿ m“0 am 1t´m´xm,m`xmupnq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Below we prove Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='2 by showing that the sequence hpnq constructed above has all the desired properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' We break the proof into three lemmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' First, we prove that hpnq is L-harmonic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' With the above definitions, we have Lhpnq “ 0 for every n P Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='3, for a fixed n P Z and all k “ 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' we have |n ˘ xk| ‰ m ` xm if m ą |n|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' It follows that hpn ˘ xkq “ hmpn ˘ xkq, where m “ |n|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Therefore, Lhpnq “ 8 ÿ k“0 pk ` hpn ` xkq ` hpn ´ xkq ˘ ´ hpnq “ 8 ÿ k“0 pk ` hmpn ` xkq ` hmpn ´ xkq ˘ ´ hmpnq (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='2) “ Lhmpnq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' We claim that Lhmpnq “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' If n “ m “ 0, then we simply have Lh0p0q “ p0 ` h0px0q ` h0p´x0q ˘ ´ h0p0q “ 2p0a0 ´ 1 “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Suppose now that n “ m ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' By the definition of hm, we have Lhmpmq “ Lhm´1pmq ` amL 1t´m´xm,m`xmupmq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Since |m ˘ xk| ‰ m ` xm if k ‰ m, and also |m ´ xm| ‰ m ` xm, we find that L 1t´m´xm,m`xmupmq “ pm 1t´m´xm,m`xmupm ` xmq ´ 1t´m´xm,m`xmupmq “ pm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Combining the above two identities and the definition am “ ´p´1 m Lhm´1pmq of am, we conclude that Lhmpmq “ Lhm´1pmq ` ampm “ 0, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='3) as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' By a similar argument (or by symmetry), we also have Lhmp´mq “ 0, and our claim is thus proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' □ In order to prove Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='2, it remains to show appropriate estimates of hpnq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Recall that for m “ 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=', pmam “ ´Lhm´1pmq “ ´ 8 ÿ k“0 pk ` hm´1pm ` xk ` hm´1pm ´ xkq ˘ ` hm´1pmq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' 20 TOMASZ GRZYWNY, MATEUSZ KWA´SNICKI Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' With the above definitions, for every ε ą 0 there is Cε such that pm|am| ď 8 ÿ k“0 pk ` |hm´1pm ` xkq| ` |hm´1pm ´ xkq| ˘ ` |hm´1pmq| ď Cεp1 ` mqε (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='4) for every m “ 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Fix ε ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Recall that pk “ 2´k´2 and xk “ 22k2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' For j large enough, say, j ą j0, we have 2j`3p1 ` jqε ď p1 ` 1 2xjqε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' We choose m0 so that 2j0`3p1 ` j0qε ď p1 ` m0qε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' With this choice, we have the following property: 2j`3p1 ` jqε ď p1 ` mqε when m ě m0 and m ě 1 2xj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='5) We choose Cε large enough, so that the desired bound (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='4) holds for m “ 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' , m0, and we prove by induction that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='4) also holds for m ą m0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Suppose that for some m ą m0 formula (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='4) holds with m replaced by any smaller number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' The first inequality in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='4) follows from the definition of am, so in order to complete the proof we only need to show the other inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' If hm´1pmq ‰ 0, then m “ j ` xj and hm´1pmq “ aj for some j ă m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Since (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='4) holds with m replaced by j, we obtain |hm´1pmq| “ |aj| ď Cεp1 ` jqε pj “ Cε2j`2p1 ` jqε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Since m ą xj, we may apply (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='5) to find that |hm´1pmq| ď Cε 2 2j`3p1 ` jqε ď Cε 2 p1 ` mqε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Similarly, if hm´1pm ˘ xkq ‰ 0, then |m ˘ xk| “ j ` xj and hm´1pm ˘ xkq “ aj for some j ă m, so that again |hm´1pm ˘ xkq| “ |aj| ď Cεp1 ` jqε pj “ Cε2j`2p1 ` jqε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='3 we find that m ě 1 2xj, and hence (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='5) again leads to |hm´1pm ˘ xkq| ď Cε 2 2j`3p1 ` jqε ď Cε 2 p1 ` mqε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' It follows that 8 ÿ k“0 pk ` |hm´1pm ` xkq| ` |hm´1pm ´ xkq| ˘ ` |hm´1pmq| ď 8 ÿ k“0 2pk ˆ Cε 2 p1 ` mqε ` Cε 2 p1 ` mqε “ Cεp1 ` mqε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' The proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' □ Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' With the above definitions, for every ε ą 0 we have lim |n|Ñ8 hpnq |n|ε “ 0 LIOUVILLE’S THEOREMS FOR LÉVY OPERATORS 21 and lim |n|Ñ8 1 |n|ε 8 ÿ k“0 pk ` |hpn ` xkq| ` |hpn ´ xkq| ˘ “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' If n ‰ 0 and |n| ‰ m ` xm for every m “ 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=', then hpnq “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' If |n| “ m ` xm for some m “ 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=', then hpnq “ am, and, by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='5, |hpnq| |n|ε “ |am| pm ` xmqε ď Cεp1 ` mqε pmpm ` xmqε “ Cε2m`2p1 ` mqε pm ` 22m2qε .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' The right-hand side clearly converges to zero as m Ñ 8, and the first part of the lemma follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' To prove the other one, we consider m “ 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' and we recall that, as in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='2) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='3), we have 8 ÿ k“0 pk ` |hpm ` xkq| ` |hpm ´ xkq| ˘ “ 8 ÿ k“0 pk ` |hmpm ` xkq| ` |hmpm ´ xkq| ˘ “ pm|am| ` 8 ÿ k“0 pkp|hm´1pm ` xkq| ` |hm´1pm ´ xkq| ˘ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='5, the right-hand side does not exceed 2Cε{2p1`mqε{2, and consequently lim mÑ8 1 mε 8 ÿ k“0 pk ` |hpm ` xkq| ` |hpm ´ xkq| ˘ “ 0, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' In a similar way (or by symmetry), lim mÑ8 1 mε 8 ÿ k“0 pk ` |hp´m ` xkq| ` |hp´m ´ xkq| ˘ “ 0, and the proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' □ Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='2 is an immediate corollary of the above series of lemmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' We prove one additional property of the sequence hpnq, which in fact proves Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='1 without the assumption that L is not concentrated on a proper closed subgroup of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' With the above definitions, the convolution kernel ˇL of the Lévy op- erator L is S 1-convolvable with the measure H “ 8 ÿ n“´8 hpnqδn, and ˇL f H “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' By the last assertion of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='2, for a given ε ą 0, there is a constant C1 such that 8 ÿ k“0 pk ` |hpn ` xkq| ` |hpn ´ xkq| ˘ ` |hpnq| ď C1p1 ` |n|qε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' 22 TOMASZ GRZYWNY, MATEUSZ KWA´SNICKI Furthermore, if ϕ, ψ P S and x P R is fixed, then there is a constant C2 such that |ϕ| ˚ |ψ|px ´ nq ď C2p1 ` |n|q´2´ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' It follows that ż R 8 ÿ n“´8 ˆ 8 ÿ k“0 pk ` |hpn ` xkq| ` |hpn ´ xkq| ˘ ` |ϕpnq| ˙ |ϕpy ´ nq||ψpx ´ yq|dy ă 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Thus, we may apply the result of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='2 and Fubini’s theorem to find that ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='0 “ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='ż ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='R ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='ÿ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='n“´8 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='´ hpnq ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='˙ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='ϕpy ´ nqψpx ´ yqdy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='“ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='ż ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='R ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='8 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='“ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='ż ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='R ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='ˆ 8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='ÿ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='k“0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='pk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='` ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='ϕpy ` xkq ` ϕpy ` xkq ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='˘ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='´ ϕpyq ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='˙ˆ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='ÿ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='n“´8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='hpnqψpx ´ y ´ nq ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='˙ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='dy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='“ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='ż ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='R ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='pˇL ˚ ϕqpyqpH ˚ ψqpx ´ yqdy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='“ pˇL ˚ ϕq ˚ pH ˚ ψqpxq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Multiplication of distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Before dealing with the case of Lévy op- erators on R, we state the following counterintuitive result about S 1-product of distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' There is a strictly positive continuous function f on R, and a nonzero tempered distribution g on R, such that f d g “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' LIOUVILLE’S THEOREMS FOR LÉVY OPERATORS 23 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Consider the one-dimensional Lévy operator L defined in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='1), its convolu- tion kernel ˇL, and the corresponding characteristic exponent Ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Thus, Ψpξq “ 2 8 ÿ k“0 pk ` 1 ´ cospxkξq ˘ “ 8 ÿ k“0 2´k´1` 1 ´ cosp22k2ξq ˘ is a Weierstrass-type nowhere differentiable function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Furthermore, let hpnq be the doubly-infinite sequence from Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='2, and let H “ 8 ÿ n“´8 hpnqδn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='7, ˇL and H are S 1-convolvable, and ˇL f H “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' The desired result essentially follows by the exchange formula: we have Ψ d FH “ ´F ˇL d FH “ FpˇL f Hq “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Clearly, Ψ is a nonnegative continuous function, and by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='2), Ψ is strictly positive everywhere except 2πZ (recall that ˇL is concentrated on Z, and it has an atom at 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' On the other hand, FH is a periodic tempered distribution with period 2π, and since hpnq is not a polynomial sequence, the support of FH is not contained in 2πZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Thus, to get the desired result, we only need to correct Ψ and ˇL so that Ψ is strictly positive everywhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' One way to do this would be to replace ˇL by ˇL ´ δ0 and repeat the construction of hpnq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' However, there is a simpler solution: it is sufficient to define f “ Ψ ` ϕ, g “ ψ ¨ FH, where ψ P S is chosen in such a way that ψ “ 0 on 2πZ and ψ¨FH is not identically zero, while ϕ P S is a nonnegative function such that ϕ ą 0 on 2πZ, but ϕ¨ψ “ 0 on R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Indeed: f is then a strictly positive continuous function, g is a nonzero tempered distribution, and f d g “ Ψ d pψ ¨ FHq ` ϕ ¨ pψ ¨ FHq “ ψ ¨ pΨ d FHq ` pϕ ¨ ψq ¨ FH “ 0, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Continuous case: proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' The proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='1 is very similar to the argument used in the discrete case, in the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Thus, we omit some details and leave them to the interested reader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' We consider a Lévy operator similar to the one given in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='1), but with an addi- tional one-dimensional Laplace operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' That is, we consider Lfpxq “ f 2pxq ` 8 ÿ k“0 pk ` fpx ` xkq ` fpx ´ xkq ´ 2fpxq ˘ “ f 2pxq ` 8 ÿ k“0 pk ` fpx ` xkq ` fpx ´ xkq ˘ ´ fpxq, where again pk “ 2´k´2, and xk is a rapidly increasing sequence to be spec- ified later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' The construction of an L-harmonic function hpxq is very similar to the construction of the sequence hpnq in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='1, but for each n P Z we re- place the single number hpnq by an appropriate compactly supported function hpxq, 24 TOMASZ GRZYWNY, MATEUSZ KWA´SNICKI x P pn ´ 1 2, n ` 1 2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Additionally, we specify the value of xk on the fly, but in any case we will have x0 “ 1, xk`1 ě 2xk and xk ě 4k for k “ 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='6) so that, in particular, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='3 applies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' We consider a nonzero smooth even function h´1 with support contained in p´ 1 2, 1 2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Then, in step m “ 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=', we define hm by appropriately modifying hm´1 on the intervals |x| P pm ` xm ´ 1 2, m ` xm ` 1 2q in such a way that hm is smooth, even, and Lhmpxq “ 0 when |x| P pm ´ 1 2, m ` 1 2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Let us describe more precisely step m “ 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' of the construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Suppose that hm´1 and x0, x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' , xm´1 have already been defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' For z P p´ 1 2, 1 2q we define hmpxq “ hm´1pxq ` ϕmpx ´ m ´ xmq ` ϕmp´x ´ m ´ xmq “ h´1pxq ` m ÿ j“0 ` ϕjpx ´ j ´ xjq ` ϕjp´x ´ j ´ xjq ˘ , where xm is specified below, ϕmpzq “ ´Lhm´1pz ` mq pm “ ´Lhm´1p´z ´ mq pm “ h2 m´1pz ` mq ` m´1 ÿ k“0 pk ` hm´1pz ` m ` xkq ` hm´1pz ` m ´ xkq ˘ ´ hm´1pz ` mq if m “ 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=', and ϕ0pzq “ ´Lh´1pzq 2p0 “ ´2h2 ´1pzq ` 2h´1pzq if m “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' For convenience, we set ϕmpzq “ 0 when |z| ě 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Then ϕm is a smooth function with compact support in p´ 1 2, 1 2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Note that in the above calculation of Lhm´1pz ` mq we truncated the series at k “ m ´ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' This is because, as we now prove, all terms corresponding to k ě m are zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Indeed: by construction, we have hm´1pxq “ 0 when |x| ě pm ´ 1q ` xm´1 ` 1 2, and if k ě m, then, by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='6), |m ˘ xk| ě xk ´ m ě xm ´ m ě 1 2xm ` xm´1 ´ m ě m ` xm´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Thus, hm´1pz ` m ˘ xkq “ 0, as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' In other words, the values of xk for k ě m are not needed in order to evaluate ϕmpzq “ ´p´1 m Lhm´1pz ` mq, as long as condi- tion (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='6) is satisfied, and this allows us to specify the value of xm only in step m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' If m ą 0, then we choose xm to be an integer large enough, so that xm ě 2xm´1, xm ě 4m, xm ě 22m2, and log xm ě supt|ϕmpzq| ` |ϕ2 mpzq| : z P p´ 1 2, 1 2qu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='7) If m “ 0, we simply let x0 “ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' We now define hpxq “ lim mÑ8 hmpxq “ h´1pxq ` 8 ÿ m“0 ` ϕmpx ´ m ´ xmq ` ϕmp´x ´ m ´ xmq ˘ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' LIOUVILLE’S THEOREMS FOR LÉVY OPERATORS 25 We stress that h is a smooth function with a very sparse support;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' namely, we have hpxq “ $ ’ ’ ’ ’ & ’ ’ ’ ’ % ϕ0pxq if x P p´ 1 2, 1 2q, ϕmpx ´ m ´ xmq if x P pm ` xm ´ 1 2, m ` xm ` 1 2q with m “ 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' , ϕmp´x ´ m ´ xmq if x P p´m ´ xm ´ 1 2, ´m ´ xm ` 1 2q with m “ 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=', 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' We now follow closely the arguments used in the discrete case in Sections 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' With the above definitions, we have Lhpxq “ 0 for every x P R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' The argument is almost exactly the same as in the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='4, except that we replace n by n ` z, with z P p´ 1 2, 1 2q, and we need to consider z “ 1 2 separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' By construction, hpn` 1 2q “ h2pn` 1 2q “ 0 for every n P Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Thus, Lhpn` 1 2q “ 0, and we only need to show that Lhpz ` nq “ 0 when n P Z and z P p´ 1 2, 1 2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='3, for every k “ 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' we have |n ˘ xk| ‰ m ` xm if m ą |n|, and thus hpz ` n ˘ xkq “ hmpz ` n ˘ xkq, where m “ |n|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Therefore, Lhpz ` nq “ h2pz ` nq ` 8 ÿ k“0 pk ` hpz ` n ` xkq ` hpz ` n ´ xkq ˘ ´ hpz ` nq “ h2 mpz ` nq ` 8 ÿ k“0 pk ` hmpz ` n ` xkq ` hmpz ` n ´ xkq ˘ ´ hmpz ` nq “ Lhmpz ` nq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' We claim that Lhmpz ` nq “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' If n “ m “ 0, then Lh0pzq “ h2 0p0q ` p0 ` h0pz ` x0q ` h0pz ´ x0q ˘ ´ h0pzq “ h2 ´1pzq ` 2p0ϕ0pzq ´ h´1pzq “ 0 by the definitions of h0 and ϕ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Suppose now that n “ m ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' By the definition of hm, we have Lhmpz ` mq “ Lhm´1pz ` mq ` Lϕmpz ´ xmq ` Lϕmpz ` 2m ` xmq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Note that |z ´ xm| ą 1 2, |z ´ xm ˘ xk| ą 1 2 if k ‰ m, and also |z ´ 2xm| ą 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Thus, Lϕmpz ´ xmq “ ϕ2 mpz ´ xmq ´ ϕmpz ´ xmq ` 8 ÿ k“0 pk ` ϕmpz ´ xm ` xkq ` ϕmpz ´ xm ´ xkq ˘ “ pmϕmpzq “ ´Lhm´1pm ` zq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' 26 TOMASZ GRZYWNY, MATEUSZ KWA´SNICKI Similarly, |z ` 2m ` xm| ą 1 2 and |z ` 2m ` xm ˘ xk| ą 1 2 for every k, and hence Lϕmpz ` 2m ` xmq “ ϕ2 mpz ` 2m ` xmq ´ ϕmpz ` 2m ` xmq ` 8 ÿ k“0 pk ` ϕmpz ` 2m ` xm ` xkq ` ϕmpz ` 2m ` xm ´ xkq ˘ “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Therefore, Lhmpz ` mq “ Lhm´1pz ` mq ´ Lhm´1pz ` mq “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' By a similar argument (or by symmetry), we also have Lhmp´z ´ mq “ 0, and our claim is thus proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' □ Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' If xk grows sufficiently fast, then, with the above definitions, for every ε ą 0 there is a constant Cε such that pm|ϕmpzq| ď |h2 m´1pz ` mq| ` 8 ÿ k“0 pk ` |hm´1pz ` m ` xkq| ` |hm´1px ´ xkq| ˘ ` |hm´1pz ` mq| ď Cεp1 ` mqε (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='8) for m “ 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' and z P p´ 1 2, 1 2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' The proof is actually simpler than the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='5, due to flexibility in the choice of xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' The first inequality in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='8) follows by the definition of ϕm, and so we are left with the proof of the other inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Let ε ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' By (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='7), there is a constant Cε such that supt|ϕmpzq| ` |ϕ2 mpzq| : z P p´ 1 2, 1 2qu ď Cεp1 ` xmqε (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='9) for m “ 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Fix m “ 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' If hm´1pz ` mq ‰ 0 for some z P p´ 1 2, 1 2q, then m “ j ` xj and hm´1pz ` mq “ ϕjpzq for some j ă m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' By (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='9), we obtain |hm´1pz ` mq| ` |h2 m´1pz ` mq| “ |ϕjpzq| ` |ϕ2 jpzq| ď Cεp1 ` xjqε ď Cεp1 ` mqε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Similarly, if hm´1pz`m˘xkq ‰ 0, then |m˘xk| “ j`xj and hm´1pz`m˘xkq “ ϕjp˘zq for some j ă m, so that again |hm´1pz ` m ˘ xkq| “ |ϕjp˘zq| ď Cεp1 ` xjqε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='3 we find that m ě 1 2xj, and hence |hm´1pz ` m ˘ xkq| ď Cεp1 ` 2mqε ď 2εCεp1 ` mqε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' It follows that |h2 m´1pz ` mq| ` 8 ÿ k“0 pk ` |hm´1pz ` m ` xkq| ` |hm´1pz ` m ´ xkq| ˘ ` |hm´1pz ` mq| ď Cεp1 ` mqε ` 2εCεp1 ` mqε ď 21`εCεp1 ` mqε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' The proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' □ Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' With the above definitions, for every ε ą 0 we have lim |x|Ñ8 |hpxq| ` |h2pxq| |x|ε “ 0, LIOUVILLE’S THEOREMS FOR LÉVY OPERATORS 27 and lim |x|Ñ8 1 |x|ε 8 ÿ k“0 pk ` |hpx ` xkq| ` |hpx ´ xkq| ˘ “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' The proof is very similar to the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' By definition, if n P Z, z P p´ 1 2, 1 2q, |n| ě 2 and hpz ` nq ‰ 0 or h2pz ` nq ‰ 0, then |n| “ m ` xm for some m “ 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=', and thus, by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='7), |hpz ` nq| ` |h2pz ` nq| |z ` n|ε “ |ϕmpzq| ` |ϕ2 mpzq| |z ` n|ε ď log xm |n|ε ď log xm xεm .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' The right-hand side clearly converges to zero as m Ñ 8, and the first part of the lemma follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' To prove the other one, we consider m “ 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' and z P p´ 1 2, 1 2q, and we recall that, as in the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='9, we have 8 ÿ k“0 pk ` |hpz ` m ` xkq| ` |hpz ` m ´ xkq| ˘ “ 8 ÿ k“0 pk ` |hmpz ` m ` xkq| ` |hmpz ` m ´ xkq| ˘ “ pm|ϕmpzq| ` 8 ÿ k“0 pkp|hm´1pz ` m ` xkq| ` |hm´1pz ` m ´ xkq| ˘ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='10, the right-hand side does not exceed Cε{2p1 ` mqε{2, and thus lim mÑ8 1 mε 8 ÿ k“0 pk ` |hpz ` m ` xkq| ` |hpz ` m ´ xkq| ˘ “ 0, uniformly with respect to z P p´ 1 2, 1 2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' By a similar argument (or by symmetry), lim mÑ8 1 mε 8 ÿ k“0 pk ` |hp´z ´ m ` xkq| ` |hp´z ´ m ´ xkq| ˘ “ 0 uniformly with respect to z P p´ 1 2, 1 2q, and the proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' □ Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' With the above definitions, the function h corresponds to a tempered distribution, which is S 1-convolvable with the convolution kernel ˇL of the Lévy operator L, and we have ˇL f h “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' The argument is virtually the same as in the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='11, for a given ε ą 0, there is a constant C1 such that |h2pzq| ` 8 ÿ k“0 pk ` |hpz ` xkq| ` |hpz ´ xkq| ˘ ` |hpzq| ď C1p1 ` |z|qε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Furthermore, if ϕ, ψ P S and x P R is fixed, then there is a constant C2 such that |ϕ| ˚ |ψ|px ´ zq ď C2p1 ` |z|q´2´ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' It follows that ż R ż R ˆ |h2pzq| ` 8 ÿ k“0 pk ` |hpz ` xkq| ` |hpz ´ xkq| ˘ ´ hpzq ˙ |ϕpy ´ zq||ψpx ´ yq|dzdy ă 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' 28 TOMASZ GRZYWNY, MATEUSZ KWA´SNICKI Thus, we may apply Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Fubini’s theorem and integration by parts to find ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='that ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='0 “ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='ż ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='R ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='ż ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='R ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='ˆ ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='ż ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='R ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='hpzqϕpyqψpx ´ y ´ zqdzdy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='“ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='ż ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='R ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='ˆ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='ϕ2pyq ` ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='ÿ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='k“0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='pk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='` ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='ϕpy ` xkq ` ϕpy ` xkq ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='˘ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='´ ϕpyq ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='˙ˆż ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='R ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='hpzqψpx ´ y ´ zqdz ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='˙ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='dy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='“ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='ż ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='R ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='pˇL ˚ ϕqpyqph ˚ ψqpx ´ yqdy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='“ pˇL ˚ ϕq ˚ ph ˚ ψqpxq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' and the proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' □ Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='1 follows directly from the above series of lemmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Signed harmonic functions In this final section of the article we prove various variants of Liouville’s theorem for signed polynomially bounded functions using Fourier transform approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' We LIOUVILLE’S THEOREMS FOR LÉVY OPERATORS 29 begin with an abstract result in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='1, and then, by choosing an appropriate Wiener-type algebra W , we obtain specific Liouville’s theorems as corollaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' General result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Recall that according to Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='5, an algebra W of con- tinuous functions on Rd is a Wiener-type algebra if every Φ P W corresponds to a tempered distribution, ϕΦ P W whenever ϕ P S and Φ P W , and the following variant of Wiener’s 1{f theorem holds: if K Ď Rd is a compact set, Φ P W and Φpξq ‰ 0 for every ξ P K, then there is ˜Φ P W such that Φpξq˜Φpξq “ 1 for every ξ P K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' A tempered distribution Ψ is said to belong to W locally on an open set U if for every compact set K Ď U there is a distribution Φ P W such that Ψ “ Φ in a neighbourhood of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' In particular, in this case the restriction of Ψ to U is given by a continuous function, and if ϕ P S has a compact support contained in U, then ϕΨ is an element of W .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Using the notation introduced in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='2, Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='6 reads as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' A tempered distribution H is said to act on W if for every Φ, Ψ P W we have: pH d Φq d Ψ “ H d pΦΨq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' The main result of this section is the following extension of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Note that we do not require ˇL to be the convolution kernel of a Lévy operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='1 (Liouville’s theorem factory).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Let W be an Wiener-type algebra of continuous functions on Rd, and let ˇL be a tempered distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Suppose that F ˇL belongs to W locally on an open set U, and assume that F ˇLpξq ‰ 0 for every ξ P U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Let h be a tempered distribution such that Fh acts on W , and such that ˇL f h “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Then the spectrum of h is contained in RdzU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' In other words: the restriction of Fh to U is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' In particular, if U “ Rdzt0u, then h is a polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Fix a compact subset K of U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Suppose that the spectrum of ϕ P S is a compact subset of U and Fϕpξq ‰ 0 for every ξ P K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Define f “ ´ˇL ˚ ϕ, Ψ “ ´F ˇL, Φ “ Ff and H “ Fh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Observe that Φ “ Ff “ ´FpˇL ˚ ϕq “ ´Fϕ ¨ F ˇL “ Fϕ ¨ Ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' By assumption, Fϕ P S , the support of Fϕ is a compact subset of U, and Ψ belongs to W locally on U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Thus, Φ P W .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Furthermore, 0 “ pˇL f hq ˚ ϕ “ h f pˇL ˚ ϕq “ h f f, and hence, by the Fourier exchange formula, 0 “ Fph f fq “ H d Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' On the other hand, Φpξq “ FϕpξqΨpξq ‰ 0 for every ξ P K, and therefore there is ˜Φ P W such that Φpξq˜Φpξq “ 1 for every ξ P K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Since H acts on W , we conclude that 0 “ pH d Φq d ˜Φ “ H d pΦ˜Φq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Recall that Φpξq˜Φpξq “ 1 for every ξ P K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' By definition, multiplication of distribu- tions is a local operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Hence, in the interior of K, we have H “ H d 1 “ H d pΦ˜Φq “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Since K is an arbitrary subset of U, we conclude that H “ 0 on U, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' 30 TOMASZ GRZYWNY, MATEUSZ KWA´SNICKI If U “ Rdzt0u, then H “ Fh is supported in RdzU “ t0u, and hence h is necessar- ily a polynomial (see Sections 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='6 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='2 in [25]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Operators with smooth symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' It is straightforward to verify that W “ S , the Schwartz class of rapidly decaying functions, is a Wiener-type algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Clearly, Ψ belongs to W locally on U if and only if Ψ is smooth on U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Finally, if Φ, Ψ P W and H is an arbitrary tempered distribution, we have pΦΨq¨H “ Φ¨pΨ¨Hq, and so every tempered distribution acts on W .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' This leads to the following statement, which is a minor extension of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='2 in [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='2 (Liouville’s theorem for operators with smooth symbols).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Let ˇL be a tempered distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Suppose that on an open set U, F ˇL corresponds to a smooth function without zeroes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Let h be a tempered distribution such that ˇLfh “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Then the spectrum of h is contained in RdzU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' In particular, if U “ Rdzt0u, then h is a polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Let α P p0, 2q and let L “ ´p´∆qα{2 be the fractional Laplace opera- tor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' If ˇL denotes the corresponding convolution kernel, then F ˇLpξq “ ´|ξ|α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Since F ˇL is smooth in Rdzt0u, we find that the only polynomially bounded L-harmonic functions (in the sense of tempered distributions) are polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' However, Lh is not well-defined if h is a polynomial of degree rαs or higher, and so all L-harmonic functions are constant when α P p0, 1s, and all L-harmonic functions are affine when α P p1, 2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' This is exactly the main result of [12] (Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='1 therein) and of [7] (Theo- rem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='3 therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Bounded L-harmonic functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Let W be the Wiener algebra: the class of Fourier transforms of integrable functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' It is straightforward to see that W satisfies the first two conditions of Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='5, and the last one is a variant of the classical Wiener’s 1{f theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' We refer to the Division Lemma in Section 3 in [8] for the proof in dimension one, and to Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='4 below for a more general statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' If ˇL is an integrable distribution, r ą 0 and ϕ P S satisfies Fϕpξq “ 1 when |ξ| ă r, then ˇL ˚ ϕ is an integrable function with Fourier transform Fϕ ¨ F ˇL in W .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Since r is arbitrary, we see that F ˇL belongs to W locally on Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' In particular, Fourier symbols of Lévy operators belong to W locally on Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Observe that if h is a bounded distribution and Φ, Ψ P W , then there are inte- grable functions f, g such that Φ “ Ff and Ψ “ Fg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Since hfpf ˚gq “ hfpf fgq “ ph f fq f g, by the exchange formula we find that Fh d pΦΨq “ pFh d Φq d Ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Thus, Fourier transforms of bounded distributions act on W .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' The above observations immediately lead to the following minor extension of the general Liouville’s theorem for bounded L-harmonic functions given in Theo- rem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='1 in [1] and in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='4 in [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='4 (Liouville’s theorem for bounded functions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Let ˇL be an integrable distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Suppose that F ˇL, which is necessarily a continuous function, has no zeroes in an open set U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Let h be a bounded distribution such that ˇL f h “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Then the spectrum of h is contained in RdzU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' In particular, if U “ Rdzt0u, then h is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' LIOUVILLE’S THEOREMS FOR LÉVY OPERATORS 31 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Operators with finite generalised moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' We define the following class of Wiener-type algebras, parameterised by auxiliary functions Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Examples of ad- missible functions Y are discussed at the end of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Here we remark that a typical choice is Y pxq « |x|α, where α ą 0 is a parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Suppose that Y is a nonnegative function on Rd with the following properties: (a) we have 1 ` Y px ` yq ď p1 ` Y pxqqp1 ` Y pyqq for every x, y P Rd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' (b) for some positive constants c, q we have Y pxq ď cp1 ` |x|qq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Define W 8 Y to be the class of Fourier transforms of integrable functions f such that also Y f is integrable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Our main goal in this section is to prove that W 8 Y is a Wiener-type algebra ac- cording to Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' The set W 8 Y defined in Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='5 is an algebra of continuous functions which satisfies conditions (a) and (b) of Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Clearly, W 8 Y is a linear space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' If Φ P W 8 Y , then Φ is the Fourier transform of an integrable function f, and hence Φ is a bounded continuous function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Thus, W 8 Y is a class of continuous functions, and every Φ P W 8 Y corresponds to a tempered distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' We claim that W 8 Y is indeed an algebra of functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' If Φ, Ψ P W 8 Y , then Φ “ Ff and Ψ “ Fg for some integrable functions f, g such that also Y f, Y g are inte- grable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' It follows that h “ f ˚ g is an integrable function, and by condition (a) in Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='5, ż Rd |p1 ` Y pxqqhpxq|dx “ ż Rd ˇˇˇˇp1 ` Y pxqq ż Rd fpyqgpx ´ yqdy ˇˇˇˇdx ď ż Rd ż Rdp1 ` Y pxqq|fpyq|gpx ´ yq|dydx “ ż Rd ż Rdp1 ` Y py ` zqq|fpyq||gpzq|dydz ď ż Rdp1 ` Y pyqq|fpyq|dy ¨ ż Rdp1 ` Y pzqq|gpzq|dz ă 8, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='1) that is, also Y h is integrable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Since ΦΨ “ Ff ¨ Fg “ Fpf ˚ gq “ Fh, we conclude that ΦΨ P W 8 Y , as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' We now show that W 8 Y contains S .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Indeed: if ϕ P S and ψ “ F ´1ϕ, then ψ P S , and so, in particular, ψ is integrable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Furthermore, by condition (b) in Defi- nition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='5, p1 ` |x|qd`1Y pxqψpxq is a bounded function of x P Rd, and so in particular Y ψ is integrable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Thus, ϕ “ Fψ indeed belongs to W 8 Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' It remains to observe that if ϕ P S and Φ P W 8 Y , then ϕ P W 8 Y , and therefore ϕΦ P W 8 Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' □ In order to prove that W 8 Y is a Wiener-type algebra, we only need to verify that W 8 Y satisfies condition (c) of Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='5, that is, a variant of Wiener’s 1{f theo- rem holds in W 8 Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' While this result is known (see [4] for further discussion and references), we provide a complete proof in order to prepare the reader for a similar, but slightly more involved argument in the next section, in the proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' 32 TOMASZ GRZYWNY, MATEUSZ KWA´SNICKI Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Let W 8 Y be the set defined in Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Suppose that K Ď Rd is a compact set, Φ P W 8 Y and Φpξq ‰ 0 for every ξ P K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Then there is ˜Φ P W 8 Y such that Φpξq˜Φpξq “ 1 for every ξ P K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' We follow the proof of the classical Wiener’s 1{f lemma given in [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Let us denote by } ¨ }p the usual norm in LppRdq, where p P r1, 8s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' We begin with the following elementary observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Clearly, if Φ “ Ff and f is integrable, then }Φ}8 ď }f}1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Conversely, for every k “ 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' there is a constant cd,k such that if Φ is smooth, both Φ and ∆kΦ are integrable, and f “ F ´1Φ, then suptp1 ` |x|2kq|fpxq| : x P Rdu ď cd,k}Φ ` p´∆qkΦ}1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' In particular, if 2k ą d, we find that f is integrable, while if 2k ą d ` q with q as in condition (b) in Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='5, then Y f is integrable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' It follows that if 2k ą d ` q, then there is a constant cd,k such that whenever Φ is smooth, and Φ and ∆kΦ are integrable, then Φ P W 8 Y and }p1 ` Y qf}1 ď cd,k}Φ ` p´∆qkΦ}1, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='2) where f “ F ´1Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' We return to the proof of the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Suppose that Φ P W 8 Y , that is, Φ “ Ff for an integrable function f such that also Y f is integrable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Suppose furthermore that K is a compact set and Φpξq ‰ 0 for every ξ P K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Let ε ą 0 be small enough, so that |Φpξq| ą 3ε for every ξ in some bounded neighbourhood U of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' With no loss of generality we assume that ε ă 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Choose ϕ P S so that the support of ϕ is a compact subset of U and ϕpξq “ 1 for ξ P K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Our goal is to prove that ˜Φpξq “ ϕpξq Φpξq is an element of W 8 Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' here, of course, ˜Φpξq “ 0 for ξ P RdzU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Once this is proved, we have Φpξq˜Φpξq “ ϕpξq “ 1 for every ξ P K, and so ˜Φ has the desired property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Define gpxq “ fpxq when |x| ă r and gpxq “ 0 otherwise, where r is large enough, so that }f ´ g}1 ` }Y f ´ Y g}1 ă ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Clearly, g and Y g are integrable, and therefore Ψ “ Fg is an element of W 8 Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Furthermore, }Φ ´ Ψ}8 ď }f ´ g}1 ă ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' In particular, for ξ P U we have |Ψpξq| ě |Φpξq| ´ |Φpξq ´ Ψpξq| ą 3ε ´ ε “ 2ε, and ˇˇˇˇ Ψpξq ´ Φpξq Ψpξq ˇˇˇˇ ă ε 2ε “ 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' It follows that for ξ P U, 1 Φpξq “ 8 ÿ n“0 pΨpξq ´ Φpξqqn pΨpξqqn`1 , LIOUVILLE’S THEOREMS FOR LÉVY OPERATORS 33 and therefore for every ξ P Rd, ˜Φpξq “ ϕpξq Φpξq “ 8 ÿ n“0 pΨpξq ´ Φpξqqn ϕpξq pΨpξqqn`1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='3) We study the terms pΨ ´ Φqn and ϕ{Ψn`1 separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Recall that Ψ ´ Φ “ Fpg ´ fq and }p1 ` Y qpg ´ fq}1 “ }g ´ f}1 ` }Y g ´ Y f}1 ă ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' If pg ´ fq˚n denotes the n-fold convolution of g ´ f, then Fpg ´ fq˚n “ pΨ ´ Φqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Furthermore, using condition (a) in Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='5 as in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='1), we find that ››p1 ` Y qpg ´ fq˚n›› 1 ď ` }p1 ` Y qpg ´ fq}1 ˘n ă εn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='4) This is the desired bound for pΨ ´ Φqn, and we turn to the estimate of ϕ{Ψn`1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Since Ψ “ Fg and g has compact support, Ψ is smooth in Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Additionally, |Ψpξq| ě 2ε ą 0 for ξ P U, and ϕpξq “ 0 for ξ outside a compact subset of U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' It follows that ϕ{Ψn`1 is smooth on Rd, and equal to zero in RdzU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' In particular, by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='2), we find that ϕ{Ψn`1 P W 8 Y , and if hn “ F ´1pϕ{Ψn`1q, then ››p1 ` Y qhn ›› 1 ď cd,k ››ϕ{Ψn`1 ` p´∆qkpϕ{Ψn`1q ›› 1, where k is a fixed sufficiently large positive integer and cd,k is an appropriate con- stant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' By applying the product rule to p´∆qkpϕ{Ψn`1q, we obtain a fixed number of terms, each of which is a product of: the derivative of ϕ of some order j, where 0 ď j ď 2k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' a finite number of derivatives of Ψ of total order 2k ´ j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Ψ´n´1´i, where 0 ď i ď 2k ´ j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' and a coefficient, which is an appropriate polynomial of n of degree at most 2k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Furthermore, |Ψpξq| ě 2ε for ξ P U, and 2ε ă 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Thus, Ψ´n´1´ipξq ď p2εq´n´1´2k when ξ P U and 0 ď i ď 2k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' It follows that there is a constant cd,k,ϕ,Ψ such that ››p1 ` Y qhn ›› 1 ď cd,k,ϕ,Ψ|U|p1 ` n2kqp2εq´n´1´2k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='5) This is the desired estimate for ϕ{Ψn`1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' We combine (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='4) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='5) as in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='1): ››p1 ` Y qpg ´ fq˚n ˚ hn ›› 1 ď ››p1 ` Y qpg ´ fq˚n›› 1 ¨ ››p1 ` Y qhn ›› 1 ď cd,k,ϕ,Ψ|U|p1 ` nkqp2εq´1´2k ¨ 2´n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' By Fubini’s theorem, we obtain ››››p1 ` Y q 8 ÿ n“0 pg ´ fq˚n ˚ hn ›››› 1 ă 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' In particular, the series ˜f “ 8 ÿ n“0 pg ´ fq˚n ˚ hn 34 TOMASZ GRZYWNY, MATEUSZ KWA´SNICKI defines an integrable function such that also Y ˜f is integrable, and again by Fubini’s theorem we find that F ˜f “ 8 ÿ n“0 F ` pg ´ fq˚n ˚ hn ˘ “ 8 ÿ n“0 F ` pg ´ fq˚n˘ Fhn “ 8 ÿ n“0 pΨ ´ Φqnpϕ{Ψn`1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Thus, by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='3), F ˜f “ ˜Φ, and therefore ˜Φ P W 8 Y , as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' □ Lemmas 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='6 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='7 immediately lead to the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' The set W 8 Y defined in Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='5 is a Wiener-type algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Suppose that Y satisfies the conditions of Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='5, ˇY pxq “ Y p´xq, and ˇL is an integrable distribution such that ˇLf ˇY is well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Then for every ϕ P S the integrable function ˇL˚ϕ is convolvable with ˇY , and so, in particular, }pˇL˚ϕq¨Y }1 ă 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Thus, Fϕ ¨ F ˇL “ FpˇL ˚ ϕq P W 8 Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' By choosing ϕ such that Fϕpξq “ 1 for ξ in a given compact set, we find that F ˇL belongs to W 8 Y locally on Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' We claim that if h is a function on Rd such that h{p1 ` ˇY q is bounded, then h cor- responds to a tempered distribution and Fh acts on W 8 Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Indeed: by condition (b) in Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='5, h is bounded by some polynomial and hence it corresponds to a tempered distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Furthermore, if Φ, Ψ P W 8 Y , then Φ “ Ff and Ψ “ Fg for some integrable functions f, g such that also Y f and Y g are integrable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Thus, by condition (a) in Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='5 and Fubini’s theorem, we find that |h| ˚ |f| ˚ |g|pxq “ ż Rd ż Rd hpx ´ y ´ zqfpyqgpzqdydz ď p1 ` Y p´xqq ż Rd ż Rd hpx ´ y ´ zq 1 ` Y p´x ` y ` zq p1 ` Y pyqqfpyqp1 ` Y pzqqgpzqdydz ď p1 ` ˇY pxqq}h{p1 ` ˇY q}8}p1 ` Y qg}1}p1 ` Y qh}1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Therefore, by Fubini’s theorem, we have h ˚ pf ˚ gq “ ph ˚ fq ˚ g, and in fact, due to condition (b) in Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='5, h f pf ˚ gq “ ph f fq f g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Applying the exchange formula, we conclude that FH d pΦ ¨ Ψq “ pFH d Φq d Ψ, which completes the proof of our claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' As an immediate corollary of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='1 and Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='8, as well as the two properties discussed above, and after exchanging the roles of Y and ˇY , we obtain the following variant of Liouville’s theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='9 (Liouville’s theorem under generalised moment condition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Suppose that Y satisfies the conditions of Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Let ˇL be an integrable distribution which is convolvable with Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Suppose that F ˇL, which is necessarily a continuous LIOUVILLE’S THEOREMS FOR LÉVY OPERATORS 35 function, has no zeroes in an open set U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Let h be a function such that h{p1 ` Y q is bounded on Rd and ˇL f h “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Then the spectrum of h is contained in RdzU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' In particular, if U “ Rdzt0u, then h is a polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' We conclude this section with examples of admissible functions Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' If α ě 0, then Y pxq “ p1 ` |x|qα ´ 1 satisfies conditions (a) and (b) in Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Indeed, p1 ` Y pxqqp1 ` Y pyqq “ p1 ` |x|qαp1 ` |y|qα ě p1 ` |x| ` |y|qα ě p1 ` |x ` y|qα “ 1 ` Y px ` yq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Thus, if L is a Lévy operator which is not concentrated on a proper closed subgroup of Rd, and the Lévy measure ν of L satisfies ż RdzB |y|ανpdyq ă 8, then every L-harmonic function h (in the sense of tempered distributions) such that p1 ` |x|q´αhpxq is a bounded function of x P Rd is necessarily a polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' When α ă 1, then it follows that h is in fact constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' For α “ 0 we thus recover Liouville’s theorem for bounded L-harmonic functions given in Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' If β ě 0, then it is easy to see that Y pxq “ plogpe2 ` |x|qqβ satisfies conditions (a) and (b) in Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Indeed: we have Y pxqY pyq “ ` logpe2 ` |x|q logpe2 ` |y|q ˘β ě ` logpe2q logpe2 ` maxt|x|, |y|uq ˘β ě ` 2 logpe2 ` 1 2p|x| ` |y|qq ˘β “ ` logpe4 ` e2p|x| ` |y|q ` 1 4p|x| ` |y|q2q ˘β ě ` logpe2 ` |x ` y|q ˘β “ Y px ` yq, and hence 1 ` Y px ` yq ď 1 ` Y pxqY pyq ď p1 ` Y pxqqp1 ` Y pyqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' It follows that if L is a Lévy operator which is not concentrated on a proper closed subgroup of Rd, and the Lévy measure ν of L satisfies ż RdzB plog |y|qβνpdyq ă 8, then every L-harmonic function h (in the sense of tempered distributions) such that plogpe ` |x|qq´βhpxq is a bounded function of x P Rd is necessarily constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Harmonic functions with finite generalised negative moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' As in the previous section, we define a class of Wiener-type algebras, again parameterised by auxiliary functions Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Once again we discuss examples of admissible functions Y at the end of this section, and here we remark that a typical example is Y pxq “ cd,αp1`|x|q´d´α, where α ą 0 is a parameter and cd,α is a sufficiently small constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Suppose that Y is an integrable function on Rd with the following properties: 36 TOMASZ GRZYWNY, MATEUSZ KWA´SNICKI (a) we have Y ˚ Y pxq ď Y pxq for every x P Rd, and if Yrpxq “ Y pxq when |x| ě r and Yrpxq “ 0 otherwise, then lim rÑ8 sup "Yr ˚ Yrpxq Y pxq : x P Rd “ 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' (b) for some positive constants c, q we have Y pxq ě cp1 ` |x|q´q for every x P Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Define W 1 Y to be the class of Fourier transforms of integrable functions f such that f{Y is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' We remark that if Y ˚ Y ď cY for some constant c, then we may replace Y by c´1Y to get a function which satisfies Y ˚ Y ď Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' We also note that condition Y ˚ Y pxq ď cY pxq is called direct jump property in [15], and we refer to that paper for further discussion and references.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Below we prove that W 1 Y is a Wiener-type algebra according to Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' The argument is similar to the one applied in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Nevertheless, since there are essential differences between the two, we provide all details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' The set W 1 Y defined in Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='12 is an algebra of continuous functions which satisfies conditions (a) and (b) of Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Clearly, W 1 Y is a linear space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' If Φ P W 1 Y , then Φ is the Fourier transform of an integrable function f, and hence Φ is a bounded continuous function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Thus, W 1 Y is a class of continuous functions, and every Φ P W 1 Y corresponds to a tempered distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' We claim that W 1 Y is indeed an algebra of functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' If Φ, Ψ P W 1 Y , then Φ “ Ff and Ψ “ Fg for some integrable functions f, g, and for some constants c1, c2 we have |fpxq| ď c1Y pxq and |gpxq| ď c2Y pxq for every x P Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' It follows that h “ f ˚ g is an integrable function, and |hpxq| “ |f ˚ gpxq| ď |f| ˚ |g|pxq ď pc1Y q ˚ pc2Y qpxq “ c1c2Y ˚ Y pxq ď c1c2Y pxq, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='6) that is, h{Y is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Since ΦΨ “ Ff ¨ Fg “ Fpf ˚ gq “ Fh, we conclude that ΦΨ P W 1 Y , as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' We now show that W 1 Y contains S .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Indeed: if ϕ P S and ψ “ F ´1ϕ, then ψ P S , and so, in particular, ψ is integrable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Furthermore, by condition (b) in Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='12, ψ{Y is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Thus, ϕ “ Fψ indeed belongs to W 1 Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' It remains to observe that if ϕ P S and Φ P W 1 Y , then ϕ P W 1 Y , and therefore ϕΦ P W 1 Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' □ As before, in order to prove that W 1 Y is a Wiener-type algebra, it remains to verify that W 1 Y satisfies condition (c) of Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='5, that is, a variant of Wiener’s 1{f theorem holds in W 1 Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Let W 1 Y be the set defined in Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Suppose that K Ď Rd is a compact set, Φ P W 1 Y and Φpξq ‰ 0 for every ξ P K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Then there is ˜Φ P W 1 Y such that Φpξq˜Φpξq “ 1 for every ξ P K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Once again we follow the proof of the classical Wiener’s 1{f lemma given in [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' We denote by } ¨ }p the usual norm in LppRdq, where p P r1, 8s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' As in the proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='7, we have the following two observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' If Φ “ Ff and f is integrable, then }Φ}8 ď }f}1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' LIOUVILLE’S THEOREMS FOR LÉVY OPERATORS 37 Conversely, if k “ 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=', Φ is smooth, both Φ and ∆kΦ are integrable, and f “ F ´1Φ, then suptp1 ` |x|2kq|fpxq| : x P Rdu ď cd,k}Φ ` p´∆qkΦ}1, where cd,k is an appropriate constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' In particular, if 2k ą d, then f is integrable, while if 2k ě q with q as in condition (b) in Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='12, then f{Y is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' It follows that if 2k ą d and 2k ě q, then there is a constant cd,k such that whenever Φ is smooth, and Φ and ∆kΦ are integrable, then Φ P W 1 Y and }f}1 ` }f{Y }8 ď cd,k}Φ ` p´∆qkΦ}1, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='7) where f “ F ´1Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' We return to the proof of the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Suppose that Φ P W 1 Y , that is, Φ “ Ff for an integrable function f such that f{Y is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Suppose furthermore that K is a compact set and Φpξq ‰ 0 for every ξ P K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Let ε ą 0 be small enough, so that |Φpξq| ą 3ε for every ξ in some bounded neighbourhood U of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Choose ϕ P S so that the support of ϕ is a compact subset of U and ϕpξq “ 1 for ξ P K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' As in the proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='7, our goal is to prove that ˜Φpξq “ ϕpξq Φpξq is an element of W 1 Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' here, of course, ˜Φpξq “ 0 for ξ P RdzU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Once this is shown, we have Φpξq˜Φpξq “ ϕpξq “ 1 for every ξ P K, and so ˜Φ has the desired property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' For r ą 0, let Br denote the ball of radius r centred at the origin, and denote Yrpxq “ Y pxq 1Brpxq, as in condition (a) in Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Choose r large enough, so that if gpxq “ fpxq 1Brpxq, then }f ´ g}1 ă ε and Λr ˚ Λrpxq ď ε2 }f{Λ}2 8 Λprq (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='8) for every x P Rd (see condition (a) in Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Clearly, g is integrable and g{Y is bounded, and therefore Ψ “ Fg is an element of W 1 Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Furthermore, }Φ ´ Ψ}8 ď }f ´ g}1 ă ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' In particular, for ξ P U we have |Ψpξq| ě |Φpξq| ´ |Φpξq ´ Ψpξq| ą 3ε ´ ε “ 2ε, and ˇˇˇˇ Ψpξq ´ Φpξq Ψpξq ˇˇˇˇ ă ε 2ε “ 1 2 , so that for ξ P U we have 1 Φpξq “ 1 Ψpξq 8 ÿ n“0 ˆΨpξq ´ Φpξq Ψpξq ˙n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' We conclude that for every ξ P Rd, ˜Φpξq “ ϕpξq Φpξq “ 8 ÿ n“0 ϕpξq pΨpξqqn`1 pΨpξq ´ Φpξqqn, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='9) 38 TOMASZ GRZYWNY, MATEUSZ KWA´SNICKI and we study the terms ϕ{Ψn`1 and pΨ ´ Φqn separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Recall that Ψ ´ Φ “ Fpg ´ fq and }g ´ f}1 ă ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Furthermore, if λ “ }f{Y }8, then |gpxq ´ fpxq| ď λYrpxq for every x P Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' If pg´fq˚n denotes the n-fold convolution of g´f, then Fpg´fq˚n “ pΨ ´ Φqn, and }pg ´ fq˚n}1 ď }g ´ f}n 1 ă εn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='10) The estimate of pg ´ fq˚n{Y is more involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' If n is even, then, by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='8), |pg ´ fq˚npxq| ď pλYrq˚npxq ď λnpYr ˚ Yrq˚n{2pxq ď λnpλ´2ε2Y q˚n{2pxq “ εnY ˚n{2pxq ď εnY pxq, and if n is odd, then, in a similar manner, |pg ´ fq˚npxq| ď pλYrq˚npxq ď λnpYr ˚ Yrq˚tn{2u ˚ Yrpxq ď λnpλ´2ε2Y q˚tn{2u ˚ Y pxq “ λεn´1Y ˚tn{2u`1pxq ď λεn´1Y pxq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Thus, for an arbitrary n, |pg ´ fq˚npxq| ď pλ ` εqεn´1Y pxq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='11) These are the desired bounds for pΨ ´ Φqn, and we now study ϕ{Ψn`1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Since Ψ “ Fg and g has compact support, Ψ is smooth in Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Additionally, |Ψpξq| ě 2ε ą 0 for ξ P U, and ϕpξq “ 0 for ξ outside a compact subset of U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' It follows that ϕ{Ψn`1 is smooth on Rd, and equal to zero in RdzU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' In particular, by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='7), we find that ϕ{Ψn`1 P W 1 Y , and if hn “ F ´1pϕ{Ψn`1q, then }hn}1 ` }hn{Y }8 ď cd,k ››ϕ{Ψn`1 ` p´∆qkpϕ{Ψn`1q ›› 1, where k is a fixed sufficiently large positive integer and cd,k is an appropriate con- stant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' By applying the product rule to p´∆qkpϕ{Ψn`1q, as in the proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='7 we find that there is a constant cd,k,ϕ,Ψ such that }hn}1 ` }hn{Y }8 ď cd,k,ϕ,Ψ|U|p1 ` nkqp2εq´n´1´2k, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='12) and this is the desired estimate for ϕ{Ψn`1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' We combine (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='10) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='12) to find that }pg ´ fq˚n ˚ hn}1 ď }pg ´ fq˚n}1}hn}1 ď ¨cd,k,ϕ,Ψ|U|p1 ` nkqp2εq´1´2k ¨ 2´n, and similarly we combine combine (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='11) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='12) as in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='6) to find that |pg ´ fq˚n ˚ hnpxq| ď cd,k,ϕ,Ψ|U|p1 ` nkqp2εq´2´2kpλ ` εq ¨ 21´nY pxq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' By Fubini’s theorem, we obtain ›››› 8 ÿ n“0 pg ´ fq˚n ˚ hn ›››› 1 ă 8, LIOUVILLE’S THEOREMS FOR LÉVY OPERATORS 39 and ›››› 1 Y 8 ÿ n“0 pg ´ fq˚n ˚ hn ›››› 8 ă 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' In particular, the series ˜f “ 8 ÿ n“0 pg ´ fq˚n ˚ hn defines an integrable function such that ˜f{Y is bounded, and as in the proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='7, by Fubini’s theorem we find that F ˜f “ 8 ÿ n“0 pΨ ´ Φqnpϕ{Ψn`1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Thus, by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='9), F ˜f “ ˜Φ, and therefore ˜Φ P W 1 Y , as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' □ As an immediate corollary of Lemmas 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='13 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='14, we obtain the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' The set W 1 Y defined in Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='12 is a Wiener-type algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Suppose that Y satisfies the conditions of Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='12, and ˇL is an integrable distribution such that for every ϕ P S the Fourier transform of pˇL ˚ ϕq{Y is a bounded function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Then, by definition, Fϕ ¨ F ˇL “ FpˇL ˚ ϕq P W 1 Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' By choosing ϕ such that Fϕpξq “ 1 for ξ in a given compact set, we find that F ˇL belongs to W 1 Y locally on Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Let ˇY pxq “ Y p´xq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' We claim that if h is a function on Rd such that ˇY h is inte- grable, then h corresponds to a tempered distribution and Fh acts on W 1 Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Indeed: by condition (b) in Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='12, h is bounded by the product of an integrable function ˇY h and a polynomial, and hence it corresponds to a tempered distribu- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Using both conditions in Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='12, we find that for some constants c, q we have Y px ´ yq ď cp1 ` |x|qqY p´yq for every x, y P Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' It follows that h and Y are convolvable, and for some constants c, q we have |h| ˚ Y pxq ď c|h| ˚ Y p0qp1 ` |x|qq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Finally, if Φ, Ψ P W 1 Y , then Φ “ Ff and Ψ “ Fg for some integrable functions f, g such that f{Y and g{Y are bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Thus, by condition (a) in Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='12 and Fubini’s theorem, we find that |h| ˚ |f| ˚ |g|pxq ď }f{Y }8}g{Y }8|h| ˚ Y ˚ Y pxq ď }f{Y }8}g{Y }8|h| ˚ Y pxq ď c}f{Y }8}g{Y }8|h| ˚ Y p0qp1 ` |x|qq (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='13) for every x P Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Therefore, by Fubini’s theorem, we have h ˚ pf ˚ gq “ ph ˚ fq ˚ g, and in fact, by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='13), h f pf ˚ gq “ ph f fq f g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Applying the exchange formula, we conclude that FH d pΦΨq “ pFH d Φq d Ψ, 40 TOMASZ GRZYWNY, MATEUSZ KWA´SNICKI which completes the proof of our claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' After exchanging the roles of Y and ˇY , Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='1 and Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='15, as well as the two properties discussed above, immediately lead to the following variant of Liouville’s theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='16 (Liouville’s theorem under generalised negative moment condi- tion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Suppose that Y satisfies the conditions of Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Let ˇL be an in- tegrable distribution such that for every ϕ P S there is a constant c such that |ˇL ˚ ϕpxq| ď cY p´xq for every x P Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Suppose that F ˇL, which is necessarily a continuous function, has no zeroes in an open set U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Let h be a function such that Y h is integrable on Rd and ˇL f h “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Then the spectrum of h is contained in RdzU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' In particular, if U “ Rdzt0u, then h is a polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' We conclude this section with examples of admissible functions Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' For this, we need the following auxiliary result, which resembles Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='1 in [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Suppose that ϕ is a positive, decreasing, continuous function on r0, 8q such that rd´1ϕprq is integrable with respect to r P r0, 8q and lim inf rÑ8 ϕp2rq ϕprq ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='14) Then, for ε ą 0 small enough, the function Y pxq “ εϕp|x|q satisfies all conditions of Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Clearly, Y is integrable over Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' We need to verify conditions (a) and (b) in Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Since ϕ is decreasing, we have min ␣ ϕp|x ´ y|q, ϕp|y|q ( “ ϕ ` maxt|x ´ y|, |y|u ˘ ď ϕp 1 2|x|q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Hence, ϕp|x ´ y|qϕp|y|q “ min ␣ ϕp|x ´ y|q, ϕp|y|q ( max ␣ ϕp|x ´ y|q, ϕp|y|q ( ď ϕp 1 2|x|q ` ϕp|x ´ y|q ` ϕp|y|q ˘ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' It follows that Y ˚ Y pxq “ ε2 ż Rd ϕp|x ´ y|qϕp|y|qdy ď ε2ϕp 1 2|x|q ż Rd ` ϕp|x ´ y|q ` ϕp|y|q ˘ dy “ 2ε2c1ϕp 1 2|x|q, where c1 is the integral of ϕp|x|q over x P Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' By assumption (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='14), there is a constant c2 such that ϕp 1 2|x|q ď c2ϕp|x|q for every x P Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' It follows that Y ˚ Y pxq ď 2c1c2εY pxq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Hence, if ε is small enough, we have Y ˚ Y pxq ď Y pxq, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' LIOUVILLE’S THEOREMS FOR LÉVY OPERATORS 41 Let Yr be defined as in condition (a) in Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='12, and let Br denote the centred ball of radius r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' By the same argument as above, we have Yr ˚ Yrpxq “ ε2 ż Rd 1RdzBrpx ´ yqϕp|x ´ y|q 1RdzBrpyqϕp|y|qdy ď ε2ϕp 1 2|x|q ż Rd 1RdzBrpx ´ yq 1RdzBrpyq ` ϕp|x ´ y|q ` ϕp|y|q ˘ dy ď c2ε2ϕp|x|q ż Rd ` 1RdzBrpx ´ yqϕp|x ´ y|q ` 1RdzBrpyqϕp|y|q ˘ dy ď 2c2εY pxq ż Rd 1RdzBrpyqϕp|y|qdy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Thus, lim rÑ8 sup "Yr ˚ Yrpxq Y pxq : x P Rd ď lim rÑ8 ˆ 2c2ε ż Rd 1RdzBrpyqϕp|y|qdy ˙ “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' We have thus proved that Y indeed satisfies condition (a) in Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Condition (b) in Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='12 is related to general theory of regular variation, but in fact it is an elementary result: if, as above, ϕp 1 2|x|q ď c2ϕp|x|q for every x P Rd and c3 is the infimum of ϕ over r0, 1s, then for x P Rdzt0u such that 2n´1 ă |x| ď 2n, where n “ 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=', we have c3 ď ϕp2´n|x|q ď cn 2ϕp|x|q “ c2p2n´1qlog c2{ log 2ϕp|x|q ď c2|x|log c2{ log 2ϕp|x|q, and hence ϕp|x|q ě pc3{c2qp1 ` |x|q´ log c2{ log 2 for every x P Rd, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' □ Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Using Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='17 it is straightforward to verify that if α ą 0, then Y pxq “ cd,αp1 ` |x|q´d´α satisfies the conditions of Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Thus, we get the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Let B denote the unit ball in Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Suppose that L is a Lévy operator which is not concentrated on a proper closed subgroup of Rd, and for some constant c the Lévy measure ν of L satisfies νpx ` Bq ď c|x|´d´α for every x P Rd such that |x| ě 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Then every L-harmonic function h such that p1 ` |x|q´d´αhpxq is an integrable function of x P Rd is necessarily a polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Suppose now that ν has a density function which is comparable with |x|´d´α for |x| large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' In this case our integrability condition on h, namely, that p1 ` |x|q´d´αhpxq is an integrable function, is automatically satisfied by every L- harmonic function in the weak sense (as defined in Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Thus, by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='2, the result given above characterises all functions h which are L- harmonic in the weak sense, with no additional conditions on h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' The above result is a variant of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='4 in [13], while the more general result given in Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='16 with this choice of Y is a minor extension of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='1 in [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' More generally, by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='17, we find that if Y pxq “ εϕp|x|q for a positive, decreasing, continuous function ϕ on r0, 8q such that ϕp|x|q is integrable with respect to x P Rd, and such that ϕprq is regularly varying as r Ñ 8, then Y satisfies the conditions of Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='12 if ε is small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' In particular, this implies the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' 42 TOMASZ GRZYWNY, MATEUSZ KWA´SNICKI Let α ą 0 and β P R, or α ą 0 and β ą 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Let L be a Lévy operator which is not concentrated on a proper closed subgroup of Rd, and such that for some constant c the Lévy measure ν of L satisfies νpx ` Bq ď c|x|´d´αplog |x|q´β for every x P Rd such that |x| ě 2 (where B is the unit ball).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Then every L-harmonic function h such that p1 ` |x|q´d´αplogpe ` |x|qq´βhpxq is an integrable function of x P Rd is necessarily a polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' As in the previous example, if ν has a density function which is comparable with |x|´d´αplog |x|q´β for |x| large enough, then L-harmonic functions in the weak sense (see Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='2) automatically satisfy the integrability condition, and hence, by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='2, the above result describes all L-harmonic functions in the weak sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Suppose that for j “ 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' , k a function Yj on Rdj satisfies the conditions of Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='12, and define d “ d1 ` d2 ` .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' ` dk and Y pxq “ k ź j“1 Yjpxjq, where x “ px1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' , xkq P Rd with xj P Rdj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' It is then immediate to check that Y satisfies the conditions of Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Combining this with Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='18, we arrive at the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Let B denote the unit ball in Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Suppose that L “ L1 ` L2 ` .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' ` Ld, where for every j “ 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' , d the operator Lj is a dj-dimensional Lévy operator which is not concentrated on a proper closed subgroup of Rdj, and for some constants cj and αj ą 0 the Lévy measure νj of L satisfies νjpx ` Bjq ď c|x|´1´αj for every x P Rdj such that |x| ě 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' here Bj is the unit ball in Rdj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' Then every L-harmonic function h such that ˆ d ź j“1 1 p1 ` |xj|qd`αj ˙ hpxq is an integrable function of x P Rd is necessarily a polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' We remark that, unlike in the previous examples, the integrability condition on h does not seem to follow automatically from the definition of an L-harmonic function in the weak sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content=' References [1] N.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='pl,mateusz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='kwasnicki@pwr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} +page_content='pl' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FAT4oBgHgl3EQfYh2h/content/2301.08540v1.pdf'} diff --git a/ctE1T4oBgHgl3EQfdwQ1/content/tmp_files/2301.03198v1.pdf.txt b/ctE1T4oBgHgl3EQfdwQ1/content/tmp_files/2301.03198v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..a2c4121290b3c267f6079bb7b0153d0a3316c870 --- /dev/null +++ b/ctE1T4oBgHgl3EQfdwQ1/content/tmp_files/2301.03198v1.pdf.txt @@ -0,0 +1,685 @@ +The Algonauts Project 2023 Challenge: How the +Human Brain Makes Sense of Natural Scenes +Alessandro T. Gifford1,*, Benjamin Lahner2, Sari Saba-Sadiya3, Martina G. Vilas3, Alex +Lascelles2, Aude Oliva2, Kendrick Kay4, Gemma Roig3,5, Radoslaw M. Cichy1,* +1 Department of Education and Psychology, Freie Universität Berlin, Germany +2 Computer Science and Artificial Intelligence Laboratory, MIT, USA +3 Department of Computer Science, Goethe Universität Frankfurt, Germany +4 Center for Magnetic Resonance Research (CMRR), University of Minnesota, USA +5 Hessian Center for AI (hessian.AI), Darmstadt, Germany +The sciences of biological and artificial intelligence +are ever more intertwined. Neural computational +principles inspire new intelligent machines, which are +in turn used to advance theoretical understanding of +the brain. To promote further exchange of ideas and +collaboration between biological and artificial +intelligence researchers, we introduce the 2023 +installment of the Algonauts Project challenge: How +the Human Brain Makes Sense of Natural Scenes +(http://algonauts.csail.mit.edu). This installment +prompts the fields of artificial and biological +intelligence to come together towards building +computational models of the visual brain using the +largest and richest dataset of fMRI responses to +visual scenes, the Natural Scenes Dataset (NSD). NSD +provides high-quality fMRI responses to ~73,000 +different naturalistic colored scenes, making it the +ideal candidate for data-driven model building +approaches promoted by the 2023 challenge. The +challenge is open to all and makes results directly +comparable and transparent through a public +leaderboard automatically updated after each +submission, +thus allowing for rapid model +development. We believe that the 2023 installment will +spark symbiotic collaborations between biological +and artificial intelligence scientists, leading to a +deeper understanding of the brain through cutting- +edge computational models and to novel ways of +engineering artificial intelligent agents through +inductive biases from biological systems. +Keywords: artificial intelligence; vision; human +neuroscience; scene understanding; fMRI; +prediction; challenge; benchmark +Introduction +In the last decade the deep learning revolution has +profoundly impacted scientific research endeavors +(Sejnowski, 2018; Baldi, 2021). During this time the +quest for solving intelligence in both its artificial and +biological form has made remarkable progress: +deep learning algorithms, originally inspired by the +visual system of the mammalian brain (Fukushima & +Miyake, 1982), are now both state-of-the-art AI +agents (Krizhevsky et al., 2017) and scientific +models of the brain itself (Yamins & DiCarlo, 2016; +Cichy & Kaiser, 2019; Kietzmann et al., 2019; +Richards et al., 2019; Saxe et al., 2021). Thus, +research in artificial and biological intelligence is +ever more intertwined. Furthermore, this synergy is +accelerated by the release of large neural datasets +that allow training deep learning models end-to-end +on brain data (Allen et al., 2022; Gifford et al., +2022), and by challenges that exploit these and +other neural datasets to develop better models of +intelligence (Schrimpf et al., 2020; Willeke et al., +2022). +We contribute to the symbiosis between +biological and artificial intelligence with the third +installment of the Algonauts Project challenge, titled +“How the Human Brain Makes Sense of Natural +Scenes”. The 2023 installment continues the spirit of +the 2019 and 2021 editions of the Algonauts Project +in its goal of predicting human visual brain +responses through computational models (Cichy et +al., 2019; Cichy et al., 2021). Yet, it goes beyond the +previous challenges in that it is based on the largest +and richest dataset of neural responses to natural +scenes, the Natural Scenes Dataset (NSD) (Allen et +al., 2022). We focus on visual scene understanding +since vision is an unresolved problem in the +sciences of artificial and biological intelligence alike +(Szegedy et al, 2014; Gheiros et al., 2019; DiCarlo +et al., 2012), and one of the areas where +collaboration between these two fields has been +most fruitful (Yamins & DiCarlo, 2016; Li et al., +2019; Safarani et al., 2021; Dapello et al., 2022). +We believe that the Algonauts Project will lead to +significant advances in both the understanding of +the brain through artificial intelligence models +(Yamins & DiCarlo, 2016; Cichy and Kaiser, 2019; +Kietzmann et al., 2019; Richards et al., 2019) and +the engineering of better AI agents through +biological intelligence constraints (Hassabis et al., +2017; Sinz et al., 2019; Ullman, 2019; Yang et al., +2022; Toneva & Wehbe, 2019; Li et al., 2019; +Safarani et al., 2021; Dapello et al., 2022), thus +contributing to the ever stronger symbiosis between +the sciences of biological and artificial intelligence. +* Correspondence: alessandro.gifford@gmail.com; rmcichy@zedat.fu-berlin.de +1 + +Lateral +Posterior +a +0 +LH +RH +LH +RH +Medial +Ventral +RH +LH +RH +LHTest images +s1 +s8 +Encoding model +NsD experiment +Predicted data +Withheld data +s1 +s8 +s1 +s8 +All vertices +All vertices +Correlate each +vertex (v) +V1 +V1 +pair across +test images +V2 +V2 +Sguared correlation scores +CN +Challenge metric: median noise- +normalized sguared correlationFigure 1. Challenge data. (a) Exemplar NSD stimuli images. All images consist of natural scenes taken from the COCO +database. (b) Depiction of the cortical surface vertices used in the challenge (purple). +Materials and Methods +Challenge goal. The goal of the Algonauts 2023 +Project challenge is to promote the development of +cutting-edge encoding models that predict neural +responses to visual stimuli, and to provide a +common platform that catalyzes collaborations +between the fields of biological and artificial +intelligence. +Primer on encoding models. Encoding models are +algorithms that predict how the brain responds to +(i.e., encodes) certain stimuli (Naselaris et al., 2011; +van Gerven, 2017). In visual neuroscience, an +encoding model typically consists in an algorithm +that takes image pixels as input, transforms them +into model features, and maps these features onto +brain data (e.g., fMRI activity), effectively predicting +the neural responses to images. +Challenge data. In this challenge participants will +leverage the unprecedented size of NSD (Allen et +al., 2022) to build encoding models of the visual +brain. NSD is a massive 8-subject dataset of high- +quality 7T fMRI responses to ~73,000 different +natural scenes (presented during central fixation) +from the Common Objects in Context (COCO) +database (Lin et al., 2014) (Fig. 1a). The challenge +uses preprocessed fMRI responses (BOLD +response amplitudes) from each subject that have +been projected onto a common cortical surface +group template (FreeSurfer's fsaverage surface). +Brain surfaces are composed of vertices, and the +challenge data consists of a subset of cortical +surface vertices in the visual cortex (a region of the +brain specialized in processing visual input) that +were maximally responsive to visual stimulation +(Fig. 1b). We provide the data in right and left +hemispheres. Further information on NSD +acquisition and preprocessing is provided on the +challenge website1 and in the NSD paper2. Since +1 http://algonauts.csail.mit.edu +2 https://www.nature.com/articles/s41593-021-00962-x +model building requires independent data splits for +training and testing, we partitioned the challenge +data into non-overlapping train and test splits +coming from, respectively, the publicly released +NSD data and the last three NSD scan sessions +from each subject (which are not publicly available). +Train split. For each of the 8 subjects we provide +[9841, 9841, 9082, 8779, 9841, 9082, 9841, 8779] +images, along with the corresponding fMRI visual +responses, z-scored within each NSD scan session +and averaged across image repeats. These data +can be used to train encoding models. +Test split. For each of the 8 subjects we provide +[159, 159, 293, 395, 159, 293, 159, 395] images +(different from the train images) but withhold the +corresponding fMRI visual responses. Challenge +participants are asked to use their encoding models +to predict the fMRI responses of these test images. +ROI indices. The visual cortex is divided into +multiple areas having different functional properties, +referred to here as regions-of-interest (ROIs). Along +with the fMRI data, we provide ROI indices for +selecting vertices belonging to specific visual ROIs; +challenge participants can optionally use these ROI +indices at their own discretion (e.g., to build different +encoding models for functionally different regions of +the visual cortex). However, the challenge +evaluation metric is computed over all available +vertices, not over any single ROI. +Development kit. We provide a Colab tutorial3 in +Python where we show how to: (i) load and visualize +the fMRI data, its ROIs and the corresponding +stimulus images; (ii) build and evaluate linearizing +encoding models (Naselaris et al., 2011) using a +pretrained AlexNet (Krizhevsky, 2014) architecture; +(iii) prepare predicted brain responses of the test +images in the right format for challenge submission. +3 https://colab.research.google.com/drive/ +1bLJGP3bAo_hAOwZPHpiSHKlt97X9xsUw?usp=sharing +2 + +Lateral +Posterior +a +0 +LH +RH +LH +RH +Medial +Ventral +RH +LH +RH +LHTest images +s1 +s8 +Encoding model +NsD experiment +Predicted data +Withheld data +s1 +s8 +s1 +s8 +All vertices +All vertices +Correlate each +vertex (v) +V1 +V1 +pair across +test images +V2 +V2 +Sguared correlation scores +CN +Challenge metric: median noise- +normalized sguared correlationChallenge submission and evaluation metric. To +quantify accuracy of encoding models, participants +submit their fMRI predictions for the test images. For +each NSD subject and hemisphere, we will correlate +the fMRI predicted data with the corresponding +ground truth (withheld) data at all vertices (across +images), square the resulting correlation scores and +normalize them with respect to the noise ceiling +(reflecting the total predictable variance given the +level of noise in the data). The resulting values will +indicate how much of the predictable variance has +been accounted for by the models. The overall +challenge evaluation metric, which quantifies the +performance of each participant's submission, is the +median noise-normalized squared correlation score +over all vertices from all subjects (Fig. 2): +where v is the index of vertices (over all subjects +and hemispheres), t is the index of the test stimuli +images, G and P correspond to, respectively, the +ground truth and predicted fMRI test data, Ḡ and P̄ +are the ground truth and predicted fMRI test data +averaged across test stimuli images, R is the +Pearson correlation coefficient between G and P, +and NC is the noise ceiling. +Baseline model. The baseline model score of the +challenge reflects a linearizing encoding model +(Naselaris et al., 2011) built using a pretrained +AlexNet. Its median noise-normalized prediction +accuracy over all subjects, hemispheres and +vertices is 40.48% of the total predictable variance. +Rules. To encourage broad participation, the +challenge has a simple submission process with +minimal rules. Participants can use any encoding +model derived from any source and trained on any +type of data, and can make a limited number of +submissions per day (the leaderboard is +automatically updated after each submission). +However, using the test split for training (in +particular brain data generated using the test +images) is prohibited. To promote open science, +participants who wish to be considered for the +challenge contest will have to submit a report to a +preprint server describing their encoding algorithm. +Furthermore, the top three entries are required to +make their code openly available and, along with +other prizes, will have the chance to present their +winning encoding models through a talk at the +Cognitive Computational Neuroscience (CCN) +conference in 2023. +3 +Figure 2. Challenge evaluation metric. Once participants +submit their predictions of the fMRI responses to the test +images of all 8 subjects (s1, …, s8), we evaluate +prediction accuracy using the withheld fMRI test data. In +detail, we concatenate the predicted data vertices (V1, V2, +…, VN) of all subjects and hemispheres, correlate them +with the corresponding withheld data vertices (across the +test stimuli images), and square the correlations, resulting +in one squared correlation score (C1, …, CN) for each +vertex. The challenge evaluation metric is the median +noise-normalized squared correlation score across all +vertices + +Lateral +Posterior +a +0 +LH +RH +LH +RH +Medial +Ventral +RH +LH +RH +LHTest images +s1 +s8 +Encoding model +NsD experiment +Predicted data +Withheld data +s1 +s8 +s1 +s8 +All vertices +All vertices +Correlate each +vertex (v) +V1 +V1 +pair across +test images +V2 +V2 +Sguared correlation scores +CN +Challenge metric: median noise- +normalized sguared correlationDiscussion +Relation to similar initiatives. The Algonauts +project relates to initiatives such as Brain-Score +(Schrimpf et al., 2020) and Sensorium 2022 (Willeke +et al., 2022), which also establish benchmarks and +leaderboards. However, the Algonauts Project +differs from these complementary efforts by +emphasizing human data, by focusing on colored +images of complex naturalistic scenes including +multiple object concepts, by leveraging a whole- +brain fMRI dataset that extensively samples +stimulus variation (Allen et al., 2022; Naselaris et al., +2021), and by incorporating educational +components (hands-on computational modeling +tutorial; talks by the top three challenge winners; +panel discussion on challenges) at a dedicated +session at CCN in 2023. +Prediction vs. explanation. Having a model that +perfectly predicts a phenomenon does not by itself +explain the phenomenon. However, prediction and +explanation are related goals and complement each +other (Cichy & Kaiser, 2019). First, successful +explanations must also provide successful +predictions (Breiman, 2001; Yarkoni & Westfall, +2017). Second, prediction accuracy can shed light +on properties that make particular models +successful, allowing testing or generating +hypotheses and guiding future engineering steps. +Finally, predictive success as an evaluation criterion +circumvents the challenges of evaluation on +qualitative or subjective grounds. +From artificial to biological intelligence. During +the last decade, interactions between biological and +artificial intelligence have profoundly affected +neuroscientific discovery, and machine/deep +learning algorithms have become state-of-the-art +models of the brain (Yamins & DiCarlo, 2016; Cichy +& Kaiser, 2019; Kietzmann et al., 2019; Richards et +al., 2019; Saxe et al., 2021). However, due to their +large parameter number, these algorithms require +massive amounts of data to properly train +(Russakovsky et al., 2015). We addressed this by +basing the 2023 installment of the Algonauts Project +on the NSD dataset (Allen et al., 2022). The +unprecedented scale of NSD, along with its +extensive sampling of stimulus variation, allows for +data-driven model building approaches such as +training deep learning architectures end-to-end to +predict neural responses to visual stimuli (Allen et +al., 2022; St-Yves et al., 2022; Khosla & Wehbe, +2022, Gifford et al., 2022). Directly infusing deep +learning models with brain data enables a novel +type of interaction between biological and artificial +intelligence, which in our opinion will catalyze +breakthroughs in neuroscientific research. +From biological to artificial intelligence. Artificial +intelligence too can benefit from interactions with +biological intelligence. Biological systems constitute +a proof of principle for how a complex computational +problem can be solved, and thus can guide the +engineering of new artificial intelligence models +(Hassabis et al., 2017; Sinz et al., 2019; Ullman, +2019; Yang et al., 2022). This research direction is +especially promising for improving artificial agents in +domains at which biological agents excel (e.g., out- +of-domain generalization, transfer learning, +adversarial robustness, few-shot learning), and even +for endowing artificial agents with cognitive faculties +idiosyncratic to humans (e.g., planning, creativity, +imagination). Artificial intelligence researchers have +been successfully exploring these possibilities for +decades. As an example, the structure of the current +state-of-the-art artificial intelligence algorithms, deep +neural networks, has been inspired by the structure +of the visual system of the mammalian brain +(Fukushima & Miyake, 1982). Furthermore, a +growing amount of literature has started to exploit +neural data representations to train natural language +processing and computer vision algorithms, +resulting in models with improved performance and +adversarial robustness (Toneva & Wehbe, 2019; Li +et al., 2019; Safarani et al., 2021; Dapello et al., +2022). The Algonauts Project fosters this exciting +area of research by promoting interactions between +the fields of artificial and biological intelligence. +The future of the project. We hope that the 2023 +installment of the Algonauts Project will continue to +inspire new challenges and collaborations at the +intersection of artificial and biological intelligence +sciences. We believe that both communities will +benefit from jointly tackling open questions on how +perception and cognition are solved in brains and +machines. We welcome researchers interested in +initiating similar initiatives or collaborating with the +Algonauts Project to contribute ideas and datasets. +Acknowledgments +This research was funded by DFG (CI-241/1-1, +CI241/1-3,CI-241/1-7) and ERC grant (ERC-2018- +StG) to RMC; NSF award (1532591) in Neural and +Cognitive Systems and the Vannevar Bush Faculty +Fellowship program funded by the ONR (N00014- +16-1-3116) to AO; the Alfons and Gertrud Kassel +foundation to GR. Collection of the NSD dataset +was supported by NSF IIS-1822683 and NSF IIS- +1822929. +References +Allen EJ, St-Yves G, Wu Y, Breedlove JL, Prince JS, +Dowdle LT, Nau M, Caron B, Pestilli F, Charest I, +Hutchinson JB, Naselaris T, Kay K. 2022. A massive +7T fMRI dataset to bridge cognitive neuroscience and +computational intelligence. 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Perspectives on Psychological Science, +12(6):1100-1122. +5 + +Lateral +Posterior +a +0 +LH +RH +LH +RH +Medial +Ventral +RH +LH +RH +LHTest images +s1 +s8 +Encoding model +NsD experiment +Predicted data +Withheld data +s1 +s8 +s1 +s8 +All vertices +All vertices +Correlate each +vertex (v) +V1 +V1 +pair across +test images +V2 +V2 +Sguared correlation scores +CN +Challenge metric: median noise- +normalized sguared correlation \ No newline at end of file diff --git a/ctE1T4oBgHgl3EQfdwQ1/content/tmp_files/load_file.txt b/ctE1T4oBgHgl3EQfdwQ1/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..509174cc0af7914a87e697129dcb5e6bbb4715a3 --- /dev/null +++ b/ctE1T4oBgHgl3EQfdwQ1/content/tmp_files/load_file.txt @@ -0,0 +1,430 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf,len=429 +page_content='The Algonauts Project 2023 Challenge: How the Human Brain Makes Sense of Natural Scenes Alessandro T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' Gifford1,*, Benjamin Lahner2, Sari Saba-Sadiya3, Martina G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' Vilas3, Alex Lascelles2, Aude Oliva2, Kendrick Kay4, Gemma Roig3,5, Radoslaw M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' Cichy1,* 1 Department of Education and Psychology, Freie Universität Berlin, Germany 2 Computer Science and Artificial Intelligence Laboratory, MIT, USA 3 Department of Computer Science, Goethe Universität Frankfurt, Germany 4 Center for Magnetic Resonance Research (CMRR), University of Minnesota, USA 5 Hessian Center for AI (hessian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content='AI), Darmstadt, Germany The sciences of biological and artificial intelligence are ever more intertwined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' Neural computational principles inspire new intelligent machines, which are in turn used to advance theoretical understanding of the brain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' To promote further exchange of ideas and collaboration between biological and artificial intelligence researchers, we introduce the 2023 installment of the Algonauts Project challenge: How the Human Brain Makes Sense of Natural Scenes (http://algonauts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content='csail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content='mit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content='edu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' This installment prompts the fields of artificial and biological intelligence to come together towards building computational models of the visual brain using the largest and richest dataset of fMRI responses to visual scenes, the Natural Scenes Dataset (NSD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' NSD provides high-quality fMRI responses to ~73,000 different naturalistic colored scenes, making it the ideal candidate for data-driven model building approaches promoted by the 2023 challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' The challenge is open to all and makes results directly comparable and transparent through a public leaderboard automatically updated after each submission, thus allowing for rapid model development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' We believe that the 2023 installment will spark symbiotic collaborations between biological and artificial intelligence scientists, leading to a deeper understanding of the brain through cutting- edge computational models and to novel ways of engineering artificial intelligent agents through inductive biases from biological systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' Keywords: artificial intelligence;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' vision;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' human neuroscience;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' scene understanding;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' fMRI;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' prediction;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' challenge;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' benchmark Introduction In the last decade the deep learning revolution has profoundly impacted scientific research endeavors (Sejnowski, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' Baldi, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' During this time the quest for solving intelligence in both its artificial and biological form has made remarkable progress: deep learning algorithms, originally inspired by the visual system of the mammalian brain (Fukushima & Miyake, 1982), are now both state-of-the-art AI agents (Krizhevsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=', 2017) and scientific models of the brain itself (Yamins & DiCarlo, 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' Cichy & Kaiser, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' Kietzmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' Richards et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' Saxe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' Thus, research in artificial and biological intelligence is ever more intertwined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' Furthermore, this synergy is accelerated by the release of large neural datasets that allow training deep learning models end-to-end on brain data (Allen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' Gifford et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=', 2022), and by challenges that exploit these and other neural datasets to develop better models of intelligence (Schrimpf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' Willeke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' We contribute to the symbiosis between biological and artificial intelligence with the third installment of the Algonauts Project challenge, titled “How the Human Brain Makes Sense of Natural Scenes”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' The 2023 installment continues the spirit of the 2019 and 2021 editions of the Algonauts Project in its goal of predicting human visual brain responses through computational models (Cichy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' Cichy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' Yet, it goes beyond the previous challenges in that it is based on the largest and richest dataset of neural responses to natural scenes, the Natural Scenes Dataset (NSD) (Allen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' We focus on visual scene understanding since vision is an unresolved problem in the sciences of artificial and biological intelligence alike (Szegedy et al, 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' Gheiros et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' DiCarlo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=', 2012), and one of the areas where collaboration between these two fields has been most fruitful (Yamins & DiCarlo, 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' Safarani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' Dapello et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' We believe that the Algonauts Project will lead to significant advances in both the understanding of the brain through artificial intelligence models (Yamins & DiCarlo, 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' Cichy and Kaiser, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' Kietzmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' Richards et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=', 2019) and the engineering of better AI agents through biological intelligence constraints (Hassabis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' Sinz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' Ullman, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' Toneva & Wehbe, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' Safarani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' Dapello et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=', 2022), thus contributing to the ever stronger symbiosis between the sciences of biological and artificial intelligence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' Correspondence: alessandro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content='gifford@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content='com;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' rmcichy@zedat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content='fu-berlin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content='de 1 Lateral Posterior a 0 LH RH LH RH Medial Ventral RH LH RH LHTest images s1 s8 Encoding model NsD experiment Predicted data Withheld data s1 s8 s1 s8 All vertices All vertices Correlate each vertex (v) V1 V1 pair across test images V2 V2 Sguared correlation scores CN Challenge metric: median noise- normalized sguared correlationFigure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' Challenge data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' (a) Exemplar NSD stimuli images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' All images consist of natural scenes taken from the COCO database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' (b) Depiction of the cortical surface vertices used in the challenge (purple).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' Materials and Methods Challenge goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' The goal of the Algonauts 2023 Project challenge is to promote the development of cutting-edge encoding models that predict neural responses to visual stimuli, and to provide a common platform that catalyzes collaborations between the fields of biological and artificial intelligence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' Primer on encoding models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' Encoding models are algorithms that predict how the brain responds to (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=', encodes) certain stimuli (Naselaris et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=', 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' van Gerven, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' In visual neuroscience, an encoding model typically consists in an algorithm that takes image pixels as input, transforms them into model features, and maps these features onto brain data (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=', fMRI activity), effectively predicting the neural responses to images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' Challenge data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' In this challenge participants will leverage the unprecedented size of NSD (Allen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=', 2022) to build encoding models of the visual brain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' NSD is a massive 8-subject dataset of high- quality 7T fMRI responses to ~73,000 different natural scenes (presented during central fixation) from the Common Objects in Context (COCO) database (Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=', 2014) (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' 1a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=" The challenge uses preprocessed fMRI responses (BOLD response amplitudes) from each subject that have been projected onto a common cortical surface group template (FreeSurfer's fsaverage surface)." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' Brain surfaces are composed of vertices, and the challenge data consists of a subset of cortical surface vertices in the visual cortex (a region of the brain specialized in processing visual input) that were maximally responsive to visual stimulation (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' 1b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' We provide the data in right and left hemispheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' Further information on NSD acquisition and preprocessing is provided on the challenge website1 and in the NSD paper2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' Since 1 http://algonauts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content='csail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content='mit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content='edu 2 https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content='nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content='com/articles/s41593-021-00962-x model building requires independent data splits for training and testing, we partitioned the challenge data into non-overlapping train and test splits coming from, respectively, the publicly released NSD data and the last three NSD scan sessions from each subject (which are not publicly available).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' Train split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' For each of the 8 subjects we provide [9841, 9841, 9082, 8779, 9841, 9082, 9841, 8779] images, along with the corresponding fMRI visual responses, z-scored within each NSD scan session and averaged across image repeats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' These data can be used to train encoding models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' Test split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' For each of the 8 subjects we provide [159, 159, 293, 395, 159, 293, 159, 395] images (different from the train images) but withhold the corresponding fMRI visual responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' Challenge participants are asked to use their encoding models to predict the fMRI responses of these test images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' ROI indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' The visual cortex is divided into multiple areas having different functional properties, referred to here as regions-of-interest (ROIs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' Along with the fMRI data, we provide ROI indices for selecting vertices belonging to specific visual ROIs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' challenge participants can optionally use these ROI indices at their own discretion (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=', to build different encoding models for functionally different regions of the visual cortex).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' However, the challenge evaluation metric is computed over all available vertices, not over any single ROI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' Development kit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' We provide a Colab tutorial3 in Python where we show how to: (i) load and visualize the fMRI data, its ROIs and the corresponding stimulus images;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' (ii) build and evaluate linearizing encoding models (Naselaris et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=', 2011) using a pretrained AlexNet (Krizhevsky, 2014) architecture;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' (iii) prepare predicted brain responses of the test images in the right format for challenge submission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' 3 https://colab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content='research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content='google.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content='com/drive/ 1bLJGP3bAo_hAOwZPHpiSHKlt97X9xsUw?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content='usp=sharing 2 Lateral Posterior a 0 LH RH LH RH Medial Ventral RH LH RH LHTest images s1 s8 Encoding model NsD experiment Predicted data Withheld data s1 s8 s1 s8 All vertices All vertices Correlate each vertex (v) V1 V1 pair across test images V2 V2 Sguared correlation scores CN Challenge metric: median noise- normalized sguared correlationChallenge submission and evaluation metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' To quantify accuracy of encoding models, participants submit their fMRI predictions for the test images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' For each NSD subject and hemisphere, we will correlate the fMRI predicted data with the corresponding ground truth (withheld) data at all vertices (across images), square the resulting correlation scores and normalize them with respect to the noise ceiling (reflecting the total predictable variance given the level of noise in the data).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' The resulting values will indicate how much of the predictable variance has been accounted for by the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=" The overall challenge evaluation metric, which quantifies the performance of each participant's submission, is the median noise-normalized squared correlation score over all vertices from all subjects (Fig." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' 2): where v is the index of vertices (over all subjects and hemispheres), t is the index of the test stimuli images, G and P correspond to, respectively, the ground truth and predicted fMRI test data, Ḡ and P̄ are the ground truth and predicted fMRI test data averaged across test stimuli images, R is the Pearson correlation coefficient between G and P, and NC is the noise ceiling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' Baseline model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' The baseline model score of the challenge reflects a linearizing encoding model (Naselaris et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=', 2011) built using a pretrained AlexNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' Its median noise-normalized prediction accuracy over all subjects, hemispheres and vertices is 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content='48% of the total predictable variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' Rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' To encourage broad participation, the challenge has a simple submission process with minimal rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' Participants can use any encoding model derived from any source and trained on any type of data, and can make a limited number of submissions per day (the leaderboard is automatically updated after each submission).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' However, using the test split for training (in particular brain data generated using the test images) is prohibited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' To promote open science, participants who wish to be considered for the challenge contest will have to submit a report to a preprint server describing their encoding algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' Furthermore, the top three entries are required to make their code openly available and, along with other prizes, will have the chance to present their winning encoding models through a talk at the Cognitive Computational Neuroscience (CCN) conference in 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' 3 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' Challenge evaluation metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' Once participants submit their predictions of the fMRI responses to the test images of all 8 subjects (s1, …, s8), we evaluate prediction accuracy using the withheld fMRI test data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' In detail, we concatenate the predicted data vertices (V1, V2, …, VN) of all subjects and hemispheres, correlate them with the corresponding withheld data vertices (across the test stimuli images), and square the correlations, resulting in one squared correlation score (C1, …, CN) for each vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' The challenge evaluation metric is the median ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content='noise-normalized ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content='squared ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content='correlation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content='score ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content='across ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content='all ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content='vertices ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content='Lateral ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content='Posterior ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content='a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content='LH ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content='RH ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content='LH ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content='RH ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content='Medial ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content='Ventral ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content='RH ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content='LH ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content='RH ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content='LHTest images ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content='s1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content='s8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content='Encoding model ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content='NsD experiment ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content='Predicted data ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content='Withheld data ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content='s1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content='s8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content='s1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content='s8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content='All vertices ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content='All vertices ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content='Correlate each ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content='vertex (v) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content='V1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content='V1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content='pair across ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content='test images ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content='V2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content='V2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content='Sguared correlation scores ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content='CN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content='Challenge metric: median noise- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content='normalized sguared correlationDiscussion ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content='Relation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content='to ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content='similar ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content='initiatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' The Algonauts project relates to initiatives such as Brain-Score (Schrimpf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=', 2020) and Sensorium 2022 (Willeke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=', 2022), which also establish benchmarks and leaderboards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' However, the Algonauts Project differs from these complementary efforts by emphasizing human data, by focusing on colored images of complex naturalistic scenes including multiple object concepts, by leveraging a whole- brain fMRI dataset that extensively samples stimulus variation (Allen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' Naselaris et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=', 2021), and by incorporating educational components (hands-on computational modeling tutorial;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' talks by the top three challenge winners;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' panel discussion on challenges) at a dedicated session at CCN in 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' Prediction vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' explanation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' Having a model that perfectly predicts a phenomenon does not by itself explain the phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' However, prediction and explanation are related goals and complement each other (Cichy & Kaiser, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' First, successful explanations must also provide successful predictions (Breiman, 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' Yarkoni & Westfall, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' Second, prediction accuracy can shed light on properties that make particular models successful, allowing testing or generating hypotheses and guiding future engineering steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' Finally, predictive success as an evaluation criterion circumvents the challenges of evaluation on qualitative or subjective grounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' From artificial to biological intelligence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' During the last decade, interactions between biological and artificial intelligence have profoundly affected neuroscientific discovery, and machine/deep learning algorithms have become state-of-the-art models of the brain (Yamins & DiCarlo, 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' Cichy & Kaiser, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' Kietzmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' Richards et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' Saxe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' However, due to their large parameter number, these algorithms require massive amounts of data to properly train (Russakovsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=', 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' We addressed this by basing the 2023 installment of the Algonauts Project on the NSD dataset (Allen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' The unprecedented scale of NSD, along with its extensive sampling of stimulus variation, allows for data-driven model building approaches such as training deep learning architectures end-to-end to predict neural responses to visual stimuli (Allen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' St-Yves et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' Khosla & Wehbe, 2022, Gifford et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' Directly infusing deep learning models with brain data enables a novel type of interaction between biological and artificial intelligence, which in our opinion will catalyze breakthroughs in neuroscientific research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' From biological to artificial intelligence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' Artificial intelligence too can benefit from interactions with biological intelligence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' Biological systems constitute a proof of principle for how a complex computational problem can be solved, and thus can guide the engineering of new artificial intelligence models (Hassabis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' Sinz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' Ullman, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' This research direction is especially promising for improving artificial agents in domains at which biological agents excel (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=', out- of-domain generalization, transfer learning, adversarial robustness, few-shot learning), and even for endowing artificial agents with cognitive faculties idiosyncratic to humans (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=', planning, creativity, imagination).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' Artificial intelligence researchers have been successfully exploring these possibilities for decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' As an example, the structure of the current state-of-the-art artificial intelligence algorithms, deep neural networks, has been inspired by the structure of the visual system of the mammalian brain (Fukushima & Miyake, 1982).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' Furthermore, a growing amount of literature has started to exploit neural data representations to train natural language processing and computer vision algorithms, resulting in models with improved performance and adversarial robustness (Toneva & Wehbe, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' Safarani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' Dapello et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' The Algonauts Project fosters this exciting area of research by promoting interactions between the fields of artificial and biological intelligence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' The future of the project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' We hope that the 2023 installment of the Algonauts Project will continue to inspire new challenges and collaborations at the intersection of artificial and biological intelligence sciences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' We believe that both communities will benefit from jointly tackling open questions on how perception and cognition are solved in brains and machines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' We welcome researchers interested in initiating similar initiatives or collaborating with the Algonauts Project to contribute ideas and datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' Acknowledgments This research was funded by DFG (CI-241/1-1, CI241/1-3,CI-241/1-7) and ERC grant (ERC-2018- StG) to RMC;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' NSF award (1532591) in Neural and Cognitive Systems and the Vannevar Bush Faculty Fellowship program funded by the ONR (N00014- 16-1-3116) to AO;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' the Alfons and Gertrud Kassel foundation to GR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' Collection of the NSD dataset was supported by NSF IIS-1822683 and NSF IIS- 1822929.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' References Allen EJ, St-Yves G, Wu Y, Breedlove JL, Prince JS, Dowdle LT, Nau M, Caron B, Pestilli F, Charest I, Hutchinson JB, Naselaris T, Kay K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' A massive 7T fMRI dataset to bridge cognitive neuroscience and computational intelligence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' Nature Neuroscience, 4 Lateral Posterior a 0 LH RH LH RH Medial Ventral RH LH RH LHTest images s1 s8 Encoding model NsD experiment Predicted data Withheld data s1 s8 s1 s8 All vertices All vertices Correlate each vertex (v) V1 V1 pair 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' Yang X, Yan J, Wang W, Li S, Hu B, Lin J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' Brain- inspired models for visual object recognition: an overview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' Artificial Intelligence Review, 1-49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' Yarkoni T, Westfall J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' Choosing prediction over explanation in psychology: Lessons from machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' Perspectives on Psychological Science, 12(6):1100-1122.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} +page_content=' 5 Lateral Posterior a 0 LH RH LH RH Medial Ventral RH LH RH LHTest images s1 s8 Encoding model NsD experiment Predicted data Withheld data s1 s8 s1 s8 All vertices All vertices Correlate each vertex (v) V1 V1 pair across test images V2 V2 Sguared correlation scores CN Challenge metric: median noise- normalized sguared correlation' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE1T4oBgHgl3EQfdwQ1/content/2301.03198v1.pdf'} diff --git a/d9AyT4oBgHgl3EQfwvm8/vector_store/index.faiss b/d9AyT4oBgHgl3EQfwvm8/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..7e37c53c8d533d5f074b32ae0e9a60405b0dbc9a --- /dev/null +++ b/d9AyT4oBgHgl3EQfwvm8/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:9489e8e665152f3a4fe147a6ea2db26961e9759489116d2a49881f403d8a7601 +size 983085 diff --git a/ddE_T4oBgHgl3EQf0xx9/content/tmp_files/2301.08331v1.pdf.txt b/ddE_T4oBgHgl3EQf0xx9/content/tmp_files/2301.08331v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..7e262fa2fec365a3a1408d4817ff71e8cdba817c --- /dev/null +++ b/ddE_T4oBgHgl3EQf0xx9/content/tmp_files/2301.08331v1.pdf.txt @@ -0,0 +1,3265 @@ +Detection of gravitational waves in circular particle accelerators +II. Response analysis and parameter estimation using synthetic data +Suvrat Rao,1, ∗ Julia Baumgarten,2 Jochen Liske,1 and Marcus Br¨uggen1 +1Hamburger Sternwarte, Universit¨at Hamburg, Gojenbergsweg 112, 21029 Hamburg, Germany +2Physics Department, Jacobs University Bremen, Campus Ring 1, 28759 Bremen, Germany +(Dated: January 23, 2023) +We simulate the response of a Storage Ring Gravitational-wave Observatory (SRGO) to astrophys- +ical gravitational waves (GWs), numerically obtaining its sensitivity curve, parameter degeneracies, +and optimal choices of some controllable experiment parameters. We also generate synthetic noisy +GW data and use Markov Chain Monte Carlo (MCMC) methods to perform parameter estimation +of the source properties. With this, we show that a single SRGO could potentially localize the GW +source in the sky using Earth’s rotation. Then, we study the source sky localization area, mass and +distance estimation errors as functions of noise, data sampling rate, and observing time. Finally, we +discuss, along with its implications, the capacity of an SRGO to detect and constrain the parameters +of millihertz (mHz) GW events. +I. +INTRODUCTION +Theoretical studies of gravitational waves (GWs) in- +teracting with storage rings (circular particle accelerators +where ion beams circulate without collisions for long peri- +ods), intending to explore the possibility of using storage +rings as GW detectors, have been conducted indepen- +dently by several authors over the past decades [1–5]. +However, these studies had only considered the sce- +nario of GWs propagating perpendicular to the plane of +the storage ring (i.e. a “face-on” orientation). This par- +ticular case maximizes the GW-induced oscillations of +the ions (test mass particles) along the ring’s radial direc- +tion. Thus, one can hope to exploit resonances with the +beam’s betatron oscillations and detect the presence of +GWs using beam position monitors. As storage ring be- +tatron frequencies generally fall in a range where no sig- +nificant astrophysical GW sources exist, and since these +radial oscillations are expected to be minuscule, this idea +did not seem very promising. Meanwhile, it can be shown +(see Appendix A) that, in general, any periodic beam or- +bit shape distortions, such as those due to GWs, can +cause deviations in the circulation times of ions (from +their expected values during no perturbations), of the +order of magnitude h2, where h is the dimensionless GW +strain which is much smaller than unity. +In our previous paper [6], henceforth referred to as +paper-I, we showed that GWs can be better detected in +storage rings by measuring the ion circulation time de- +viation, which, in general, is proportional to h, being +caused by a GW-induced change in the velocities of the +ions, and only for the specific configuration of the GWs +propagating face-on to the storage ring does it become +proportional to h2 (and therefore, negligible compared +to h), where the GWs can only cause a distortion of the +beam orbit shape. Moreover, we showed that, although +the circulation time deviation has a periodicity equal to +∗ suvrat.rao@uni-hamburg.de +that of the GWs, its peak value is proportional to the +GW period, suggesting that it builds up over time dur- +ing the first half of a GW period and then wanes during +its second half. +This is true in the regime where the +observation time is much greater than the GW period. +As a result, it makes such a detector more sensitive to +lower frequency GWs. Importantly, we showed in paper- +I that, due to an overlap of several conditions, a Storage +Ring Gravitational-wave Observatory (SRGO) would be +most sensitive to the yet undetected millihertz (mHz) +GWs from astrophysical sources, that are also targeted +by future space-based GW detectors such as Laser Inter- +ferometer Space Antenna (LISA) [7, 8]. +The quantity measured by an SRGO would hence be +analogous to the “timing residuals” measured by pul- +sar timing arrays (PTAs), which are the GW-induced +arrival-time deviations of radio pulses (produced by mil- +lisecond pulsars) from their expected, highly regular ar- +rival times in the absence of GWs and noise sources [9]. +Moreover, SRGO ions being the moving test masses in- +fluenced by GWs, is similar to the idea of GW detection +by atom interferometry, where ballistic atoms are used +as test masses [10–14]. +Recently, D’Agnolo et al. independently found results +that are parametrically in agreement with our main cal- +culations from paper-I, under the condition that no RF +(radio frequency) system is present in the storage ring +[15]. Therefore, the general relativistic calculations for an +SRGO, done from two different reference frames (metric +formalism in our paper-I, and Riemann tensor formalism +by D’Agnolo et al.), give effectively, the same result. +In paper-I, we conducted a case study on the Large +Hadron Collider (LHC) at CERN as an existing facility +that could potentially be turned into an SRGO. How- +ever, LHC is not the ideal facility to realize the SRGO +detector because the presence of an RF system in the +storage ring can dampen the GW signal that we hope to +detect (D’Agnolo et al.). Instead, rings capable of stor- +ing coasting (meaning without RF system) “crystalline +ion beams” [16–18] or rings that could potentially store +arXiv:2301.08331v1 [gr-qc] 18 Jan 2023 + +2 +a single circulating ion [19] may be better laboratories +for detecting GWs. Moreover, there are better options +for the ion time-tagging detector technology than that +proposed in paper-I, such as “beam arrival time moni- +tors” [20], which are electro-optic charge centroid mon- +itors providing femtosecond timing precision. Also, im- +proving the vacuum quality inside storage rings would en- +able sustaining stable, coasting ion beams or single ions +for longer periods, allowing for longer SRGO observation +runs. +II. +REVIEW AND REVISION OF SRGO BASICS +Using the metric formalism of general relativity, in +paper-I, we derived the circulation time deviation of test +masses in a storage ring due to GWs. Here, we revise +some of these results and display them in a neater form. +We recall from Eq. (10) of paper-I, that the circulation +time deviation was given by the general expression, +∆TGW = +� +1− v2 +0 +2c2 +� � t0+T +t0 +� +hθφψ(t, α(t))−hθφψ(t0, α0) +� +dt, +(1) +where v0 is the speed of the ions in the absence of GWs, +c is the speed of light, t0 is the start of the observing time, +T is the duration of the observing time, and hθφψ has a +complex expression, being a function of the GW strain +amplitudes, h+,×(t); the time-varying orientation of the +GW with respect to the ring caused by Earth’s rotation, +θ(t), φ(t), ψ(t); and the angular trajectory of the ions in +the ring, α(t). +However, as it is not possible to know a priori whether +GWs are present at any given time, therefore, v0 can- +not be measured in practice. So instead, we reformulate +Eq. (1) in terms of v(t0) = vi (the instantaneous initial +speed of the ions), which can be measured. +We start +from Eq. (6) of paper-I, and keeping the term v(t0) as +it is, we follow through with the remaining derivation as +done in paper-I, to get a reformulated expression for the +circulation time deviation, +∆TGW = +� t0+T +t0 +�� +1 − v2 +i +2c2 +� +hθφψ(t, α(t)) +− 1 +2 +� +1 − v2 +i +c2 +� +hθφψ(t0, α0) +� +dt. +(2) +Now we make the substitution vi ≈ c, since ions in +storage rings are usually ultrarelativistic. +This would +make the second term within the square brackets negligi- +ble compared to the first, even for long observation times. +Although it is currently unclear, but using ultrarelativis- +tic ions might be beneficial (Schmirander et al., in prepa- +ration). This is because a single ion with sufficiently high +energy would, in principle, by its sheer momentum, ap- +preciably attenuate the effect of many stochastic and de- +terministic noise sources that would directly perturb the +ion (some of which were explored in paper-I). This would +also allow a coasting ion to deflect or deviate less from its +ideal orbit, and last longer in a non-ideal vacuum, thus +potentially allowing for longer SRGO observation runs. +Meanwhile, the first term is a function of three differ- +ent frequencies, viz. the GW frequency, Earth’s rotation +frequency and the revolution frequency of the storage +ring ions. The latter frequency is always much greater +than the former two in the case of ultrarelativistic ions +and mHz GWs. This allows us to analytically integrate +out the rapidly oscillating terms corresponding to the ion +revolution frequency. From paper-I, we recall, +hθφψ(t, α) = h+(t) +� +f + +s sin2 α + f + +c cos2 α + f + +sc sin 2α +� ++ +h×(t) +� +f × +s sin2 α + f × +c cos2 α + f × +sc sin 2α +� +, +(3) +where α(t) = α0 + v0 +R (t − t0), R being the radius of +the storage ring. +The terms sin2 α, cos2 α and sin 2α +integrate out to yield the constant factors +1 +2, +1 +2 and 0 +respectively. +The ‘f’-terms inside the parentheses are +functions of cosines and sines of the angles θ, φ, ψ (see +paper-I, Sect. 2). +Therefore, in the end, we obtain the reformulation (for +ultrarelativistic ions in an SRGO), +∆TGW = −1 +4 +� t0+T +t0 +� +F+h+ + F×h× +� +dt, +(4) +where h+(t) and h×(t) are the plus and cross polar- +ization GW strain components. F+ and F× are the plus +and cross polarization antenna pattern functions of an +SRGO, with F+ = sin2 θ cos 2ψ and F× = sin2 θ sin 2ψ. +Note that the integrand in Eq. (4) now has the same +form as the GW response of the Laser Interferometer +Gravitational-wave Observatory (LIGO) [21], but the re- +sponse signal in our case is a time-integral of this in- +tegrand. +Also, while the antenna pattern functions of +SRGO are very different compared to those of LIGO, +interestingly, they happen to be exactly the same as +those of a bar detector [22–25] whose longitudinal axis +is aligned perpendicular to the plane of the storage ring +(see Sect. 4.2.1 of [26]). Further, the SRGO antenna pat- +tern shown in Fig. 2 of paper-I (averaged over all polar- +ization angles), which was derived after making several +approximations, happens to be exactly valid even for the +general case derived here. +In paper-I, Appendix B, we noted that in the mHz +regime, the effect of Earth’s rotation must be taken into +account, which would cause the angles φ, θ, ψ (the az- +imuth, inclination and polarization angles respectively, +which orient an observer who is initially stood at the cen- +ter of the storage ring, along the GW propagation direc- +tion and the GW polarization axes) to be periodic func- +tions of time with a period of a sidereal Earth day. Here, +we revise the relation from paper-I, which establishes the + +3 +connection between the time-varying φ(t), θ(t), ψ(t), and +the parameters as measured from the equatorial celestial +coordinate system, in which the orientation of the GW +is fixed:- +� +� +cos ψ − sin ψ 0 +sin ψ +cos ψ +0 +0 +0 +1 +� +� +� +� +cos θ +0 sin θ +0 +1 +0 +− sin θ 0 cos θ +� +� +� +� +cos φ − sin φ 0 +sin φ +cos φ +0 +0 +0 +1 +� +� = +� +� +cos ψeq − sin ψeq 0 +sin ψeq +cos ψeq +0 +0 +0 +1 +� +�∗ +� +� +− sin +� +δsrc +� +0 +cos +� +δsrc +� +0 +1 +0 +− cos +� +δsrc +� +0 − sin +� +δsrc +� +� +� ∗ +� +� +− cos ωe +� +αsrc − l0 − (t − t0) +� +sin ωe +� +αsrc − l0 − (t − t0) +� +0 +− sin ωe +� +αsrc − l0 − (t − t0) +� +− cos ωe +� +αsrc − l0 − (t − t0) +� +0 +0 +0 +1 +� +�∗ +� +� +sin +� +θlat +� +0 − cos +� +θlat +� +0 +1 +0 +cos +� +θlat +� +0 +sin +� +θlat +� +� +�∗ +� +� +cos φ0 − sin φ0 0 +sin φ0 +cos φ0 +0 +0 +0 +1 +� +� . +(5) +ωe is the angular speed of Earth’s rotation. αsrc and +δsrc are the right ascension and declination of the GW +source [27]. +ψeq is the polarization of the GW in the +equatorial celestial coordinates. φ0 [28] is the angle be- +tween the line joining the center of the storage ring to the +timing detector, and the longitude passing through the +center of the storage ring, measured using the right-hand +curl rule starting from the detector position. l0 and θlat +are respectively, the local sidereal time at the start of the +observing time, t0, and the latitude of the center of the +storage ring. +III. +MODELS AND NUMERICAL +PROCEDURES +We obtain all the numerical results in this work from +a computer code written in Python and freely available +online on GitHub [29]. Below, we detail the mathemat- +ical models and numerical procedures programmed into +the code to obtain our results: +A. +GW source model +We consider the simplest realistic models for mHz GWs +from astrophysical sources viz. the dominant harmonic +of GWs from the quasi-circular inspiral phase of non- +spinning compact objects [30], accounting for the redshift +correction to the GW frequency and chirp mass. +The inspiral phase of a non-spinning binary system can +be modeled using post-Newtonian analysis [31], which +provides relatively simple analytical expressions for the +time-varying GW strain amplitudes corresponding to the +plus and cross polarizations: +h+ = 4 +dL +�GM +c2 +� 5 +3 �πf +c +� 2 +3 1 + cos2 (i) +2 +cos (2πft + δ0), +(6) +h× = 4 +dL +�GM +c2 +� 5 +3 �πf +c +� 2 +3 +cos (i) sin (2πft + δ0). (7) +If m1 and m2 are the masses of the objects in the bi- +nary, then M = (1+z)(m1m2) +3 +5 +(m1+m2) +1 +5 +is the redshift-corrected +chirp mass. +dL is the luminosity distance of the GW +source and z is its redshift. +i is the inclination angle +between the observer’s line of sight to the GW source +and the angular momentum vector of the GW source. δ0 +is the initial phase of the GW at the start of the ob- +serving time, t0. +The redshift-corrected, time-varying +GW frequency is f = (1 + z)−1 � +f +− 8 +3 +0 +− 8 +3k(t − t0) +�− 3 +8 = +(1 + z)−1 � +G(m1 + m2)/πr +3 +2 , where r is the separation be- +tween the objects in the binary; f0 is the GW frequency +at t = t0, corresponding to an initial separation of r = r0; +and k = 96 +5 π +8 +3 (GM/c3) +5 +3 . G and c are respectively, the +gravitational constant and the speed of light. +We use +an approximate analytical relation between dL and z for +ΛCDM cosmology, from [32]. +We use an approximation of the innermost stable cir- +cular orbit, risco = 6G max (m1,m2) +c2 +, to numerically mark +the end of the inspiral phase, upon reaching which, or at +the end of the user-provided observation time (whichever +comes earlier), the computer code is halted. +B. +Storage ring model +We consider a hypothetical circular storage ring having +a 100 m circumference and containing a single ultrarela- +tivistic ion that is coasting stably at close to the speed +of light, with no RF system and a single timing detector +present within the ring. The timing detector is placed +to the south of the storage ring’s center, such that it lies +on the longitude that passes through the center of the +storage ring i.e. φ0 = 0. + +4 +FIG. 1: The blue curve is the expected response of an SRGO to mHz GWs, for the given parameters. The pink dots +are a demonstration of discrete, noisy data points, created by adding artificial Gaussian noise to the SRGO response +signal. The orange ring is the initial SRGO position, while the black lines show the GW propagation direction and +plus polarization axes. +FIG. 2: Spectrogram of the SRGO response (left panel) and the time evolution of the Euler angles, φ, θ, ψ (right +panel), for the case in Fig. 1. For the spectrogram, the observation time begins earlier than in Fig. 1. +As per the Nyquist-Shannon sampling theorem, the +minimum sampling (data measurement) rate must be +greater than twice the highest expected GW frequency, +while the maximum possible sampling rate would corre- +spond to the timing detector making one detection per +revolution of the ion bunch, i.e. every time it arrives at +the detector. In our code, we choose the total number of +data points to be powers of two, as this allows for faster +computation of the Fast Fourier Transform (FFT) done +during the MCMC GW parameter estimation. +We assume that deterministic sources of noise, such as +gravity gradient noise, seismic activity, etc. are techno- +logically eliminated or accounted for, and only stochastic +noise sources remain in the experiment. The net resid- +ual noise is assumed to be Gaussian (an assumption also +made by LIGO [33]), with a standard deviation between +0.1 to 100 times the peak GW signal. We consider this +range of PSNR (peak signal-to-noise ratio) as values be- +yond this range may give trivial results. A better noise +model for the current study cannot be assumed until an + +1e-15 +Expected SRGO signal without noise +Synthetic noisy data points (201 data points, +SRGO latitude = 51.0° +PSNR = 3, Noise Std. Dev. = 5.1e-16 s) +GW source right ascension =_ oh om 0.0s +Sidereal day +GW source declination = 0.0° +GW polarization angle = 0.0. +GW initial phase = 0.0° +Equal mass (~106Mo) binary at z = 0.1 +End of inspiral phase at 75.0 hrs +Observer-SMBBH inclination angle = 0.0° +ATGW +2 +24 +30 +36 +48 +54 +60 +66 +72 +12 +18 +42 +Observation Time [hrs] +1 unit = 1.0 hrs1e-26 +0.0025 +2.00 +- 1.75 +0.0020 +- 1.50 +[ZH] + 1.25 +0.0015 +Frequency +- 1.00 +0.0010 +0.75 +- 0.50 +0.0005 +- 0.25 +0.0000 +0.00 +30 +36 +48 +54 +6 +12 +18 +24 +42 +60 +66 +72 +78 +84 +90 +96 +102 +108 +114 +120 +126 +132 +138 +144150 +Time [hrs]180 +150 +120 +90 +60 +30 +Angle [ +0 +-30 +-60 +06- +-120 +-150 +0 +-180 +6 +12 +18 +24 +30 +36 +54 +42 +48 +60 +66 +72 +0 +Time [hrs]5 +SRGO facility is actually established, or detailed studies +of noise sources have been done. +C. +Numerical solution for finding φ(t), θ(t), ψ(t) +Upon analytically multiplying all the matrices on the +left hand side of Eq. (5), and naming the final matrix on +the right hand side as R (which must be obtained by nu- +merically multiplying the five matrices on the right hand +side), we arrive at the following relations by comparing +the matrix elements on both sides: +φ(t) = arctan (R21, −R20), +θ(t) = arccos (R22), +ψ(t) = arctan (R12, −R02). +(8) +Note that in the above relations, numerically, we must +use the “arctan2” function to obtain the angles in their +correct quadrants. The right panel of Fig. 2 shows the +time evolution of the angles for the case corresponding +to Fig. 1. +D. +Numerical integration procedure +We use Boole’s rule quadrature [34] over a timestep to +compute its contribution to the integral of Eq. (4). Start- +ing from the initial value of the integral (equal to zero), +the contribution of each timestep is added to the integral, +and its cumulative value is saved after each timestep. We +perform this procedure till the halting condition (men- +tioned in Sect. III A) is reached. Thus, we numerically +obtain the SRGO response signal as a time series. +E. +Markov Chain Monte Carlo (MCMC) fitting +procedure +MCMC is a Bayesian inference tool for numerically +obtaining the joint posterior probability distribution of +unknown model parameters by directly drawing samples +from the posterior. MCMC works by following an algo- +rithm to find and explore around the regions in parameter +space that correspond to the maximum likelihood of the +parameters being a fit for the given data, given model +and a prior probability distribution of the unknown fit- +ting parameters [35–37]. +MCMC methods are preferred over the conventional +matched filtering algorithm [26] for thorough and effi- +cient GW parameter estimation, because the typical GW +models contain around 15 to 17 fitting parameters, and +making a grid in parameter space of such a high dimen- +sionality would require an impossibly long computation +time. +In our study, we do GW parameter estimation +using MCMC methods, keeping all possible unknown pa- +rameters as the fitting parameters. +These are nine in +number, namely, the GW source masses m1, m2 ; the ini- +tial separation between the masses, r0 ; the GW source +inclination angle, i ; the GW source redshift, z ; the ini- +tial phase of the GW, δ0 ; the right ascension, αsrc and +declination, δsrc of the GW source ; and the polarization +angle, ψeq of the GW in the reference frame of equatorial +celestial coordinates. +In our code, we first create synthetic noisy data points +by adding Gaussian noise to the SRGO response com- +puted for user-provided parameters. The noisy data is +then transformed to Fourier space via a Fast Fourier +Transform (FFT) and passed to the likelihood function. +We use flat priors for the unknown fitting parameters +and a two-dimensional Gaussian noise “Whittle” likeli- +hood function, as also done by LIGO for GW param- +eter estimation [33]. +The priors corresponding to an- +gular parameters are bounded between −180° and 180°, +whereas priors corresponding to non-angular parameters +are bounded between zero and double of their true pa- +rameter values for computational efficiency. We employ +the Differential Evolution Markov Chain (DE-MC) algo- +rithm [38] for the MCMC chains, and run per case, 1000 +parallel chains which draw 1250 samples each. Since we +are purely interested in parameter estimation, and not +in showing the convergence of chains to the region of +maximum likelihood, we do not discard 25% of the ini- +tial traces as the “burn-in” phase. Instead, we allow the +chains to start from the true parameter values and then +explore around. Each case is repeated 10 times by regen- +erating the noisy data points, to obtain the statistical +variation of the joint posterior. In all cases, we choose +the 3−sigma (99.7%) highest posterior density (HPD) +region for the parameter estimation, and discard those +cases where the true parameter values do not lie within +this region. MCMC is carried out in our code using the +PyMC3 Python module [39]. +IV. +RESULTS: SRGO RESPONSE ANALYSIS +A. +Response signal analysis +In Fig. 1, we show a demonstration of the response +of an SRGO to mHz GWs from an SMBBH (supermas- +sive binary black hole) inspiral, for an arbitrarily cho- +sen configuration of parameters. In general, we notice +that the response has an envelope which periodically re- +peats every sidereal day due to the effect of Earth’s ro- +tation. Further, unlike LIGO, the chirping of the GW +strain amplitude does not reflect in SRGO’s response +spectrogram (Fig. 2 left panel), whose amplitude actu- +ally diminishes with time, as the SMBBH inspirals and +the GW frequency increases. +This is because, the re- +sponse amplitude has an inverse relation with the GW +frequency, which overpowers the contribution due to an +increase in the GW strain amplitude with time. The ex- +pected orders of magnitude of the SRGO response signal +amplitude (due to astrophysical sources) are discussed in + +6 +Sect. IV C. +In Fig. 3, we notice first that, in the regime where +f Tobs ≫ 1, the PSNR decreases with increasing GW +frequency. Thus, the right side tails in the plots agree +with our analytical results from paper-I. At very low +GW frequencies, where f Tobs ≪ 1, the timing deviation +buildup is slow, and the GW strain amplitude is also +quite small. Therefore, the plots have tails on the left +side as well. Finally, in the regime where f Tobs ∼ 1, the +PSNR decreases with decreasing observation time. This +is because, when the observation time is large and also +close to the GW period, a large timing deviation can be +accumulated by the circulating ions, as opposed to cases +with smaller observation times. +The dynamically changing peaks and valleys in the se- +ries of plots can be somewhat elucidated: they are the +result of the interplay between the GW period, initial +GW phase, the orientation of SRGO relative to the GW +source, Earth’s rotation period and the observation pe- +riod. The orientation of SRGO relative to the GW source +and Earth’s rotation period together determine the en- +velope seen in the response signal of Fig. 1. During the +observation time, large peaks in these plots occur when +peaks in the GW strain waveform align with the envelope +peaks, and valleys occur when the GW strain peaks align +with the envelope valleys (or vice versa, whichever pro- +duces a greater response signal). For a given observation +time, as the initial GW frequency is increased, a number +of successive GW strain peaks and valleys occur during +the observation period, which may or may not align with +the envelope peaks and valleys. An aligned GW strain +peak (resulting in a local maxima in the plot curves), +upon increasing the initial GW frequency, will become +misaligned, resulting in a drop in the curve until the next +peak starts becoming aligned, causing then a rise in the +curve. This succeeding local maxima may be higher or +lower than its predecessor, depending on the location of +the global maxima, which occurs when the initial GW +period is close to the observation time. +This explains +the spiky sections of the plot curves, prominently seen in +Fig. 3c, but also present in the other plots. +In Figs. 3a, 3b and 3c, the noisy regions in the right +side tails of the plots have purely numerical origins, being +caused by the abrupt halting of the code one timestep af- +ter the inspiral phase has been crossed. Therefore, they +exist only at the ends of the right side tails (correspond- +ing to high initial GW frequencies, meaning that the bi- +nary compact objects begin close to the end of their in- +spiral phases). As the observation time is reduced, the +end of the inspiral phase would be reached within the +observation time at higher initial frequencies. Therefore, +the noisy regions shift towards higher frequencies and +eventually disappear in Figs. 3d, 3e and 3f. +In Fig. 4, we plot the PSNR as a function of the source +position in the sky, scaled such that the arbitrary case +corresponding to Fig. 1 has a PSNR of unity. +While +changing the source position, all other parameters re- +main the same as those in Fig. 1, but the initial SMBBH +separation is 1 AU, and signal is computed over an obser- +vation time of 1 day. We repeat this for different latitudes +of the SRGO location on Earth. +We observe that the complex patterns in the plots are +a modified projection of the SRGO antenna pattern on +the sky, as expected from Eq. 4. Other parameters being +fixed, for a given SRGO latitude and source declination, +the horizontal variation is determined by the initial hour +angle, the GW frequency, initial GW phase, GW polar- +ization angle and Earth’s rotation frequency. +Maxima +in the horizontal variation occur during the observation +time when peaks in the GW strain waveform occur ex- +actly at a time when there are peaks in the response sig- +nal envelope. Minima occur when the GW strain peaks +align with the envelope valleys (or vice versa, whichever +produces a greater response signal). We note that the +pattern depends on the initial hour angle i.e. the right +ascension of the GW source relative to the initial SRGO +local sidereal time (and not on their absolute values). +However, the same is not true between the SRGO lati- +tude and source declination. That symmetry is broken by +the Earth’s spin axis, and therefore, the SRGO latitude +variation produces different patterns in the plots. The +bottom-right plot corresponds to SRGO being placed at +the pole. For this case, if the GW source is also at one +of the poles, then the orientation always remains face-on, +and no response signal is produced. For all other cases, a +non-zero response signal is produced over 24 hours, due +to a changing orientation of the GW source relative to +the ring. +Further, we observe that the plots show antipodal sym- +metry, since placing the ring and/or the GW source at +antipodal positions, and/or having the ions circulating +in the opposite direction, would all produce the same +SRGO response. The plots also appear to have a left- +right symmetry, because two GW sources at the same +declination, initially located on either side of the ring +and having the same hour angle magnitude, would pro- +duce the same value of the initial SRGO response. The +response signals over a day’s time would, however, not +be exactly the same due to Earth’s rotation, as SRGO +would move towards the source in one case and away from +the source in the other case. But over 24 hours, SRGO +would cover all possible azimuthal orientations relative to +the two sources. Therefore, the peak signal values would +be similar between such cases, although the peak sig- +nal may be achieved at different times. This symmetry +would break at higher GW frequencies, if the response +signal amplitude drops quick enough within one day due +to the chirping of the GW frequency. There is also a lati- +tudinal symmetry, as having SRGO located at equal and +opposite latitudes would simply flip the plots about the +horizontal axis. +The SRGO latitude variation indicates that placing the +ring near the equatorial latitudes on Earth would be more +advantageous than placing the ring near the polar lati- +tudes, offering an increase of the maximum PSNR by +around 3 times, of the average PSNR by around 4 times, + +7 +(a) Tobs = 24 hrs +(b) Tobs = 12 hrs +(c) Tobs = 3 hrs +(d) Tobs = 1 hour +(e) Tobs = 10 mins +(f) Tobs = 1 min +FIG. 3: Scaled values of the peak signal-to-noise ratio as a function of the initial GW frequency, for different values +of the observation time, Tobs. All other parameters are equal to those shown in Fig. 1, except the initial separation +between the masses, which is 1 AU, and the black hole masses, which have been chosen to be 3 M⊙ each, so that the +chosen GW frequency range can be covered during their inspiral phase. The scaling is done such that the largest +peak among all plots has a value of unity. +and of the minimum PSNR by a factor of unity from zero, +between the polar and equatorial SRGO latitudes. +B. +SRGO sensitivity curve +The sensitivity curve of a GW detector may be de- +fined as the curve in the plane of the GW strain ampli- +tude spectral density versus the GW frequency, where +the signal-to-noise ratio is unity. +In this study, we use our code to numerically compute +the SRGO sensitivity curve. We start by equating the +root mean square value of the response signal to the ef- +fective noise, +[∆TGW ]rms = +σnoise +� +fsampleTobs +, +(9) +where Tobs is the total observation time and fsample = +npv0 +2πR is the data sampling rate, with np being the num- +ber of circulating test masses that are timed, which may +be individual ions or bunches of ions. We then expand +the left hand side using Eq. (4), substituting h+ and h× +from Eq. (6) and Eq. (7), respectively. Then we set k = 0 +in the GW frequency evolution to get continuous GWs +(i.e. GWs with a constant frequency). Finally, we club +together all the common time-independent terms within +the integral, identifying this quantity as the GW strain +amplitude, ˜h(f). Rearranging the equation, we thus ob- +tain, +˜h(f) = +4σnoise +� +fsampleTobs +. +�� t0+T +t0 +� +F+.1 + cos2 (i) +2 +cos (2πft + δ0) ++ F×. cos (i) sin (2πft + δ0) +� +dt +�−1 +rms +. +(10) +The GW strain amplitude spectral density is plotted as +a function of the GW frequency, f, and it is numerically +averaged over all the GW source parameters within the +integral of Eq. 10. It is given by, +ASD = +˜h(f) +√Tobs +. +(11) +The result shown in Fig. 5 numerically confirms that +an SRGO would be sensitive to mHz GWs by design. In +the log scale, the SRGO sensitivity curve has a linear + +1.0 +Tobs = 24.0 hrs +GW period equals Tobs +0.8 +Scaled PSNR +0.6 +0.4 +0.2 +0.0 +10-7 +10-6 +10-5 +10-3 +10-2 +100 +10-4 +10-1 +Initial GW frequency [Hz]0.5 +Tobs = 12.0 hrs +GW period equals Tobs +0.4 +Scaled PSNR +0.3 +0.2 +0.1 +0.0 +10-7 +10-6 +10-5 +10-4 +10-3 +10-2 +100 +10-1 +Initial GW frequency [Hz]0.12 +Tobs = 3.0 hrs +GW period equals Tobs +0.10 +0.08 +Scaled PSNR +0.06 +0.04 +0.02 +0.00 +10-7 +10-6 +10-5 +10-4 +10-3 +10-2 +10-1 +100 +Initial GW frequency [Hz]Tobs = 1.0 hrs +0.08 +GW period equals Tobs +0.07. +0.06 +Scaled PSNR +0.05 +0.04 +0.03 +0.02 +0.01 +0.00 +10-6 +10-5 +10-4 +10-3 +10-2 +10-1 +100 +10-7 +Initial GW frequency [Hz]Tobs = 10.0 mins + GW period equals Tobs +0.05 - +0.04 - +Scaled PSNR +0.03 +0.02 +0.01 : +0.00 +10-7 +10-6 +10-5 +10-4 +10-3 +10-2 +10-1 +100 +Initial GW frequency [Hz]Tobs = 1.0 mins +0.025 +GW period equals Tobs +0.020 +Scaled PSNR +0.015 +0.010 +0.005 +0.000 +10-7 +10-6 +10-5 +10-3 +10-2 +10-1 +100 +10-4 +Initial GW frequency [Hz]8 +(a) SRGO latitude = 0° +(b) SRGO latitude = 30° +(c) SRGO latitude = 51° +(d) SRGO latitude = 90° +FIG. 4: Scaled values of the peak signal-to-noise ratio as a function of the GW source’s initial position in the sky +relative to SRGO. All other parameters are equal to those shown in Fig. 1, except the initial separation between the +masses which is 1 AU, and the observation time which is 1 day. The top-left, top-right, bottom-left and +bottom-right figures respectively correspond to an SRGO latitude of 0°, 30°, 51° and 90°. The scaling is done relative +to the case corresponding to the center of the bottom-left plot. +behaviour in the mHz frequency regime, as analytically +predicted in paper-I [40]. The sensitivity deteriorates at +higher frequencies. This is because, as the GW period +becomes smaller, the SRGO response amplitude also de- +creases, since the ions spend lesser time accumulating a +timing deviation during every half-cycle of the GW. This +aspect is discussed in the previous section and shown in +Fig. 3. Therefore, a larger strain amplitude would be re- +quired to detect high frequency GWs. This would greatly +exceed the predicted strain amplitudes from astrophysi- +cal sources in the decihertz or kilohertz ranges (i.e. for +“LIGO-like” sources). +At very low frequencies also, the sensitivity curve rises. +We may perform a thought-experiment to analyse this +situation: A zero frequency GW would be equivalent to +an anisotropic spacetime having a constant distortion, +and not necessarily a flat spacetime. In such a spacetime, +if the instantaneous initial speed of the circulating test +mass is measured and used to predict the expected future +arrival times of the test mass at the timing detector, then +the observed arrival times would deviate from the predic- +tions, as the test mass traverses an anisotropic spacetime. +This is why, if we input f = 0 in Eq. (4), we still get a +finite value of the response signal. Therefore, unlike laser +interferometers and atom interferometers (which use test +masses that can only move linearly) which require both +the temporal and spatial components of GW spacetime +in order to probe it, an SRGO (which utilizes circulating +test masses) would, in principle, be able to probe purely +the spatial anisotropy of GW spacetime even at very low +GW frequencies and finite observation times. However, +for low frequency GWs, as the GW period far exceeds +the total observation time, Tobs, the peak response signal +value would start decreasing, as discussed in the previ- +ous section and shown in Fig. 3. That is why the SRGO +sensitivity curve rises again at low frequencies. Also, for +most resolvable astrophysical GW sources, the strain am- +plitude at near-zero frequencies would be near-zero. +The sensitivity curve should corroborate with Fig. 3c, +as both are computed for an observation time of 3 hours +and we expect them to be inversely related. Although +the general shape of the two curves agree with each other, + + 3.0 +SRGO initial zenith +SRGO zenith's track in 24.0 hrs +Scaled by PSNR at center for 51° SRGO latitude +SCALED PSNR VS. SOURCE POSITION +Maxima = 3.04 +75° +Average = 1.98 +Minima = 1.08 + 2.5 +60° +45° +30° + 2.0 +15° +1.5 +D +-15° +-30° +1.0 +-45° +-60° + 0.5 +-75° +INITIAL HOUR ANGLE [HRS] +0.(3.0 + SRGO initial zenith +SRGO zenith's track in 24.0 hrs +Scaled by PSNR at center for 51° SRGO latitude +SCALED PSNR VS. SOURCE POSITION +Maxima = 2.83 +75° +Average = 1.94 +Minima = 1.12 + 2.5 +60° +45° +30° + 2.0 +15° +1.5 +D +-15° +-30° +1.0 +-45° +-60° + 0.5 +-75° +INITIAL HOUR ANGLE [HRS] +0.(- 3.0 +SRGO initial zenith +SRGO zenith's track in 24.0 hrs +Scaled by PSNR at center for 51° SRGO latitude +SCALED PSNR VS. SOURCE POSITION +Maxima = 2.46 +75° +Average = 1.66 +Minima = 0.6 + 2.5 +60° +45° +30° + 2.0 +15° +C +1.5 +-15° +-30° + 1.0 +-45° +-60° + 0.5 +-75° +INITIAL HOUR ANGLE [HRS]3.0 +SRGO initial zenith +SRGO zenith's track in 24.0 hrs +Scaled by PSNR at center for 51° SRGO latitude +SCALED PSNR VS. SOURCE POSITION +Maxima = 1.09 +75° +Average = 0.54 +Minima = 0.0 +2.5 +60° +45° +30° +2.( +15° +1.5 +-15° +30° +-45° +60° +0.5 +-75° +INITIAL HOUR ANGLE [HRS]9 +FIG. 5: The numerically computed sensitivity curve of +an SRGO for the given parameter values, and averaged +over all other parameter values. +their details are different, since the sensitivity curve has +been computed by averaging over several parameters, +whereas Fig. 3c corresponds to a fixed set of parame- +ters. The GW frequency of the sensitivity curve minima +matches the GW frequency of the maxima in Fig. 3c. In +general, combining the insights from Figs. 3 and 5, we de- +duce that the minima of the sensitivity curve would occur +at a GW frequency close to the inverse of the observation +time, and that this minima would be smaller for longer +observation times. Based on the predicted astrophysical +mHz GW sources, we can conclude that the minimum ob- +servation time for an SRGO experiment to be maximally +sensitive to the entire mHz GW regime, would be of a few +hours. This can be seen in Fig. 5, where the minima of +the sensitivity curve lies close to the low-frequency edge +of the predicted mHz GW regime. +C. +SRGO observational range +From the ninth catalogue of spectroscopic binary orbits +(SB9) [41], cross-referenced with the Gaia Data Release +3 [42–44], we find that the nearest spectroscopic binaries +within our galaxy, including white dwarf (WD) binaries +with masses ∼ 0.5M⊙ and periods of a few days, are lo- +cated at distances of a few tens of parsecs (pc). Whereas, +the nearest spectroscopic binaries with periods of a few +hours are located at distances of several tens of parsecs. +Hence, we choose a distance of 50 pc to mark the nearest +WD binaries that would emit mHz GWs. +The nearest neutron star (NS) binaries [45–49] with +masses ∼ 1M⊙ and periods of a few days, are located at +distances of a few hundreds of parsecs [50]. Whereas, the +nearest known double neutron star system with a period +of a few hours is located at around 600 pc [51, 52]. Hence, +we choose this distance to mark the nearest NS binaries +FIG. 6: The maximum effective noise allowed in an +SRGO to make a detection, or equivalently, the largest +SRGO response amplitude expected in the best-case +(optimum parameter choice) scenarios, due to GW +sources at various distances. The colored dash-dotted +lines indicate the nearest location of a particular type of +source i.e. these GW sources are absent to the left of +the colored line corresponding to them. All +computations are done for a fixed observation time of 3 +hours, and an initial GW frequency which is at the +expected lower limit of the mHz regime, as this would +maximise the SRGO response amplitude. +that would emit mHz GWs. +It is estimated that our Milky Way galaxy contains mil- +lions of stellar mass black holes [53]. From binary black +hole population simulations for Milky Way-like galax- +ies [54], it is estimated that binary black holes may be +present as close as 1 kpc from Earth, although most of +them would be present 8 kpc away, near the galactic cen- +ter. This also happens to agree with the recent unam- +biguous detection via astrometric microlensing, of an iso- +lated stellar mass black hole [55], located at 1.58 kpc from +Earth. Hence, we choose this distance as an estimate of +the nearest stellar mass binary black holes with masses +∼ 10M⊙. +Intermediate mass black holes (IMBHs) of 102 – +105M⊙ are expected to be found in globular clusters +and massive star clusters, but would be more numerous +within galactic bulges of large galaxies and within dwarf +galaxies [56, 57]. In galaxies like ours, globular clusters +containing IMBHs of 103 – 104M⊙ are expected to be +numerous at distances of 10 kpc from the galactic center, +and these IMBHs can emit mHz GWs by merging with +stellar mass black holes [58, 59]. Hence, we use the dis- +tance to the nearest known globular cluster “M4”, of 2.2 +kpc, to estimate the whereabouts of the nearest ∼ 10M⊙ +& 104M⊙ extreme mass ratio inspirals (EMRIs). +As per [60], a typical large galaxy can contain several +“wandering” SMBHs of ∼ 106M⊙, spread out across the + +10-10 +Massive binaries +Extreme mass-ratio inspirals +10-12. +Resolvable galactic binaries +Unresolvable galactic binaries +SRGO sensitivity curve +Tobs = 3.0hrs, Onoise = 1ps, fsample = 2.998MHz +10-14 +10-16 +10-18 +10-20 +10-22 +10-24 +10-26 +10-7 +10-5 +10-3 +101 +10-1 +Frequency [Hz]GW source redshift (z) +100 +101 +10-9 +10-8 +10-7 +10-6 +10-4 +10-5 +10-3 +10-2 +10-1 +10-6± +fgw = 5 × 10-4 Hz; Tobs = 3.0 hrs + nearest ~0.5M。 WD binaries + nearest ～1M。 NS binaries + nearest ～1OMo BH binaries +S + nearest ～10M。 104M。 EMRIs +10- +9 +PSNR=1 I + nearest ～103M。& 106M。EMRIs + nearest ～103M。& 108Mo EMRIs +nearest ～107M。 SMBH binaries + first ～1o7M。 SMBH binaries +for +10-12 + noise +10-15 +lax. +M +10-21 +102 +103 +105 +107 +101 +104 +106 +108 +109 +1010 +1011 +GW source distance [pc]10 +galactic halo, from near the galactic center to within the +dwarf satellite galaxies and anywhere in between. Milky +Way’s central supermassive black hole (SMBH), Sgr A*, +also happens to be of ∼ 106M⊙ [61]. According to [62], +the most promising mHz GW scenario in the IMBH – +light SMBH mass range, is of ∼ 103M⊙ IMBHs merging +with ∼ 106M⊙ SMBHs. Lastly, the nearest galaxy to us, +M31 (Andromeda), contains a ∼ 108M⊙ central SMBH +[63]. For all these reasons, we choose a distance of 8 kpc +(distance from Earth to Milky Way’s center, as well as +to the closest dwarf galaxy, Canis Major) to represent +the location of the nearest ∼ 103M⊙ & 106M⊙ EMRIs. +We choose 0.778 Mpc (distance from Earth to M31 An- +dromeda galaxy) to represent the location of the nearest +∼ 103M⊙ & 108M⊙ EMRIs. +The redshift evolution of the SMBH mass function [64– +66] tells us that, on average, 107M⊙ SMBH mergers may +be the most frequent. The nearest detected inspiralling +SMBHs due to a galaxy merger, are located at a distance +of 27.4 Mpc [67]. +As per [68], the first SMBH merg- +ers happened at around z = 10, when the first galaxies +started merging in the early universe. We input this in- +formation in Fig. 6 +From Fig. 6, we see that the largest SRGO response +signals would correspond to mHz GWs from SMBH bi- +naries in galaxy mergers. EMRIs involving an SMBH, +typically the central SMBH of galaxies, or even wander- +ing SMBHs interacting with smaller black holes, would +also be significant sources. +WD binaries, NS binaries +and stellar mass BH binaries within our galaxy would +not produce great responses, even if they were individu- +ally resolvable sources and located as close to Earth as +possible. IMBH EMRIs within globular clusters in our +galaxy may also give decent response amplitudes. +Fig. 6 also tells us that, an SRGO should aim for an ef- +fective residual stochastic noise of ∼ 1ps or better. Since +the effective noise depends not only on the true noise, but +also on the data sampling rate and observing time, the +true residual stochastic noise may be greater than ∼ 1ps, +but can be effectively cut down by collecting more timing +data points during the observation run (see Sect. IV B). +At this level of noise (or better), an SRGO could po- +tentially detect mHz GW events involving supermassive +black holes starting from within our galaxy, up to galaxy +merger events at high redshifts. +V. +RESULTS: GW PARAMETER ESTIMATION +The antenna pattern of a GW detector is typically om- +nidirectional (see [69] for the LIGO antenna pattern, and +paper-I for the SRGO antenna pattern). Therefore, even +with a high signal-to-noise ratio, a single GW detector +like LIGO, in principle, cannot pinpoint the position of +the GW source in the sky. However, three or more detec- +tors working together can triangulate the source position +via the relative time-delays between their detections from +the same source. But unlike LIGO, even a single SRGO, +being a potential Earth-based mHz GW detector (where +the GW signal may last for hours, days, or much longer), +in principle, should be able to make use of Earth’s rota- +tion (which would cause the GW source to sweep across +its antenna pattern and produce a unique envelope in +its response) to pinpoint the source position. The GW +source sky localization area, however, would certainly de- +pend on the signal-to-noise ratio. +A single LIGO de- +tector, on the other hand, being sensitive to kHz GWs, +would not be able to fully make use of Earth’s rotation, +because the response signal duration of LIGO would be +much shorter compared to SRGO for the same effective +PSNR and observation time. This hypothesis is verified +by our simulation results, where MCMC methods have +been used to do GW parameter estimation on noisy data +points that were created by adding Gaussian noise to the +SRGO response signal. +An example of our results, the sky localization map +for a case corresponding to 32 effective data points taken +over 12 hours at a PSNR of 100, is shown in Fig. 8a. We +note here our method of computing the sky localization +area: We take the joint posterior of the right ascension +and declination of the GW source (Fig. 8b), and calculate +the ratio of the colored to the total pixels. +Then we +multiply this with the range of the right ascension and the +range of the declination. Finally, we apply a correction +for the spherical projection onto the sky. The formula +becomes, +Asky +� +deg2� += colored pixels +total pixels +180 +π ∆αsrc ∆ sin δsrc (12) +Fig. 7 corresponds to the same case as Fig. 8. It is +an example of the MCMC chain traces, and the diagonal +elements of the 9 × 9 joint posterior corner-plot (shown +in Appendix B, Fig. 10). We see that in general, at a +decent PSNR, the MCMC chains converge around the +true parameter values and explore around this location +in the 9-dimensional parameter space. +A. +Variation of parameter estimation quantities +We explore the variation of the posteriors as functions +of some controllable experiment parameters (such as the +observation time, data sampling rate and PSNR), for 5 +out of the 9 fitting parameters in our model. These are: +the GW source component masses, m1 & m2 ; the GW +source redshift, z ; and the GW source sky position, αsrc +& δsrc. The results for αsrc & δsrc jointly correspond +to the sky localization area, shown in Figs. 9a and 9b. +By symmetry, the results for m1 & m2 are the same, +and correspond to Figs. 9c and 9d. +The results for z +have been translated into the GW source luminosity dis- +tance, and correspond to Figs. 9e and 9f. +We choose +these 5 parameters as they are the most relevant ones +for multi-messenger astronomy, being the first to be es- +timated upon a GW detection. + +11 +FIG. 7: On the left are the marginalized posteriors of +the fitting parameters and on the right are the +corresponding MCMC traces, consisting of 1000 parallel +chains with 1250 samples each. The true parameter +values for this case are the ones in Fig. 1, except the +initial separation between the masses which is 1 AU. 32 +data points with artifical noise added (PSNR = 100) +are taken over an observing time of 12 hours. +We generate 16, 32 and 64 data points in this study for +a multitude of reasons: First, as explained in Sect. III E, +we use powers of two as this allows for faster computa- +tion of the Fast Fourier Transform (FFT). Furthermore, +in our computations, 16 data points happens to be the +lower limit for the Shannon-Nyquist condition to hold. +Hence, this would give us an upper limit on the parame- +ter estimation for a given PSNR. Also, although in reality +it would be possible for a timing detector to make sev- +eral million measurements within an observation time of +hours to days, we use only a small number of data points +for computational efficiency. This is justified because a +smaller number of data points at a given PSNR may be +interpreted as binning a large number of data points that +correspond to a lower true PSNR, thus giving the same +(a) GW source localization sky map +(b) δsrc vs. αsrc joint posterior +FIG. 8: The GW source sky localization i.e. the +3-sigma (99.7%) HPD region on the joint posterior of +the GW source’s right ascension and declination, shown +on a sky map. This figure corresponds to the case +shown in Fig. 7. This shows that a single SRGO can +potentially use Earth’s rotation to localize the GW +source in the sky. +effective PSNR. +In Fig. 9a, we see that, for a given data sampling +rate and observation time, the sky localization area de- +creases with decreasing noise, and saturates at around +a few tens of deg2 for high effective PSNR. This is due +to parameter degeneracies that cannot be resolved fur- +ther, unless multiple SRGOs are utilized or better mod- +els are utilized that, for instance, account for higher-order +harmonic modes of GWs [70]. Furthermore, at a given +PSNR, the sky localization improves upon increasing the +data sampling rate. This is because the effective noise +is inversely proportional to the square root of the total +number of data points, or in other words, the square root +of the data sampling rate times the observing time. The +error bars show the statistical variation of the parame- +ter estimation, and they increase with increasing noise. +Thus, parameter estimation becomes unreliable at high + +GW source mass #l posterior [Mo] +1e6 +GW source mass #1 posterior [Mo] +1.05 +1.00 +0.95 +0.90 +1.02 +200 +400 +600 +800 +1000 +1200 +0.90 +0.92 +0.94 +0.96 +0.98 +1.00 +1.04 +0 +1e6 +GW source mass #2 posterior [Mo] +1e6 +GW source mass #2 posterior [Mo] +1.05 +1.00 +0.95 +0.90. +1.075 +0.925 +0.950 +1.000 +1.025 +200 +400 +1000 +0.900 +0.975 +1.050 +600 +800 +1200 +1e6 +Initial binary separation posterior [au] +Initial binary separation posterior [au] +1.00 +0.98 +200 +400 +0.980 +0.985 +0.990 +0.995 +1.000 +1.005 +1.010 +0 +600 +800 +1000 +1200 +0.975 +GW source inclination posterior [°] +GW source inclination posterior [°] +10 +0 +-10 +0 +200 +400 +600 +800 +1000 +1200 +10 +-5 +10 +5 +GW source redshift posterior +GW source redshift posterior +0.11 +0.10 +001t'0. S20t'0 0501'0 5201'0 0001'0 s260'0 0560'0. 5260'0 0060'0 +200 +400 +600 +800 +1000 +1200 +GW initial phase posterior [°] +GW initial phase posterior [°] +20 +0 +20 +10 +0 +200 +400 +600 +800 +1000 +-20 +-10 +0 +20 +1200 +GW source RA posterior [°] +GW source RA posterior [°] +400 +200 +600 +800 +1000 +1200 +-2 +GW source DEC posterior [°] +GW source DEC posterior [°] +0 +200 +400 +600 +800 +1000 +1200 +-4 +-2 +0 +2 +GW polarization posterior [°] +GW polarization posterior [°] +10 +0 +-10 +0 +200 +400 +600 +800 +1000 +1200 +-10 +-5 +0 +5 +10GW source position +X +3-sigma HPD region +Localization area = 28.6451 deg3 +75° +.09 +45° +30° +15° +10h +08h +06h +04h +02h +22h +20h +18h +16h +0Qh +14h +0° +D +-15° +-30° +-45° +.09- +-75° +RA2 +GW source DEC posterior [°] +O +2 +-4 +-4 +-2 +0 +2 +4 +GW source RA posterior []12 +(a) Sky localization area vs. PSNR +(b) Sky localization area vs. Tobs +(c) Mass estimation error vs. PSNR +(d) Mass estimation error vs. Tobs +(e) Distance estimation error vs. PSNR +(f) Distance estimation error vs. Tobs +FIG. 9: The parameter estimations of three parameters are shown: GW source sky localization area, relative errors +of distance and mass estimation. The left column shows their variation with PSNR for different data sampling rates. +The right column shows their variation with observation time at a fixed data sampling rate, for different values of +PSNR. + +16 data points over 1 day +32 data points over 1 day +64 data points over 1 day +104 +103 +102 +101 +10-1 +100 +101 +102 +PsNR (peak signal-to-noise ratio)PSNR = 1.0 +PSNR = 10.0 +104 +PSNR = 100.0 +103 +Sky localization area +102 +101 +15 +21 +0 +3 +6 +9 +12 +18 +24 +27 +Observation time [hrs]Relative error of mass estimation [%] +102 +16 data points over 1 day +32 data points over 1 day +64 data points over 1 day +101 +10-1 +100 +101 +102 +PsNR (peak signal-to-noise ratio)Relative error of mass estimation [%] +102 +PSNR = 1.0 +PSNR = 1O.0 +PSNR = 100.0 +101 +15 +21 +0 +3 +6 +6 +12 +18 +24 +27 +Observation time [hrs] +(fsample = 2.666 data points per hour)Relative error of distance estimation [%] +102 +16 data points over 1 day +32 data points over 1 day +64 data points over 1 day +10-1 +100 +101 +102 +PsNR (peak signal-to-noise ratio) estimation [%] +102 +Relative error of distance +PSNR = 1.0 +PSNR = 10.0 +PSNR = 100.0 +3 +12 +15 +21 +6 +6 +18 +24 +27 +0 +Observation time [hrs]13 +levels of noise, with the sky localization area covering al- +most the entire sky for PNSR values lower than ∼ 0.1. +An effective PSNR of ∼ 80 seems to be the threshold for +a single SRGO to achieve its best possible sky localiza- +tion. Comparing values from a curve of constant PSNR +in Fig. 9b with the corresponding values at the same +PSNR in Fig. 9a, we see that in general, for the same +effective PSNR (i.e. +same PSNR and number of data +points), increasing the observation time improves sky lo- +calization. This is because, the effect of Earth’s rotation +can be exploited to a greater extent to break some param- +eter degeneracies. An exception to this trend may occur +when the Shannon-Nyquist condition is violated, i.e. for +a fixed number of data points, a smaller observation time +may result in better parameter estimation if increasing +the observation time (reducing the data sampling rate) +results in aliasing. This scenario would typically not be +relevant for a realistic SRGO experiment, where the data +sampling rate would be orders of magnitude higher than +the GW frequency. Finally, beyond 24 hours observing +time, the Earth’s rotation cannot break any more pa- +rameter degeneracies in principle, and therefore even at +high effective PSNR values in Figs. 9a and 9b, the sky +localization tends to saturate at around 10 deg2. +In Figs. 9c, 9d, 9e and 9f, we observe similar trends for +the GW source mass and distance estimation as observed +for the sky localization. The relative errors of mass and +distance estimation saturate at 200% for low PSNR val- +ues only because of the bounded flat priors that we use +in the MCMC algorithm, mentioned in Sect. III E. Even +at very high PSNR values, the mass and distance estima- +tion remain finite, up to a few tens of percent, and would +also likely saturate because of parameter degeneracies. +For the same reasons mentioned previously, comparing +values from a given curve in Fig. 9d (9f) with the cor- +responding values at the same PSNR in Fig. 9c (9e), +we notice the trend that for the same effective PSNR, +increasing the observation time improves parameter es- +timation. However, unlike the sky localization, it seems +that the mass and distance estimations would saturate at +an even higher effective PSNR values than the sky local- +ization, since the curves in Figs. 9c and 9e do not flatten +out towards the right side ends. +B. +Parameter degeneracies +In Appendix B, we show an example of the 36 joint +posterior pair-plots for our 9 model parameters. These +correspond to a case where 32 noisy data points were +taken over 12 hours, at a PNSR of 100. The true param- +eter values for this case are the same as in Fig. 1, except +that the initial binary separation is 1 AU. At such a high +PSNR, the joint posterior correlations would show the +degeneracies between the parameters. Here, we try to +explain the observed correlations based on the model de- +tails described in Sect. III A: +The joint posterior of the two binary masses (Fig. 10a) +shows an anti-correlation, because upon increasing one of +the masses, the other must be decreased to have the same +chirp mass. The two binary masses are also positively +correlated with the initial binary separation (Figs. 10b +and 10i), since increasing the mass increases the GW +strain amplitude and also affects the frequency evolution, +which can be countered by increasing the initial binary +separation. The positive correlation between the masses +and the GW source redshift (Figs. 10h and 10k) exists +because increasing the redshift increases the GW source +distance while decreasing the observed GW frequency, +both of which decrease the GW strain amplitude. How- +ever, increasing the redshift also increases the observed +chirp mass, but this is not sufficient and therefore, a fur- +ther mass increase is required to counter the effect of a +decrease in the GW strain amplitude. Instead of increas- +ing the mass, we can also counter this by increasing the +initial binary separation. That is why the joint poste- +rior between the source redshift and the initial binary +separation also shows positive correlation (Fig. 10q). +An interesting joint posterior to analyze is the source +redshift, z vs. the source inclination angle, i (Fig. 10v). +The degeneracy between the source distance and inclina- +tion angle is well known in GW astronomy. Therefore, +we expect them to be anti-correlated, since increasing the +source distance decreases the GW strain amplitude, but +decreasing the inclination angle can counter this. How- +ever, this is only true for linearly polarized GWs. In our +case, since i = 0, this corresponds to a face-on orien- +tation of the binary system towards Earth, resulting in +circularly polarized GWs. Changing the inclination an- +gle changes the relative amplitudes of the plus and cross +polarization strain components, producing elliptically po- +larized GWs. Furthermore, in our simulations, the source +distance is not a separate parameter, but is instead cal- +culated from the cosmological redshift, which is a model +parameter that affects not only the source distance, but +also the chirp mass and GW frequency. Therefore, in our +results, the z vs. i joint posterior shows no correlation, +since these two parameters control very different aspects +of the response signal. +Another noteworthy joint posterior is of the GW polar- +ization angle vs. the GW initial phase (Fig. 10gg), which +shows a strong linear correlation. This can be derived an- +alytically for our special case of i = 0. For this case, the +response signal Eq. (4), after substituting all the terms +and collecting the common GW strain amplitude terms +into a factor h0, can be re-written as, +∆TGW = −1 +4 +� t0+T +t0 +h0 sin2 θ +� +cos (2ψ) cos (2πft + δ0) ++ sin (2ψ) sin (2πft + δ0) +� +dt += −1 +4 +� t0+T +t0 +h0 sin2 θ cos (2πft + δ0 − 2ψ)dt. +(13) +This perfectly explains why the joint posterior of ψ + +14 +vs. δ0 has a slope of 1/2, and it implies that for special +cases, changing the GW polarization angle is effectively +the same as changing the initial phase of the GW. If the +Earth were stationary, this would be the same as begin- +ning the observation run at a different time. +The re- +maining observed correlations in some of the joint poste- +rior pair-plots cannot be interpreted analytically. Finally, +the rest of the joint posteriors are uncorrelated, because +the corresponding model parameters control widely dif- +ferent aspects of the response signal, and cannot produce +degeneracies. +VI. +DISCUSSION +What are the limitations and caveats of this study? +One of the primary limitations of this study is the sim- +plistic static Gaussian noise model for the residual sys- +tem noise, based on the assumption that our hypotheti- +cal storage ring facility is capable of attenuating most of +the (yet un-studied) noise sources, similar to LIGO. We +cannot yet model noise sources in detail, especially their +frequency and time dependence, until a thorough study +is conducted (Schmirander et al., in preparation). +Next, our GW waveform models do not account for the +spins of the compact objects in the binary, the eccentric- +ity of their orbit, and other parameters corresponding to +realistic binary systems. While our model contains 9 un- +known fitting parameters, realistic GW models contain +around 15 to 17. However, as a first step towards estab- +lishing a novel experiment concept, for the sake of consis- +tency and ease of analysis, it is better to use a realistic toy +model for making order-of-magnitude estimations, rather +than to use complex and detailed models from the very +beginning, which can make analysis quite difficult in a +topic that has not been explored to such an extent prior +to this study. Although our GW source and ring models +are simple, they are realistic enough to provide correct +orders of magnitude of the estimates. Perhaps, incorpo- +rating detailed GW waveforms and storage ring models +is the next logical step in this series of works. +We also do not model the merger and ringdown phases, +and cover only the inspiral phase of the binaries. How- +ever, because the response signal tends to decrease with +increasing GW frequency (Fig. 3), the merger and ring- +down phases would likely not produce an SRGO response +signal as large as the one during the inspiral phase. +Furthermore, our GW source models, which are de- +rived from post-Newtonain (PN) analysis, cannot accu- +rately model EMRIs (extreme and intermediate mass ra- +tio inspirals). In Sect. IV C, Fig. 6, some GW sources +that are actually EMRIs, have been estimated with our +post-Newtonian GW waveform model, which is not op- +timal. But since we are interested in order-of-magnitude +estimates and since we do not expect the difference be- +tween our model and an EMRI GW waveform to be +orders-of-magnitude greater, we regard this as a justi- +fiable simplification. +This is supported by [71], where +it can be seen that the simple PN waveform models are +accurate enough to model EMRIs for small observation +times of a few hours or days, as considered in our study. +Due to the specifics of the MCMC setup, described in +Sect. III E, we miss the antipodal sky localization region, +which should exist because placing the ring and/or the +GW source at antipodal positions, and/or having the ions +circulating in the opposite direction, would all produce +the same SRGO response. Although we provide flat pri- +ors and allow the MCMC chains to explore over the full +range of the angular parameters, the chains seem to con- +verge and explore only around the true parameter values. +This may be due to the nature of the DE-MC algirithm +used, which is known for converging quickly to a solu- +tion in the parameter space and staying around it. An +antipodal sky localization region would double the area, +but would not change the shape of Figs. 9a, 9b. Hence, +our results would not change beyond their estimated er- +rors, and thus the interpreted conclusions would remain +the same. +Many of our results have been generated by averaging +over as many parameters as possible, so that the con- +clusions interpreted from them may remain accurate and +general. However, some of our conclusions are extrapo- +lated based on results for a specific and arbitrary combi- +nation of parameters, corresponding to Fig. 1 (with some +variations which are described in the sections pertaining +to each result). This was done for cases where averag- +ing over parameters was very difficult or computation- +ally expensive. These include results in Sects. IV A and +V. However, we do not expect the parameter-averaged +results to be different in order-of-magnitude for these +cases, and hence expect them to be sufficient for first +estimates and general conclusions. For instance, the re- +sults in Sect. IV A, which are based on scaled values of +the PSNR, are intrinsically independent of some param- +eters to a great extent, such as the GW source mass and +distance. Moreover, we can make estimates of how some +of the results would change for a different set of param- +eters. For example, the results of Sect. V, for a different +set of true parameters, can be estimated by combining +the results of Sects. IV A and V A, which should at least +be correct in order of magnitude. +Lastly, we have not yet accounted in the SRGO re- +sponse formulation, the effect of the GW on the storage +ring magnetic field, which may possibly boost the re- +sponse signal. This would be included in future works +(Schmirander et al, in preparation). +How to measure the instantaneous initial ion speed, vi? +Two timing detectors placed close by would detect a +passing ion with a delay. Dividing the known distance +covered by the ion with this timing delay would give us +vi. This measurement could be made more accurate by +repeating this procedure over the first several revolutions +and then taking an average value. However, performing +this procedure with a single timing detector (i.e. dividing +the orbit circumference by the time interval between two +successive detections of the same ion by a single detec- + +15 +tor) would be less accurate, because although the time- +varying quantities would change negligibly during a sin- +gle ion revolution, but the ion would still be affected by +the anisotropy of the spacetime. Hence, compared to the +former procedure, this way would give us a slightly worse +substitute for the quantity that we wish to measure. +How to measure ∆TGW ? +Using vi and the circumference of the ion orbit, we +can predict the expected arrival times of the ion to the +timing detector. +These must then be subtracted from +the actual ion arrival times that are measured by the +timing detector. The result will constitute the discrete +noisy data points ∆TGW , which when plotted against the +expected arrival times, will look like Fig. 1. This is why +the second term within the integral of Eq. (4) differs from +that of Eq. (1), when we measure vi = v0 +� +1 + hθφψ(0) +2 +� +instead of v0. Since the speed is used to predict the times +when the ion would arrive at the detector, a different +speed would change the predicted ion arrival times, and +thus, also the signal (which is the observed arrival time +minus the predicted arrival time of the ion). +Do GWs affect the atomic clock of the timing detector? +Since the storage ring ion clock and the atomic/optical +clock of the timing detector would be located next to each +other, they would both be affected in the same way due to +the temporal component of the GW metric (or any other +spacetime metric). Therefore, in principle, the temporal +component of the spacetime metric cannot be measured +by a comparison between the ion clock and atomic clock +geodesics (the working principle of SRGO). However, +since the location of the atomic clock would be station- +ary in the reference frame, while the ion would revolve +in an anisotropic GW spacetime, the spatial components +of the GW metric would affect the storage ring ion clock +differently as compared to the atomic/optical clock. This +difference would result in the response signal that can, in +principle, be measured by an SRGO. This is the reason +why, as explained in Sect. IV C, laser and atom inter- +ferometer GW detectors cannot probe the anisotropy of +a static distorted spacetime (such as very low frequency +GW spacetimes over short observation times), even in +principle. Whereas, this would possible in principle with +an SRGO, even in the absence of Earth’s rotation. How- +ever, practically, this might never be tested because of +the stochastic gravitational wave background (SGWB), +which exists due to an overlap of a large number of un- +resolved and incoherent astrophysical GW sources at low +frequencies [72–74]. +Why did we choose MCMC methods over Fisher Infor- +mation for parameter estimation? +The Fisher Information Matrix (FIM) can be described +as the inverse of the covariance matrix of some distri- +bution. It may also be interpreted as the curvature of +the log-likelihood graph. The FIM can be calculated an- +alytically, requiring only the model that generates the +response signal. This makes the FIM a fast and simple +method of obtaining the precision of the parameter esti- +mation pipeline without actually having to make a mea- +surement of artificial noisy data. However, the FIM does +have limitations: It assumes a model with linearly cor- +related parameters, a detector with Gaussian noise, and +a high SNR. It has been shown that for a non-spinning +binary GW source model with 9 unknown parameters +such as ours, at total binary mass higher than 10.0M⊙, +the standard deviation predicted by the FIM does not +agree with the standard deviation of a fully calculated +posterior by MCMC methods [75]. +What are the implications for the FCC (Future Circu- +lar Collider)? +FCC [76] is a proposed circular particle accelerator +which will be able to accelerate ultrarelativistic ions at +even higher energies than the LHC. This could increase +the natural attenuation of any stochastic noise sources di- +rectly acting on the ions, due to the ions having a higher +relativistic mass or momentum. However, the proposed +100 km circumference of the FCC would have implica- +tions for noise levels from sources such as seismic noise, +gravity gradient noise and others, which unlike the ex- +pected SRGO response signal, would likely be sensitive to +the ring size. Currently, it is unclear whether a larger or +smaller ring size would be more suitable for an SRGO ex- +periment. It is hoped that, upon detailed computational +modeling of the noise sources, an optimal configuration +within the parameter space can be found, which reveals +the optimal ring size (Schmirander et al., in preparation). +What are the implications for multi-messenger astron- +omy? +The yet undetected mHz GW events are also predicted +to be associated with the emission of electromagnetic ra- +diation and neutrinos [8, 77–79]. +For transient astro- +physical events that correspond to high frequency GWs +such as those detected by LIGO, the usual case for multi- +messenger observations is of the event first being detected +by the omnidirectional GW detectors, which then per- +form fast parameter estimations and send out real-time +alerts to other observatories, providing the estimated +GW source component types, masses, spins and impor- +tantly, the sky localization region. An effort is then made +to quickly and simultaneously observe the GW event via +the other messenger signals, using the alert information. +However, for mHz GW events, fast alert response would +be of lesser concern, because most of these events would +be long-lasting. Therefore, improving parameter estima- +tion, especially the sky localization, would be most im- +portant for multi-messenger studies of mHz GW events. +Other than improving detector sensitivities, this is best +achieved by collaboration between multiple mHz GW de- +tectors. It is estimated that a proposed mHz GW detec- +tor such as LISA, by itself, would not be good enough to +pinpoint the host galaxies of mHz GW sources [77, 80]. +On this front, it is clear that the successful realization +of an SRGO would greatly complement other mHz GW +detectors such as LISA, and improve the GW alerts for +multi-messenger observations. +Assuming that a mHz GW event is detected simul- +taneously by LISA and SRGO, and further assuming + +16 +that the realized SRGO has effectively the same capabil- +ities as the hypothetical system considered in Sect. III B +of this study, then we can make a rough estimation of +the improvement in the GW source sky localization due +to a combination of SRGO and LISA. The LISA sky +localization for massive black holes is estimated to be +1 − 100 deg2, and LISA would be lagging the Earth orbit +by 20° [7]. In the optimistic case, assuming that a sin- +gle SRGO on Earth manages to localize the same GW +event up to 1 − 20 deg2 as obtained in Sect. V A, then +by combining this data via simple 3D geometry, we can +roughly estimate that the improved sky localization may +be as good as sub–deg2, and as bad as a few tens of deg2. +Overall, this would be a very good improvement, and it +could be made even better by having multiple SRGOs at +different locations on Earth. +VII. +SUMMARY AND CONCLUSION +In Sect. I, we discuss previous studies on storage rings +as GW detectors, highlighting what they missed, and ex- +plaining the novelty of our idea. We provide comparisons +and analogies between an SRGO and other known GW +detection techniques. We also discuss references that sup- +port our findings and throw light on potential ways for +realizing an SRGO. In Sect. II, we provide a review of the +theory behind an SRGO, and revise important formulae +to display them in a better format. Sect. III describes +the mathematical models and numerical procedures of +our simulation code. +In Sect. IV A, we study the variation of the response +signal with the experiment parameters, obtaining useful +physical insights about how an SRGO works. Our results +suggest that the response signal would be maximised by +placing an SRGO at equatorial latitudes on Earth and by +having long observation times. In Sect. IV B, we numer- +ically obtain the SRGO sensitivity curve, which shows +that an SRGO would be intrinsically sensitive to the mHz +GW regime. The sensitivity curve also suggests that a +minimum observation time (run time of the storage ring) +of at least a few hours would be required for an SRGO +experiment. In Sect. IV C, we find that a typical SRGO +may have maximum response signal amplitudes of up to +∼ 1ps due to astrophysical mHz GW sources. Therefore, +an SRGO should aim to have, at worse, similar effective +noise levels to make a detection. At this level of noise +(or better), an SRGO could potentially detect mHz GW +events involving supermassive black holes starting from +within our galaxy, up to galaxy merger events at high +redshifts. +The results of Sect. V prove that even a single SRGO +can, in principle, perform accurate GW parameter esti- +mation, being able to provide a closed region on a sky +map for the GW source localization, which would im- +prove with increasing PSNR. In Sect. V A, we find that +an effective PSNR (i.e. true PSNR times the square root +of the total number of data points) of at least ∼ 80 would +be required to achieve decent parameter estimation with +a single SRGO, which may be achieved by a combination +of noise reduction and increasing the data measurement +rate. At this effective PSNR or higher, a single SRGO +would be capable of constraining the GW source param- +eters (such as the sky localization area, relative distance +and mass estimations, etc.) to within a few tens of per- +cent of their true values. In Sect. V B, we obtain more +physical insights by studying the parameter degeneracies +of an SRGO experiment. +Finally, in Sect. VI, we discuss the limitations of this +study; justify some approaches we have taken in this +study; answer fundamental questions about the working +principle of an SRGO; and discuss future implications of +realizing an SRGO. +In conclusion, SRGO seems promising as a near-future +Earth-based GW detector sensitive to the yet undetected +mHz GWs. +It could complement space-based detec- +tors such as LISA, or even make detections prior to the +launch of LISA, assuming that rapid technological devel- +opment during this decade allows a functional SRGO to +be built. The main effort required in this direction would +be detailed studies, techniques and technologies to han- +dle noise sources; finding the optimum operation mode of +a storage ring for an SRGO experiment; techniques and +technologies for the timing data readout. Further studies +of single ion storage rings and improvement in vacuum +technology would also help. +ACKNOWLEDGMENTS +We acknowledge Saloni Priya, Florian Gr¨uner, Wolf- +gang Hillert, Roman Schnabel, Mikhail Korobko, Thor- +ben Schmirander and Velizar Miltchev for fruitful dis- +cussions. This research was supported by the Deutsche +Forschungsgemeinschaft (DFG, German Research Foun- +dation) under Germany’s Excellence Strategy – EXC +2121 Quantum Universe – 390833306. +This work has +made use of data from the European Space Agency +(ESA) mission Gaia (https://www.cosmos.esa.int/ +gaia), processed by the Gaia Data Processing and Analy- +sis Consortium (DPAC, https://www.cosmos.esa.int/ +web/gaia/dpac/consortium). +Funding for the DPAC +has been provided by national institutions, in particu- +lar the institutions participating in the Gaia Multilateral +Agreement. +Appendix A: Contribution to ∆TGW from beam +orbit shape distortions +Consider a circular ion beam of radius R, which gets +distorted into, say, an ellipse with axes R ± ∆R, where +∆R +R += h represents the GW strain amplitude, which is +much smaller than unity. +The perimeter of a near-circular ellipse is approxi- +mated to an excellent accuracy by Ramanujan’s formula + +17 +[81], +Cellipse = π(a + b) +� +1 + +3λ2 +10 + +√ +4 − 3λ2 +� +. +(A1) +Here, a = R+∆R, b = R−∆R, λ = (a−b) +(a+b) = h. The er- +ror in Ramanujan’s approximation is O(h10). Over many +revolutions, the ion circulation time deviation will be pro- +portional to a time integral over the difference between +the perimeters of the distorted and ideal orbit shapes, +∆Torbit ∝ +� t0+T +t0 +(Cellipse − 2πR) dt ∝ h2. +(A2) +This result is, in general, also true for more complex +beam orbit shape distortions caused by other sources +(such as seismic activity), as long as the corresponding +quantity equivalent to h is small. +Appendix B: Corner plot shown as individual joint +posterior plots +Due to space constraints, we show in Fig. 10 the 36 +individual joint posterior pair-plots corresponding to the +non-diagonal elements of the 9 × 9 corner plot. The di- +agonal elements of the corner plot have been shown in +Fig. 7. +In each pair-plot, we show the 1-sigma (68%) +and 3-sigma (99.7%) highest posterior density (HPD) re- +gions. 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Dai, An Arena for Multi- +Messenger Astrophysics: Inspiral and Tidal Disruption +of White Dwarfs by Massive Black Holes, Bulletin of the +AAS 51 (2019), https://baas.aas.org/pub/2020n3i010. +[80] W.-H. Ruan, C. Liu, Z.-K. Guo, Y.-L. Wu, and R.-G. Cai, +The lisa–taiji network, Nature Astronomy 4, 108 (2020). +[81] M. B. Villarino, Ramanujan’s perimeter of an ellipse +(2005), arXiv:math/0506384. + +20 +(a) m1 and m2 +(b) m1 and r0 +(c) m1 and i +(d) m1 and z +(e) m1 and δ0 +(f) m1 and αsrc +(g) m1 and δsrc +(h) m1 and ψeq +(i) m2 and r0 +(j) m2 and i +(k) m2 and z +(l) m2 and δ0 +(m) m2 and αsrc +(n) m2 and δsrc +(o) m2 and ψeq +(p) r0 and i +(q) r0 and z +(r) r0 and δ0 +(s) r0 and αsrc +(t) r0 and δsrc + +1e6 +1.08 +1.06 +posterior [Mo] +1.04 +1.02 +2 +1.00 +# +mass +0.98 +GW source +0.96 +0.94 +0.92 +0.90 +0.92 +0.94 +0.96 +0.98 +1.00 +1.02 +1.04 +GW source mass #l posterior [Mo] +1e61.010 +/ separation posterior [au] +1.005 +1.000 +0.995 +Initial binary +0.990 +0.985 +0.980 +0.90 +0.92 +0.94 +0.96 +0.98 +1.00 +1.02 +1.04 +GW source mass #1 posterior [Mo] +1e610 +GW source inclination posterior [° +-10 +0.90 +0.92 +0.94 +0.96 +0.98 +1.00 +1.02 +1.04 +GW source mass #1 posterior [Mo] +1e63 +0.1100 +0.1075 +GW source redshift posterior +0.1050 +0.1025 +0.1000 +0.0975 +0.0950 +0.0925 +0.90 +0.92 +0.94 +0.96 +0.98 +1.00 +1.02 +1.04 +GW source mass #l posterior [Mo] +1e620 +15 +10 +GW initial phase posterior [°] +5 +5 +-10 +-15 +-20 +0.90 +0.92 +0.94 +0.96 +0.98 +1.00 +1.02 +1.04 +GW source mass #l posterior [Mo] +1e64 +GW source RA posterior [°] +-2 +-4 +0.90 +0.92 +0.94 +0.96 +0.98 +1.00 +1.02 +1.04 +GW source mass #l posterior [Mo] +1e62 +GW source DEC posterior [°] +-4 +0.90 +0.92 +0.94 +0.96 +0.98 +1.00 +1.02 +1.04 +GW source mass #l posterior [Mo] +1e610 +GW polarization posterior [°] +5 +.5 +-10 +0.90 +0.92 +0.94 +0.96 +0.98 +1.00 +1.02 +1.04 +GW source mass #l posterior [Mo] +1e61.010 + separation posterior [au] +1.005 +1.000 +0.995 +Initial binary +0.990 +0.985 +0.980 +0.92 +0.94 +0.96 +0.98 +1.00 +1.02 +1.04 +1.06 +1.08 +GW source mass #2 posterior [Mo] +1e610 +GW source inclination posterior [°] +5 +-10 +0.92 +0.94 +0.96 +0.98 +1.00 +1.02 +1.04 +1.06 +1.08 +GW source mass #2 posterior [Mo] +1e60.1100 +0.1075 +GW source redshift posterior +0.1050 +0.1025 +0.1000 +0.0975 +0.0950 +0.0925 +0.92 +0.94 +0.96 +0.98 +1.00 +1.02 +1.04 +1.06 +1.08 +GW source mass #2 posterior [Mo] +1e620 +15 +10 +GW initial phase posterior [°] +5 +0 +.5 +-10 +-15 +-20 +0.92 +0.94 +0.96 +0.98 +1.00 +1.02 +1.04 +1.06 +1.08 +GW source mass #2 posterior [Mo] +1e64 +GW source RA posterior [° +2 +0 +-2 +-4 +0.92 +0.94 +0.96 +0.98 +1.00 +1.02 +1.04 +1.06 +1.08 +GW source mass #2 posterior [Mo] +1e62 +GW source DEC posterior [°] +2 +-4 +0.92 +0.94 +0.96 +0.98 +1.00 +1.02 +1.04 +1.06 +1.08 +GW source mass #2 posterior [Mo] +1e610 +GW polarization posterior [°] +5 +.5 +-10 +0.92 +0.94 +0.96 +0.98 +1.00 +1.02 +1.04 +1.06 +1.08 +GW source mass #2 posterior [Mo] +1e610 +GW source inclination posterior [°] +5 +O +-5 +-10 +0.980 +0.985 +0.990 +0.995 +1.000 +1.005 +1.010 +Initial binary separation posterior [au]0.1100 +0.1075 +GW source redshift posterior +0.1050 +0.1025 +0.1000 +0.0975 +0.0950 +0.0925 +0.980 +0.985 +0.990 +0.995 +1.000 +1.005 +1.010 +Initial binary separation posterior [au]20 +15 +10 +GW initial phase posterior [°] +5 +0 +-10 +-15 +-20 +0.980 +0.985 +0.990 +0.995 +1.000 +1.005 +1.010 +Initial binary separation posterior [au]4 +GW source RA posterior [°] +2 +-2 +-4 +0.980 +0.985 +0.990 +0.995 +1.000 +1.005 +1.010 +Initial binary separation posterior [au]2 +GW source DEC posterior [° +O +-4 +0.980 +0.985 +0.990 +0.995 +1.000 +1.005 +1.010 +Initial binary separation posterior [au]21 +(u) r0 and ψeq +(v) i and z +(w) i and δ0 +(x) i and αsrc +(y) i and δsrc +(z) i and ψeq +(aa) z and δ0 +(bb) z and αsrc +(cc) z and δsrc +(dd) z and ψeq +(ee) δ0 and αsrc +(ff) δ0 and δsrc +(gg) δ0 and ψeq +(hh) αsrc and δsrc +(ii) αsrc and ψeq +(jj) δsrc and ψeq +FIG. 10: Individually shown joint posteriors of the 9 × 9 corner plot obtained after MCMC parameter estimation, +for true parameter values described in Appendix B. + +2 +GW source DEC posterior [°] +O +2 +-4 +-4 +-2 +0 +2 +4 +GW source RA posterior []10 +GW polarization posterior [°] +5 +.5 +-10 +0.980 +0.985 +0.990 +0.995 +1.000 +1.005 +1.010 +Initial binary separation posterior [au]0.1100 +0.1075 +GW source redshift posterior +0.1050 +0.1025 +0.1000 +0.0975 +0.0950 +0.0925 +-10 +10 +5 +0 +GW source inclination posterior []20 +15 +10 +GW initial phase posterior [°] +-5 +-10 +-15 +-20 +-10 +5 +10 +GW source inclination posterior []4 +GW source RA posterior [° +2 +-2 +-4 +-10 +0 +10 +5 +GW source inclination posterior []2 +GW source DEC posterior [°] +2 +-4 +-10 +0 +10 +5 +GW source inclination posterior []10 +GW polarization posterior [°] +5 +-5 +-10 +-10 +-5 +5 +10 +GW source inclination posterior []20 +15 +10 +GW initial phase posterior [°] +5 +-5 +-10 +-15 +一20 +GW source redshift posterior4 +GW source RA posterior [°] +2 +-2 +-4 +0.0925 0.0950 0.0975 0.1000 0.1025 0.1050 0.1075 0.1100 +GW source redshift posterior2 +GW source DEC posterior [° +0 +2 +-4 +0.0925 0.0950 0.0975 0.1000 0.1025 0.1050 0.1075 0.1100 +GW source redshift posterior10 +5 +GW polarization posterior [ +-5 +:2 +-10 +GW source redshift posterior4 +. +GW source RA posterior [°] +-2 +-4 +-20 +-15 +-10 +5 +10 +15 +20 +GW initial phase posterior []2 +GW source DEC posterior [°] +0 +-4 +-20 +-15 +-10 +.5 +0 +10 +15 +20 +5 +GW initial phase posterior []10 +GW polarization posterior [°] +5 +0 +-5 +-10 +-20 +-15 +-10 +-5 +0 +5 +10 +15 +20 +GW initial phase posterior []10 +GW polarization posterior [°j +.5 +-10 +-4 +-2 +0 +4 +GW source RA posterior []10 +GW polarization posterior [°] +5 +5 +-10 +-4 +0 +2 +GW source DEC posterior [] \ No newline at end of file diff --git a/ddE_T4oBgHgl3EQf0xx9/content/tmp_files/load_file.txt b/ddE_T4oBgHgl3EQf0xx9/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d680b15db6556ab1f68f3db224c52c021d75e066 --- /dev/null +++ b/ddE_T4oBgHgl3EQf0xx9/content/tmp_files/load_file.txt @@ -0,0 +1,2016 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf,len=2015 +page_content='Detection of gravitational waves in circular particle accelerators II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Response analysis and parameter estimation using synthetic data Suvrat Rao,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' ∗ Julia Baumgarten,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='2 Jochen Liske,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='1 and Marcus Br¨uggen1 1Hamburger Sternwarte,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Universit¨at Hamburg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Gojenbergsweg 112,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' 21029 Hamburg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Germany 2Physics Department,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Jacobs University Bremen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Campus Ring 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' 28759 Bremen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Germany (Dated: January 23,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' 2023) We simulate the response of a Storage Ring Gravitational-wave Observatory (SRGO) to astrophys- ical gravitational waves (GWs),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' numerically obtaining its sensitivity curve,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' parameter degeneracies,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' and optimal choices of some controllable experiment parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' We also generate synthetic noisy GW data and use Markov Chain Monte Carlo (MCMC) methods to perform parameter estimation of the source properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' With this, we show that a single SRGO could potentially localize the GW source in the sky using Earth’s rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Then, we study the source sky localization area, mass and distance estimation errors as functions of noise, data sampling rate, and observing time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Finally, we discuss, along with its implications, the capacity of an SRGO to detect and constrain the parameters of millihertz (mHz) GW events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' INTRODUCTION Theoretical studies of gravitational waves (GWs) in- teracting with storage rings (circular particle accelerators where ion beams circulate without collisions for long peri- ods), intending to explore the possibility of using storage rings as GW detectors, have been conducted indepen- dently by several authors over the past decades [1–5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' However, these studies had only considered the sce- nario of GWs propagating perpendicular to the plane of the storage ring (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' a “face-on” orientation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' This par- ticular case maximizes the GW-induced oscillations of the ions (test mass particles) along the ring’s radial direc- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Thus, one can hope to exploit resonances with the beam’s betatron oscillations and detect the presence of GWs using beam position monitors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' As storage ring be- tatron frequencies generally fall in a range where no sig- nificant astrophysical GW sources exist, and since these radial oscillations are expected to be minuscule, this idea did not seem very promising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Meanwhile, it can be shown (see Appendix A) that, in general, any periodic beam or- bit shape distortions, such as those due to GWs, can cause deviations in the circulation times of ions (from their expected values during no perturbations), of the order of magnitude h2, where h is the dimensionless GW strain which is much smaller than unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' In our previous paper [6],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' henceforth referred to as paper-I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' we showed that GWs can be better detected in storage rings by measuring the ion circulation time de- viation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' which,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' in general,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' is proportional to h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' being caused by a GW-induced change in the velocities of the ions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' and only for the specific configuration of the GWs propagating face-on to the storage ring does it become proportional to h2 (and therefore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' negligible compared to h),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' where the GWs can only cause a distortion of the beam orbit shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Moreover, we showed that, although the circulation time deviation has a periodicity equal to ∗ suvrat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='rao@uni-hamburg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='de that of the GWs, its peak value is proportional to the GW period, suggesting that it builds up over time dur- ing the first half of a GW period and then wanes during its second half.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' This is true in the regime where the observation time is much greater than the GW period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' As a result, it makes such a detector more sensitive to lower frequency GWs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Importantly, we showed in paper- I that, due to an overlap of several conditions, a Storage Ring Gravitational-wave Observatory (SRGO) would be most sensitive to the yet undetected millihertz (mHz) GWs from astrophysical sources, that are also targeted by future space-based GW detectors such as Laser Inter- ferometer Space Antenna (LISA) [7, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' The quantity measured by an SRGO would hence be analogous to the “timing residuals” measured by pul- sar timing arrays (PTAs), which are the GW-induced arrival-time deviations of radio pulses (produced by mil- lisecond pulsars) from their expected, highly regular ar- rival times in the absence of GWs and noise sources [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Moreover, SRGO ions being the moving test masses in- fluenced by GWs, is similar to the idea of GW detection by atom interferometry, where ballistic atoms are used as test masses [10–14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Recently, D’Agnolo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' independently found results that are parametrically in agreement with our main cal- culations from paper-I, under the condition that no RF (radio frequency) system is present in the storage ring [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Therefore, the general relativistic calculations for an SRGO, done from two different reference frames (metric formalism in our paper-I, and Riemann tensor formalism by D’Agnolo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' ), give effectively, the same result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' In paper-I, we conducted a case study on the Large Hadron Collider (LHC) at CERN as an existing facility that could potentially be turned into an SRGO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' How- ever, LHC is not the ideal facility to realize the SRGO detector because the presence of an RF system in the storage ring can dampen the GW signal that we hope to detect (D’Agnolo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Instead, rings capable of stor- ing coasting (meaning without RF system) “crystalline ion beams” [16–18] or rings that could potentially store arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='08331v1 [gr-qc] 18 Jan 2023 2 a single circulating ion [19] may be better laboratories for detecting GWs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Moreover, there are better options for the ion time-tagging detector technology than that proposed in paper-I, such as “beam arrival time moni- tors” [20], which are electro-optic charge centroid mon- itors providing femtosecond timing precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Also, im- proving the vacuum quality inside storage rings would en- able sustaining stable, coasting ion beams or single ions for longer periods, allowing for longer SRGO observation runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' REVIEW AND REVISION OF SRGO BASICS Using the metric formalism of general relativity, in paper-I, we derived the circulation time deviation of test masses in a storage ring due to GWs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Here, we revise some of these results and display them in a neater form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' We recall from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' (10) of paper-I, that the circulation time deviation was given by the general expression, ∆TGW = � 1− v2 0 2c2 � � t0+T t0 � hθφψ(t, α(t))−hθφψ(t0, α0) � dt, (1) where v0 is the speed of the ions in the absence of GWs, c is the speed of light, t0 is the start of the observing time, T is the duration of the observing time, and hθφψ has a complex expression, being a function of the GW strain amplitudes, h+,×(t);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' the time-varying orientation of the GW with respect to the ring caused by Earth’s rotation, θ(t), φ(t), ψ(t);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' and the angular trajectory of the ions in the ring, α(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' However, as it is not possible to know a priori whether GWs are present at any given time, therefore, v0 can- not be measured in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' So instead, we reformulate Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' (1) in terms of v(t0) = vi (the instantaneous initial speed of the ions), which can be measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' We start from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' (6) of paper-I, and keeping the term v(t0) as it is, we follow through with the remaining derivation as done in paper-I, to get a reformulated expression for the circulation time deviation, ∆TGW = � t0+T t0 �� 1 − v2 i 2c2 � hθφψ(t, α(t)) − 1 2 � 1 − v2 i c2 � hθφψ(t0, α0) � dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' (2) Now we make the substitution vi ≈ c, since ions in storage rings are usually ultrarelativistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' This would make the second term within the square brackets negligi- ble compared to the first, even for long observation times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Although it is currently unclear, but using ultrarelativis- tic ions might be beneficial (Schmirander et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=', in prepa- ration).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' This is because a single ion with sufficiently high energy would, in principle, by its sheer momentum, ap- preciably attenuate the effect of many stochastic and de- terministic noise sources that would directly perturb the ion (some of which were explored in paper-I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' This would also allow a coasting ion to deflect or deviate less from its ideal orbit, and last longer in a non-ideal vacuum, thus potentially allowing for longer SRGO observation runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Meanwhile, the first term is a function of three differ- ent frequencies, viz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' the GW frequency, Earth’s rotation frequency and the revolution frequency of the storage ring ions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' The latter frequency is always much greater than the former two in the case of ultrarelativistic ions and mHz GWs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' This allows us to analytically integrate out the rapidly oscillating terms corresponding to the ion revolution frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' From paper-I, we recall, hθφψ(t, α) = h+(t) � f + s sin2 α + f + c cos2 α + f + sc sin 2α � + h×(t) � f × s sin2 α + f × c cos2 α + f × sc sin 2α � , (3) where α(t) = α0 + v0 R (t − t0), R being the radius of the storage ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' The terms sin2 α, cos2 α and sin 2α integrate out to yield the constant factors 1 2, 1 2 and 0 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' The ‘f’-terms inside the parentheses are functions of cosines and sines of the angles θ, φ, ψ (see paper-I, Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Therefore, in the end, we obtain the reformulation (for ultrarelativistic ions in an SRGO), ∆TGW = −1 4 � t0+T t0 � F+h+ + F×h× � dt, (4) where h+(t) and h×(t) are the plus and cross polar- ization GW strain components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' F+ and F× are the plus and cross polarization antenna pattern functions of an SRGO, with F+ = sin2 θ cos 2ψ and F× = sin2 θ sin 2ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Note that the integrand in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' (4) now has the same form as the GW response of the Laser Interferometer Gravitational-wave Observatory (LIGO) [21], but the re- sponse signal in our case is a time-integral of this in- tegrand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Also, while the antenna pattern functions of SRGO are very different compared to those of LIGO, interestingly, they happen to be exactly the same as those of a bar detector [22–25] whose longitudinal axis is aligned perpendicular to the plane of the storage ring (see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='1 of [26]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Further, the SRGO antenna pat- tern shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' 2 of paper-I (averaged over all polar- ization angles), which was derived after making several approximations, happens to be exactly valid even for the general case derived here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' In paper-I, Appendix B, we noted that in the mHz regime, the effect of Earth’s rotation must be taken into account, which would cause the angles φ, θ, ψ (the az- imuth, inclination and polarization angles respectively, which orient an observer who is initially stood at the cen- ter of the storage ring, along the GW propagation direc- tion and the GW polarization axes) to be periodic func- tions of time with a period of a sidereal Earth day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Here,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' we revise the relation from paper-I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' which establishes the 3 connection between the time-varying φ(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' θ(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' ψ(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' and the parameters as measured from the equatorial celestial coordinate system,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' in which the orientation of the GW ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='is fixed:- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='cos ψ − sin ψ 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='sin ψ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='cos ψ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='� ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='− sin θ 0 cos θ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='cos φ − sin φ 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='sin φ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='cos φ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='� = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='cos ψeq − sin ψeq 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='sin ψeq ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='cos ψeq ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='0 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='δsrc ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='− cos ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='δsrc ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='0 − sin ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='δsrc ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='� ∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='− cos ωe ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='αsrc − l0 − (t − t0) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='sin ωe ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='αsrc − l0 − (t − t0) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='− sin ωe ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='αsrc − l0 − (t − t0) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='− cos ωe ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='αsrc − l0 − (t − t0) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='�∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='sin ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='θlat ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='0 − cos ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='θlat ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='cos ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='θlat ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='sin ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='θlat ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='�∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='cos φ0 − sin φ0 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='sin φ0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='cos φ0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' (5) ωe is the angular speed of Earth’s rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' αsrc and δsrc are the right ascension and declination of the GW source [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' ψeq is the polarization of the GW in the equatorial celestial coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' φ0 [28] is the angle be- tween the line joining the center of the storage ring to the timing detector, and the longitude passing through the center of the storage ring, measured using the right-hand curl rule starting from the detector position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' l0 and θlat are respectively, the local sidereal time at the start of the observing time, t0, and the latitude of the center of the storage ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' MODELS AND NUMERICAL PROCEDURES We obtain all the numerical results in this work from a computer code written in Python and freely available online on GitHub [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Below, we detail the mathemat- ical models and numerical procedures programmed into the code to obtain our results: A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' GW source model We consider the simplest realistic models for mHz GWs from astrophysical sources viz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' the dominant harmonic of GWs from the quasi-circular inspiral phase of non- spinning compact objects [30], accounting for the redshift correction to the GW frequency and chirp mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' The inspiral phase of a non-spinning binary system can be modeled using post-Newtonian analysis [31], which provides relatively simple analytical expressions for the time-varying GW strain amplitudes corresponding to the plus and cross polarizations: h+ = 4 dL �GM c2 � 5 3 �πf c � 2 3 1 + cos2 (i) 2 cos (2πft + δ0), (6) h× = 4 dL �GM c2 � 5 3 �πf c � 2 3 cos (i) sin (2πft + δ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' (7) If m1 and m2 are the masses of the objects in the bi- nary, then M = (1+z)(m1m2) 3 5 (m1+m2) 1 5 is the redshift-corrected chirp mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' dL is the luminosity distance of the GW source and z is its redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' i is the inclination angle between the observer’s line of sight to the GW source and the angular momentum vector of the GW source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' δ0 is the initial phase of the GW at the start of the ob- serving time, t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' The redshift-corrected, time-varying GW frequency is f = (1 + z)−1 � f − 8 3 0 − 8 3k(t − t0) �− 3 8 = (1 + z)−1 � G(m1 + m2)/πr 3 2 , where r is the separation be- tween the objects in the binary;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' f0 is the GW frequency at t = t0, corresponding to an initial separation of r = r0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' and k = 96 5 π 8 3 (GM/c3) 5 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' G and c are respectively, the gravitational constant and the speed of light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' We use an approximate analytical relation between dL and z for ΛCDM cosmology, from [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' We use an approximation of the innermost stable cir- cular orbit, risco = 6G max (m1,m2) c2 , to numerically mark the end of the inspiral phase, upon reaching which, or at the end of the user-provided observation time (whichever comes earlier), the computer code is halted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Storage ring model We consider a hypothetical circular storage ring having a 100 m circumference and containing a single ultrarela- tivistic ion that is coasting stably at close to the speed of light, with no RF system and a single timing detector present within the ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' The timing detector is placed to the south of the storage ring’s center, such that it lies on the longitude that passes through the center of the storage ring i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' φ0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' 4 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' 1: The blue curve is the expected response of an SRGO to mHz GWs, for the given parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' The pink dots are a demonstration of discrete, noisy data points, created by adding artificial Gaussian noise to the SRGO response signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' The orange ring is the initial SRGO position, while the black lines show the GW propagation direction and plus polarization axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' 2: Spectrogram of the SRGO response (left panel) and the time evolution of the Euler angles, φ, θ, ψ (right panel), for the case in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' For the spectrogram, the observation time begins earlier than in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' As per the Nyquist-Shannon sampling theorem, the minimum sampling (data measurement) rate must be greater than twice the highest expected GW frequency, while the maximum possible sampling rate would corre- spond to the timing detector making one detection per revolution of the ion bunch, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' every time it arrives at the detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' In our code, we choose the total number of data points to be powers of two, as this allows for faster computation of the Fast Fourier Transform (FFT) done during the MCMC GW parameter estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' We assume that deterministic sources of noise, such as gravity gradient noise, seismic activity, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' are techno- logically eliminated or accounted for, and only stochastic noise sources remain in the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' The net resid- ual noise is assumed to be Gaussian (an assumption also made by LIGO [33]), with a standard deviation between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='1 to 100 times the peak GW signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' We consider this range of PSNR (peak signal-to-noise ratio) as values be- yond this range may give trivial results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' A better noise model for the current study cannot be assumed until an 1e-15 Expected SRGO signal without noise Synthetic noisy data points (201 data points, SRGO latitude = 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='0° PSNR = 3, Noise Std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Dev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='1e-16 s) GW source right ascension =_ oh om 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='0s Sidereal day GW source declination = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='0° GW polarization angle = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' GW initial phase = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='0° Equal mass (~106Mo) binary at z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='1 End of inspiral phase at 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='0 hrs Observer-SMBBH inclination angle = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='0° ATGW 2 24 30 36 48 54 60 66 72 12 18 42 Observation Time [hrs] 1 unit = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='0 hrs1e-26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='0025 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='0020 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='50 [ZH] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='0015 Frequency 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='0010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='0005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='0000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='00 30 36 48 54 6 12 18 24 42 60 66 72 78 84 90 96 102 108 114 120 126 132 138 144150 Time [hrs]180 150 120 90 60 30 Angle [ 0 30 60 06- 120 150 0 180 6 12 18 24 30 36 54 42 48 60 66 72 0 Time [hrs]5 SRGO facility is actually established, or detailed studies of noise sources have been done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Numerical solution for finding φ(t), θ(t), ψ(t) Upon analytically multiplying all the matrices on the left hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' (5), and naming the final matrix on the right hand side as R (which must be obtained by nu- merically multiplying the five matrices on the right hand side), we arrive at the following relations by comparing the matrix elements on both sides: φ(t) = arctan (R21, −R20), θ(t) = arccos (R22), ψ(t) = arctan (R12, −R02).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' (8) Note that in the above relations, numerically, we must use the “arctan2” function to obtain the angles in their correct quadrants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' The right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' 2 shows the time evolution of the angles for the case corresponding to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Numerical integration procedure We use Boole’s rule quadrature [34] over a timestep to compute its contribution to the integral of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Start- ing from the initial value of the integral (equal to zero), the contribution of each timestep is added to the integral, and its cumulative value is saved after each timestep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' We perform this procedure till the halting condition (men- tioned in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' III A) is reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Thus, we numerically obtain the SRGO response signal as a time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Markov Chain Monte Carlo (MCMC) fitting procedure MCMC is a Bayesian inference tool for numerically obtaining the joint posterior probability distribution of unknown model parameters by directly drawing samples from the posterior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' MCMC works by following an algo- rithm to find and explore around the regions in parameter space that correspond to the maximum likelihood of the parameters being a fit for the given data, given model and a prior probability distribution of the unknown fit- ting parameters [35–37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' MCMC methods are preferred over the conventional matched filtering algorithm [26] for thorough and effi- cient GW parameter estimation, because the typical GW models contain around 15 to 17 fitting parameters, and making a grid in parameter space of such a high dimen- sionality would require an impossibly long computation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' In our study, we do GW parameter estimation using MCMC methods, keeping all possible unknown pa- rameters as the fitting parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' These are nine in number, namely, the GW source masses m1, m2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' the ini- tial separation between the masses, r0 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' the GW source inclination angle, i ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' the GW source redshift, z ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' the ini- tial phase of the GW, δ0 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' the right ascension, αsrc and declination, δsrc of the GW source ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' and the polarization angle, ψeq of the GW in the reference frame of equatorial celestial coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' In our code, we first create synthetic noisy data points by adding Gaussian noise to the SRGO response com- puted for user-provided parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' The noisy data is then transformed to Fourier space via a Fast Fourier Transform (FFT) and passed to the likelihood function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' We use flat priors for the unknown fitting parameters and a two-dimensional Gaussian noise “Whittle” likeli- hood function, as also done by LIGO for GW param- eter estimation [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' The priors corresponding to an- gular parameters are bounded between −180° and 180°, whereas priors corresponding to non-angular parameters are bounded between zero and double of their true pa- rameter values for computational efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' We employ the Differential Evolution Markov Chain (DE-MC) algo- rithm [38] for the MCMC chains, and run per case, 1000 parallel chains which draw 1250 samples each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Since we are purely interested in parameter estimation, and not in showing the convergence of chains to the region of maximum likelihood, we do not discard 25% of the ini- tial traces as the “burn-in” phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Instead, we allow the chains to start from the true parameter values and then explore around.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Each case is repeated 10 times by regen- erating the noisy data points, to obtain the statistical variation of the joint posterior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' In all cases, we choose the 3−sigma (99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='7%) highest posterior density (HPD) region for the parameter estimation, and discard those cases where the true parameter values do not lie within this region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' MCMC is carried out in our code using the PyMC3 Python module [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' RESULTS: SRGO RESPONSE ANALYSIS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Response signal analysis In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' 1, we show a demonstration of the response of an SRGO to mHz GWs from an SMBBH (supermas- sive binary black hole) inspiral, for an arbitrarily cho- sen configuration of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' In general, we notice that the response has an envelope which periodically re- peats every sidereal day due to the effect of Earth’s ro- tation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Further, unlike LIGO, the chirping of the GW strain amplitude does not reflect in SRGO’s response spectrogram (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' 2 left panel), whose amplitude actu- ally diminishes with time, as the SMBBH inspirals and the GW frequency increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' This is because, the re- sponse amplitude has an inverse relation with the GW frequency, which overpowers the contribution due to an increase in the GW strain amplitude with time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' The ex- pected orders of magnitude of the SRGO response signal amplitude (due to astrophysical sources) are discussed in 6 Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' IV C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' 3, we notice first that, in the regime where f Tobs ≫ 1, the PSNR decreases with increasing GW frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Thus, the right side tails in the plots agree with our analytical results from paper-I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' At very low GW frequencies, where f Tobs ≪ 1, the timing deviation buildup is slow, and the GW strain amplitude is also quite small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Therefore, the plots have tails on the left side as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Finally, in the regime where f Tobs ∼ 1, the PSNR decreases with decreasing observation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' This is because, when the observation time is large and also close to the GW period, a large timing deviation can be accumulated by the circulating ions, as opposed to cases with smaller observation times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' The dynamically changing peaks and valleys in the se- ries of plots can be somewhat elucidated: they are the result of the interplay between the GW period, initial GW phase, the orientation of SRGO relative to the GW source, Earth’s rotation period and the observation pe- riod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' The orientation of SRGO relative to the GW source and Earth’s rotation period together determine the en- velope seen in the response signal of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' During the observation time, large peaks in these plots occur when peaks in the GW strain waveform align with the envelope peaks, and valleys occur when the GW strain peaks align with the envelope valleys (or vice versa, whichever pro- duces a greater response signal).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' For a given observation time, as the initial GW frequency is increased, a number of successive GW strain peaks and valleys occur during the observation period, which may or may not align with the envelope peaks and valleys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' An aligned GW strain peak (resulting in a local maxima in the plot curves), upon increasing the initial GW frequency, will become misaligned, resulting in a drop in the curve until the next peak starts becoming aligned, causing then a rise in the curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' This succeeding local maxima may be higher or lower than its predecessor, depending on the location of the global maxima, which occurs when the initial GW period is close to the observation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' This explains the spiky sections of the plot curves, prominently seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' 3c, but also present in the other plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' In Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' 3a, 3b and 3c, the noisy regions in the right side tails of the plots have purely numerical origins, being caused by the abrupt halting of the code one timestep af- ter the inspiral phase has been crossed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Therefore, they exist only at the ends of the right side tails (correspond- ing to high initial GW frequencies, meaning that the bi- nary compact objects begin close to the end of their in- spiral phases).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' As the observation time is reduced, the end of the inspiral phase would be reached within the observation time at higher initial frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Therefore, the noisy regions shift towards higher frequencies and eventually disappear in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' 3d, 3e and 3f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' 4, we plot the PSNR as a function of the source position in the sky, scaled such that the arbitrary case corresponding to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' 1 has a PSNR of unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' While changing the source position, all other parameters re- main the same as those in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' 1, but the initial SMBBH separation is 1 AU, and signal is computed over an obser- vation time of 1 day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' We repeat this for different latitudes of the SRGO location on Earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' We observe that the complex patterns in the plots are a modified projection of the SRGO antenna pattern on the sky, as expected from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Other parameters being fixed, for a given SRGO latitude and source declination, the horizontal variation is determined by the initial hour angle, the GW frequency, initial GW phase, GW polar- ization angle and Earth’s rotation frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Maxima in the horizontal variation occur during the observation time when peaks in the GW strain waveform occur ex- actly at a time when there are peaks in the response sig- nal envelope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Minima occur when the GW strain peaks align with the envelope valleys (or vice versa, whichever produces a greater response signal).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' We note that the pattern depends on the initial hour angle i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' the right ascension of the GW source relative to the initial SRGO local sidereal time (and not on their absolute values).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' However, the same is not true between the SRGO lati- tude and source declination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' That symmetry is broken by the Earth’s spin axis, and therefore, the SRGO latitude variation produces different patterns in the plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' The bottom-right plot corresponds to SRGO being placed at the pole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' For this case, if the GW source is also at one of the poles, then the orientation always remains face-on, and no response signal is produced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' For all other cases, a non-zero response signal is produced over 24 hours, due to a changing orientation of the GW source relative to the ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Further, we observe that the plots show antipodal sym- metry, since placing the ring and/or the GW source at antipodal positions, and/or having the ions circulating in the opposite direction, would all produce the same SRGO response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' The plots also appear to have a left- right symmetry, because two GW sources at the same declination, initially located on either side of the ring and having the same hour angle magnitude, would pro- duce the same value of the initial SRGO response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' The response signals over a day’s time would, however, not be exactly the same due to Earth’s rotation, as SRGO would move towards the source in one case and away from the source in the other case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' But over 24 hours, SRGO would cover all possible azimuthal orientations relative to the two sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Therefore, the peak signal values would be similar between such cases, although the peak sig- nal may be achieved at different times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' This symmetry would break at higher GW frequencies, if the response signal amplitude drops quick enough within one day due to the chirping of the GW frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' There is also a lati- tudinal symmetry, as having SRGO located at equal and opposite latitudes would simply flip the plots about the horizontal axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' The SRGO latitude variation indicates that placing the ring near the equatorial latitudes on Earth would be more advantageous than placing the ring near the polar lati- tudes, offering an increase of the maximum PSNR by around 3 times, of the average PSNR by around 4 times, 7 (a) Tobs = 24 hrs (b) Tobs = 12 hrs (c) Tobs = 3 hrs (d) Tobs = 1 hour (e) Tobs = 10 mins (f) Tobs = 1 min FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' 3: Scaled values of the peak signal-to-noise ratio as a function of the initial GW frequency, for different values of the observation time, Tobs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' All other parameters are equal to those shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' 1, except the initial separation between the masses, which is 1 AU, and the black hole masses, which have been chosen to be 3 M⊙ each, so that the chosen GW frequency range can be covered during their inspiral phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' The scaling is done such that the largest peak among all plots has a value of unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' and of the minimum PSNR by a factor of unity from zero, between the polar and equatorial SRGO latitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' SRGO sensitivity curve The sensitivity curve of a GW detector may be de- fined as the curve in the plane of the GW strain ampli- tude spectral density versus the GW frequency, where the signal-to-noise ratio is unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' In this study, we use our code to numerically compute the SRGO sensitivity curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' We start by equating the root mean square value of the response signal to the ef- fective noise, [∆TGW ]rms = σnoise � fsampleTobs , (9) where Tobs is the total observation time and fsample = npv0 2πR is the data sampling rate, with np being the num- ber of circulating test masses that are timed, which may be individual ions or bunches of ions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' We then expand the left hand side using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' (4), substituting h+ and h× from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' (6) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' (7), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Then we set k = 0 in the GW frequency evolution to get continuous GWs (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' GWs with a constant frequency).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Finally, we club together all the common time-independent terms within the integral, identifying this quantity as the GW strain amplitude, ˜h(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Rearranging the equation, we thus ob- tain, ˜h(f) = 4σnoise � fsampleTobs .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' �� t0+T t0 � F+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='1 + cos2 (i) 2 cos (2πft + δ0) + F×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' cos (i) sin (2πft + δ0) � dt �−1 rms .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' (10) The GW strain amplitude spectral density is plotted as a function of the GW frequency, f, and it is numerically averaged over all the GW source parameters within the integral of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' It is given by, ASD = ˜h(f) √Tobs .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' (11) The result shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' 5 numerically confirms that an SRGO would be sensitive to mHz GWs by design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' In the log scale, the SRGO sensitivity curve has a linear 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='0 Tobs = 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='0 hrs GW period equals Tobs 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='8 Scaled PSNR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='0 10-7 10-6 10-5 10-3 10-2 100 10-4 10-1 Initial GW frequency [Hz]0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='5 Tobs = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='0 hrs GW period equals Tobs 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='4 Scaled PSNR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='0 10-7 10-6 10-5 10-4 10-3 10-2 100 10-1 Initial GW frequency [Hz]0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='12 Tobs = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='0 hrs GW period equals Tobs 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='08 Scaled PSNR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='00 10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 Initial GW frequency [Hz]Tobs = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='0 hrs 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='08 GW period equals Tobs 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='07.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='06 Scaled PSNR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='00 10-6 10-5 10-4 10-3 10-2 10-1 100 10-7 Initial GW frequency [Hz]Tobs = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='0 mins GW period equals Tobs 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='05 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='04 - Scaled PSNR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='01 : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='00 10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 Initial GW frequency [Hz]Tobs = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='0 mins 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='025 GW period equals Tobs 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='020 Scaled PSNR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='000 10-7 10-6 10-5 10-3 10-2 10-1 100 10-4 Initial GW frequency [Hz]8 (a) SRGO latitude = 0° (b) SRGO latitude = 30° (c) SRGO latitude = 51° (d) SRGO latitude = 90° FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' 4: Scaled values of the peak signal-to-noise ratio as a function of the GW source’s initial position in the sky relative to SRGO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' All other parameters are equal to those shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' 1, except the initial separation between the masses which is 1 AU, and the observation time which is 1 day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' The top-left, top-right, bottom-left and bottom-right figures respectively correspond to an SRGO latitude of 0°, 30°, 51° and 90°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' The scaling is done relative to the case corresponding to the center of the bottom-left plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' behaviour in the mHz frequency regime, as analytically predicted in paper-I [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' The sensitivity deteriorates at higher frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' This is because, as the GW period becomes smaller, the SRGO response amplitude also de- creases, since the ions spend lesser time accumulating a timing deviation during every half-cycle of the GW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' This aspect is discussed in the previous section and shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Therefore, a larger strain amplitude would be re- quired to detect high frequency GWs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' This would greatly exceed the predicted strain amplitudes from astrophysi- cal sources in the decihertz or kilohertz ranges (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' for “LIGO-like” sources).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' At very low frequencies also, the sensitivity curve rises.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' We may perform a thought-experiment to analyse this situation: A zero frequency GW would be equivalent to an anisotropic spacetime having a constant distortion, and not necessarily a flat spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' In such a spacetime, if the instantaneous initial speed of the circulating test mass is measured and used to predict the expected future arrival times of the test mass at the timing detector, then the observed arrival times would deviate from the predic- tions, as the test mass traverses an anisotropic spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' This is why, if we input f = 0 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' (4), we still get a finite value of the response signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Therefore, unlike laser interferometers and atom interferometers (which use test masses that can only move linearly) which require both the temporal and spatial components of GW spacetime in order to probe it, an SRGO (which utilizes circulating test masses) would, in principle, be able to probe purely the spatial anisotropy of GW spacetime even at very low GW frequencies and finite observation times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' However, for low frequency GWs, as the GW period far exceeds the total observation time, Tobs, the peak response signal value would start decreasing, as discussed in the previ- ous section and shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' That is why the SRGO sensitivity curve rises again at low frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Also, for most resolvable astrophysical GW sources, the strain am- plitude at near-zero frequencies would be near-zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' The sensitivity curve should corroborate with Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' 3c, as both are computed for an observation time of 3 hours and we expect them to be inversely related.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Although the general shape of the two curves agree with each other, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content="0 SRGO initial zenith SRGO zenith's track in 24." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='0 hrs Scaled by PSNR at center for 51° SRGO latitude SCALED PSNR VS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' SOURCE POSITION Maxima = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='04 75° Average = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='98 Minima = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='08 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='5 60° 45° 30° 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='0 15° 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='5 D 15° 30° 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='0 45° 60° 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='5 75° INITIAL HOUR ANGLE [HRS] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content="0 SRGO initial zenith SRGO zenith's track in 24." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='0 hrs Scaled by PSNR at center for 51° SRGO latitude SCALED PSNR VS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' SOURCE POSITION Maxima = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='83 75° Average = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='94 Minima = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='12 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='5 60° 45° 30° 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='0 15° 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='5 D 15° 30° 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='0 45° 60° 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='5 75° INITIAL HOUR ANGLE [HRS] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' (- 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content="0 SRGO initial zenith SRGO zenith's track in 24." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='0 hrs Scaled by PSNR at center for 51° SRGO latitude SCALED PSNR VS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' SOURCE POSITION Maxima = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='46 75° Average = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='66 Minima = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='5 60° 45° 30° 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='0 15° C 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='5 15° 30° 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='0 45° 60° 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='5 75° INITIAL HOUR ANGLE [HRS]3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content="0 SRGO initial zenith SRGO zenith's track in 24." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='0 hrs Scaled by PSNR at center for 51° SRGO latitude SCALED PSNR VS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' SOURCE POSITION Maxima = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='09 75° Average = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='54 Minima = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='5 60° 45° 30° 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' ( 15° 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='5 15° 30° 45° 60° 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='5 75° INITIAL HOUR ANGLE [HRS]9 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' 5: The numerically computed sensitivity curve of an SRGO for the given parameter values, and averaged over all other parameter values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' their details are different, since the sensitivity curve has been computed by averaging over several parameters, whereas Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' 3c corresponds to a fixed set of parame- ters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' The GW frequency of the sensitivity curve minima matches the GW frequency of the maxima in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' 3c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' In general, combining the insights from Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' 3 and 5, we de- duce that the minima of the sensitivity curve would occur at a GW frequency close to the inverse of the observation time, and that this minima would be smaller for longer observation times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Based on the predicted astrophysical mHz GW sources, we can conclude that the minimum ob- servation time for an SRGO experiment to be maximally sensitive to the entire mHz GW regime, would be of a few hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' This can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' 5, where the minima of the sensitivity curve lies close to the low-frequency edge of the predicted mHz GW regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' SRGO observational range From the ninth catalogue of spectroscopic binary orbits (SB9) [41], cross-referenced with the Gaia Data Release 3 [42–44], we find that the nearest spectroscopic binaries within our galaxy, including white dwarf (WD) binaries with masses ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='5M⊙ and periods of a few days, are lo- cated at distances of a few tens of parsecs (pc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Whereas, the nearest spectroscopic binaries with periods of a few hours are located at distances of several tens of parsecs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Hence, we choose a distance of 50 pc to mark the nearest WD binaries that would emit mHz GWs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' The nearest neutron star (NS) binaries [45–49] with masses ∼ 1M⊙ and periods of a few days, are located at distances of a few hundreds of parsecs [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Whereas, the nearest known double neutron star system with a period of a few hours is located at around 600 pc [51, 52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Hence, we choose this distance to mark the nearest NS binaries FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' 6: The maximum effective noise allowed in an SRGO to make a detection, or equivalently, the largest SRGO response amplitude expected in the best-case (optimum parameter choice) scenarios, due to GW sources at various distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' The colored dash-dotted lines indicate the nearest location of a particular type of source i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' these GW sources are absent to the left of the colored line corresponding to them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' All computations are done for a fixed observation time of 3 hours, and an initial GW frequency which is at the expected lower limit of the mHz regime, as this would maximise the SRGO response amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' that would emit mHz GWs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' It is estimated that our Milky Way galaxy contains mil- lions of stellar mass black holes [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' From binary black hole population simulations for Milky Way-like galax- ies [54], it is estimated that binary black holes may be present as close as 1 kpc from Earth, although most of them would be present 8 kpc away, near the galactic cen- ter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' This also happens to agree with the recent unam- biguous detection via astrometric microlensing, of an iso- lated stellar mass black hole [55], located at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='58 kpc from Earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Hence, we choose this distance as an estimate of the nearest stellar mass binary black holes with masses ∼ 10M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Intermediate mass black holes (IMBHs) of 102 – 105M⊙ are expected to be found in globular clusters and massive star clusters, but would be more numerous within galactic bulges of large galaxies and within dwarf galaxies [56, 57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' In galaxies like ours, globular clusters containing IMBHs of 103 – 104M⊙ are expected to be numerous at distances of 10 kpc from the galactic center, and these IMBHs can emit mHz GWs by merging with stellar mass black holes [58, 59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Hence, we use the dis- tance to the nearest known globular cluster “M4”, of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='2 kpc, to estimate the whereabouts of the nearest ∼ 10M⊙ & 104M⊙ extreme mass ratio inspirals (EMRIs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' As per [60], a typical large galaxy can contain several “wandering” SMBHs of ∼ 106M⊙, spread out across the 10-10 Massive binaries Extreme mass-ratio inspirals 10-12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Resolvable galactic binaries Unresolvable galactic binaries SRGO sensitivity curve Tobs = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='0hrs, Onoise = 1ps, fsample = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='998MHz 10-14 10-16 10-18 10-20 10-22 10-24 10-26 10-7 10-5 10-3 101 10-1 Frequency [Hz]GW source redshift (z) 100 101 10-9 10-8 10-7 10-6 10-4 10-5 10-3 10-2 10-1 10-6± fgw = 5 × 10-4 Hz;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Tobs = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='0 hrs nearest ~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='5M。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' WD binaries nearest ～1M。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' NS binaries nearest ～1OMo BH binaries S nearest ～10M。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' 104M。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' EMRIs 10- 9 PSNR=1 I nearest ～103M。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='& 106M。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='EMRIs nearest ～103M。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='& 108Mo EMRIs nearest ～107M。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' SMBH binaries first ～1o7M。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' SMBH binaries for 10-12 noise 10-15 lax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' M 10-21 102 103 105 107 101 104 106 108 109 1010 1011 GW source distance [pc]10 galactic halo, from near the galactic center to within the dwarf satellite galaxies and anywhere in between.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Milky Way’s central supermassive black hole (SMBH), Sgr A*, also happens to be of ∼ 106M⊙ [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' According to [62], the most promising mHz GW scenario in the IMBH – light SMBH mass range, is of ∼ 103M⊙ IMBHs merging with ∼ 106M⊙ SMBHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Lastly, the nearest galaxy to us, M31 (Andromeda), contains a ∼ 108M⊙ central SMBH [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' For all these reasons, we choose a distance of 8 kpc (distance from Earth to Milky Way’s center, as well as to the closest dwarf galaxy, Canis Major) to represent the location of the nearest ∼ 103M⊙ & 106M⊙ EMRIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' We choose 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='778 Mpc (distance from Earth to M31 An- dromeda galaxy) to represent the location of the nearest ∼ 103M⊙ & 108M⊙ EMRIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' The redshift evolution of the SMBH mass function [64– 66] tells us that, on average, 107M⊙ SMBH mergers may be the most frequent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' The nearest detected inspiralling SMBHs due to a galaxy merger, are located at a distance of 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='4 Mpc [67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' As per [68], the first SMBH merg- ers happened at around z = 10, when the first galaxies started merging in the early universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' We input this in- formation in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' 6 From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' 6, we see that the largest SRGO response signals would correspond to mHz GWs from SMBH bi- naries in galaxy mergers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' EMRIs involving an SMBH, typically the central SMBH of galaxies, or even wander- ing SMBHs interacting with smaller black holes, would also be significant sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' WD binaries, NS binaries and stellar mass BH binaries within our galaxy would not produce great responses, even if they were individu- ally resolvable sources and located as close to Earth as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' IMBH EMRIs within globular clusters in our galaxy may also give decent response amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' 6 also tells us that, an SRGO should aim for an ef- fective residual stochastic noise of ∼ 1ps or better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Since the effective noise depends not only on the true noise, but also on the data sampling rate and observing time, the true residual stochastic noise may be greater than ∼ 1ps, but can be effectively cut down by collecting more timing data points during the observation run (see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' IV B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' At this level of noise (or better), an SRGO could po- tentially detect mHz GW events involving supermassive black holes starting from within our galaxy, up to galaxy merger events at high redshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' RESULTS: GW PARAMETER ESTIMATION The antenna pattern of a GW detector is typically om- nidirectional (see [69] for the LIGO antenna pattern, and paper-I for the SRGO antenna pattern).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Therefore, even with a high signal-to-noise ratio, a single GW detector like LIGO, in principle, cannot pinpoint the position of the GW source in the sky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' However, three or more detec- tors working together can triangulate the source position via the relative time-delays between their detections from the same source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' But unlike LIGO, even a single SRGO, being a potential Earth-based mHz GW detector (where the GW signal may last for hours, days, or much longer), in principle, should be able to make use of Earth’s rota- tion (which would cause the GW source to sweep across its antenna pattern and produce a unique envelope in its response) to pinpoint the source position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' The GW source sky localization area, however, would certainly de- pend on the signal-to-noise ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' A single LIGO de- tector, on the other hand, being sensitive to kHz GWs, would not be able to fully make use of Earth’s rotation, because the response signal duration of LIGO would be much shorter compared to SRGO for the same effective PSNR and observation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' This hypothesis is verified by our simulation results, where MCMC methods have been used to do GW parameter estimation on noisy data points that were created by adding Gaussian noise to the SRGO response signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' An example of our results, the sky localization map for a case corresponding to 32 effective data points taken over 12 hours at a PSNR of 100, is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' 8a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' We note here our method of computing the sky localization area: We take the joint posterior of the right ascension and declination of the GW source (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' 8b), and calculate the ratio of the colored to the total pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Then we multiply this with the range of the right ascension and the range of the declination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Finally, we apply a correction for the spherical projection onto the sky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' The formula becomes, Asky � deg2� = colored pixels total pixels 180 π ∆αsrc ∆ sin δsrc (12) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' 7 corresponds to the same case as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' It is an example of the MCMC chain traces, and the diagonal elements of the 9 × 9 joint posterior corner-plot (shown in Appendix B, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' We see that in general, at a decent PSNR, the MCMC chains converge around the true parameter values and explore around this location in the 9-dimensional parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Variation of parameter estimation quantities We explore the variation of the posteriors as functions of some controllable experiment parameters (such as the observation time, data sampling rate and PSNR), for 5 out of the 9 fitting parameters in our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' These are: the GW source component masses, m1 & m2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' the GW source redshift, z ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' and the GW source sky position, αsrc & δsrc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' The results for αsrc & δsrc jointly correspond to the sky localization area, shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' 9a and 9b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' By symmetry, the results for m1 & m2 are the same, and correspond to Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' 9c and 9d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' The results for z have been translated into the GW source luminosity dis- tance, and correspond to Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' 9e and 9f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' We choose these 5 parameters as they are the most relevant ones for multi-messenger astronomy, being the first to be es- timated upon a GW detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' 11 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' 7: On the left are the marginalized posteriors of the fitting parameters and on the right are the corresponding MCMC traces, consisting of 1000 parallel chains with 1250 samples each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' The true parameter values for this case are the ones in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' 1, except the initial separation between the masses which is 1 AU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' 32 data points with artifical noise added (PSNR = 100) are taken over an observing time of 12 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' We generate 16, 32 and 64 data points in this study for a multitude of reasons: First, as explained in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' III E, we use powers of two as this allows for faster computa- tion of the Fast Fourier Transform (FFT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Furthermore, in our computations, 16 data points happens to be the lower limit for the Shannon-Nyquist condition to hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Hence, this would give us an upper limit on the parame- ter estimation for a given PSNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Also, although in reality it would be possible for a timing detector to make sev- eral million measurements within an observation time of hours to days, we use only a small number of data points for computational efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' This is justified because a smaller number of data points at a given PSNR may be interpreted as binning a large number of data points that correspond to a lower true PSNR, thus giving the same (a) GW source localization sky map (b) δsrc vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' αsrc joint posterior FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' 8: The GW source sky localization i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' the 3-sigma (99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='7%) HPD region on the joint posterior of the GW source’s right ascension and declination, shown on a sky map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' This figure corresponds to the case shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' This shows that a single SRGO can potentially use Earth’s rotation to localize the GW source in the sky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' effective PSNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' 9a, we see that, for a given data sampling rate and observation time, the sky localization area de- creases with decreasing noise, and saturates at around a few tens of deg2 for high effective PSNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' This is due to parameter degeneracies that cannot be resolved fur- ther, unless multiple SRGOs are utilized or better mod- els are utilized that, for instance, account for higher-order harmonic modes of GWs [70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Furthermore, at a given PSNR, the sky localization improves upon increasing the data sampling rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' This is because the effective noise is inversely proportional to the square root of the total number of data points, or in other words, the square root of the data sampling rate times the observing time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' The error bars show the statistical variation of the parame- ter estimation, and they increase with increasing noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Thus, parameter estimation becomes unreliable at high GW source mass #l posterior [Mo] 1e6 GW source mass #1 posterior [Mo] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='95 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='90 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='02 200 400 600 800 1000 1200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='92 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='94 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='98 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='04 0 1e6 GW source mass #2 posterior [Mo] 1e6 GW source mass #2 posterior [Mo] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='95 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='925 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='950 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='025 200 400 1000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='900 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='975 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='050 600 800 1200 1e6 Initial binary separation posterior [au] Initial binary separation posterior [au] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='98 200 400 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='980 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='985 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='990 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='995 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='005 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='010 0 600 800 1000 1200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='975 GW source inclination posterior [°] GW source inclination posterior [°] 10 0 10 0 200 400 600 800 1000 1200 10 5 10 5 GW source redshift posterior GW source redshift posterior 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content="10 001t'0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=" S20t'0 0501'0 5201'0 0001'0 s260'0 0560'0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=" 5260'0 0060'0 " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='600 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='800 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='1200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='GW initial phase posterior [°] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='GW initial phase posterior [°] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='600 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='800 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='1200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='GW source RA posterior [°] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='GW source RA posterior [°] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='600 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='800 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='1200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='GW source DEC posterior [°] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='GW source DEC posterior [°] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='600 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='800 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='1200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='GW polarization posterior [°] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='GW polarization posterior [°] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='600 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='800 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='1200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='10GW source position ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='X ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='3-sigma HPD region ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='Localization area = 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='6451 deg3 75° .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='09 45° 30° 15° 10h 08h 06h 04h 02h 22h 20h 18h 16h 0Qh 14h 0° D 15° 30° 45° .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='09- 75° RA2 GW source DEC posterior [°] O 2 4 4 2 0 2 4 GW source RA posterior []12 (a) Sky localization area vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' PSNR (b) Sky localization area vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Tobs (c) Mass estimation error vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' PSNR (d) Mass estimation error vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Tobs (e) Distance estimation error vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' PSNR (f) Distance estimation error vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Tobs FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' 9: The parameter estimations of three parameters are shown: GW source sky localization area, relative errors of distance and mass estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' The left column shows their variation with PSNR for different data sampling rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' The right column shows their variation with observation time at a fixed data sampling rate, for different values of PSNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' 16 data points over 1 day 32 data points over 1 day 64 data points over 1 day 104 103 102 101 10-1 100 101 102 PsNR (peak signal-to-noise ratio)PSNR = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='0 PSNR = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='0 104 PSNR = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='0 103 Sky localization area 102 101 15 21 0 3 6 9 12 18 24 27 Observation time [hrs]Relative error of mass estimation [%] 102 16 data points over 1 day 32 data points over 1 day 64 data points over 1 day 101 10-1 100 101 102 PsNR (peak signal-to-noise ratio)Relative error of mass estimation [%] 102 PSNR = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='0 PSNR = 1O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='0 PSNR = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='0 101 15 21 0 3 6 6 12 18 24 27 Observation time [hrs] (fsample = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='666 data points per hour)Relative error of distance estimation [%] 102 16 data points over 1 day 32 data points over 1 day 64 data points over 1 day 10-1 100 101 102 PsNR (peak signal-to-noise ratio) estimation [%] 102 Relative error of distance PSNR = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='0 PSNR = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='0 PSNR = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='0 3 12 15 21 6 6 18 24 27 0 Observation time [hrs]13 levels of noise, with the sky localization area covering al- most the entire sky for PNSR values lower than ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' An effective PSNR of ∼ 80 seems to be the threshold for a single SRGO to achieve its best possible sky localiza- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Comparing values from a curve of constant PSNR in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' 9b with the corresponding values at the same PSNR in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' 9a, we see that in general, for the same effective PSNR (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' same PSNR and number of data points), increasing the observation time improves sky lo- calization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' This is because, the effect of Earth’s rotation can be exploited to a greater extent to break some param- eter degeneracies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' An exception to this trend may occur when the Shannon-Nyquist condition is violated, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' for a fixed number of data points, a smaller observation time may result in better parameter estimation if increasing the observation time (reducing the data sampling rate) results in aliasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' This scenario would typically not be relevant for a realistic SRGO experiment, where the data sampling rate would be orders of magnitude higher than the GW frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Finally, beyond 24 hours observing time, the Earth’s rotation cannot break any more pa- rameter degeneracies in principle, and therefore even at high effective PSNR values in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' 9a and 9b, the sky localization tends to saturate at around 10 deg2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' In Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' 9c, 9d, 9e and 9f, we observe similar trends for the GW source mass and distance estimation as observed for the sky localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' The relative errors of mass and distance estimation saturate at 200% for low PSNR val- ues only because of the bounded flat priors that we use in the MCMC algorithm, mentioned in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' III E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Even at very high PSNR values, the mass and distance estima- tion remain finite, up to a few tens of percent, and would also likely saturate because of parameter degeneracies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' For the same reasons mentioned previously, comparing values from a given curve in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' 9d (9f) with the cor- responding values at the same PSNR in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' 9c (9e), we notice the trend that for the same effective PSNR, increasing the observation time improves parameter es- timation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' However, unlike the sky localization, it seems that the mass and distance estimations would saturate at an even higher effective PSNR values than the sky local- ization, since the curves in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' 9c and 9e do not flatten out towards the right side ends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Parameter degeneracies In Appendix B, we show an example of the 36 joint posterior pair-plots for our 9 model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' These correspond to a case where 32 noisy data points were taken over 12 hours, at a PNSR of 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' The true param- eter values for this case are the same as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' 1, except that the initial binary separation is 1 AU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' At such a high PSNR, the joint posterior correlations would show the degeneracies between the parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Here, we try to explain the observed correlations based on the model de- tails described in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' III A: The joint posterior of the two binary masses (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' 10a) shows an anti-correlation, because upon increasing one of the masses, the other must be decreased to have the same chirp mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' The two binary masses are also positively correlated with the initial binary separation (Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' 10b and 10i), since increasing the mass increases the GW strain amplitude and also affects the frequency evolution, which can be countered by increasing the initial binary separation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' The positive correlation between the masses and the GW source redshift (Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' 10h and 10k) exists because increasing the redshift increases the GW source distance while decreasing the observed GW frequency, both of which decrease the GW strain amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' How- ever, increasing the redshift also increases the observed chirp mass, but this is not sufficient and therefore, a fur- ther mass increase is required to counter the effect of a decrease in the GW strain amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Instead of increas- ing the mass, we can also counter this by increasing the initial binary separation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' That is why the joint poste- rior between the source redshift and the initial binary separation also shows positive correlation (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' 10q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' An interesting joint posterior to analyze is the source redshift, z vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' the source inclination angle, i (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' 10v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' The degeneracy between the source distance and inclina- tion angle is well known in GW astronomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Therefore, we expect them to be anti-correlated, since increasing the source distance decreases the GW strain amplitude, but decreasing the inclination angle can counter this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' How- ever, this is only true for linearly polarized GWs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' In our case, since i = 0, this corresponds to a face-on orien- tation of the binary system towards Earth, resulting in circularly polarized GWs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Changing the inclination an- gle changes the relative amplitudes of the plus and cross polarization strain components, producing elliptically po- larized GWs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Furthermore, in our simulations, the source distance is not a separate parameter, but is instead cal- culated from the cosmological redshift, which is a model parameter that affects not only the source distance, but also the chirp mass and GW frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Therefore, in our results, the z vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' i joint posterior shows no correlation, since these two parameters control very different aspects of the response signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Another noteworthy joint posterior is of the GW polar- ization angle vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' the GW initial phase (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' 10gg), which shows a strong linear correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' This can be derived an- alytically for our special case of i = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' For this case, the response signal Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' (4), after substituting all the terms and collecting the common GW strain amplitude terms into a factor h0, can be re-written as, ∆TGW = −1 4 � t0+T t0 h0 sin2 θ � cos (2ψ) cos (2πft + δ0) + sin (2ψ) sin (2πft + δ0) � dt = −1 4 � t0+T t0 h0 sin2 θ cos (2πft + δ0 − 2ψ)dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' (13) This perfectly explains why the joint posterior of ψ 14 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' δ0 has a slope of 1/2, and it implies that for special cases, changing the GW polarization angle is effectively the same as changing the initial phase of the GW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' If the Earth were stationary, this would be the same as begin- ning the observation run at a different time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' The re- maining observed correlations in some of the joint poste- rior pair-plots cannot be interpreted analytically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Finally, the rest of the joint posteriors are uncorrelated, because the corresponding model parameters control widely dif- ferent aspects of the response signal, and cannot produce degeneracies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' DISCUSSION What are the limitations and caveats of this study?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' One of the primary limitations of this study is the sim- plistic static Gaussian noise model for the residual sys- tem noise, based on the assumption that our hypotheti- cal storage ring facility is capable of attenuating most of the (yet un-studied) noise sources, similar to LIGO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' We cannot yet model noise sources in detail, especially their frequency and time dependence, until a thorough study is conducted (Schmirander et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=', in preparation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Next, our GW waveform models do not account for the spins of the compact objects in the binary, the eccentric- ity of their orbit, and other parameters corresponding to realistic binary systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' While our model contains 9 un- known fitting parameters, realistic GW models contain around 15 to 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' However, as a first step towards estab- lishing a novel experiment concept, for the sake of consis- tency and ease of analysis, it is better to use a realistic toy model for making order-of-magnitude estimations, rather than to use complex and detailed models from the very beginning, which can make analysis quite difficult in a topic that has not been explored to such an extent prior to this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Although our GW source and ring models are simple, they are realistic enough to provide correct orders of magnitude of the estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Perhaps, incorpo- rating detailed GW waveforms and storage ring models is the next logical step in this series of works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' We also do not model the merger and ringdown phases, and cover only the inspiral phase of the binaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' How- ever, because the response signal tends to decrease with increasing GW frequency (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' 3), the merger and ring- down phases would likely not produce an SRGO response signal as large as the one during the inspiral phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Furthermore, our GW source models, which are de- rived from post-Newtonain (PN) analysis, cannot accu- rately model EMRIs (extreme and intermediate mass ra- tio inspirals).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' In Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' IV C, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' 6, some GW sources that are actually EMRIs, have been estimated with our post-Newtonian GW waveform model, which is not op- timal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' But since we are interested in order-of-magnitude estimates and since we do not expect the difference be- tween our model and an EMRI GW waveform to be orders-of-magnitude greater, we regard this as a justi- fiable simplification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' This is supported by [71], where it can be seen that the simple PN waveform models are accurate enough to model EMRIs for small observation times of a few hours or days, as considered in our study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Due to the specifics of the MCMC setup, described in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' III E, we miss the antipodal sky localization region, which should exist because placing the ring and/or the GW source at antipodal positions, and/or having the ions circulating in the opposite direction, would all produce the same SRGO response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Although we provide flat pri- ors and allow the MCMC chains to explore over the full range of the angular parameters, the chains seem to con- verge and explore only around the true parameter values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' This may be due to the nature of the DE-MC algirithm used, which is known for converging quickly to a solu- tion in the parameter space and staying around it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' An antipodal sky localization region would double the area, but would not change the shape of Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' 9a, 9b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Hence, our results would not change beyond their estimated er- rors, and thus the interpreted conclusions would remain the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Many of our results have been generated by averaging over as many parameters as possible, so that the con- clusions interpreted from them may remain accurate and general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' However, some of our conclusions are extrapo- lated based on results for a specific and arbitrary combi- nation of parameters, corresponding to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' 1 (with some variations which are described in the sections pertaining to each result).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' This was done for cases where averag- ing over parameters was very difficult or computation- ally expensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' These include results in Sects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' IV A and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' However, we do not expect the parameter-averaged results to be different in order-of-magnitude for these cases, and hence expect them to be sufficient for first estimates and general conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' For instance, the re- sults in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' IV A, which are based on scaled values of the PSNR, are intrinsically independent of some param- eters to a great extent, such as the GW source mass and distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Moreover, we can make estimates of how some of the results would change for a different set of param- eters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' For example, the results of Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' V, for a different set of true parameters, can be estimated by combining the results of Sects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' IV A and V A, which should at least be correct in order of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Lastly, we have not yet accounted in the SRGO re- sponse formulation, the effect of the GW on the storage ring magnetic field, which may possibly boost the re- sponse signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' This would be included in future works (Schmirander et al, in preparation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' How to measure the instantaneous initial ion speed, vi?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Two timing detectors placed close by would detect a passing ion with a delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Dividing the known distance covered by the ion with this timing delay would give us vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' This measurement could be made more accurate by repeating this procedure over the first several revolutions and then taking an average value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' However, performing this procedure with a single timing detector (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' dividing the orbit circumference by the time interval between two successive detections of the same ion by a single detec- 15 tor) would be less accurate, because although the time- varying quantities would change negligibly during a sin- gle ion revolution, but the ion would still be affected by the anisotropy of the spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Hence, compared to the former procedure, this way would give us a slightly worse substitute for the quantity that we wish to measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' How to measure ∆TGW ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Using vi and the circumference of the ion orbit, we can predict the expected arrival times of the ion to the timing detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' These must then be subtracted from the actual ion arrival times that are measured by the timing detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' The result will constitute the discrete noisy data points ∆TGW , which when plotted against the expected arrival times, will look like Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' This is why the second term within the integral of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' (4) differs from that of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' (1), when we measure vi = v0 � 1 + hθφψ(0) 2 � instead of v0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Since the speed is used to predict the times when the ion would arrive at the detector, a different speed would change the predicted ion arrival times, and thus, also the signal (which is the observed arrival time minus the predicted arrival time of the ion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Do GWs affect the atomic clock of the timing detector?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Since the storage ring ion clock and the atomic/optical clock of the timing detector would be located next to each other, they would both be affected in the same way due to the temporal component of the GW metric (or any other spacetime metric).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Therefore, in principle, the temporal component of the spacetime metric cannot be measured by a comparison between the ion clock and atomic clock geodesics (the working principle of SRGO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' However, since the location of the atomic clock would be station- ary in the reference frame, while the ion would revolve in an anisotropic GW spacetime, the spatial components of the GW metric would affect the storage ring ion clock differently as compared to the atomic/optical clock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' This difference would result in the response signal that can, in principle, be measured by an SRGO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' This is the reason why, as explained in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' IV C, laser and atom inter- ferometer GW detectors cannot probe the anisotropy of a static distorted spacetime (such as very low frequency GW spacetimes over short observation times), even in principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Whereas, this would possible in principle with an SRGO, even in the absence of Earth’s rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' How- ever, practically, this might never be tested because of the stochastic gravitational wave background (SGWB), which exists due to an overlap of a large number of un- resolved and incoherent astrophysical GW sources at low frequencies [72–74].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Why did we choose MCMC methods over Fisher Infor- mation for parameter estimation?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' The Fisher Information Matrix (FIM) can be described as the inverse of the covariance matrix of some distri- bution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' It may also be interpreted as the curvature of the log-likelihood graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' The FIM can be calculated an- alytically, requiring only the model that generates the response signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' This makes the FIM a fast and simple method of obtaining the precision of the parameter esti- mation pipeline without actually having to make a mea- surement of artificial noisy data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' However, the FIM does have limitations: It assumes a model with linearly cor- related parameters, a detector with Gaussian noise, and a high SNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' It has been shown that for a non-spinning binary GW source model with 9 unknown parameters such as ours, at total binary mass higher than 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='0M⊙, the standard deviation predicted by the FIM does not agree with the standard deviation of a fully calculated posterior by MCMC methods [75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' What are the implications for the FCC (Future Circu- lar Collider)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' FCC [76] is a proposed circular particle accelerator which will be able to accelerate ultrarelativistic ions at even higher energies than the LHC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' This could increase the natural attenuation of any stochastic noise sources di- rectly acting on the ions, due to the ions having a higher relativistic mass or momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' However, the proposed 100 km circumference of the FCC would have implica- tions for noise levels from sources such as seismic noise, gravity gradient noise and others, which unlike the ex- pected SRGO response signal, would likely be sensitive to the ring size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Currently, it is unclear whether a larger or smaller ring size would be more suitable for an SRGO ex- periment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' It is hoped that, upon detailed computational modeling of the noise sources, an optimal configuration within the parameter space can be found, which reveals the optimal ring size (Schmirander et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=', in preparation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' What are the implications for multi-messenger astron- omy?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' The yet undetected mHz GW events are also predicted to be associated with the emission of electromagnetic ra- diation and neutrinos [8, 77–79].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' For transient astro- physical events that correspond to high frequency GWs such as those detected by LIGO, the usual case for multi- messenger observations is of the event first being detected by the omnidirectional GW detectors, which then per- form fast parameter estimations and send out real-time alerts to other observatories, providing the estimated GW source component types, masses, spins and impor- tantly, the sky localization region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' An effort is then made to quickly and simultaneously observe the GW event via the other messenger signals, using the alert information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' However, for mHz GW events, fast alert response would be of lesser concern, because most of these events would be long-lasting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Therefore, improving parameter estima- tion, especially the sky localization, would be most im- portant for multi-messenger studies of mHz GW events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Other than improving detector sensitivities, this is best achieved by collaboration between multiple mHz GW de- tectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' It is estimated that a proposed mHz GW detec- tor such as LISA, by itself, would not be good enough to pinpoint the host galaxies of mHz GW sources [77, 80].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' On this front, it is clear that the successful realization of an SRGO would greatly complement other mHz GW detectors such as LISA, and improve the GW alerts for multi-messenger observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Assuming that a mHz GW event is detected simul- taneously by LISA and SRGO, and further assuming 16 that the realized SRGO has effectively the same capabil- ities as the hypothetical system considered in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' III B of this study, then we can make a rough estimation of the improvement in the GW source sky localization due to a combination of SRGO and LISA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' The LISA sky localization for massive black holes is estimated to be 1 − 100 deg2, and LISA would be lagging the Earth orbit by 20° [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' In the optimistic case, assuming that a sin- gle SRGO on Earth manages to localize the same GW event up to 1 − 20 deg2 as obtained in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' V A, then by combining this data via simple 3D geometry, we can roughly estimate that the improved sky localization may be as good as sub–deg2, and as bad as a few tens of deg2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Overall, this would be a very good improvement, and it could be made even better by having multiple SRGOs at different locations on Earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' SUMMARY AND CONCLUSION In Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' I, we discuss previous studies on storage rings as GW detectors, highlighting what they missed, and ex- plaining the novelty of our idea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' We provide comparisons and analogies between an SRGO and other known GW detection techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' We also discuss references that sup- port our findings and throw light on potential ways for realizing an SRGO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' In Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' II, we provide a review of the theory behind an SRGO, and revise important formulae to display them in a better format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' III describes the mathematical models and numerical procedures of our simulation code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' In Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' IV A, we study the variation of the response signal with the experiment parameters, obtaining useful physical insights about how an SRGO works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Our results suggest that the response signal would be maximised by placing an SRGO at equatorial latitudes on Earth and by having long observation times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' In Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' IV B, we numer- ically obtain the SRGO sensitivity curve, which shows that an SRGO would be intrinsically sensitive to the mHz GW regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' The sensitivity curve also suggests that a minimum observation time (run time of the storage ring) of at least a few hours would be required for an SRGO experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' In Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' IV C, we find that a typical SRGO may have maximum response signal amplitudes of up to ∼ 1ps due to astrophysical mHz GW sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Therefore, an SRGO should aim to have, at worse, similar effective noise levels to make a detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' At this level of noise (or better), an SRGO could potentially detect mHz GW events involving supermassive black holes starting from within our galaxy, up to galaxy merger events at high redshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' The results of Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' V prove that even a single SRGO can, in principle, perform accurate GW parameter esti- mation, being able to provide a closed region on a sky map for the GW source localization, which would im- prove with increasing PSNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' In Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' V A, we find that an effective PSNR (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' true PSNR times the square root of the total number of data points) of at least ∼ 80 would be required to achieve decent parameter estimation with a single SRGO, which may be achieved by a combination of noise reduction and increasing the data measurement rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' At this effective PSNR or higher, a single SRGO would be capable of constraining the GW source param- eters (such as the sky localization area, relative distance and mass estimations, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=') to within a few tens of per- cent of their true values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' In Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' V B, we obtain more physical insights by studying the parameter degeneracies of an SRGO experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Finally, in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' VI, we discuss the limitations of this study;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' justify some approaches we have taken in this study;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' answer fundamental questions about the working principle of an SRGO;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' and discuss future implications of realizing an SRGO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' In conclusion, SRGO seems promising as a near-future Earth-based GW detector sensitive to the yet undetected mHz GWs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' It could complement space-based detec- tors such as LISA, or even make detections prior to the launch of LISA, assuming that rapid technological devel- opment during this decade allows a functional SRGO to be built.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' The main effort required in this direction would be detailed studies, techniques and technologies to han- dle noise sources;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' finding the optimum operation mode of a storage ring for an SRGO experiment;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' techniques and technologies for the timing data readout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Further studies of single ion storage rings and improvement in vacuum technology would also help.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' ACKNOWLEDGMENTS We acknowledge Saloni Priya, Florian Gr¨uner, Wolf- gang Hillert, Roman Schnabel, Mikhail Korobko, Thor- ben Schmirander and Velizar Miltchev for fruitful dis- cussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' This research was supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foun- dation) under Germany’s Excellence Strategy – EXC 2121 Quantum Universe – 390833306.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' This work has made use of data from the European Space Agency (ESA) mission Gaia (https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='cosmos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='esa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='int/ gaia), processed by the Gaia Data Processing and Analy- sis Consortium (DPAC, https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='cosmos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='esa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='int/ web/gaia/dpac/consortium).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Funding for the DPAC has been provided by national institutions, in particu- lar the institutions participating in the Gaia Multilateral Agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Appendix A: Contribution to ∆TGW from beam orbit shape distortions Consider a circular ion beam of radius R, which gets distorted into, say, an ellipse with axes R ± ∆R, where ∆R R = h represents the GW strain amplitude, which is much smaller than unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' The perimeter of a near-circular ellipse is approxi- mated to an excellent accuracy by Ramanujan’s formula 17 [81], Cellipse = π(a + b) � 1 + 3λ2 10 + √ 4 − 3λ2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' (A1) Here, a = R+∆R, b = R−∆R, λ = (a−b) (a+b) = h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' The er- ror in Ramanujan’s approximation is O(h10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Over many revolutions, the ion circulation time deviation will be pro- portional to a time integral over the difference between the perimeters of the distorted and ideal orbit shapes, ∆Torbit ∝ � t0+T t0 (Cellipse − 2πR) dt ∝ h2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' (A2) This result is, in general, also true for more complex beam orbit shape distortions caused by other sources (such as seismic activity), as long as the corresponding quantity equivalent to h is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Appendix B: Corner plot shown as individual joint posterior plots Due to space constraints, we show in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' 10 the 36 individual joint posterior pair-plots corresponding to the non-diagonal elements of the 9 × 9 corner plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' The di- agonal elements of the corner plot have been shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' In each pair-plot, we show the 1-sigma (68%) and 3-sigma (99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='7%) highest posterior density (HPD) re- gions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' The true parameter values for this case correspond to those in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' 1, except the initial binary separation which is 1 AU.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Derdzinski, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' D’Orazio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=', Multimessenger science opportuni- ties with mHz gravitational waves, Bulletin of the AAS 51 (2019), https://baas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='aas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='org/pub/2020n3i123.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' [79] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Eracleous, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Gezari, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Sesana, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Bogdanovic, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' MacLeod, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Roth, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Dai, An Arena for Multi- Messenger Astrophysics: Inspiral and Tidal Disruption of White Dwarfs by Massive Black Holes, Bulletin of the AAS 51 (2019), https://baas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='aas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='org/pub/2020n3i010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' [80] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Ruan, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Liu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='-K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Guo, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Wu, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Cai, The lisa–taiji network, Nature Astronomy 4, 108 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' [81] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' Villarino, Ramanujan’s perimeter of an ellipse (2005), arXiv:math/0506384.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' 20 (a) m1 and m2 (b) m1 and r0 (c) m1 and i (d) m1 and z (e) m1 and δ0 (f) m1 and αsrc (g) m1 and δsrc (h) m1 and ψeq (i) m2 and r0 (j) m2 and i (k) m2 and z (l) m2 and δ0 (m) m2 and αsrc (n) m2 and δsrc (o) m2 and ψeq (p) r0 and i (q) r0 and z (r) r0 and δ0 (s) r0 and αsrc (t) r0 and δsrc 1e6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='08 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='06 posterior [Mo] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='04 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='02 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='00 # mass 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='98 GW source 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='02 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='04 GW source mass #l posterior [Mo] 1e61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='010 / separation posterior [au] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='005 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='995 Initial binary 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='990 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='985 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='980 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='92 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='94 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='98 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='02 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='04 GW source mass #1 posterior [Mo] 1e610 GW source inclination posterior [° 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='90 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='1100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='1075 GW source redshift posterior 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='1050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='1025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='1000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='0975 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='02 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='04 GW source mass #l posterior [Mo] 1e620 15 10 GW initial phase posterior [°] 5 5 10 15 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='92 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='94 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='96 0.' metadata={'source': 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source RA posterior [°] 2 2 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='980 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='985 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='990 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='995 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='005 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='010 Initial binary separation posterior [au]21 (u) r0 and ψeq (v) i and z (w) i and δ0 (x) i and αsrc (y) i and δsrc (z) i and ψeq (aa) z and δ0 (bb) z and αsrc (cc) z and δsrc (dd) z and ψeq (ee) δ0 and αsrc (ff) δ0 and δsrc (gg) δ0 and ψeq (hh) αsrc and δsrc (ii) αsrc and ψeq (jj) δsrc and ψeq FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' 10: Individually shown joint posteriors of the 9 × 9 corner plot obtained after MCMC parameter estimation, for true parameter values described in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content=' 2 GW source DEC posterior [°] O 2 4 4 2 0 2 4 GW source RA posterior []10 GW polarization posterior [°] 5 .' metadata={'source': 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[]2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='GW source DEC posterior [°] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='GW source inclination posterior []10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='GW polarization posterior [°] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE_T4oBgHgl3EQf0xx9/content/2301.08331v1.pdf'} 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P. Gnatenko1∗, +Ivan Franko National University of Lviv, +Professor Ivan Vakarchuk Department for Theoretical Physics, +12, Drahomanov St., Lviv, 79005, Ukraine +SoftServe Inc., 2d Sadova St., 79021 Lviv, Ukraine +January 11, 2023 +Abstract +The geometric measure of entanglement of variational quantum +states is studied on the basis of its relation with the mean value of +spin. We examine n-qubit quantum states prepared by a variational +circuit with a layer formed by the rotational gates and two-qubit con- +trolled phase gates. The variational circuit is a generalization of that +used for preparing quantum Generative Adversarial Network states. +The entanglement of a qubit with other qubits in the variational quan- +tum states is determined by the angles of rotational gates that act on +the qubit and qubits entangled with it by controlled phase gates and +also their parameters. In the case of one layer variational circuit, the +states can be associated with graphs with vertices representing qubits +and edges corresponding to two-qubit gates. The geometric measure +of entanglement of a qubit with other qubits in the quantum graph +state depends on the properties of the vertex that represents it in the +graph, namely it depends on the vertex degree. The dependence of +the geometric measure of entanglement of variational quantum states +on their parameters is quantified on IBM’s quantum computer. +∗khrystyna.gnatenko@gmail.com +1 + +1 +Introduction +Variational quantum circuits have received much attention because of their +wide usage in various quantum algorithms among them Quantum Machine +Learning, error correction, algorithms for solving optimization problems, and +many others (see, for instance, [1, 2, 3, 4, 5, 6, 7] and references therein). +A critical and festinating resource in quantum computing is entanglement +[8, 9, 10, 11]. The geometric measure of entanglement of quantum states has a +clear definition that is based on the geometric idea of computing the minimal +distance between the entangled state and a set of non-entangled states. The +measure of entanglement can be found as E(| ψ⟩) = min|ψs⟩ d2 +F S(|ψs⟩, |ψ⟩), +where | ψ⟩ is an entangled state, | ψs⟩ is a set of non-entangled states, +dF S = +� +1 − |⟨ψ|ψs⟩|2 is the Fubini-Study distance between the states | ψs⟩, +| ψ⟩ [9]. The procedure of finding the minimal distance requires a lot of +computational resources. In the paper, [12] it was found that to detect the +geometric measure of entanglement of a spin with an arbitrary quantum +system in a pure state it is sufficient to calculate the mean value of the spin. +Namely, the geometric measure of entanglement of a spin with quantum +system that are in a state | ψ⟩ = a |↑⟩ | Φ1⟩ + b |↓⟩ | Φ2⟩, can be found as +E(| ψ⟩) = 1 +2(1 − +� +⟨σ⟩2). +(1) +Here | Φ1⟩, | Φ2⟩ are states of a quantum system (⟨Φi | Φi⟩ = 1) that in +general case can be nonorthogonal, a, b are constants, +� +⟨σ⟩2 = +� +⟨σx⟩2 + ⟨σy⟩2 + ⟨σz⟩2, +(2) +σx, σy, σz are Pauli matrixes [12]. It is important to mention that relation +(1) opens a possibility to quantify the geometric measure of entanglement of +quantum states on a quantum devices. +We study the geometric measure of entanglement of n-qubit quantum +states prepared by a variational circuit with a layer formed by the rotational +gates and controlled phase gates on the basis of analytical calculations and +programming on IBM’s quantum computer and IBM’s quantum simulator. +2 + +2 +Detection of the geometric measure of en- +tanglement of variational quantum states +on IBM’s quantum computer +Let us consider variational quantum circuit with a layer formed by the rota- +tional gates RY (θi,j) and controlled phase gates CP(ϕi,j) presented in Fig. +1. Note that in the particular case of ϕi,j = π the variational n+1-qubit form +q[2] +q[1] +q[0] +q[n] +Ry +( +0,0) +Ry +( +1,0) +Ry +( +2,0) +Ry +( +n,0) +Ry +( +0,1) +Ry +( +1,1) +Ry +( +2,1) +Ry +( +n,1) +Ry +( +1,k) +Ry +( +0,k) +Ry +( +2,k) +Ry +( +n,k) +... + ... +... +... +θ +θ +θ +θ +θ +θ +θ +θ +θ +θ +θ +θ +P +( +2,1) +ϕ +P +( +0,1) +ϕ +P +( +1,1) +ϕ +P +( +1,k) +ϕ +P +( +1,k) +ϕ +P +( +0,k) +ϕ +Figure 1: Variational quantum circuit of a depth k. The layer formed by the +rotational gates RY (θi,j) and controlled phase gates CP(ϕi,j). +corresponds to that used as a quantum generator in the quantum Generative +Adversarial Networks (qGANs) [1]. +To detect the geometric measure of entanglement of a qubit with other +qubits in a state prepared with a variational circuit we quantify the mean +value of operators σx, σy, σz in the corresponding variational quantum state. +For instance, to evaluate the entanglement of qubit q[1] with other qubits in +the state we consider quantum protocol presented in Fig. 2 +q[2] +q[1] +q[0] +q[n] +Ry +( +0,0) +Ry +( +1,0) +Ry +( +2,0) +Ry +( +n,0) +Ry +( +0,1) +Ry +( +1,1) +Ry +( +2,1) +Ry +( +n,1) +... + ... +... +... +Rα +θ +θ +θ +θ +θ +θ +θ +θ +P +( +2,1) +ϕ +P +( +0,1) +ϕ +P +( +1,1) +ϕ +P +( +1,k) +ϕ +P +( +1,k) +ϕ +P +( +0,k) +ϕ +z +c +Figure 2: Quantum protocol for detection of the geometric measure of entan- +glement of qubit q[1] with other qubits in a state prepared by the variational +quantum circuit with the depth k. +3 + +In the quantum protocol it is taken into account that ⟨ψ|σx|ψ⟩ = ⟨ ˜ψy|σz| ˜ψy⟩ = +|⟨ ˜ψy|0⟩|2 − |⟨ ˜ψy|1⟩|2, ⟨ψ|σy|ψ⟩ = ⟨ ˜ψx|σz| ˜ψx⟩ = |⟨ ˜ψx|0⟩|2 − |⟨ ˜ψx|1⟩|2, where +| ˜ψx⟩ = exp(−iπσx/4)|ψ⟩, | ˜ψy⟩ = exp(iπσy/4)|ψ⟩. Therefore to quantify the +mean value ⟨σx⟩ the Rα = RY (−π/2) gate has to be applied to the state of +a qubit before the measurement in the standard basis. To detect the mean +value ⟨σy⟩ the Rα = RX(π/2) gate has to be used. Mean value of σz opera- +tor can be calculated, using the results of measurement in the standard basis +⟨ψ|σz|ψ⟩ = |⟨ψ|0⟩|2 − |⟨ψ|1⟩|2. +It is worth mentioning that quantum states generated by variational quan- +tum circuit with the depth k = 1 are graph states. These states can be +associated with undirected graphs G(E, V ). The qubits are represented by +vertices V in the graph and the edges E correspond to two-qubit gates. The +geometric measure of entanglement of a qubit with other qubits in the graph +state | G⟩ = � +(a,b)∈E CPab(φ) � +i RYi(θ) | 0⟩⊗n, (here RYi(θ), CPij(φ) are +rotational and controlled phase gates acting on the states of qubits q[i], q[j]) +depends on the graph properties [13, 14]. Namely, the entanglement of qubit +q[l] with other qubits in n-qubit graph state depends on the the degree of +vertex nl that represents it. Performing analytical calculations, on the basis +of the result (1), one finds [13] +El = 1 +2 − 1 +2 +� +sin2 θ +� +cos2 φ +2 + sin2 φ +2 cos2 θ +�nl ++ cos2 θ. +(3) +Using relation of the geometric measure of the entanglement with the +mean value of spin (1), we obtain the entanglement of qubit q[1] with other +qubits in the quantum Generative Adversarial Network state prepared by +the n + 1 qubit variational quantum circuit Fig. 1 with depth k = 1, and +φi,1 = π. It reads +E(θ0,0, θ1,0, θ2,0) = 1 +2 +� +1 − +� +cos2 θ0,0 cos2 θ2,0 sin2 θ1,0 + cos2 θ1,0 +� +. +(4) +Note that the entanglement of qubit q[1] with other qubits in the state de- +pends only on the parameters θ0,0, θ1,0, θ2,0 of RY gates that act on the +qubits entangled with q[1] by CZ01, CZ12 gates and do not depend on other +parameters of the variational circuit. +To study the geometric measure of entanglement of the quantum Gener- +ative Adversarial Network states on a quantum device we realized quantum +4 + +protocol Fig. 2 on ibmq-manila and ibmq-qasm-simulator for k = 1, φi,1 = π, +θi,0 = θ changing from 0 to 2π with the step π/32. The results of calculations +are presented in Fig. 3 (a). +Let us also examine the geometric measure of entanglement of qubit q[1] +with other qubits in the quantum Generative Adversarial Network state gen- +erated by variational circuit with depth k = 2. In the case of θi,0 = π/2 the +entanglement reads +E(θ0,1, θ1,1, θ2,1) = 1 +2(1 − 1 +4| cos(θ0,1 + θ1,1 − θ2,1) + ++ cos(θ0,1 − θ1,1 + θ2,1) + cos(−θ0,1 + θ1,1 + θ2,1) + cos(θ0,1 + θ1,1 + θ2,1)|). (5) +We realized protocol Fig. 2 for k = 2, θi,0 = π/2, θi,1 = θ1, φi,1 = φi,2 = +π and θ1 changing from 0 to 2π with the step π/32 on quantum device +ibmq-manila and ibmq-qasm-simulator. The results of quantum calculations +and the theoretical result for the entanglement in this case E(θ1) = (1 − +| cos3 θ1|)/2 are plotted in Fig. 3 (b). Quantum protocol Fig. 2 was also +implemented for, k = 2, θi,0 = θ0, θi,1 = π/2, φi,1 = φi,2 = π and θ0 +changing from 0 to 2π with the step π/32 on the quantum computer and +simulator. +The results of quantum calculations and the analytical result +E(θ0) = (2 − | sin(2θ0)|)/2 are presented in Fig. 3 (c). +In addition, to detect the dependence of the geometric measure of entan- +glement on the parameters of phase gates we realized quantum protocol Fig. +2 for different values of φi,1 = φ changing from 0 to 4π with the step π/32 and +θi,0 = π/2 (the depth is k = 1) Fig. 2 (a) and for different values of φi,1 = φ1, +changing from 0 to 4π with the step π/32, θi,0 = π/2, φi,2 = π (the depth is +k = 2) Fig. 2 (b). The results of quantum calculations and the analytical +results E(φ) = (1 − cos2(φ/2))/2, E(φ1) = (1 − | cos(φ1)(1 + cos(φ1))|)/2 are +presented in Fig. 2, cases (a) and (b), respectively. Note that for k = 1, the +results of quantum calculations are in good correspondence with the theo- +retical ones. Not so good agreement of the results was obtained for circuits +with the depth k = 2 because of accumulating of the gate errors. +3 +Conclusions +We have studied the geometric measure of entanglement of a variational +quantum states prepared by rotational gates and entangled blocks formed +by controlled phase gates Fig. 1. The studies have been done on the basis +5 + +Figure 3: Results of calculations of entanglement of qubit q[1] with other +qubits in variational quantum states (a) for different values of θi,0 = θ, φi,1 = +π and k = 1 (left plot); (b) for different values of θi,1 = θ1 and θi,0 = π/2, +φi,1 = φi,2 = π, k = 2 (middle plot); (c) for different values of θi,0 = θ0 +and θi,1 = π/2, φi,1 = φi,2 = π, k = 2 (right plot), obtained on ibmq-manila +(marked by black crosses), ibmq-qasm-simulator (marked by red circles), and +analytical results (line). +Figure 4: Results of calculations of entanglement of qubit q[1] with other +qubits in variational quantum states (a) for different values of φi,1 = φ, +θi,0 = π/2, and the depth k = 1 (left plot); (b) for different values of φi,1 = φ1 +and θi,0 = π/2, φi,2 = π, and the depth k = 2 (right plot), obtained on the +on ibmq-manila (marked by black crosses), ibmq-qasm-simulator (marked by +red circles), and analytical results (line) +of the relation of the entanglement with the mean value of spin (1). In the +case of variational circuit of one layer the variational quantum state can be +associated with graph and the the geometric measure of entanglement of a +qubit with other qubits in the variational quantum state with parameters +θi,0 = θ, φi,0 = φ is related with the degree of vertex representing it in the +graph (3). +6 + +The entanglement of a qubit with other qubits in the variational quan- +tum state is determined by the angles of rotational gates in the variational +circuit that act on the qubits entangled with it by controlled phase gates and +their parameters. The relation of the geometric measure of the entanglement +with mean value of spin opens a possibility to quantify the entanglement on +quantum devices, realizing quantum protocol Fig. 2. On the basis of the +relation the dependencies of the variational quantum states entanglement on +the parameters of the variational quantum circuit have been calculated on +IBM’s quantum computer ibmq-manila and ibmq-qasm-simulator Figs. 3, 4. +Acknowledgements +The author thanks Prof. Tkachuk V. M. for valuable discussions in the filed +of studies. +References +[1] Zoufal, Ch. Lucchi, A. Woerner, S.: Quantum Generative Adversarial +Networks for learning and loading random distribution. npj Quantum +Information 5, 103 (2019). +[2] Cerezo, M. Arrasmith, A Babbush, R. et al: Variational quantum algo- +rithms. Nature Reviews Physics 3, 625-644 (2021). +[3] Du, Yuxuan Huang, Tao You, Shan Hsieh, Min-Hsiu Tao, Dacheng: +Quantum circuit architecture search for variational quantum algorithms, +npj Quantum Information 8, 62 (2022). +[4] Bravo-Prieto, C. Lumbreras-Zarapico, J. Tagliacozzo, L. Latorre, J. I.: +Scaling of variational quantum circuit depth for condensed matter sys- +tems. Quantum 4, 272 (2020). +[5] Xu, Xiaosi Benjamin, Simon C. Yuan, Xiao: Variational circuit compiler +for quantum error correction, Phys. Rev. Applied 15, 034068 (2021). +[6] Moll, N. Barkoutsos, P. Bishop, L. S. Quantum optimization using varia- +tional algorithms on near-term quantum devices. Quantum Sci. Technol. +3, 030503 (2018). +7 + +[7] Wecker, D. Hastings, M. B., Troyer M.: Progress towards practical quan- +tum variational algorithms. Phys. Rev. A 92, 042303 (2015). +[8] Horodecki, R. Horodecki, P. Horodecki, M. Horodecki, K.: Quantum +entanglement. Rev. Mod. Phys. 81, 865 (2009). +[9] Shimony A.: Degree of entanglement. Ann. N.Y. Acad. Sci. 755, 675 +(1995). +[10] di Pierro, A. Mancini, S. Memarzadeh, L. Mengoni, R. Homological +analysis of multi-qubit entanglement. EPL (Europhys. Lett.) 123 30006 +(2018). +[11] Behera, B. K. Seth, S. Das, A. Panigrahi, P. K.: Demonstration of +entanglement purification and swapping protocol to design quantum re- +peater in IBM quantum computer. Quantum Information Processing 18, +108 (2019). +[12] Frydryszak, A. M. Samar, M. I. Tkachuk, V. M.: Quantifying geometric +measure of entanglement by mean value of spin and spin correlations with +application to physical systems. Eur. Phys. J. D 71, 233 (2017). +[13] Gnatenko, Kh. P. Susulovska, N. A.: Geometric measure of entangle- +ment of multi-qubit graph states and its detection on a quantum com- +puter. EPL (Europhys. Lett.) 136, 40003 (2021). +[14] Gnatenko, Kh. P. Tkachuk, V. M.: Entanglement of graph states of +spin system with Ising interaction and its quantifying on IBM’s quantum +computer. Phys. Lett. A. 396, 127248 (2021). +[15] IBM Quntum computing. +8 + diff --git a/e9E2T4oBgHgl3EQfbgdS/content/tmp_files/load_file.txt b/e9E2T4oBgHgl3EQfbgdS/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..38a2c201d6a6dc65ce8597ff4c9e2d64518a6a68 --- /dev/null +++ b/e9E2T4oBgHgl3EQfbgdS/content/tmp_files/load_file.txt @@ -0,0 +1,203 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfbgdS/content/2301.03885v1.pdf,len=202 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfbgdS/content/2301.03885v1.pdf'} +page_content='03885v1 [quant-ph] 10 Jan 2023 Evaluation of variational quantum states entanglement on a quantum computer by the mean value of spin Kh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfbgdS/content/2301.03885v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfbgdS/content/2301.03885v1.pdf'} +page_content=' Gnatenko1∗, Ivan Franko National University of Lviv, Professor Ivan Vakarchuk Department for Theoretical Physics, 12, Drahomanov St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfbgdS/content/2301.03885v1.pdf'} +page_content=', Lviv, 79005, Ukraine SoftServe Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfbgdS/content/2301.03885v1.pdf'} +page_content=', 2d Sadova St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfbgdS/content/2301.03885v1.pdf'} +page_content=', 79021 Lviv, Ukraine January 11, 2023 Abstract The geometric measure of entanglement of variational quantum states is studied on the basis of its relation with the mean value of spin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfbgdS/content/2301.03885v1.pdf'} +page_content=' We examine n-qubit quantum states prepared by a variational circuit with a layer formed by the rotational gates and two-qubit con- trolled phase gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfbgdS/content/2301.03885v1.pdf'} +page_content=' The variational circuit is a generalization of that used for preparing quantum Generative Adversarial Network states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfbgdS/content/2301.03885v1.pdf'} +page_content=' The entanglement of a qubit with other qubits in the variational quan- tum states is determined by the angles of rotational gates that act on the qubit and qubits entangled with it by controlled phase gates and also their parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfbgdS/content/2301.03885v1.pdf'} +page_content=' In the case of one layer variational circuit, the states can be associated with graphs with vertices representing qubits and edges corresponding to two-qubit gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfbgdS/content/2301.03885v1.pdf'} +page_content=' The geometric measure of entanglement of a qubit with other qubits in the quantum graph state depends on the properties of the vertex that represents it in the graph, namely it depends on the vertex degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfbgdS/content/2301.03885v1.pdf'} +page_content=' The dependence of the geometric measure of entanglement of variational quantum states on their parameters is quantified on IBM’s quantum computer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfbgdS/content/2301.03885v1.pdf'} +page_content=' ∗khrystyna.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfbgdS/content/2301.03885v1.pdf'} +page_content='gnatenko@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfbgdS/content/2301.03885v1.pdf'} +page_content='com 1 1 Introduction Variational quantum circuits have received much attention because of their wide usage in various quantum algorithms among them Quantum Machine Learning, error correction, algorithms for solving optimization problems, and many others (see, for instance, [1, 2, 3, 4, 5, 6, 7] and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfbgdS/content/2301.03885v1.pdf'} +page_content=' A critical and festinating resource in quantum computing is entanglement [8, 9, 10, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfbgdS/content/2301.03885v1.pdf'} +page_content=' The geometric measure of entanglement of quantum states has a clear definition that is based on the geometric idea of computing the minimal distance between the entangled state and a set of non-entangled states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfbgdS/content/2301.03885v1.pdf'} +page_content=' The measure of entanglement can be found as E(| ψ⟩) = min|ψs⟩ d2 F S(|ψs⟩, |ψ⟩), where | ψ⟩ is an entangled state, | ψs⟩ is a set of non-entangled states, dF S = � 1 − |⟨ψ|ψs⟩|2 is the Fubini-Study distance between the states | ψs⟩, | ψ⟩ [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfbgdS/content/2301.03885v1.pdf'} +page_content=' The procedure of finding the minimal distance requires a lot of computational resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfbgdS/content/2301.03885v1.pdf'} +page_content=' In the paper, [12] it was found that to detect the geometric measure of entanglement of a spin with an arbitrary quantum system in a pure state it is sufficient to calculate the mean value of the spin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfbgdS/content/2301.03885v1.pdf'} +page_content=' Namely, the geometric measure of entanglement of a spin with quantum system that are in a state | ψ⟩ = a |↑⟩ | Φ1⟩ + b |↓⟩ | Φ2⟩, can be found as E(| ψ⟩) = 1 2(1 − � ⟨σ⟩2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfbgdS/content/2301.03885v1.pdf'} +page_content=' (1) Here | Φ1⟩, | Φ2⟩ are states of a quantum system (⟨Φi | Φi⟩ = 1) that in general case can be nonorthogonal, a, b are constants, � ⟨σ⟩2 = � ⟨σx⟩2 + ⟨σy⟩2 + ⟨σz⟩2, (2) σx, σy, σz are Pauli matrixes [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfbgdS/content/2301.03885v1.pdf'} +page_content=' It is important to mention that relation (1) opens a possibility to quantify the geometric measure of entanglement of quantum states on a quantum devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfbgdS/content/2301.03885v1.pdf'} +page_content=' We study the geometric measure of entanglement of n-qubit quantum states prepared by a variational circuit with a layer formed by the rotational gates and controlled phase gates on the basis of analytical calculations and programming on IBM’s quantum computer and IBM’s quantum simulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfbgdS/content/2301.03885v1.pdf'} +page_content=' 2 2 Detection of the geometric measure of en- tanglement of variational quantum states on IBM’s quantum computer Let us consider variational quantum circuit with a layer formed by the rota- tional gates RY (θi,j) and controlled phase gates CP(ϕi,j) presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfbgdS/content/2301.03885v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfbgdS/content/2301.03885v1.pdf'} +page_content=' Note that in the particular case of ϕi,j = π the variational n+1-qubit form q[2] q[1] q[0] q[n] Ry ( 0,0) Ry ( 1,0) Ry ( 2,0) Ry ( n,0) Ry ( 0,1) Ry ( 1,1) Ry ( 2,1) Ry ( n,1) Ry ( 1,k) Ry ( 0,k) Ry ( 2,k) Ry ( n,k) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfbgdS/content/2301.03885v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfbgdS/content/2301.03885v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfbgdS/content/2301.03885v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfbgdS/content/2301.03885v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfbgdS/content/2301.03885v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfbgdS/content/2301.03885v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfbgdS/content/2301.03885v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfbgdS/content/2301.03885v1.pdf'} +page_content=' θ θ θ θ θ θ θ θ θ θ θ θ P ( 2,1) ϕ P ( 0,1) ϕ P ( 1,1) ϕ P ( 1,k) ϕ P ( 1,k) ϕ P ( 0,k) ϕ Figure 1: Variational quantum circuit of a depth k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfbgdS/content/2301.03885v1.pdf'} +page_content=' The layer formed by the rotational gates RY (θi,j) and controlled phase gates CP(ϕi,j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfbgdS/content/2301.03885v1.pdf'} +page_content=' corresponds to that used as a quantum generator in the quantum Generative Adversarial Networks (qGANs) [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfbgdS/content/2301.03885v1.pdf'} +page_content=' To detect the geometric measure of entanglement of a qubit with other qubits in a state prepared with a variational circuit we quantify the mean value of operators σx, σy, σz in the corresponding variational quantum state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfbgdS/content/2301.03885v1.pdf'} +page_content=' For instance, to evaluate the entanglement of qubit q[1] with other qubits in the state we consider quantum protocol presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfbgdS/content/2301.03885v1.pdf'} +page_content=' 2 q[2] q[1] q[0] q[n] Ry ( 0,0) Ry ( 1,0) Ry ( 2,0) Ry ( n,0) Ry ( 0,1) Ry ( 1,1) Ry ( 2,1) Ry ( n,1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfbgdS/content/2301.03885v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfbgdS/content/2301.03885v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfbgdS/content/2301.03885v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfbgdS/content/2301.03885v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfbgdS/content/2301.03885v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfbgdS/content/2301.03885v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfbgdS/content/2301.03885v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfbgdS/content/2301.03885v1.pdf'} +page_content=' Rα θ θ θ θ θ θ θ θ P ( 2,1) ϕ P ( 0,1) ϕ P ( 1,1) ϕ P ( 1,k) ϕ P ( 1,k) ϕ P ( 0,k) ϕ z c Figure 2: Quantum protocol for detection of the geometric measure of entan- glement of qubit q[1] with other qubits in a state prepared by the variational quantum circuit with the depth k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfbgdS/content/2301.03885v1.pdf'} +page_content=' 3 In the quantum protocol it is taken into account that ⟨ψ|σx|ψ⟩ = ⟨ ˜ψy|σz| ˜ψy⟩ = |⟨ ˜ψy|0⟩|2 − |⟨ ˜ψy|1⟩|2, ⟨ψ|σy|ψ⟩ = ⟨ ˜ψx|σz| ˜ψx⟩ = |⟨ ˜ψx|0⟩|2 − |⟨ ˜ψx|1⟩|2, where | ˜ψx⟩ = exp(−iπσx/4)|ψ⟩, | ˜ψy⟩ = exp(iπσy/4)|ψ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfbgdS/content/2301.03885v1.pdf'} +page_content=' Therefore to quantify the mean value ⟨σx⟩ the Rα = RY (−π/2) gate has to be applied to the state of a qubit before the measurement in the standard basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfbgdS/content/2301.03885v1.pdf'} +page_content=' To detect the mean value ⟨σy⟩ the Rα = RX(π/2) gate has to be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfbgdS/content/2301.03885v1.pdf'} +page_content=' Mean value of σz opera- tor can be calculated, using the results of measurement in the standard basis ⟨ψ|σz|ψ⟩ = |⟨ψ|0⟩|2 − |⟨ψ|1⟩|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfbgdS/content/2301.03885v1.pdf'} +page_content=' It is worth mentioning that quantum states generated by variational quan- tum circuit with the depth k = 1 are graph states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfbgdS/content/2301.03885v1.pdf'} +page_content=' These states can be associated with undirected graphs G(E, V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfbgdS/content/2301.03885v1.pdf'} +page_content=' The qubits are represented by vertices V in the graph and the edges E correspond to two-qubit gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfbgdS/content/2301.03885v1.pdf'} +page_content=' The geometric measure of entanglement of a qubit with other qubits in the graph state | G⟩ = � (a,b)∈E CPab(φ) � i RYi(θ) | 0⟩⊗n, (here RYi(θ), CPij(φ) are rotational and controlled phase gates acting on the states of qubits q[i], q[j]) depends on the graph properties [13, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfbgdS/content/2301.03885v1.pdf'} +page_content=' Namely, the entanglement of qubit q[l] with other qubits in n-qubit graph state depends on the the degree of vertex nl that represents it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfbgdS/content/2301.03885v1.pdf'} +page_content=' Performing analytical calculations, on the basis of the result (1), one finds [13] El = 1 2 − 1 2 � sin2 θ � cos2 φ 2 + sin2 φ 2 cos2 θ �nl + cos2 θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfbgdS/content/2301.03885v1.pdf'} +page_content=' (3) Using relation of the geometric measure of the entanglement with the mean value of spin (1), we obtain the entanglement of qubit q[1] with other qubits in the quantum Generative Adversarial Network state prepared by the n + 1 qubit variational quantum circuit Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfbgdS/content/2301.03885v1.pdf'} +page_content=' 1 with depth k = 1, and φi,1 = π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfbgdS/content/2301.03885v1.pdf'} +page_content=' It reads E(θ0,0, θ1,0, θ2,0) = 1 2 � 1 − � cos2 θ0,0 cos2 θ2,0 sin2 θ1,0 + cos2 θ1,0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfbgdS/content/2301.03885v1.pdf'} +page_content=' (4) Note that the entanglement of qubit q[1] with other qubits in the state de- pends only on the parameters θ0,0, θ1,0, θ2,0 of RY gates that act on the qubits entangled with q[1] by CZ01, CZ12 gates and do not depend on other parameters of the variational circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfbgdS/content/2301.03885v1.pdf'} +page_content=' To study the geometric measure of entanglement of the quantum Gener- ative Adversarial Network states on a quantum device we realized quantum 4 protocol Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfbgdS/content/2301.03885v1.pdf'} +page_content=' 2 on ibmq-manila and ibmq-qasm-simulator for k = 1, φi,1 = π, θi,0 = θ changing from 0 to 2π with the step π/32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfbgdS/content/2301.03885v1.pdf'} +page_content=' The results of calculations are presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfbgdS/content/2301.03885v1.pdf'} +page_content=' 3 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfbgdS/content/2301.03885v1.pdf'} +page_content=' Let us also examine the geometric measure of entanglement of qubit q[1] with other qubits in the quantum Generative Adversarial Network state gen- erated by variational circuit with depth k = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfbgdS/content/2301.03885v1.pdf'} +page_content=' In the case of θi,0 = π/2 the entanglement reads E(θ0,1, θ1,1, θ2,1) = 1 2(1 − 1 4| cos(θ0,1 + θ1,1 − θ2,1) + + cos(θ0,1 − θ1,1 + θ2,1) + cos(−θ0,1 + θ1,1 + θ2,1) + cos(θ0,1 + θ1,1 + θ2,1)|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfbgdS/content/2301.03885v1.pdf'} +page_content=' (5) We realized protocol Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfbgdS/content/2301.03885v1.pdf'} +page_content=' 2 for k = 2, θi,0 = π/2, θi,1 = θ1, φi,1 = φi,2 = π and θ1 changing from 0 to 2π with the step π/32 on quantum device ibmq-manila and ibmq-qasm-simulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfbgdS/content/2301.03885v1.pdf'} +page_content=' The results of quantum calculations and the theoretical result for the entanglement in this case E(θ1) = (1 − | cos3 θ1|)/2 are plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfbgdS/content/2301.03885v1.pdf'} +page_content=' 3 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfbgdS/content/2301.03885v1.pdf'} +page_content=' Quantum protocol Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfbgdS/content/2301.03885v1.pdf'} +page_content=' 2 was also implemented for, k = 2, θi,0 = θ0, θi,1 = π/2, φi,1 = φi,2 = π and θ0 changing from 0 to 2π with the step π/32 on the quantum computer and simulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfbgdS/content/2301.03885v1.pdf'} +page_content=' The results of quantum calculations and the analytical result E(θ0) = (2 − | sin(2θ0)|)/2 are presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfbgdS/content/2301.03885v1.pdf'} +page_content=' 3 (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfbgdS/content/2301.03885v1.pdf'} +page_content=' In addition, to detect the dependence of the geometric measure of entan- glement on the parameters of phase gates we realized quantum protocol Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfbgdS/content/2301.03885v1.pdf'} +page_content=' 2 for different values of φi,1 = φ changing from 0 to 4π with the step π/32 and θi,0 = π/2 (the depth is k = 1) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfbgdS/content/2301.03885v1.pdf'} +page_content=' 2 (a) and for different values of φi,1 = φ1, changing from 0 to 4π with the step π/32, θi,0 = π/2, φi,2 = π (the depth is k = 2) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfbgdS/content/2301.03885v1.pdf'} +page_content=' 2 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfbgdS/content/2301.03885v1.pdf'} +page_content=' The results of quantum calculations and the analytical results E(φ) = (1 − cos2(φ/2))/2, E(φ1) = (1 − | cos(φ1)(1 + cos(φ1))|)/2 are presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfbgdS/content/2301.03885v1.pdf'} +page_content=' 2, cases (a) and (b), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfbgdS/content/2301.03885v1.pdf'} +page_content=' Note that for k = 1, the results of quantum calculations are in good correspondence with the theo- retical ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfbgdS/content/2301.03885v1.pdf'} +page_content=' Not so good agreement of the results was obtained for circuits with the depth k = 2 because of accumulating of the gate errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfbgdS/content/2301.03885v1.pdf'} +page_content=' 3 Conclusions We have studied the geometric measure of entanglement of a variational quantum states prepared by rotational gates and entangled blocks formed by controlled phase gates Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfbgdS/content/2301.03885v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfbgdS/content/2301.03885v1.pdf'} +page_content=' The studies have been done on the basis 5 Figure 3: Results of calculations of entanglement of qubit q[1] with other qubits in variational quantum states (a) for different values of θi,0 = θ, φi,1 = π and k = 1 (left plot);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfbgdS/content/2301.03885v1.pdf'} +page_content=' (b) for different values of θi,1 = θ1 and θi,0 = π/2, φi,1 = φi,2 = π, k = 2 (middle plot);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfbgdS/content/2301.03885v1.pdf'} +page_content=' (c) for different values of θi,0 = θ0 and θi,1 = π/2, φi,1 = φi,2 = π, k = 2 (right plot), obtained on ibmq-manila (marked by black crosses), ibmq-qasm-simulator (marked by red circles), and analytical results (line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfbgdS/content/2301.03885v1.pdf'} +page_content=' Figure 4: Results of calculations of entanglement of qubit q[1] with other qubits in variational quantum states (a) for different values of φi,1 = φ, θi,0 = π/2, and the depth k = 1 (left plot);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfbgdS/content/2301.03885v1.pdf'} +page_content=' (b) for different values of φi,1 = φ1 and θi,0 = π/2, φi,2 = π, and the depth k = 2 (right plot), obtained on the on ibmq-manila (marked by black crosses), ibmq-qasm-simulator (marked by red circles), and analytical results (line) of the relation of the entanglement with the mean value of spin (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfbgdS/content/2301.03885v1.pdf'} +page_content=' In the case of variational circuit of one layer the variational quantum state can be associated with graph and the the geometric measure of entanglement of a qubit with other qubits in the variational quantum state with parameters θi,0 = θ, φi,0 = φ is related with the degree of vertex representing it in the graph (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfbgdS/content/2301.03885v1.pdf'} +page_content=' 6 The entanglement of a qubit with other qubits in the variational quan- tum state is determined by the angles of rotational gates in the variational circuit that act on the qubits entangled with it by controlled phase gates and their parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfbgdS/content/2301.03885v1.pdf'} +page_content=' The relation of the geometric measure of the entanglement with mean value of spin opens a possibility to quantify the entanglement on quantum devices, realizing quantum protocol Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfbgdS/content/2301.03885v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfbgdS/content/2301.03885v1.pdf'} +page_content=' On the basis of the relation the dependencies of the variational quantum states entanglement on the parameters of the variational quantum circuit have been calculated on IBM’s quantum computer ibmq-manila and ibmq-qasm-simulator Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfbgdS/content/2301.03885v1.pdf'} +page_content=' 3, 4.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfbgdS/content/2301.03885v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfbgdS/content/2301.03885v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfbgdS/content/2301.03885v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfbgdS/content/2301.03885v1.pdf'} +page_content=' 396, 127248 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfbgdS/content/2301.03885v1.pdf'} +page_content=' [15] IBM Quntum computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfbgdS/content/2301.03885v1.pdf'} +page_content=' 8' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfbgdS/content/2301.03885v1.pdf'} diff --git 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mode 100644 index 0000000000000000000000000000000000000000..013ab42c6cfc444c29b32864066b84b6fd600554 --- /dev/null +++ b/ftE2T4oBgHgl3EQfxwj2/content/tmp_files/2301.04115v1.pdf.txt @@ -0,0 +1,319 @@ +Sensing the Environment with 5G Scattered Signals +(5G-CommSense): A Feasibility Analysis +Sandip Jana +Dept. of Electrical Engineering +Indian Institute of Technology +Hyderabad, India +ee20resch11013@iith.ac.in +Amit Kumar Mishra +Dept. of Electrical Engineering +University of Cape Town +Cape Town, South Africa +akmishra@ieee.org +Mohammed Zafar Ali Khan +Dept. of Electrical Engineering +Indian Institute of Technology +Hyderabad, India +zafar@ee.iith.ac.in +Abstract—By making use of the sensors and AI (SensAI) +algorithms for a specialized task, Application Specific INstrumen- +tation (ASIN) framework uses less computational overhead and +gives a good performance. This work evaluates the feasibility of +the ASIN framework dependent Communication based Sensing +(CommSense) system using 5th Generation New Radio (5G NR) +infrastructure. Since our proposed system is backed up by 5G +NR infra, this system is termed as 5G-CommSense. In this paper, +we have used NR channel models specified by the 3rd Generation +Partnership Project (3GPP) and added white Gaussian noise +(AWGN) to vary the signal to noise ratio at the receiver. Finally, +from our simulation result, we conclude that the proposed system +is practically feasible. +Index Terms—ASIN, CommSense, 5G New Radio, Tapped +Delay Line, Cluster Delay Line +I. INTRODUCTION +Inspired by thousands of years of inter-species co-evolution +in nature, the CommSense system takes advantage of exist- +ing communication infrastructure1 to sense the environment. +CommSense system is based on ASIN framework [3], which +is dedicated to a specific task and might use crude resolution +sensors that require less computational power. A CommSense +system exploits the reference symbols used in communication +and senses the immediate environment, which may not require +high resolution, unlike conventional radar, where we compute +the precise range and Doppler of a target. In our paper, we +are using the upcoming 5G NR telecommunication standard +to explore the possibility of sensing using a 5G NR system. +To do this, we simulate the sensing results using the ASIN +framework for the 5G NR channel models [4]. +For 4th generation Long Term Evolution (4G LTE), there +were six channel models defined by International Telecom- +munication Union (ITU) [5], which were similar to TDL +models. Whereas for 5G NR, 3GPP defines [4] three different +cluster delay line (CDL) channel models (i.e. CDL-A, CDL-B, +and CDL-C) for non-line of sight (NLOS) scenarios and two +models (CDL-D and CDL-E) for line of sight (LOS) scenarios, +these CDL models are useful for Multiple Input Multiple +Output (MIMO) channel modeling, which is an integral part of +the NR. Till now CDL channels have not been used for sensing +1In recent literature [1], [2] the similar concept is being referred to as +Channel Impulse Response (CIR) or Impulse Radio (IR) based sensing. +applications. For simplified evaluations e.g. non MIMO evalu- +ations [4], 3GPP defines three tapped delay line (TDL) models +(i.e. TDL-A, TDL-B, and TDL-C) for NLOS and two models +(i.e. TDL-D and TDL-E) for LOS, but these five TDL channels +have more number of “Taps” as compared to ITU specified +LTE channels. In this paper, first, we have generated multiple +data points for each of the specified channel models. Then by +using channel equalization blocks our proposed system can +distinguish each of the five channel models from both the +CDL and TDL models. Moreover, we are varying the signal +to noise ratio (SNR) to verify the performance of the system. +The result we got from the simulation is as expected: each of +the clusters generated using different channel models is well +distinguishable by visual inspection. As the SNR increases, +we get more compact clusters and an increase in inter-cluster +separations. +The rest of the paper is arranged as follows: Section II will +give the necessary concepts of a CommSense system, TDL +model, and CDL model; then Section III gives the simulation +results and discusses the feasibility of the proposed system. +Finally, conclusion and future scope is given in Section IV. +II. THEORY +A. CommSense System +In most communication standards, to filter out the channel +modulation, we estimate the fading and non-ergodic wireless +CSI from UEs +ASIN framework to estimate the event +Phenomenological data for an event +Detection of event +Fig. 1. +CommSense flow-diagram to detect an event of interest +arXiv:2301.04115v1 [eess.SP] 10 Jan 2023 + +Fig. 2. An example Tapped Delay Line (TDL). ai and τi denotes scaling factor and delay of the signal corresponding to ith path. In (a), we are considering +three scattered paths from gNodeB to the user equipment (UE) multipath power profile is given. TDL representation of the given wireless channel is given +in (b) +channel by sending pilot symbols. As illustrated in Fig.1, +we extract the channel state information (CSI) from channel +equalization blocks at the receiver or user equipment (UE). In +the ASIN framework, first, we gather information about the +event of interest, then this stored data and extracted CSI is +used for training and pattern classification; then it gives the +prediction (i.e. if the event of interest has occurred or not). +B. Tapped Delay Line (TDL) +A “Tap” refers to a point in the delay line that introduces +a certain amount of delay and optionally scales the signal +[6]. To model the multi-path nature of the wireless channel +shown in Fig.2(a), the signals from each tap are summed. The +difference equation of the TDL from Fig.2(b) can be given by, +y(t) = a0s(t − τ0) + a1s(t − τ1) + a2s(t − τ2) +(1) +The corresponding transfer function from eq.1 is given as, +H(z) = a0z−τ0 + a1z−τ1 + a2z−τ2 +(2) +C. Cluster Delay Line (CDL) +Sub-paths in wireless channels might get clustered around a +particular delay [7]. A simple illustration of CDL is shown in +Fig.3, where three clusters are present and each cluster consists +of several scatterers. The power profile for each sub-path and +each cluster is shown in terms of the impulse response. +For our simulation we are using the 3GPP standard [4]; for +the TDL channel models normalized delay and scaling factor +for each tap, while for the CDL channels normalized delay, +power, angle of departure (AoD), angle of arrival (AoA), and +other angle measurements for each sub-paths are mentioned +in details. +III. SIMULATED RESULTS AND FEASIBILITY +At the carrier frequency of 4GHz, for both the TDL +and CDL model we generated 50 samples using MATLAB +for each of the five channel models according to the 3GPP +specifications; then we added AWGN noise to vary the SNR at +the receiver; we are considering SNR of 0dB, 10dB and 20dB. +After the pilot symbol is received at the UE, we use MMSE +Fig. 3. +An example of CDL with three clusters; θis represent Angles of +Departure (AoD) with respect to the Line of Sight Direction of Departure +based equalizer to estimate the channel. This high-dimensional +CSI is projected to two dimensions using PCA [8], then we +are using the SVM classifier to evaluate the accuracy. +Fig.4 shows the results by varying the SNR for five TDL +models and accuracy using SVM based classifier; it achieves +the accuracy of 96%, 97.3% and 100% at the SNR of 0dB, +10dB and 20dB respectively. Fig.5 shows the results for CDL +models; here we achieve an accuracy of 93.33%, 100% and +100% for the SNR of 0dB, 10dB and 20dB respectively. In +lower SNR, data points in CDL clusters are more dispersed +compared to TDL due to the presence of correlated multipaths, +but as the SNR increases the classification accuracy goes up +as expected. + +(（）） +s(t) +T。-(t,-To)(t2-ti) +a. +a, +(a +Base Station) +Ia,12 +(az, T,) +y(t) +(UE) +T. T, T2 +(Tapped Delay Line (TDL) representation) +Multi-path power profile) +(a) +(b)Intra-cluster power profile +(cluster - 3) +(sub-paths) +（（（ +0 +(cluster - l) +Inter-cluster +power profile) +(cluster - 2) +scatterers(a) SNR = 0dB +(b) SNR = 10dB +(c) SNR = 20dB +Fig. 4. Projection of data (generated using TDL channel models specified by 3GPP) onto the first two Principal Components’ direction using PCA and SNR +is varied to check the robustness of the proposed system. In (a) For each of the TDL channel model, SNR of each symbol is 0dB, in (b) and (c) SNR is 10dB +and 20dB respectively. By using SVM classifier, we get accuracy of 96% for (a), 97.3% for (b) and 100% for (c) +(a) SNR = 0dB +(b) SNR = 10dB +(c) SNR = 20dB +Fig. 5. Projection of CDL data onto first two Principal Components’ direction using PCA, In (a) For each of the CDL channel models, SNR of each symbol +is increased from (a) to (c). By using SVM classifier, we get sensing accuracy of 93.33% for (a), 100% for (b) and 100% for (c) +IV. CONCLUSION AND FUTURE WORK +Our work analyzed the feasibility of environment sensing +using 5G NR infrastructure, especially for the CDL channels +which was not done before. The performance from our simula- +tion shows that our proposed system is feasible using 5G NR. +Since this system can co-exist with the communication system, +it will be a cost-effective solution for sensing applications. +Here, we have considered one UE in the simulations. In the +future study, we will consider multiple UEs and develop an +optimal decision fusion algorithm. +ACKNOWLEDGMENT +This work was mainly supported by University Grants +Commission (UGC) and partly by project VAJRA, Department +of Science and Technology (DST), Govt. of India. +REFERENCES +[1] M. De Sanctis, A. Conte, T. Rossi, S. Di Domenico, and E. Cianca, “Cir- +based device-free people counting via uwb signals,” Sensors, vol. 21, +no. 9, p. 3296, 2021. +[2] X. Yang, W. Yin, L. Li, and L. Zhang, “Dense people counting using +ir-uwb radar with a hybrid feature extraction method,” IEEE Geoscience +and Remote Sensing Letters, vol. 16, pp. 30–34, 2019. +[3] A. K. Mishra, “Application specific instrumentation (asin): A bio-inspired +paradigm to instrumentation by fusing sensors and ai (sensai),” in +Proceedings of the International Conference on Data Science, Machine +Learning and Artificial Intelligence, ser. DSMLAI ’21’. +New York, NY, +USA: Association for Computing Machinery, 2021, p. 1–6. [Online]. +Available: https://doi.org/10.1145/3484824.3484921 +[4] “Tr 138 901 - v15.0.0 - 5g; study on channel model for frequencies from +from 0.5 to 100 ghz.” [Online]. Available: https://www.etsi.org/deliver/ +etsi TR/138900 138999/138901/15.00.00 60/tr 138901v150000p.pdf +[5] S. Sardar, A. K. Mishra, and M. Z. A. Khan, “Lte-commsense system and +its feasibility analysis,” in 2017 IEEE AFRICON, 2017, pp. 1564–1568. +[6] J. +O. +Smith, +Physical +Audio +Signal +Processing. +http://- +ccrma.stanford.edu/˜jos/pasp/, accessed 28 August, 2022, +online book, 2010 edition. +[7] F. Darbari, R. W. Stewart, and I. A. Glover, “Mimo channel modelling,” +in Signal Processing, S. Miron, Ed. +Rijeka: IntechOpen, 2010, ch. 5. +[Online]. Available: https://doi.org/10.5772/8530 +[8] S. Jana, A. K. Mishra, and M. Z. A. Khan, “Evaluation of visualiza- +tion algorithms for commsense system,” in 2022 IEEE 95th Vehicular +Technology Conference: (VTC2022-Spring), 2022, pp. 1–5. + +PCA +125 +TDL-A +TDL-B +TDL-C +100 +TDL-D +TDL-E +Second Principal Component +-25 +-50 +100 +100 +First Principal ComponentPCA +TDL-A +150 +TDL-B +TDL-C +TDL-D +TDL-E +100 +-50 +100 +-50 +100 +150 +200 +First Principal ComponentPCA +TDL-A +150 +TDL-B +TDL-C +TDL-D +TDL-E +-50 +100 +-50 +100 +150 +200 +First Principal ComponentPCA +CDL-A +CDL-B +Ot +CDL-C +CDL-D +CDL-E +★ +★ +★ +20 +100 +50 +100 +150 +200 +First Principal ComponentPCA +CDL-A +CDL-B +CDL-C +CDL-D +30 +CDL-E +Second Principal Component +10 +20 +100 +100 +200 +First Principal ComponentPCA +CDL-A +CDL-B +30 +CDL-C +CDL-D +CDL-E +-10 +20 +-200 +100 +100 +200 +OOE +First Principal Component \ No newline at end of file diff --git a/ftE2T4oBgHgl3EQfxwj2/content/tmp_files/load_file.txt b/ftE2T4oBgHgl3EQfxwj2/content/tmp_files/load_file.txt new file mode 100644 index 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Abstract—By making use of the sensors and AI (SensAI) algorithms for a specialized task, Application Specific INstrumen- tation (ASIN) framework uses less computational overhead and gives a good performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfxwj2/content/2301.04115v1.pdf'} +page_content=' This work evaluates the feasibility of the ASIN framework dependent Communication based Sensing (CommSense) system using 5th Generation New Radio (5G NR) infrastructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfxwj2/content/2301.04115v1.pdf'} +page_content=' Since our proposed system is backed up by 5G NR infra, this system is termed as 5G-CommSense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfxwj2/content/2301.04115v1.pdf'} +page_content=' In this paper, we have used NR channel models specified by the 3rd Generation Partnership Project (3GPP) and added white Gaussian noise (AWGN) to vary the signal to noise ratio at the receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfxwj2/content/2301.04115v1.pdf'} +page_content=' Finally, from our simulation result, we conclude that the proposed system is practically feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfxwj2/content/2301.04115v1.pdf'} +page_content=' Index Terms—ASIN, CommSense, 5G New Radio, Tapped Delay Line, Cluster Delay Line I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfxwj2/content/2301.04115v1.pdf'} +page_content=' INTRODUCTION Inspired by thousands of years of inter-species co-evolution in nature, the CommSense system takes advantage of exist- ing communication infrastructure1 to sense the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfxwj2/content/2301.04115v1.pdf'} +page_content=' CommSense system is based on ASIN framework [3], which is dedicated to a specific task and might use crude resolution sensors that require less computational power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfxwj2/content/2301.04115v1.pdf'} +page_content=' A CommSense system exploits the reference symbols used in communication and senses the immediate environment, which may not require high resolution, unlike conventional radar, where we compute the precise range and Doppler of a target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfxwj2/content/2301.04115v1.pdf'} +page_content=' In our paper, we are using the upcoming 5G NR telecommunication standard to explore the possibility of sensing using a 5G NR system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfxwj2/content/2301.04115v1.pdf'} +page_content=' To do this, we simulate the sensing results using the ASIN framework for the 5G NR channel models [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfxwj2/content/2301.04115v1.pdf'} +page_content=' For 4th generation Long Term Evolution (4G LTE), there were six channel models defined by International Telecom- munication Union (ITU) [5], which were similar to TDL models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfxwj2/content/2301.04115v1.pdf'} +page_content=' Whereas for 5G NR, 3GPP defines [4] three different cluster delay line (CDL) channel models (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfxwj2/content/2301.04115v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfxwj2/content/2301.04115v1.pdf'} +page_content=' CDL-A, CDL-B, and CDL-C) for non-line of sight (NLOS) scenarios and two models (CDL-D and CDL-E) for line of sight (LOS) scenarios, these CDL models are useful for Multiple Input Multiple Output (MIMO) channel modeling, which is an integral part of the NR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfxwj2/content/2301.04115v1.pdf'} +page_content=' Till now CDL channels have not been used for sensing 1In recent literature [1], [2] the similar concept is being referred to as Channel Impulse Response (CIR) or Impulse Radio (IR) based sensing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfxwj2/content/2301.04115v1.pdf'} +page_content=' applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfxwj2/content/2301.04115v1.pdf'} +page_content=' For simplified evaluations e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfxwj2/content/2301.04115v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfxwj2/content/2301.04115v1.pdf'} +page_content=' non MIMO evalu- ations [4], 3GPP defines three tapped delay line (TDL) models (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfxwj2/content/2301.04115v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfxwj2/content/2301.04115v1.pdf'} +page_content=' TDL-A, TDL-B, and TDL-C) for NLOS and two models (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfxwj2/content/2301.04115v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfxwj2/content/2301.04115v1.pdf'} +page_content=' TDL-D and TDL-E) for LOS, but these five TDL channels have more number of “Taps” as compared to ITU specified LTE channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfxwj2/content/2301.04115v1.pdf'} +page_content=' In this paper, first, we have generated multiple data points for each of the specified channel models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfxwj2/content/2301.04115v1.pdf'} +page_content=' Then by using channel equalization blocks our proposed system can distinguish each of the five channel models from both the CDL and TDL models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfxwj2/content/2301.04115v1.pdf'} +page_content=' Moreover, we are varying the signal to noise ratio (SNR) to verify the performance of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfxwj2/content/2301.04115v1.pdf'} +page_content=' The result we got from the simulation is as expected: each of the clusters generated using different channel models is well distinguishable by visual inspection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfxwj2/content/2301.04115v1.pdf'} +page_content=' As the SNR increases, we get more compact clusters and an increase in inter-cluster separations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfxwj2/content/2301.04115v1.pdf'} +page_content=' The rest of the paper is arranged as follows: Section II will give the necessary concepts of a CommSense system, TDL model, and CDL model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfxwj2/content/2301.04115v1.pdf'} +page_content=' then Section III gives the simulation results and discusses the feasibility of the proposed system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfxwj2/content/2301.04115v1.pdf'} +page_content=' Finally, conclusion and future scope is given in Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfxwj2/content/2301.04115v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfxwj2/content/2301.04115v1.pdf'} +page_content=' THEORY A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfxwj2/content/2301.04115v1.pdf'} +page_content=' CommSense System In most communication standards, to filter out the channel modulation, we estimate the fading and non-ergodic wireless CSI from UEs ASIN framework to estimate the event Phenomenological data for an event Detection of event Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfxwj2/content/2301.04115v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfxwj2/content/2301.04115v1.pdf'} +page_content=' CommSense flow-diagram to detect an event of interest arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfxwj2/content/2301.04115v1.pdf'} +page_content='04115v1 [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfxwj2/content/2301.04115v1.pdf'} +page_content='SP] 10 Jan 2023 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfxwj2/content/2301.04115v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfxwj2/content/2301.04115v1.pdf'} +page_content=' An example Tapped Delay Line (TDL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfxwj2/content/2301.04115v1.pdf'} +page_content=' ai and τi denotes scaling factor and delay of the signal corresponding to ith path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfxwj2/content/2301.04115v1.pdf'} +page_content=' In (a), we are considering three scattered paths from gNodeB to the user equipment (UE) multipath power profile is given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfxwj2/content/2301.04115v1.pdf'} +page_content=' TDL representation of the given wireless channel is given in (b) channel by sending pilot symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfxwj2/content/2301.04115v1.pdf'} +page_content=' As illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfxwj2/content/2301.04115v1.pdf'} +page_content='1, we extract the channel state information (CSI) from channel equalization blocks at the receiver or user equipment (UE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfxwj2/content/2301.04115v1.pdf'} +page_content=' In the ASIN framework, first, we gather information about the event of interest, then this stored data and extracted CSI is used for training and pattern classification;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfxwj2/content/2301.04115v1.pdf'} +page_content=' then it gives the prediction (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfxwj2/content/2301.04115v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfxwj2/content/2301.04115v1.pdf'} +page_content=' if the event of interest has occurred or not).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfxwj2/content/2301.04115v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfxwj2/content/2301.04115v1.pdf'} +page_content=' Tapped Delay Line (TDL) A “Tap” refers to a point in the delay line that introduces a certain amount of delay and optionally scales the signal [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfxwj2/content/2301.04115v1.pdf'} +page_content=' To model the multi-path nature of the wireless channel shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfxwj2/content/2301.04115v1.pdf'} +page_content='2(a), the signals from each tap are summed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfxwj2/content/2301.04115v1.pdf'} +page_content=' The difference equation of the TDL from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfxwj2/content/2301.04115v1.pdf'} +page_content='2(b) can be given by, y(t) = a0s(t − τ0) + a1s(t − τ1) + a2s(t − τ2) (1) The corresponding transfer function from eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfxwj2/content/2301.04115v1.pdf'} +page_content='1 is given as, H(z) = a0z−τ0 + a1z−τ1 + a2z−τ2 (2) C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfxwj2/content/2301.04115v1.pdf'} +page_content=' Cluster Delay Line (CDL) Sub-paths in wireless channels might get clustered around a particular delay [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfxwj2/content/2301.04115v1.pdf'} +page_content=' A simple illustration of CDL is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfxwj2/content/2301.04115v1.pdf'} +page_content='3, where three clusters are present and each cluster consists of several scatterers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfxwj2/content/2301.04115v1.pdf'} +page_content=' The power profile for each sub-path and each cluster is shown in terms of the impulse response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfxwj2/content/2301.04115v1.pdf'} +page_content=' For our simulation we are using the 3GPP standard [4];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfxwj2/content/2301.04115v1.pdf'} +page_content=' for the TDL channel models normalized delay and scaling factor for each tap, while for the CDL channels normalized delay, power, angle of departure (AoD), angle of arrival (AoA), and other angle measurements for each sub-paths are mentioned in details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfxwj2/content/2301.04115v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfxwj2/content/2301.04115v1.pdf'} +page_content=' SIMULATED RESULTS AND FEASIBILITY At the carrier frequency of 4GHz, for both the TDL and CDL model we generated 50 samples using MATLAB for each of the five channel models according to the 3GPP specifications;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfxwj2/content/2301.04115v1.pdf'} +page_content=' then we added AWGN noise to vary the SNR at the receiver;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfxwj2/content/2301.04115v1.pdf'} +page_content=' we are considering SNR of 0dB, 10dB and 20dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfxwj2/content/2301.04115v1.pdf'} +page_content=' After the pilot symbol is received at the UE, we use MMSE Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfxwj2/content/2301.04115v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfxwj2/content/2301.04115v1.pdf'} +page_content=' An example of CDL with three clusters;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfxwj2/content/2301.04115v1.pdf'} +page_content=' θis represent Angles of Departure (AoD) with respect to the Line of Sight Direction of Departure based equalizer to estimate the channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfxwj2/content/2301.04115v1.pdf'} +page_content=' This high-dimensional CSI is projected to two dimensions using PCA [8], then we are using the SVM classifier to evaluate the accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfxwj2/content/2301.04115v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfxwj2/content/2301.04115v1.pdf'} +page_content='4 shows the results by varying the SNR for five TDL models and accuracy using SVM based classifier;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfxwj2/content/2301.04115v1.pdf'} +page_content=' it achieves the accuracy of 96%, 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfxwj2/content/2301.04115v1.pdf'} +page_content='3% and 100% at the SNR of 0dB, 10dB and 20dB respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfxwj2/content/2301.04115v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfxwj2/content/2301.04115v1.pdf'} +page_content='5 shows the results for CDL models;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfxwj2/content/2301.04115v1.pdf'} +page_content=' here we achieve an accuracy of 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfxwj2/content/2301.04115v1.pdf'} +page_content='33%, 100% and 100% for the SNR of 0dB, 10dB and 20dB respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfxwj2/content/2301.04115v1.pdf'} +page_content=' In lower SNR, data points in CDL clusters are more dispersed compared to TDL due to the presence of correlated multipaths, but as the SNR increases the classification accuracy goes up as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfxwj2/content/2301.04115v1.pdf'} +page_content=' (（）） s(t) T。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfxwj2/content/2301.04115v1.pdf'} +page_content='-(t,-To)(t2-ti) a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfxwj2/content/2301.04115v1.pdf'} +page_content=' a, (a Base Station) Ia,12 (az, T,) y(t) (UE) T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfxwj2/content/2301.04115v1.pdf'} +page_content=' T, T2 (Tapped Delay Line (TDL) representation) Multi-path power profile) (a) (b)Intra-cluster power profile (cluster - 3) (sub-paths) （（（ 0 (cluster - l) Inter-cluster power profile) (cluster - 2) scatterers(a) SNR = 0dB (b) SNR = 10dB (c) SNR = 20dB Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfxwj2/content/2301.04115v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfxwj2/content/2301.04115v1.pdf'} +page_content=' Projection of data (generated using TDL channel models specified by 3GPP) onto the first two Principal Components’ direction using PCA and SNR is varied to check the robustness of the proposed system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfxwj2/content/2301.04115v1.pdf'} +page_content=' In (a) For each of the TDL channel model, SNR of each symbol is 0dB, in (b) and (c) SNR is 10dB and 20dB respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfxwj2/content/2301.04115v1.pdf'} +page_content=' By using SVM classifier, we get accuracy of 96% for (a), 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfxwj2/content/2301.04115v1.pdf'} +page_content='3% for (b) and 100% for (c) (a) SNR = 0dB (b) SNR = 10dB (c) SNR = 20dB Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfxwj2/content/2301.04115v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfxwj2/content/2301.04115v1.pdf'} +page_content=' Projection of CDL data onto first two Principal Components’ direction using PCA, In (a) For each of the CDL channel models, SNR of each symbol is increased from (a) to (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfxwj2/content/2301.04115v1.pdf'} +page_content=' By using SVM classifier, we get sensing accuracy of 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfxwj2/content/2301.04115v1.pdf'} +page_content='33% for (a), 100% for (b) and 100% for (c) IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfxwj2/content/2301.04115v1.pdf'} +page_content=' CONCLUSION AND FUTURE WORK Our work analyzed the feasibility of environment sensing using 5G NR infrastructure, especially for the CDL channels which was not done before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfxwj2/content/2301.04115v1.pdf'} +page_content=' The performance from our simula- tion shows that our proposed system is feasible using 5G NR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfxwj2/content/2301.04115v1.pdf'} +page_content=' Since this system can co-exist with the communication system, it will be a cost-effective solution for sensing applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfxwj2/content/2301.04115v1.pdf'} +page_content=' Here, we have considered one UE in the simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfxwj2/content/2301.04115v1.pdf'} +page_content=' In the future study, we will consider multiple UEs and develop an optimal decision fusion algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfxwj2/content/2301.04115v1.pdf'} +page_content=' ACKNOWLEDGMENT This work was mainly supported by University Grants Commission (UGC) and partly by project VAJRA, Department of Science and Technology (DST), Govt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfxwj2/content/2301.04115v1.pdf'} +page_content=' of India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfxwj2/content/2301.04115v1.pdf'} +page_content=' REFERENCES [1] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfxwj2/content/2301.04115v1.pdf'} +page_content=' De Sanctis, A.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfxwj2/content/2301.04115v1.pdf'} +page_content=' DSMLAI ’21’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfxwj2/content/2301.04115v1.pdf'} +page_content=' New York, NY, USA: Association for Computing Machinery, 2021, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfxwj2/content/2301.04115v1.pdf'} +page_content=' 1–6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfxwj2/content/2301.04115v1.pdf'} +page_content=' [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfxwj2/content/2301.04115v1.pdf'} +page_content=' Available: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfxwj2/content/2301.04115v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfxwj2/content/2301.04115v1.pdf'} +page_content='1145/3484824.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfxwj2/content/2301.04115v1.pdf'} +page_content='3484921 [4] “Tr 138 901 - v15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfxwj2/content/2301.04115v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfxwj2/content/2301.04115v1.pdf'} +page_content='0 - 5g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfxwj2/content/2301.04115v1.pdf'} +page_content=' study on channel model for frequencies from from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE2T4oBgHgl3EQfxwj2/content/2301.04115v1.pdf'} +page_content='5 to 100 ghz.” [Online].' metadata={'source': 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a/hdAzT4oBgHgl3EQf4f6J/content/tmp_files/2301.01845v1.pdf.txt b/hdAzT4oBgHgl3EQf4f6J/content/tmp_files/2301.01845v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..ea32a770b7f6343e03295e1a1cf5a9931fdb2b40 --- /dev/null +++ b/hdAzT4oBgHgl3EQf4f6J/content/tmp_files/2301.01845v1.pdf.txt @@ -0,0 +1,6329 @@ +arXiv:2301.01845v1 [math.AP] 4 Jan 2023 +SIGN-CHANGING BUBBLE TOWER SOLUTIONS FOR SINH-POISSON TYPE +EQUATIONS ON PIERCED DOMAINS +PABLO FIGUEROA +Abstract. For asymmetric sinh-Poisson type problems with Dirichlet boundary condition arising +as a mean field equation of equilibrium turbulence vortices with variable intensities of interest in +hydrodynamic turbulence, we address the existence of sign-changing bubble tower solutions on a +pierced domain Ωǫ := Ω \ B(ξ, ǫ), where Ω is a smooth bounded domain in IR2 and B(ξ, ǫ) is a ball +centered at ξ ∈ Ω with radius ǫ > 0. Precisely, given a small parameter ρ > 0 and any integer m ≥ 2, +there exist a radius ǫ = ǫ(ρ) > 0 small enough such that each sinh-Poisson type equation, either in +Liouville form or mean field form, has a solution uρ with an asymptotic profile as a sign-changing +tower of m singular Liouville bubbles centered at the same ξ and with ǫ(ρ) → 0+ as ρ approaches to +zero. +1. Introduction +Let Ω be a smooth bounded domain in IR2. Given ǫ > 0 and ξ ∈ Ω, define Ωǫ := Ω\B(ξ, ǫ), a pierced +domain, where B(ξ, ǫ) is a ball centered at ξ with radius ǫ. Inspired by results in [1, 13, 17, 20, 30], +we are interested in constructing sign-changing bubble tower solutions to sinh-Poisson type equations, +either in Liouville form or mean field form, with variable intensities and Dirichlet boundary conditions +on a pierced domain Ωǫ. Precisely, on one hand, we consider the following problem in Liouville form +� +−∆u = ρ(V0(x)eu − νV1(x)e−τu) +in Ωǫ +u = 0 +on ∂Ωǫ +, +(1.1) +and, on the other hand, we also study the problem in mean field form + + + +−∆u = λ0V0(x)eu +� +Ωǫ V0eu − λ1τV1(x)e−τu +� +Ωǫ V1e−τu +in Ωǫ +u = 0 +on ∂Ωǫ +, +(1.2) +where ρ > 0 is small, λ0, λ1 > 0, V0, V1 > 0 are smooth potentials in Ω, τ > 0, ǫ > 0 is a small number +and ν ≥ 0. Our aim is to construct for each problem a family of solutions uρ for a suitable choice +of ǫ = ǫ(ρ), with an asymptotic profile as a sum of positive and negative singular Liouville bubbles +centered at the same point ξ as ρ → 0, on the line of [20, 30]. +These equations and related ones have attracted a lot of attention in recent years due to its relevance +in the statistical mechanics description of 2D-turbulence, as initiated by Onsager [28]. Precisely, in +this context Caglioti, Lions, Marchioro, Pulvirenti [6] and Sawada, Suzuki [35] derive the following +equation: +− ∆u = λ +� +[−1,1] +αeαu +� +Ω +eαudxdP(α) +in Ω, +u = 0 on ∂Ω, +(1.3) +where Ω is a bounded domain in R2, u is the stream function of the flow, λ > 0 is a constant related +to the inverse temperature and P is a Borel probability measure in [−1, 1] describing the point-vortex +intensities distribution. We observe that (1.3) is obtained under a deterministic assumption on the +distribution of the vortex circulations. +Date: January 6, 2023. +2020 Mathematics Subject Classification. 35B44, 35J25, 35J60. +Key words and phrases. sinh-Poisson type equation, pierced domain, tower of bubbles. +1 + +2 +P. FIGUEROA +Equation (1.3) includes several well-known problems depending on a suitable choice of P. For instance, +if P = δ1 is concentrated at 1, then (1.3) corresponds to the classical mean field equation +− ∆u = λ +eu +� +Ω +eudx +in Ω, +u = 0 on ∂Ω. +(1.4) +Since there are plenty of results in the literature concerning (1.4), let us just quote [2, 7, 8, 9, 12, 26]. +When P = σδ1 + (1 − σ)δ−τ with τ ∈ [−1, 1] and σ ∈ [0, 1], equation (1.3) becomes +− ∆u = λ +� +σ +eu +� +Ω +eudx − (1 − σ)τ +e−τu +� +Ω +e−τudx +� +in Ω, +u = 0 on ∂Ω. +(1.5) +Setting λ0 = λσ, λ1 = λ(1−σ) and V0 = V1 = 1 problem (1.5) can be rewritten as (1.2) replacing Ωǫ by +Ω. If τ = 1 and V0 = V1 ≡ 1 problem (1.2) reduces to mean field equation of the equilibrium turbulence, +see [5, 21, 24, 27, 31] or its related sinh-Poisson version, see [3, 4, 20, 23, 25], which have received a +considerable interest in recent years. Recently, sign-changing solutions have been constructed in Ω for +the sinh-Poisson equation with Robin boundary condition in [19]. +Concerning results for τ > 0, Pistoia and Ricciardi built in [29] sequences of blowing-up solutions +to (1.2) (in Ω instead Ωǫ) when λ0, λ1τ 2 are close to 8π. Ricciardi and Takahashi in [32] provided +a complete blow-up picture for solution sequences of (1.2) and successively in [33] Ricciardi et al. +constructed min-max solutions when λ0 → 8π+ and λ1 → 0 on a multiply connected domain (in this +case the nonlinearity e−τu may be treated as a lower-order term with respect to the main term eu). A +blow-up analysis and some existence results are obtained when τ > 0 in a compact Riemann surface +in [22, 34]. Bubbling solutions in a compact Riemann surface has been recently constructed in [18]. +On the other hand, on pierced domains, Ahmedou and Pistoia in [1] proved that on Ωǫ, there exists +a solution to the classical mean field equation (1.4) which blows-up at ξ as ǫ → 0 for any λ > 8π +(extra symmetric conditions are required when λ ∈ 8πN). In [13] the authors constructed a family of +solutions to the mean field equation with variable intensities (1.2) blowing-up positively and negatively +at ξ1, . . . , ξm1 and ξm1+1, . . . , ξm, respectively, as ǫ1, . . . , ǫm → 0 on a pierced domain with several +holes (Ωǫ is replaced by in Ω \ ∪m +i=1B(ξi, ǫi) ), in the super-critical regime λ0 > 8πm1 and λ1τ 2 > +8π(m − m1) with m1 ∈ {0, 1, . . ., m}. Recently, in the same spirit of [13], the author in [17] addressed +the sinh-Poisson type equation with variable intensities (1.1) on a pierced domain with several holes +Ω \ ∪m +i=1B(ξi, ǫi). Equation (1.1) is related, but not equivalent, to problem (1.2) by using the change +ρ = +λ0 +� +Ωǫ V0eu +and +ρν = +λ1τ +� +Ωǫ V1e−τu . +To the extent of our knowledge, there are by now just few results concerning non-simple blow-up for +sinh-Poisson type problems. Precisely, sign-changing solutions with non-simple blow-up has been built +in [15] for the Neumann sinh-Poisson equation. Grossi and Pistoia built in [20] a sign-changing bubble +tower solutions for the sinh-Poisson version (τ = 1) in a symmetric domain Ω with respect to a fixed +point ξ ∈ Ω. After that, Pistoia and Ricciardi in [30] extend this construction to a sinh-Poisson type +equation with asymmetric exponents (τ ̸= 1) under a symmetric assumption on Ω depending on either +τ ∈ Q or τ /∈ Q. In both situations [20, 30] the number of bubbles can be arbitrary large. +A matter of interest to us is whether do there exist sign-changing bubble tower solutions to (1.1) +for small values of ρ or to (1.2) for some values of the parameters λ0, λ1, τ > 0 on a pierced domain +Ωǫ, with bubbles centered at ξ. Our first result in this direction without symmetry assumptions reads +as follows. +Theorem 1.1. Let m ≥ 2 be a positive integer. There exists ρ0 > 0 such that for all 0 < ρ < ρ0 there +is ǫ = ǫ(ρ) small enough such that problem (1.1) has a sign-changing solution uρ in Ωǫ blowing-up at +ξ in the sense that +uρ = (−1)m+1 2π +τ ν(m) (αm + 2)G(·, ξ) + o(1) +(1.6) + +SIGN-CHANGING BUBBLE TOWER SOLUTIONS ON PIERCED DOMAINS +3 +locally uniformly in ¯Ω \ {ξ} as ρ → 0+. +Here, for simplicity, we denote +ν(i) = 1 + (−1)i +2 += +� +0 +if i is odd +1 +if i is even +and +σ(i) = 1 − (−1)i +2 += +� +1 +if i is odd +0 +if i is even , +(1.7) +αm is given by (2.4) with i = m and G(x, y) = − 1 +2π log |x − y| + H(x, y) is the Green’s function of −∆ +in Ω, where the regular part H is a harmonic function in Ω so that H(x, y) = +1 +2π log |x − y| on ∂Ω. +Nevertheless, the latter result may not tell us whether (1.2) has sign-changing bubble tower solutions +for some values of the parameters λ0, λ1, τ > 0. Therefore, we perform directly to problem (1.2) a +similar procedure. Conversely, Theorem 1.2 below may not tell us whether (1.1) has sign-changing +bubble tower solutions for all small ρ > 0. Assume that λ0 and λ1 decompose for some α1 > 2 (see +(2.3)), as either +λ0 = 2πm +� +α1 + (m − 2) +� +1 + 1 +τ +�� +, +λ1τ 2 = 2πm [α1τ + m(1 + τ)] , +if m even +(1.8) +or +λ0 = 2π(m+ 1) +� +α1 + (m − 1) +� +1 + 1 +τ +�� +, λ1τ 2 = 2π(m− 1) [α1τ + (m − 1)(1 + τ)] , if m odd. (1.9) +In particular, we choose α1 > 2 and α1 /∈ 2IN if τ = 1. +Our second result (without symmetry +assumptions) is the following +Theorem 1.2. If either (1.8) or (1.9) holds for a positive integer m ≥ 2, then there exists a radius +ǫ > 0 small enough such that problem (1.2) has a sign-changing solution uǫ in Ωǫ blowing-up at ξ in +the sense of (1.6) locally uniformly in ¯Ω \ {ξ} as ǫ → 0+. +Our solutions correspond to a superposition of highly concentrated vortex configurations of alternating +orientation around the hole B(ξ, ǫ) and they extend some known results [20, 30] for symmetric domains. +We also point out that a delicate point in the paper concerns the linear theory developed in Section 5. +A bit more complicated analysis is necessary in comparison with linear theories developed in previous +works [1, 11, 13, 14, 16, 20, 30]. +Without loss of generality, we shall assume in the rest of the paper that ξ = 0 ∈ Ω, so that Ωǫ = +Ω \ B(0, ǫ) where ǫ > 0 is small and ν = 1, since we can replace νV2 by V2. However, we need the +presence of ν when we compare (1.1) with equation (1.2). +Finally, we point out some comments about the proofs of the theorems. Following the ideas presented +in [20, 30] about bubble tower solutions for sinh-Poisson type equations and in [1, 13, 16] about +construction of solutions on pierced domains, we find a solution uρ using a perturbative approach, +precisely, we look for a solution of (1.1) as +uρ = U + φ, +(1.10) +where U is a suitable ansatz built using the projection operator Pǫ onto H1 +0(Ωǫ)(see (2.1)) and φ ∈ +H1 +0(Ωǫ) is a small remainder term. +Letting wδ,α(x) = log +2α2δα +(δα+|x|α)2 be a solution to the singular +Liouville equation +∆u + |x|α−2eu = 0 +in IR2, +� +IR2 |x|α−2eu < +∞, +the ansatz U is defined as follows +U(x) = +� +i odd +Pǫwi(x) − 1 +τ +� +i even +Pǫwi(x) = +m +� +i=1 +(−1)i+1 +τ ν(i) +Pǫwi(x), +(1.11) +where wj := wδj,αj for j = 1, . . . , m. A careful choice of the parameters δj’s, αi’s and the radius ǫ, +depending on ρ > 0, is made in section 2 (see (2.2), (2.3)-(2.4) and (2.21)) in order to make U be a +good approximated solution. Indeed, the error term R for (1.1) given by +R = ∆U + ρ(V0(x)eU − V1(x)e−τU) +(1.12) + +4 +P. FIGUEROA +is small in Lp-norm for p > 1 close to 1 (see Lemma 3.1). A linearization procedure around U leads +us to re-formulate (1.1) in terms of a nonlinear problem for φ (see equation (3.13)). We will prove +the existence of such a solution φ to (3.13) by using a fixed point argument, thanks to some estimates +in subsection 3.2 (see (3.20) and (3.21)). The corresponding solution uρ in (1.10) behaves as a sign- +changing tower of m singular Liouville bubbles thanks to the asymptotic properties of its main order +term U (see (2.23) in Lemma 2.6). In Section 5 we will prove the invertibility of the linear operator +naturally associated to the problem (see (3.14)) stated in Proposition 3.1. To conclude Theorem 1.2, +the same procedure with the same ansatz is performed to equation (1.2), assuming (1.8)-(1.9) and +ǫ = ǫ(ρ), where the error term is given by +R = ∆U + λ0 +V0(x)eU +� +Ωǫ V0eU − λ1τ V1(x)e−τU +� +Ωǫ V1e−τU . +(1.13) +2. Approximation of the solution +In this section we shall make a choice of the parameters αi’s, δj’s and ǫ = ǫ(ρ) in order to make U a +good approximation. Introduce the function Pǫw as the unique solution of +� ∆Pǫw = ∆w +in Ωǫ +Pǫw = 0, +on ∂Ωǫ. +(2.1) +For simplicity, we will denote h0 = H(0, 0). We have the following asymptotic expansion of Pwδ,α as +δ → 0, as shown in [1, Lemma 2.1] (see also [13, Lemma 2.1]): +Lemma 2.1. The function Pǫwδ,α satisfies +Pǫwδ,α(x) = wδ,α(x) − log(2α2δα) + 4παH(x, 0) − γα +δ,ǫG(x, 0) + O +� +δα + +�ǫ +δ +�α ++ +� +1 + +���log δ +log ǫ +��� +� +ǫ +� +, +uniformly in Ωǫ, where γα +δ,ǫ is given by γα +δ,ǫ = −2α log δ+4παh0 +− 1 +2π log ǫ+h0 +. In particular, there holds +Pǫwδ,α(x) = [4πα − γα +δ,ǫ]G(x, 0) + O +� +δα + +� ǫ +δ +�α ++ +� +1 + +���log δ +log ǫ +��� +� +ǫ +� +as ǫ → 0 locally uniformly in Ω \ {0}. +Given m ∈ IN with m ≥ 2, consider δj > 0 and αj > 2, j = 1, . . . , m, so that our approximating +solution U is defined by (1.11), parametrized by δj’s and α′ +is with δj = δj(ρ, α1) (it also depends on τ, +h0, V0(0) and V1(0)), where ν(i) is defined in (1.7) and Pǫ the projection operator defined by (2.1) for +a suitable choice of ǫ. In order to have a good approximation, for any i = 1, . . . , m we will assume that +δαi +i += diρβi, +(2.2) +for small ρ > 0, where αi’s, βi’s, and di’s will be specified below. We choose +α1 > 2, +with +α1 ∈ + + + + + + + + + + + + + + + +(m−2)/2 +� +k=0 +� +2IN − 4k +τ +�c � � m/2 +� +k=1 +�2 +τ IN − 4k + 2 +�c� +if m is even +(m−1)/2 +� +k=0 +� +2IN − 4k +τ +�c � � (m−1)/2 +� +k=1 +�2 +τ IN − 4k + 2 +�c� +if m is odd +, +(2.3) +and for i ≥ 2 +αi = + + + + + +α1 + 2(i − 1) + 2(i − 1) +τ +if i is odd +α1τ + 2(i − 1)τ + 2(i − 1) +if i is even +. +(2.4) +Note that +αi = +� +α1 + 2i − 2 + 2i − 2 +τ +� +τ ν(i), +for i ≥ 2 +(2.5) +and αi > 0 for all i = 1, . . . , m. Furthermore, several identities and properties of αi’s, βi’s, di’s and ǫ +will be proven in order to have a good approximation. From de definition (2.4), it is readily checked + +SIGN-CHANGING BUBBLE TOWER SOLUTIONS ON PIERCED DOMAINS +5 +that {α2k+1}k and {α2k}k are increasing in k. Since α1 > 2 and α2 = α1τ + 2τ + 2 > 2, it follows +that αi > 2 for all i = 1, . . . , m. Notice that all these sets are countable : 2IN − 4k +τ for k = 0, . . . , m−2 +2 +and 2 +τ IN − 4k + 2 for k = 1, . . . , m +2 with m even; 2IN − 4k +τ for k = 0, . . . , m−1 +2 +and 2 +τ IN − 4k + 2 for +k = 1, . . . , m−1 +2 +with m odd. Hence, its union is also countable set and the complement of its union is +dense in IR. Therefore, there exist α1 ∈ IR satisfying (2.3). +Lemma 2.2. If αi’s are given by (2.4) then +αj+1 = (αj + 2)τ(−1)j+1 + 2 = + + + + + +(αj + 2)τ + 2 +if j is odd +αj + 2 +τ ++ 2 +if j is even +(2.6) +and +j +� +i=1 +(−1)i+1 +τ ν(i) +αi = +� +−j − j +τ +if j is even +α1 + j − 1 + j−1 +τ +if j is odd +(2.7) +holds for all j ≥ 1. If α1 satisfies (2.3) then αi /∈ 2IN for all i = 1, . . . , m. +Proof: Assume first that i is odd. Then, i + 1 is even, from (2.4) for i odd, αi = α1 + 2i − 2 + 2i−2 +τ +and direct computations lead us to obtain that +(αi + 2)τ + 2 = +� +α1 + 2i + 2i − 2 +τ +� +τ + 2 = (α1 + 2i)τ + 2i. +On the other hand, from (2.4) for i + 1 even, we find that αi+1 = α1τ + 2iτ + 2i, so that αi+1 = +(αi + 2)τ + 2. Direct computations in case i is even allows us to conclude (2.6). +From the choice of αi’s and (2.5), it follows that +j +� +i=1 +(−1)i+1 +τ ν(i) +αi = +j +� +i=1 +(−1)i+1 +� +α1 + 2i − 2 + 2i − 2 +τ +� += α1 +j +� +i=1 +(−1)i+1 + 2 +j +� +i=1 +(−1)i+1(i − 1) + 2 +τ +j +� +i=1 +(−1)i+1(i − 1) +and we conclude (2.7), in view of +j +� +i=1 +(−1)i+1(i − 1) = +j +� +i=1 +(−1)i+1i + +j +� +i=1 +(−1)i+1 = +� +− j +2 + 0 +if j is even +j+1 +2 +− 1 +if j is odd . +Finally, assume that m is even. +If α1 satisfies (2.3) then we have that α1 ∈ +� +2IN − 4k +τ +�c, k = +0, 1, . . ., m−2 +2 +and α1 ∈ +� 2 +τ IN−4k+2 +�c, k = 1, 2, . . . , m +2 . That is to say, α1 + 4k +τ /∈ 2IN, k = 0, 1 . . . , m−2 +2 +and τ(α1 + 4k − 2) /∈ 2IN, k = 1, . . . , m +2 . Therefore, α2k+1 = α1 + 4k + 4k +τ +/∈ 2IN, k = 0, . . . , m−2 +2 +and +α2k = α1τ +(4k−2)τ +4k−2 /∈ 2IN, k = 1, . . . , m +2 . Similar argument lead us to conclude that αi /∈ 2IN +for all i = 1, . . . , m if m is odd. +□ +In particular, we have that α1 > 2, α2 = (α1 + 2)τ + 2, α3 = +α2+2 +τ ++ 2, α4 = (α3 + 2)τ + 2, +α5 = α4+2 +τ ++ 2, . . . Furthermore, we have that +4π +m +� +i=1 +(−1)i+1 +τ ν(i) +αi − 2π(α1 − 2) = (−1)m+1 2π +τ ν(m) (αm + 2). +(2.8) +Now, we define βi as follows, if m is even then +βl = τ ν(l) +� +m − l + m − l + 1 +τ +� += + + + + + +(m − l)τ + m − l + 1, +if l is even +m − l + m − l + 1 +τ +if l is odd +, +(2.9) + +6 +P. FIGUEROA +for l = 1, 2, . . . , m. If m is odd then +βl = τ ν(l) +� +m − l + 1 + m − l +τ +� += + + + + + +(m − l + 1)τ + m − l, +if l is even +m − l + 1 + m − l +τ +if l is odd +(2.10) +for l = 1, 2, . . . , m. In any case, either m is odd or even, it is readily checked that βm = 1 and βi > 0 +for all i = 1, . . . , m. Note that we can rewrite +βl = τ ν(l) +�m − l + 1 +τ ν(m) ++ +m − l +τ ν(m+1) +� +. +(2.11) +Hence, we shall prove the following useful properties from the definition (2.9) and (2.10). Note that +1 − σ(i) = ν(i) = σ(i + 1) and σ(i) = 1 − ν(i) = ν(i + 1). +Lemma 2.3. If βi’s are given by either (2.9) or (2.10) then for any l = 2, . . . , m +βl = + + + + + +τβl−1 − τ − 1, +if l is even +βl−1 +τ +− 1 − 1 +τ +if l is odd +(2.12) +and for any l = 1, . . . , m − 1 +βl − 1 +2τ ν(l) = +m +� +j=l+1 +(−1)j+1 +τ ν(j) +βj. +(2.13) +Furthermore, it holds that βl +αl < βl−1 +αl−1 for any l = 2, . . . , m, so that, δi +δj → 0 as ρ → 0 for any i < j. +Proof: First, assume that m is even. So, βl’s are given by (2.9). If l is even, then we have that l − 1 +is odd, so that, +βl−1 = m − l + 1 + m − l + 2 +τ +and +τβl−1 − τ − 1 = (m − l)τ + τ + m − l + 2 − τ − 1 = βl. +Next, assume that l is odd (still m is even). Similarly as above, it follows that βl−1 +τ +− 1 − 1 +τ = βl. +Arguing in the same way for βl’s given by (2.10) when m is odd, we conclude (2.12). +Now, for any m, we shall prove that +βl = + + + + + + + + + + + + + +1 + 2 +m−l +� +i=1 +(−1)i+1τ σ(i)βl+i, +if l is even +1 + 2 +m−l +� +i=1 +(−1)i+1 +τ σ(i) +βl+i +if l is odd +. +(2.14) +Assume that m is even. If l is even then ν(i + l) = 1 − σ(i) and from (2.11) we get that +m−l +� +i=1 +(−1)i+1τ σ(i)βl+i = +m−l +� +i=1 +(−1)i+1τ +�βl +τ − +� +1 + 1 +τ +� +i +� += βl +m−l +� +i=1 +(−1)i+1 − (τ + 1) +m−l +� +i=1 +(−1)i+1i +and (2.14) follows, in view of m − l is even, �m−l +i=1 (−1)i+1 = 0 and �m−l +i=1 (−1)i+1i = − m−l +2 . Next, +assume that l is odd (still m is even) so that ν(i + l) = σ(i) and similarly as above +m−l +� +i=1 +(−1)i+1 +τ σ(i) +βl+i = +m−l +� +i=1 +(−1)i+1 +� +βl − +� +1 + 1 +τ +� +i +� += βl +m−l +� +i=1 +(−1)i+1 − +� +1 + 1 +τ +� m−l +� +i=1 +(−1)i+1i = βl − 1 +2 +, +in view of m − l is odd, �m−l +i=1 (−1)i+1 = 1 and �m−l +i=1 (−1)i+1i = m−l+1 +2 +. Arguing in the same way for +βl’s given by (2.10) when m is odd we conclude (2.14). Now, from (2.14) taking j = l + i we have that + +SIGN-CHANGING BUBBLE TOWER SOLUTIONS ON PIERCED DOMAINS +7 +if l is even then +βl = 1 + 2 +m−l +� +i=1 +(−1)i+1τ σ(i)βl+i = 1 + 2 +m +� +j=l+1 +(−1)j+1 +τ +τ ν(j) βi +in view of σ(j − l) = σ(j) = 1 − ν(j), and if l is odd then +βl = 1 + 2 +m−l +� +i=1 +(−1)i+1 +τ σ(i) +βl+i = 1 + 2 +m +� +j=l+1 +(−1)j +τ ν(j) βi, +in view of σ(j − l) = ν(j). Thus, we deduce (2.13). +Now, we know that 0 < αl−1 + αl−1τ + 2βl−1 + 2βl−1τ for any l. Hence, for l even we have that +βl = τβl−1 − τ − 1 and αl = (αl−1 + 2)τ + 2 by using (2.12) and (2.6) respectively, so that +αl−1[τβl−1 − τ − 1] < [(αl−1 + 2)τ + 2]βl−1 ⇐⇒ αl−1βl < αlβl−1 +By using (2.6) and (2.12), for l odd we have that βl = βl−1 +τ +− 1 +τ − 1 and αl = αl−1+2 +τ ++ 2, and it is +readily checked that αl−1βl < αlβl−1. Thus, we deduce that δi +δj → 0 as ρ → 0 for any i < j, in view of +δi +δj += d1/αi +i +ρβi/αi +d1/αj +j +ρβj/αj = d1/αi +i +d1/αj +j +ρ +βi +αi − +βj +αj +and +βj +αj +< βi +αi +. +This finishes the proof. +□ +Now, we define di’s by +log dm = am +and +log dl = al + 2τ ν(l) +m +� +i=l+1 +ai +τ ν(i) = + + + + + + + + + + + +al + 2 +m +� +i=l+1 +aiτ σ(i), +if l is even +al + 2 +m +� +i=l+1 +ai +τ ν(i) +if l is odd +(2.15) +for l = 1, 2, . . . , m − 1, where +al = log +�τ ν(l)Vν(l)(0) +2α2 +l +� ++ (−1)l +τ σ(l) 2π(αm + 2)h0 = + + + + + + + +log +�τV1(0) +2α2 +l +� ++ 2π(αm + 2)h0, +if l is even +log +�V0(0) +2α2 +l +� +− 2π +τ (αm + 2)h0 +if l is odd +(2.16) +when m even, while when m is odd +al = log +�τ ν(l)Vν(l)(0) +2α2 +l +� ++ (−1)l+1τ ν(l)2π(αm + 2)h0 += + + + + + + + +log +�τV1(0) +2α2 +l +� +− 2π(αm + 2)τh0, +if l is even +log +�V0(0) +2α2 +l +� ++ 2π(αm + 2)h0 +if l is odd +. +(2.17) +Lemma 2.4. If di’s are given by (2.15) then for any l = 1, . . . , m − 1 +log dl = + + + + + + + + + + + + + +al + 2 +m−l +� +i=1 +(−1)i+1τ σ(i) log dl+i, +if l is even +al + 2 +m−l +� +i=1 +(−1)i+1 +τ σ(i) +log dl+i +if l is odd +(2.18) + +8 +P. FIGUEROA +and consequently, +log dl − al +2τ ν(l) += +m +� +j=l+1 +(−1)j+1 +τ ν(j) +log dj. +(2.19) +Proof: First, assume that m is even. So, al’s are given by (2.16). If l is even, then we have that +ν(i + l) = 1 − σ(i), +τ +τ ν(j) = τσ(j) and m − l is even, so that, σ(m − l) = 0 and +m−l +� +i=1 +(−1)i+1τ σ(i) log dl+i = +m−l−1 +� +i=1 +(−1)i+1τ σ(i) +� +al+i + 2τ ν(l+i) +m +� +j=l+i+1 +aj +τ ν(j) +� +− am += +m−l−1 +� +i=1 +(−1)i+1τ σ(i)al+i + 2τ +m−l−1 +� +i=1 +m +� +j=l+i+1 +(−1)i+1 aj +τ ν(j) − am += +m−1 +� +j=l+1 +(−1)j+1τ σ(j)aj + +m +� +j=l+2 +[1 + (−1)j]ajτ σ(j) − am = +m +� +j=l+1 +ajτ σ(j) +and (2.18) follows. Similarly as above, if l is odd, then we have that ν(i + l) = σ(i), +τ +τ ν(j) = τσ(j) and +m − l + 1 is even, so that, σ(m − l + 1) = 0 and +m−l +� +i=1 +(−1)i+1 +τ σ(i) +log dl+i = +m−l−1 +� +i=1 +(−1)i+1 +τ σ(i) +al+i + 2 +m−l−1 +� +i=1 +m +� +j=l+i+1 +(−1)i+1 aj +τ ν(j) + am +τ += +m−1 +� +j=l+1 +(−1)j +τ ν(j) aj + +m +� +j=l+2 +[1 + (−1)j+1] aj +τ ν(j) + am +τ += +m +� +j=l+1 +aj +τ ν(j) +and we conclude (2.18). Similar arguments work out in case m is odd. We deduce (2.19) by using the +change j = l + i. This conclude the proof. +□ +Now, ǫ = ǫ(ρ) is chosen so that +m +� +i=1 +(−1)i+1 +τ ν(i) +γi = 2π(α1 − 2), +where +γj = γαj +δj,ǫ, +j = 1, . . . , m +(2.20) +and γα +δ,ǫ is given in Lemma 2.1. Precisely, ǫ = ǫ(ρ) is given by +ǫα1−2 = + + + + + + + + + + + + + +[V0(0)]m[τV1(0)] +m +τ +�m +i=1 α4/τ ν(i) +i +e +2π +τ (αm+2)h0�ρ +2 +�m+ m +τ +if m is even +[V0(0)]m+1[τV1(0)] +m−1 +τ +�m +i=1 α4/τ ν(i) +i +e2π(αm+2)h0�ρ +2 +�m+1+ m−1 +τ +if m is odd +(2.21) +It is readily checked that ǫ(ρ) → 0+ as ρ → 0+. Moreover, ǫα1−2 ∼ ρβ1+1, in view of +β1 = +� +m − 1 + m +τ +if m is even +m + m−1 +τ +if m is odd +(2.22) +Lemma 2.5. If ǫ is given by (2.21) then it holds (2.20) and +ǫ +δ1 → 0 as ρ → 0. +Proof: First, assume that m is even so that, from (2.21) it follows that +(α1 − 2) log ǫ = m log V0(0) + m +τ log[τV1(0)] + 2π +τ (αm + 2)h0 − 4 +m +� +i=1 +log αi +τ ν(i) + +� +m + m +τ +� +log +�ρ +2 +� +. +On the other hand, from the definition of δi and γi it follows that +� +− 1 +2π log ǫ + h0 +� +γi = −2 log di − 2βi log ρ + 4παih0 + +SIGN-CHANGING BUBBLE TOWER SOLUTIONS ON PIERCED DOMAINS +9 +so that, +� +− 1 +2π log ǫ + h0 +� m +� +i=1 +(−1)i+1 +τ ν(i) +γi = −2 +m +� +i=1 +(−1)i+1 +τ ν(i) +log di − 2 log ρ +m +� +i=1 +(−1)i+1 +τ ν(i) +βi + 4πh0 +m +� +i=1 +(−1)i+1 +τ ν(i) +αi. +Hence, we compute the sums involved as follows. From Lemma 2.4 we have that for any m, either odd +or even, +log d1 = a1 + 2 +m +� +j=2 +(−1)j +τ ν(j) log dj. +Then by using (2.15) and m = 2k for some k ∈ IN we get that +m +� +j=1 +(−1)j +τ ν(j) log dj = a1 + log d1 +2 += +m +� +i=1 +ai +τ ν(i) = +k +� +i=1 +1 +τ +� +log[τV1(0)] + 2π(αm + 2)h0 − log(2α2 +2i) +� ++ +k +� +i=1 +� +log V0(0) − 2π +τ (αm + 2)h0 − log(2α2 +2i−1) +� += m +2 log V0(0) + m +2τ log[τV1(0)] − +� +m + m +τ +�log 2 +2 +− 2 +m +� +i=1 +log αi +τ ν(i) . +Also, from the choice of βi’s in (2.9), it is readily checked that +m +� +i=1 +(−1)i+1 +τ ν(i) +βi = +m +� +i=1 +(−1)i+1 +� +m + m + 1 +τ +− +� +1 + 1 +τ +� +i +� += m +2 + m +2τ . +Therefore, by using (2.7) we obtain that +� +− 1 +2π log ǫ + h0 +� m +� +i=1 +(−1)i+1 +τ ν(i) +γi = −m log V0(0) − m +τ log[τV1(0)] + +� +m + m +τ +� +log 2 + 4 +m +� +i=1 +log αi +τ ν(i) +− +� +m + m +τ +� +log ρ + 4πh0 +� +− m − m +τ +� += −(α1 − 2) log ǫ + 2π +τ (αm + 2)h0 − 4πh0 +� +m + m +τ +� += −(α1 − 2) log ǫ + 2π(α1 − 2)h0, +in view of αm+2 +τ +− 2m − 2m +τ += α1 − 2 and (2.20) follows. Similarly, (2.20) is proven if m is odd. +Finally, taking into account (2.22), (2.21) and δα1 +1 += d1ρβ1, we deduce that +ǫ +δ1 → 0 ⇐⇒ +β1+1 +α1−2 > β1 +α1 , +in view of ǫα1−2 ∼ ρβ1+1. Indeed, it is readily checked that β1+1 +α1−2 > β1+1 +α1 +> β1 +α1 . This completes the +proof. +□ +Let us stress that the behavior of ǫ = ǫ(ρ) and δj = δj(ρ)’s with respect to ρ is given by +ǫ +δj +, +δi +δj +→ 0 +as +ρ → 0 for i < j and +ǫ → 0 as +ρ → 0. +Assume that δi’s are given by (2.2) with αi’s, βi’s and di’s defined in (2.4), (2.9)-(2.10) and (2.15), +respectively, and ǫ is given by (2.21). Notice that log δj +log ǫ = O(1) as ρ → 0, for all j = 1, . . . , m. Now, +define the shrinking annuli +Ai = {x ∈ Ω | +� +δi−1δi < |x| ≤ +� +δiδi+1}, +j = 1, . . . , m, +where for simplicity we denote δ0 = +ǫ2 +δ1 and δm+1 = +M2 +0 +δm with M0 = sup{|x| : x ∈ Ω}, so that, +Ωǫ = ∪m +j=1Aj, ∩m +j=1Aj = ∅ and Aj +δj approaches to IR2 as ρ → 0 for each j = 1, . . . , m. + +10 +P. FIGUEROA +Lemma 2.6. There exist η > 0 such that the following expansions hold +U(δjy) = (−1)j+1 +τ ν(j) +� +−2 log δj − aj − log ρ + log +|y|αj−2 +(1 + |y|αj)2 +� ++ (−1)m+1 +2π +τ ν(m) (αm + 2)H(δjy, 0) + O (ρη) +(2.23) +for any j = 1, . . . , m, uniformly for δjy ∈ Aj, j = 1, . . . , m. +Proof: From the expansions of Pǫwj, j = 1, . . . , m and the definition of γj in Lemma 2.1, (2.8) and +(2.20) we obtain that +U(x) = +m +� +i=1 +(−1)i+1 +τ ν(i) +� +wi − log(2α2 +i δαi +i ) +� ++ 1 +2π +m +� +i=1 +(−1)i+1 +τ ν(i) +γi log |x| ++ +m +� +i=1 +(−1)i+1 +τ ν(i) +[4παi − γi] H(x, 0) + O + + +m +� +j=1 +� +δαj +j ++ +� ǫ +δj +�αj� ++ ǫ + + += +m +� +i=1 +(−1)i+1 +τ ν(i) +log +1 +(δαi +i ++ |x|αi)2 + (α1 − 2) log |x| + (−1)m+1 2π +τ ν(m) (αm + 2)H(x, 0) ++ O (ρη) , +(2.24) +where we choose +0 < η ≤ min +� +1, β1 + 1 +α1 − 2, min +� +αj +�β1 + 1 +α1 − 2 − βj +αj +� +: j = 1, . . . , m +�� +, +(2.25) +in view βj are decreasing so that δαj +j += O(ρ) for all j = 1, . . . , m, +� ǫ +δj +�αj = O +� +ρ +αj( β1+1 +α1−2 − +βj +αj )� +and +ǫ = O +� +ρ +β1+1 +α1−2 � +. Now, from the choice of δi’s in (2.2), (2.13) and (2.19) we have that for j odd +2 +m +� +i=j+1 +(−1)i +τ (ν(i) αi log δi = 2 +m +� +i=j+1 +(−1)i +τ (ν(i) log di + 2 log ρ +m +� +i=j+1 +(−1)i +τ (ν(i) βi += 2log dj − aj +2 ++ 2βj − 1 +2 +log ρ = αj log δj − aj − log ρ +and similarly, for j even 2 �m +i=j+1 +(−1)i +τ (ν(i) αi log δi = − 1 +τ +� +αj log δj − aj − log ρ +� +. Thus, we rewrite as +2 +m +� +i=j+1 +(−1)i +τ (ν(i) αi log δi = (−1)j+1 +τ ν(j) +� +αj log δj − aj − log ρ +� +. +(2.26) + +SIGN-CHANGING BUBBLE TOWER SOLUTIONS ON PIERCED DOMAINS +11 +On the other hand, for any y ∈ A1 +δ1 it holds that +U(δ1y) = − 2α1 log δ1 + log +1 +(1 + |y|α1)2 + (α1 − 2) log(δ1|y|) ++ +m +� +i=2 +(−1)i+1 +τ ν(i) +log +1 +(δαi +i ++ δαi +1 |y|αi)2 + (−1)m+1 +2π +τ ν(m) (αm + 2)H(δ1y, 0) + O (ρη) += − (α1 + 2) log δ1 + +m +� +i=2 +(−1)i 2αi +τ ν(i) log δi + log +|y|α1−2 +(1 + |y|α1)2 ++ +m +� +i=2 +(−1)i +τ ν(i) 2 log +� +1 + +�δ1|y| +δi +�αi� ++ (−1)m+1 +2π +τ ν(m) (αm + 2)H(δ1y, 0) + O (ρη) += − 2 log δ1 − a1 − log ρ + log +|y|α1−2 +(1 + |y|α1)2 + (−1)m+1 +2π +τ ν(m) (αm + 2)H(δ1y, 0) ++ O + + +m +� +j=2 +�δ1 +δj +� αj +2 + ρη + + , +(2.27) +in view of (2.26) and +log +� +1 + +�δ1|y| +δi +�αi� += O +��δ1|y| +δi +�αi� += O +��δ1 +δi +� αi +2 � +, +i = 2, . . . , m. +Choosing η > 0 satisfying (2.25) and +0 < η ≤ 1 +2 min +� +αi +� β1 +α1 +− βi +αi +� +: +1 < i +� +(2.28) +we get that +� δ1 +δi +� αi +2 = O (ρη), i = 2, . . . , m. Similarly, for y ∈ Aj +δj , 1 < j < m we find that +U(δjy) = +j−1 +� +i=1 +(−1)i+1 +τ ν(i) +� +− 2αi log(δj|y|) − 2 log +� +1 + +� δi +δj|y| +�αi� � ++ (α1 − 2) log(δj|y|) ++ (−1)j+1 +τ ν(j) +� +− 2αj log δj + log +1 +(1 + |y|αj)2 +� ++ +m +� +i=j+1 +(−1)i +τ ν(i) 2αi log δi ++ +m +� +i=j+1 +(−1)i +τ ν(i) 2 log +� +1 + +�δj|y| +δi +�αi� ++ (−1)m+1 +2π +τ ν(m) (αm + 2)H(δjy, 0) + O (ρη) += −2 log δj +j +� +i=1 +(−1)i+1 +τ ν(i) +αi + 2 +m +� +i=j+1 +(−1)i +τ ν(i) αi log δi + (α1 − 2) log δj + (α1 − 2) log |y| +− 2 log |y| +j−1 +� +i=1 +(−1)i+1 +τ ν(i) +αi + (−1)j+1 +τ ν(j) +log +1 +(1 + |y|αj)2 ++ (−1)m+1 +2π +τ ν(m) (αm + 2)H(δjy, 0) + O +� j−1 +� +i=1 +� δi +δj +� αi +2 + +m +� +i=j+1 +�δj +δi +� αi +2 + ρη +� +, +(2.29) +in view of +log +� +1 + +� δi +δj|y| +�αi� += O +�� δi +δj|y| +�αi� += O +�� δi +δj +� αi +2 � +, +i < j, +log +� +1 + +�δj|y| +δi +�αi� += O +��δj|y| +δi +�αi� += O +��δj +δi +� αi +2 � +, +j < i, + +12 +P. FIGUEROA +and by using again (2.20). Now, we choose η satisfying (2.25), (2.28) and smaller, if necessary, so that +0 < η ≤ 1 +2 min +� +αi +� βi +αi +− βj +αj +� +: +i < j +� +, +j = 2, . . . , m +and +0 < η ≤ 1 +2 min +� +αi +�βj +αj +− βi +αi +� +: +j < i +� +, +j = 1, . . . , m − 1 +thus, +� δi +δj +� αi +2 = O (ρη), i < j and +� δj +δi +� αi +2 = O (ρη), j < i. Hence, by using (2.7) and (2.26) for j even +we get that +U(δjy) = −2 log δj +� +− j − j +τ +� +− 1 +τ [αj log δj − aj − log ρ] + (α1 − 2) log δj + (α1 − 2) log |y| +− 2 log |y| +� +α1 + j − 2 + j − 2 +τ +� +− 1 +τ log +1 +(1 + |y|αj)2 + (−1)m+1 2π +τ ν(m) (αm + 2)H(δjy, 0) + O (ρη) +and we conclude (2.23). Similarly, for j odd we get that +U(δjy) = −2 log δj +� +α1 + j − 1 + j − 1 +τ +� ++ αj log δj − aj − log ρ + (α1 − 2) log δj + (α1 − 2) log |y| +− 2 log |y| +� +−j + 1 − j − 1 +τ +� ++ log +1 +(1 + |y|αj)2 + (−1)m+1 2π +τ ν(m) (αm + 2)H(δjy, 0) + O(ρη) +and we obtain (2.23). Also, we have for any y ∈ Am +δm +U(δmy) = +m−1 +� +i=1 +(−1)i+1 +τ ν(i) +� +−2αi log(δm|y|) − 2 log +� +1 + +� +δi +δm|y| +�αi�� ++ (α1 − 2) log(δm|y|) ++ (−1)m+1 +τ ν(m) +� +−2αm log δm + log +1 +(1 + |y|αm)2 +� ++ (−1)m+1 2π +τ ν(m) (αm + 2)H(δjy, 0) + O(ρη) += (−1)m+1 +τ ν(m) +� +−2 log δm − am − log ρ + log +|y|αm−2 +(1 + |y|αm)2 + 2π(αm + 2)H(δmy, 0) +� ++ O(ρη). +(2.30) +This completes the proof. +□ +3. Problem in Liouville form (1.1) +3.1. Error estimate. In order to evaluate how well the approximation U satisfies the equation (1.1) +(and get the invertibility of the linearized operator), we will use the norms +∥h∥p = +�� +Ωǫ +|h(x)|p dx +�1/p +and +∥h∥ = +�� +Ωǫ +|∇h(x)|2 dx +�1/2 +, +the usual norms in the Banach spaces Lp(Ωǫ) and H1 +0(Ωǫ), respectively. Assume that δi’s are given +by (2.2) with αi’s, βi’s and di’s defined in (2.4), (2.9)-(2.10) and (2.15), respectively, and ǫ is given by +(2.21). Let us evaluate the approximation rate of U in ∥ · ∥p, encoded in (1.12) +Lemma 3.1. There exist ρ0 > 0, a constant C > 0 and 1 < p0 < 2, such that for any ρ ∈ (0, ρ0) and +p ∈ (1, p0) it holds +∥R∥p ≤ Cρηp +(3.1) +for some ηp > 0. +Proof: First, note that ∆U = �m +i=1 +(−1)i +τ ν(i) |x|αi−2ewi so that, for any 1 < j < m and y ∈ Aj +δj we have +that +∆U(δjy) = (−1)j +τ ν(j) +2α2 +j|y|αj−2 +δ2 +j (1 + |y|αj)2 + O +� j−1 +� +i=1 +� δi +δj +�αi +1 +δ2 +j |y|αi+2 + +m +� +i=j+1 +�δj +δi +�αi |y|αi−2 +δ2 +j +� +. +(3.2) + +SIGN-CHANGING BUBBLE TOWER SOLUTIONS ON PIERCED DOMAINS +13 +Similarly, we obtain that +∆U(δ1y) = − 2α2 +j|y|αj−2 +δ2 +j (1 + |y|αj)2 + O +� m +� +i=2 +�δj +δi +�αi |y|αi−2 +δ2 +j +� +, +y ∈ A1 +δ1 +(3.3) +and +∆U(δmy) = (−1)m +τ ν(m) +2α2 +j|y|αj−2 +δ2 +j (1 + |y|αj)2 + O +� m−1 +� +i=1 +� δi +δj +�αi +1 +δ2 +j |y|αi+2 +� +, +y ∈ Am +δm +(3.4) +By using (2.16) and (2.23) (from Lemma 2.6) for y ∈ Aj +δj and any j odd we have that +ρV0(δjy)eU(δjy) = +2α2 +j|y|αj−2 +δ2 +j (1 + |y|αj)2 +V0(δjy) +V0(0) exp +� +(−1)m+1 +2π +τ ν(m) (αm + 2)[H(δjy, 0) − h0] + O(ρη) +� += +2α2 +j|y|αj−2 +δ2 +j (1 + |y|αj)2 [1 + O(δj|y| + ρη)] +(3.5) +and +ρV1(δjy)e−τU(δjy) = ρV1(δjy) exp +� +− τ +� +− 2 log δj − aj − log ρ + log +|y|αj−2 +δ2 +j (1 + |y|αj)2 ++ (−1)m+1 2π +τ ν(m) (αm + 2)H(δjy, 0) +� ++ O(ρη) +� += O +� +ρ1+τδ2τ +j +(1 + |y|αj)2τ +|y|(αj−2)τ +� +, +(3.6) +Similarly, by using (2.17) and 1 − ν(m) = σ(m) for y ∈ Aj +δj and j even +ρV1(δjy)e−τU(δjy) = +2α2 +j|y|αj−2 +δ2 +j τ(1 + |y|αj)2 +V1(δjy) +V1(0) exp +� +(−1)m2πτ σ(m)(αm + 2)[H(δjy, 0) − h0] + O(ρη) +� += +2α2 +j|y|αj−2 +δ2 +j τ(1 + |y|αj)2 [1 + O(δj|y| + ρη)] +(3.7) +and +ρV0(δjy)eU(δjy) = O +� +ρ1+1/τδ2/τ +j +(1 + |y|αj)2/τ +|y|(αj−2)/τ +� +. +(3.8) +Hence, by using (3.3), (3.5) and (3.6) for δ1y ∈ A1 we get that +R(δ1y) = 1 +δ2 +1 +2α2 +1|y|α1−2 +(1 + |y|α1)2 O(δ1|y| + ρη) + O +� +ρ1+τδ2τ +1 +(1 + |y|α1)2τ +|y|(α1−2)τ ++ +m +� +i=2 +�δ1 +δi +�αi |y|αi−2 +δ2 +1 +� +, +(3.9) +by using (3.2), (3.5) and (3.6) for 1 < j < m, j odd and δjy ∈ Aj +R(δjy) = 1 +δ2 +j +2α2 +j|y|αj−2 +(1 + |y|αj)2 O(δj|y| + ρη) ++ O +� +ρ1+τδ2τ +j +(1 + |y|αj)2τ +|y|(αj−2)τ ++ +j−1 +� +i=1 +� δi +δj +�αi +1 +δ2 +j |y|αi+2 + +m +� +i=j+1 +�δj +δi +�αi |y|αi−2 +δ2 +j +� +, +(3.10) +by using (3.2), (3.7) and (3.8) for 1 < j < m, j even and δjy ∈ Aj +R(δjy) = 1 +δ2 +j τ +2α2 +j|y|αj−2 +(1 + |y|αj)2 O(δj|y| + ρη) ++ O +� +ρ1+1/τδ2/τ +j +(1 + |y|αj)2/τ +|y|(αj−2)/τ ++ +j−1 +� +i=1 +� δi +δj +�αi +1 +δ2 +j |y|αi+2 + +m +� +i=j+1 +�δj +δi +�αi |y|αi−2 +δ2 +j +� +(3.11) + +14 +P. FIGUEROA +and by (3.4) for δmy ∈ Am +R(δmy) = +1 +δ2mτ ν(m) +2α2 +m|y|αm−2 +(1 + |y|αm)2 O(δm|y| + ρη) ++ O +� +ρ1+τ (−1)m+1 +δ2τ (−1)m+1 +m +(1 + |y|αm)2τ (−1)m+1 +|y|(αm−2)τ (−1)m+1 ++ +m−1 +� +i=1 +� δi +δm +�αi +1 +δ2m|y|αi+2 +� +(3.12) +By (3.9)-(3.12), we finally get that there exist ρ0 > 0 small enough and p0 > 1 close to 1, so that for +all 0 < ρ ≤ ρ0 and 1 < p ≤ p0 +� +Ωǫ +|R(x)|p dx = +m +� +j=1 +� +Aj +|R(x)|p dx = +m +� +j=1 +δ2 +j +� +Aj +|R(δjy)|p dy = O (ρpηp) , +for some ηp > 0, see Appendix, Lemma A.1 for more details. This completes the proof. +□ +3.2. The nonlinear problem and proof of Theorem 1.1. In this subsection, we will look for a +solution u of (1.1) in the form u = U + φ, for some small remainder term φ. In terms of φ, the problem +(1.1) is equivalent to find φ ∈ H1 +0(Ωǫ) so that +� +L(φ) = −[R + Λ(φ) + N(φ)], +in Ωǫ, +φ = 0, +on ∂Ωǫ. +(3.13) +where the linear operators L and Λ are defined as +L(φ) = ∆φ + K(x)φ, +K(x) = +m +� +i=1 +|x|αi−2ewi +(3.14) +and +Λ(φ) = ρ[V0(x)eU + τV1(x)e−τU]φ − Kφ +(3.15) +The nonlinear part N is given by +N(φ) = ρV0(x)eU� +eφ − φ − 1 +� +− ρV1(x)e−τU� +e−τφ + τφ − 1 +� +. +(3.16) +Proposition 3.1. For any p > 1 there exists ρ0 > 0 so that for all 0 < ρ ≤ ρ0, h ∈ Lp(Ωǫ) there is a +unique solution φ ∈ H1 +0(Ωǫ) of +� L(φ) = h +in Ωǫ +φ = 0 +on ∂Ωǫ. +(3.17) +Moreover, there is a constant C > 0 independent of ρ such that +∥φ∥ ≤ C| log ρ|∥h∥p. +(3.18) +The latter proposition implies that the unique solution φ = T (h) of (3.17) defines a continuous +linear map from Lp(Ωǫ) into H1 +0(Ωǫ), with norm bounded by C| log ρ|. We are now in position to study +the nonlinear problem. +Proposition 3.2. There exist p0 > 1 and ρ0 > 0 so that for any 1 < p < p0 and all 0 < ρ ≤ ρ0, the +problem +� L(φ) = −[R + Λ(φ) + N(φ)], +in Ωǫ +φ = 0, +on ∂Ωǫ +(3.19) +admits a unique solution φ(ρ) ∈ H1 +0(Ωǫ), where N, Λ and R are given by (3.16), (3.15) and (1.12), +respectively. Moreover, there is a constant C > 0 such that for some ηp > 0 +∥φ∥∞ ≤ Cρηp| log ρ| +Here, ηp is the same as in (3.1). We shall use the following estimates. +Lemma 3.2. There exist p0 > 1 and ρ0 > 0 so that for any 1 < p < p0 and all 0 < ρ ≤ ρ0 it holds +∥Λ(φ)∥p ≤ Cρη′ +p∥φ∥, +(3.20) +for all φ ∈ H1 +0(Ωǫ) with ∥φ∥ ≤ Mρηp| log ρ|, for some η′ +p > 0. + +SIGN-CHANGING BUBBLE TOWER SOLUTIONS ON PIERCED DOMAINS +15 +Proof: Arguing in the same way as in [13, Lemma 3.3], for simplicity, denote W = ρ[V0eU +τV1e−τU], +so that the linear operator Λ is re-written as Λ(φ) = (W − K)φ. By using (3.5)-(3.6) and (3.7)-(3.8), +we find that for i odd +δ2 +i W(δiy) = 2α2 +i |y|αi−2 +(1 + |y|α +i )2 [1 + O(|δiy| + ρη)] + O +� +ρ1+τδ2+2τ +i +(1 + |y|αi)2τ +|y|(αi−2)τ +� +uniformly for δiy ∈ Ai and for i even +δ2 +i W(δiy) = 2α2 +i |y|αi−2 +(1 + |y|α +i )2 [1 + O(|δiy| + ρη)] + O +� +ρ1+1/τδ2+2/τ +i +(1 + |y|αi)2/τ +|y|(αi−2)/τ +� +uniformly for x ∈ Ai and i even. Hence, for any q ≥ 1 and for i odd there holds +∥W − K∥q +Lq(Ai) ≤ C +� +δ2−q +i +� +Ai +δi +���� +2α2 +i |y|αi−1 +(1 + |y|αi)2 +���� +q +dy + ρηqδ2−2q +i +� +Ai +δi +���� +2α2 +i |y|αi−2 +(1 + |y|αi)2 +���� +q +dy ++ ρ(1+τ)qδ2+2τq +i +� +Ai +δi +���� +(1 + |y|αi)2τ +|y|(αi−2)τ +���� +q +dy + +� +ji +δ2−2q +i +� δi +δj +�αjq � +Ai +δi +|y|(αj−2)qdy +� +≤ Cρqη′ +1,q +for some η′ +1,q > 0, see Lemma A.1. Similarly, there holds ∥W − K∥q +Lq(Ai) ≤ Cρqη′ +2,q for any q ≥ 1 and +for i even for some η′ +2,q > 0. Hence, we get that +∥Λ(φ)∥p ≤ ∥(W − K) φ∥p ≤ ∥W − K∥pr ∥φ∥ps ≤ Cρη′ +p∥φ∥, +where η′ +p = min +� +η′ +1,pr, η′ +2,ps +� +with r, s satisfying 1 +r + 1 +s = 1. Furthermore, we have used the H¨older’s +inequality ∥uv∥q ≤ ∥u∥qr∥v∥qs with 1 +r + 1 +s = 1 and the inclusions Lp(Ωǫ) ֒→ Lpr(Ωǫ) for any r > 1 and +H1 +0(Ωǫ) ֒→ Lq(Ωǫ) for any q > 1. Let us stress that we can choose p, r and s close enough to 1 such +that η′ +p > 0. +□ +Lemma 3.3. There exist p0 > 1 and ρ0 > 0 so that for any 1 < p < p0 and all 0 < ρ ≤ ρ0 it holds +∥N(φ1) − N(φ2)∥p ≤ Cρη′′ +p ∥φ1 − φ2∥ +(3.21) +for all φi ∈ H1 +0(Ωǫ) with ∥φi∥ ≤ Mρηp| log ρ|, i = 1, 2, and for some η′′ +p > 0. In particular, we have +that +∥N(φ)∥p ≤ Cρη′′ +p ∥φ∥ +(3.22) +for all φ ∈ H1 +0(Ωǫ) with ∥φ∥ ≤ Mρηp| log ρ|. +Proof: We will argue in the same way as in [1, Lemma 5.1], see also [13, Lemma 3.4]. First, denoting +fi(φ) = Vi(x)e(−τ)i(U+φ) we point out that +N(φ) = +1 +� +i=0 +ρ {fi(φ) − fi(0) − f ′ +i(0)[φ]} +and +N(φ1) − N(φ2) = +1 +� +i=0 +ρf ′′ +i (˜φµi)[φθi, φ1 − φ2], +(3.23) +by the mean value theorem, where φθi = θiφ1 + (1 − θi)φ2, ˜φµi = µiφθi for some θi, µi ∈ [0, 1], i = 0, 1, +and +f ′′ +i (φ)[ψ, v] = τ 2iVi(x)e(−τ)i(U+φ)ψv. Using H¨older’s inequalities we get that +∥f ′′ +i (φ)[ψ, v]∥p ≤ |τ|2i∥Vie(−τ)i(U+φ)∥pri∥ψ∥psi∥v∥pti +(3.24) +with +1 +ri + 1 +si + 1 +ti = 1. We have used the H¨older’s inequality and ∥uvw∥q ≤ ∥u∥qr∥v∥qs∥w∥qt with +1 +r + 1 +s + 1 +t = 1. Now, let us estimate ∥Vie(−τ)i(U+φ)∥pri with φ = ˜φµi, i = 0, 1. By (3.5) and the change +of variable x = δiy let us estimate +� +Ai +��V0eU��q dx = O +� +ρ−qδ2−2q +i +� +Ai +δi +���� +|y|αi−2 +(1 + |y|αi)2 +���� +q � +1 + O +� +ρη + δi|y| +��q +� += O +� +ρ +βi +αi (2−2q)−q� +(3.25) + +16 +P. FIGUEROA +for any i odd and similarly, by (3.7) we get that +� +Ai +��V1e−τU��q dx = O +� +ρ +βi +αi (2−2q)−q� +(3.26) +for any i even, in view of (2.2). By (3.6) we get the estimate +� +Ai +��V0eU��q dx += +O +� +ρ +q +τ δ +2+ 2q +τ +i +� +Ai +δi +� +|y|αi−2 +(1 + |y|αi)2 +�− q +τ dy +� += O +� +ρ−q+qηq� +for all i even, in view of Lemma A.1. Similarly, by (3.8) we deduce that +� +Ai +��V1e−τU��q dx = O(ρ−q+qηq) +(3.27) +for i odd, again in view of Lemma A.1. Therefore, by using (3.25) and (3.27) and that βi +αi are decreasing, +we deduce that +��V0eU��q +q = +m +� +i=1 +i odd +O +� +ρ +βi +αi (2−2q)−q� ++ O +� +ρ−q+qηq� += O +� +ρ +β1 +α1 (2−2q)−q� +for any q ≥ 1 +(3.28) +and, by using (3.26) and (3.27) we obtain that +��V1e−τU��q +q = O +� +ρ +β1 +α1 (2−2q)−q� +for any q ≥ 1. +(3.29) +On the other hand, using the estimate |ea − 1| ≤ C|a| uniformly for any a in compact subsets of IR, +H¨older’s inequality with +1 +s′ +i + 1 +t′ +i = 1, i = 0, 1, ∥˜φµi∥ ≤ Mρηp| log ρ| ≤ C, i = 0, 1 and triangle inequality +we find that +��Vie(−τ)i(U+ ˜φµi )�� +q = O +� +ρ +η0,qs′ +i −1+ηp| log ρ| + ρη0,q−1� +, +(3.30) +where for q > 1 we denote η0,q = β1(2−2q) +α1q +(on the line of [17, eq.(3.14) proof of Lemma 3.3]). Note that +η0,q < 0 for any q > 1. Also, choosing q and s′ +i, i = 0, 1, close enough to 1, we get that 0 < ηp + η0,qs′ +i. +Now, we can conclude the estimate by using (3.23)-(3.30) to get +∥N(φ1) − N(φ2)∥p ≤ C +1 +� +i=0 +ρ∥Vie(−τ)i(U+ ˜φµi )∥pri∥φθi∥∥φ1 − φ2∥ ≤ C +1 +� +i=0 +ρηp+η0,pri| log ρ|∥φ1 − φ2∥ +and (3.21) follows, where η′′ +p = +1 +2 min{ηp + η0,pri +: +i = 0, 1} > 0 choosing ri close to 1 so that +ηp + η0,pri > 0 for i = 0, 1. Let us stress that p > 1 is chosen so that ηp > 0 and η′ +p > 0. +□ +Proof of the Proposition 3.2. Taking into account Proposition 3.1, estimates (3.1), (3.20), (3.21), +(3.22) and standard arguments it turns out that for all ρ > 0 sufficiently small A is a contraction +mapping of FM (for M large enough), and therefore a unique fixed point of A exists in FM, where +A(φ) := −T (R + Λ(φ) + N(φ)) and FM = {φ ∈ H1 +0(Ωǫ) : ∥φ∥ ≤ Mρηp| log ρ|}. See the proof of +Proposition 3.2 in [13] for more details. +□ +Proof of the Theorem 1.1. Taking into account (1.10) and the definition of U, the existence of a +solution to equation (1.1) follows directly by Proposition 3.2. The asymptotic behavior of uρ as ρ → 0+ +follows from (2.23) in Lemma 2.6 and estimate for φ in Proposition 3.2. Furthermore, we have that uρ +has the desired concentration properties (1.6) as ρ → 0+ locally uniformly in ¯Ω \ {0}. +□ +4. Problem in mean field form (1.2) +4.1. Error estimate. Assume that δi’s are given by (2.2) with αi’s, βi’s and di’s defined in (2.4), +(2.9)-(2.10) and (2.15), respectively, and ǫ is given by (2.21). Let us evaluate the error term in ∥ · ∥p, +encoded in (1.13). + +SIGN-CHANGING BUBBLE TOWER SOLUTIONS ON PIERCED DOMAINS +17 +Lemma 4.1. There exist ρ0 > 0, a constant C > 0 and 1 < p0 < 2, such that for any ρ ∈ (0, ρ0) and +p ∈ (1, p0) it holds +∥R∥p ≤ Cρηp +(4.1) +for some ηp > 0. +Proof: By using (3.5) and (3.8) we have that +� +Ωǫ +V0(x)eU = +m +� +j=1 +� +Aj +δj +δ2 +j V0(δjy)eU(δjy)dy = +m +� +j=1 +j odd +1 +ρ +� +Aj +δj +2α2 +j|y|αj−2 +δ2 +j (1 + |y|αj)2 [1 + O(δj|y| + ρη)] dy ++ +m +� +j=1 +j even +O +� +ρ1/τδ2+2/τ +j +� +Aj +δj +(1 + |y|αj)2/τ +|y|(αj−2)/τ +dy +� += 1 +ρ +� +4π +m +� +j=1 +j odd +αj + O(ρη) +� +(4.2) +and similarly, from (3.6) and (3.7) we get that +� +Ωǫ +V1(x)e−τU = 1 +ρτ +� +4π +m +� +j=1 +j even +αj + O(ρη) +� +, +(4.3) +in view of +� +R2 +|y|αi−2 +(1+|y|αi)2 dy = 2π +αi . From assumptions (1.8)-(1.9) it follows that λ0 = 4π �m +j=1 +j odd αj and +λ1τ 2 = 4π �m +j=1 +j even αj so that, by using (4.2)-(4.3) we find that for y ∈ Aj +δj and any j odd +λ0 +V0(δjy)eU(δjy) +� +Ωǫ V0(x)eU += +2α2 +j|y|αj−2 +δ2 +j (1 + |y|αj)2 [1 + O(δj|y| + ρη)] +(4.4) +and +λ1τ V1(δjy)e−τU(δjy) +� +Ωǫ V1(x)e−τU += O +� +ρ1+τδ2τ +j +(1 + |y|αj)2τ +|y|(αj−2)τ +� +. +(4.5) +Similarly, for any y ∈ Aj +δj and j even +λ0 +V0(δjy)eU(δjy) +� +Ωǫ V0(x)eU += O +� +ρ1+1/τδ2/τ +j +(1 + |y|αj)2/τ +|y|(αj−2)/τ +� +(4.6) +and +λ1τ V1(δjy)e−τU(δjy) +� +Ωǫ V1(x)e−τU += +2α2 +j|y|αj−2 +δ2 +j τ(1 + |y|αj)2 [1 + O(δj|y| + ρη)] . +(4.7) +Hence, similar to the proof of Lemma 3.1 by using (3.2)-(3.4) and (4.4)-(4.7) for δ1y ∈ A1 we get that +R(δ1y) = 1 +δ2 +1 +2α2 +1|y|α1−2 +(1 + |y|α1)2 O(δ1|y| + ρη) + O +� +ρ1+τδ2τ +1 +(1 + |y|α1)2τ +|y|(α1−2)τ ++ +m +� +i=2 +�δ1 +δi +�αi |y|αi−2 +δ2 +1 +� +, +(4.8) +for 1 < j < m, j odd and δjy ∈ Aj +R(δjy) = 1 +δ2 +j +2α2 +j|y|αj−2 +(1 + |y|αj)2 O(δj|y| + ρη) ++ O +� +ρ1+τδ2τ +j +(1 + |y|αj)2τ +|y|(αj−2)τ ++ +j−1 +� +i=1 +� δi +δj +�αi +1 +δ2 +j |y|αi+2 + +m +� +i=j+1 +�δj +δi +�αi |y|αi−2 +δ2 +j +� +(4.9) +for 1 < j < m, j even and δjy ∈ Aj +R(δjy) = 1 +δ2 +j τ +2α2 +j|y|αj−2 +(1 + |y|αj)2 O(δj|y| + ρη) ++ O +� +ρ1+1/τδ2/τ +j +(1 + |y|αj)2/τ +|y|(αj−2)/τ ++ +j−1 +� +i=1 +� δi +δj +�αi +1 +δ2 +j |y|αi+2 + +m +� +i=j+1 +�δj +δi +�αi |y|αi−2 +δ2 +j +� +(4.10) + +18 +P. FIGUEROA +and for δmy ∈ Am +R(δmy) = +1 +δ2mτ ν(m) +2α2 +m|y|αm−2 +(1 + |y|αm)2 O(δm|y| + ρη) ++ O +� +ρ1+τ (−1)m+1 +δ2τ (−1)m+1 +m +(1 + |y|αm)2τ (−1)m+1 +|y|(αm−2)τ (−1)m+1 ++ +m−1 +� +i=1 +� δi +δm +�αi +1 +δ2m|y|αi+2 +� +(4.11) +By (4.8)-(4.11) and similar ideas to get (3.1) we conclude the proof. +□ +4.2. The nonlinear problem and proof of Theorem 1.2. In this section we shall study the +following nonlinear problem: � L(φ) = −[R + Λ0(φ) + N(φ)] +in Ωǫ +φ = 0, +on ∂Ωǫ, +(4.12) +where the linear operators L, Λ0 are defined as +L(φ) = ∆φ + K0 +� +φ − 1 +λ0 +� +Ωǫ +K0φdx +� ++ K1 +� +φ − +1 +λ1τ 2 +� +Ωǫ +K1φdx +� +(4.13) +and +Λ0(φ) = λ0 +V0eU +� +Ωǫ V0eUdx +� +φ − +� +Ωǫ V0eUφdx +� +Ωǫ V0eUdx +� ++ λ1τ 2 +V1e−τU +� +Ωǫ V1e−τUdx +� +φ − +� +Ωǫ V1e−τUφdx +� +Ωǫ V1(x)e−τUdx +� +− K0 +� +φ − 1 +λ0 +� +Ωǫ +K0φdx +� +− K1 +� +φ − +1 +λ1τ 2 +� +Ωǫ +K1φdx +� +(4.14) +with +K0 = +m +� +k=1 +k odd +|x|αk−2ewk, +K1 = +m +� +k=1 +k even +|x|αk−2ewk. +(4.15) +The nonlinear term N(φ) is given by +N(φ) =λ0 +� +V0eU+φ +� +Ωǫ V0eU+φdx − +V0eU +� +Ωǫ V0eUdx − +V0eU +� +Ωǫ V0eUdx +� +φ − +� +Ωǫ V0eUφdx +� +Ωǫ V0eUdx +�� +− λ1τ +� +V1e−τ(U+φ)dx +� +Ωǫ V1e−τ(U+φ)dx − +V1e−τU +� +Ωǫ V1e−τUdx + τ +V1e−τU +� +Ωǫ V1e−τUdx +� +φ − +� +Ωǫ V1e−τUφdx +� +Ωǫ V1e−τUdx +�� +. +(4.16) +It is readily checked that φ is a solution to (3.13) if and only if uǫ given by (1.11) is a solution to (??). +In Section 5 we will prove the following result. +Proposition 4.1. For any p > 1, there exists ρ0 > 0 and C > 0 such that for any ρ ∈ (0, ρ0) and +h ∈ Lp(Ωǫ) there exists a unique φ ∈ H1 +0(Ωǫ) solution of +L(φ) = h in Ωǫ, +φ = 0 on ∂Ωǫ, +(4.17) +which satisfies +∥φ∥ ≤ C| log ρ| ∥h∥p. +(4.18) +Lemma 4.2. There exist p0 > 1 and ρ0 > 0 so that for any 1 < p < p0 and all 0 < ρ ≤ ρ0 it holds +∥Λ0(φ)∥p ≤ Cρσ′ +p∥φ∥, +(4.19) +for all φ ∈ H1 +0(Ωǫ) with ∥φ∥ ≤ Mρσp| log ρ|, for some σ′ +p > 0. +Proof: Arguing in the same way as in [13, Lemma 3.3], denote Wi = +λiτ 2iVie(−τ)iU +� +Ωǫ Vie(−τ)iU dx for i = 0, 1. By us- +ing (4.4), (4.6), Lemma A.1 and similar computations to prove (3.20), we find that ∥W0 − K0∥q +Lq(Ai) ≤ +Cρqσ′ +0,q for any i for some σ′ +0,q. Similarly, we find that ∥W1 − K1∥q +Lq(Ai) ≤ Cρqσ′ +1,q for any i for some +σ′ +1,q. It is possible to see that taking q > 1 close enough to 1, we get that σ′ +i,q > 0 for i = 0, 1. + +SIGN-CHANGING BUBBLE TOWER SOLUTIONS ON PIERCED DOMAINS +19 +Notice that Λ0 is a linear operator and we re-write Λ0(φ) as +Λ0(φ) = +1 +� +i=0 +� +(Wi − Ki) φ − +1 +λiτ 2i (Wi − Ki) +� +Ωǫ +Wiφ + +1 +λiτ 2i Ki +� +Ωǫ +(Ki − Wi) φ +� +. +Hence, we get that +∥Λ0(φ)∥p ≤ +1 +� +i=0 +� +∥Wi − Ki∥pri0 ∥φ∥psi0 + ∥Wi∥ri1 +λiτ 2i +∥Wi − Ki∥p ∥φ∥si1 + +∥Ki∥p +λiτ 2(i−1) ∥Ki − Wi∥ri2 ∥φ∥si2 +� +≤ C +1 +� +i=0 +� +ρσ′ +i,pri0 ∥φ∥ + ρσ′ +i,p+σ3,ri1 ∥φ∥ + ρσ′ +i,ri2 +σ3,p∥φ∥ +� +and (4.19) follows, where σ′ +p = min +� +σ′ +i,pri0; σ′ +i,p + σ3,ri1; σ′ +i,ri2 + σ3,p | i = 0, 1 +� +with rij, sij, i = 0, 1, +j = 0, 1, 2 satisfying +1 +rij + +1 +sij = 1. We have used that +∥W0∥r01 +r01 ≤ C +� +m +� +j=1 +j odd +δ2−2r01 +j +� +Aj +δj +����� +2α2 +j|y|αj−2 +(1 + |y|αi)2 +����� +r01 ++ +m +� +j=1 +j even +ρ(1+1/τ)r01δ +2+2 r01 +τ +j +� +Aj +δj +���� +(1 + |y|αj)2/τ +|y|(αj−2)/τ +���� +r01� +≤ Cρσ3,r01 +and similarly, ∥W1∥r11 +r11 ≤ Cρσ3,r11 , where σ3,q = β1 +α1 (2−2q) and similarly that ∥Ki∥p +p ≤ Cρσ3,p, i = 0, 1. +Note that 2−2q +αjq < 1 for any j = 1, . . . , m. Furthermore, we have used the H¨older’s inequality ∥uv∥q ≤ +∥u∥qr∥v∥qs with 1 +r + 1 +s = 1 and the inclusions Lp(Ωǫ) ֒→ Lpr(Ωǫ) for any r > 1 and H1 +0(Ωǫ) ֒→ Lq(Ωǫ) +for any q > 1. Let us stress that we can choose p, rij and sij, i = 1, 2, j = 0, 1, 2, close enough to 1 +such that σ′ +p > 0. +□ +Lemma 4.3. There exist p0 > 1 and ρ0 > 0 so that for any 1 < p < p0 and all 0 < ρ ≤ ρ0 it holds +∥N(φ1) − N(φ2)∥p ≤ Cρσ′′ +p ∥φ1 − φ2∥ +(4.20) +for all φi ∈ H1 +0(Ωǫ) with ∥φi∥ ≤ Mρσp| log ρ|, i = 1, 2, and for some σ′′ +p > 0. In particular, we have +that +∥N(φ)∥p ≤ Cρσ′′ +p ∥φ∥ +(4.21) +for all φ ∈ H1 +0(Ωǫ) with ∥φ∥ ≤ Mρσp| log ρ|. +Proof: Arguing in the same way as in [1, Lemma 5.1], we denote gi(φ) = +Vi(x)e(−τ)i(U+φ) +� +Ωǫ Vi(x)e(−τ)i(U+φ) and point +out that +N(φ) = +1 +� +i=0 +λi(−τ)i {gi(φ) − gi(0) − g′ +i(0)[φ]} +and by the mean value theorem we get that +N(φ1) − N(φ2) = +2 +� +i=1 +λi(−τ)ig′′ +i (˜φµi)[φθi, φ1 − φ2], +(4.22) +where φθi = θiφ1 + (1 − θi)φ2, ˜φµi = µiφθi for some θi, µi ∈ [0, 1], i = 0, 1, and +g′′ +i (φ)[ψ, v] = τ 2i +� Vi(x)euiψv +� +Ωǫ Vi(x)eui − +Vi(x)euiv +� +Ωǫ Vi(x)euiψ +� � +Ωǫ Vi(x)eui�2 +− +Vi(x)euiψ +� +Ωǫ Vi(x)euiv +� � +Ωǫ Vi(x)eui�2 +− +Vi(x)eui � +Ωǫ Vi(x)euiψv +� � +Ωǫ Vi(x)eui�2 ++ 2 +Vi(x)eui � +Ωǫ Vi(x)euiv +� +Ωǫ Vi(x)euiψ +� � +Ωǫ Vi(x)eui�3 +� +, + +20 +P. FIGUEROA +where for simplicity we denote ui = (−τ)i(U + φ). Using H¨older’s inequalities we get that +∥g′′ +i (φ)[ψ, v]∥p ≤ C +� +∥Vieui∥pri +∥Vieui∥1 ++ ∥Vieui∥2 +pri +∥Vieui∥2 +1 ++ ∥Vieui∥3 +pri +∥Vieui∥3 +1 +� +∥ψ∥∥v∥, +(4.23) +with +1 +ri + 1 +si + 1 +ti = 1, +1 +ri + 1 +qi = 1, +1 +pri + 1 +˜ri = 1. We have used the H¨older’s inequality, the inclusions +presented in the previous Lemma and ∥uvw∥q ≤ ∥u∥qr∥v∥qs∥w∥qt with 1 +r + 1 +s + 1 +t = 1. Now, let us +estimate +∥Vieui ∥pri +∥Vieui ∥1 +with φ = ˜φµi, i = 0, 1. Taking into account (3.28)-(3.29) and (4.2)-(4.3), we obtain +that for i = 0, 1 +���Vie(−τ)iU��� +q +q = O +� +ρ +β1 +α1 (2−2q)−q� +for any q ≥ 1 +and +���Vie(−τ)iU��� +1 ≥ Ci +ρ +for some Ci > 0. +Note that σ3,q − 1 ≤ +2−(αi+2)q +αiq +for any +i = 1, . . . , m. +By the previous estimates we find that +��Vie(−τ)iU+ ˜φµi �� +q = O +� +ρ +η0,qs′ +i −1+ηp| log ρ| + ρη0,q−1� +(on the line of [17, eq. (3.14) proof of Lemma 3.3]. +Also, choosing si, i = 0, 1, close enough to 1, we get that σp + η0,si > 0 and +���Vie(−τ)i(U+ ˜φµi )��� +1 ≥ Ci +ρ − Cρη0,si −1+σp| log ρ| ≥ 1 +ρ +� +Ci − Cρη0,si +σp | log ρ| +� +≥ Ci +2ρ. +Taking q = pri, we obtain the estimate for i = 0, 1 +∥Vie(−τ)i(U+ ˜φµi )∥pri +∥Vie(−τ)i(U+ ˜φµi )∥1 += O +� +ρσ3,pri +� +ρσp+σ3,prisi −σpri| log ρ| + 1 +�� += O (ρσ3,pri ) +(4.24) +choosing si > 1 close enough to 1 so that σp + σ3,prisi − σpri > 0, i = 1, 2. Now, we can conclude the +estimate by using (4.22)-(4.24) to get +∥N(φ1) − N(φ2)∥p ≤ C +1 +� +i=0 +λiρσp+3σ3,pri | log ρ|∥φ1 − φ2∥ ≤ Cρσ′′ +p ∥φ1 − φ2∥, +where σ′′ +p = 1 +2 min{σp + 3σ3,pri +: +i = 0, 1} > 0 choosing ri close to 1 so that σp + 3σ3,pri > 0 for +i = 0, 1. Let us stress that p > 1 is chosen so that σp > 0. +□ +Taking into account previous estimates (4.1), (4.19), (4.20) and (4.21), and by using the same +argument as in the proof of Proposition 3.2 we conclude the following +Proposition 4.2. There exist p0 > 1 and ρ0 > 0 so that for any 1 < p < p0 and all 0 < ρ ≤ ρ0, the +problem (4.12) admits a unique solution φ(ρ) ∈ H1 +0(Ωǫ), where L, R, Λ0(φ) and N are given by (4.13), +(1.13), (4.14) and (4.16), respectively. Moreover, there is a constant C > 0 such that +∥φ∥∞ ≤ Cρσp| log ρ|, +for some σp > 0. +Proof of the Theorem 1.2. Arguing in the same way as in the proof of Theorem 1.1, the existence +of a solution (1.10) to equation (1.2) follows directly by Proposition 4.2 and the definition of U in +(1.11). +□ +5. The linear theory +In this section we present the invertibility of the linear operators L and L defined in (3.14) and (4.13) +respectively. Roughly speaking in the scale annulus Aj +δj the operator L apporaches to the following +linear operator in IR2 +Lj(φ) = ∆φ + 2α2 +j|y|αj−2 +(1 + |y|αj)2 φ, +j = 1, . . . , m. + +SIGN-CHANGING BUBBLE TOWER SOLUTIONS ON PIERCED DOMAINS +21 +It is well known that, in case αj ∈ 2IN, the bounded solutions of Lj(φ) = 0 in IR2 are precisely linear +combinations of the functions +Y1j(y) = +|y| +αj +2 +1 + |y|αj cos +�αj +2 θ +� +, +Y2j(y) = +|y| +αj +2 +1 + |y|αj sin +�αj +2 θ +� +and +Y0j(y) = 1 − |y|αj +1 + |y|αj , +which are written in polar coordinates for j = 1, . . . , m. See [10] for a proof. In our case, we will +consider solutions of Lj(φ) = 0 with αj /∈ 2IN for all j = 1, . . . , m such that +� +IR2 |∇φ(y)|2 dy < +∞, +which are multiples of Y0j, see [1, Theorem A.1] for a proof. Another key element in the study of L, +which shows technical details, is to get rid of the presence of +˜cj(φ) = − +1 +λjτ 2j +� +Ωǫ +Kjφ +j = 0, 1. +(5.1) +Let us introduce the following Banach spaces for j = 1, 2, . . . , m +Lαj(IR2) = +� +u ∈ W 1,2 +loc (IR2) : +� +IR2 +|y|αj−2 +(1 + |y|αj)2 |u(y)|2 dy < +∞ +� +(5.2) +and +Hαj(IR2) = +� +u ∈ W 1,2 +loc (IR2) : +� +IR2 |∇u(y)|2 dy + +� +IR2 +|y|αj−2 +(1 + |y|αj)2 |u(y)|2 dy < +∞ +� +(5.3) +endowed with the norms +∥u∥Lαj := +�� +IR2 +|y|αj−2 +(1 + |y|αj)2 |u(y)|2 dy +�1/2 +and +∥u∥Hαj := +�� +IR2 |∇u(y)|2 dy + +� +IR2 +|y|αj−2 +(1 + |y|αj)2 |u(y)|2 dy +�1/2 +. +We point out the compactness of the embedding iαj : Hαj(IR2) → Lαj(IR2), (see for example [20]). +Proof of Proposition 3.1. Let us assume the existence of p > 1, sequences ρn → 0, ǫn = ǫ(ρn) → 0, +functions hn ∈ Lp(Ωǫn), φn ∈ H1 +0(Ωǫn) such that +L(φn) = hn in Ωǫn, +φn = 0 on ∂Ωǫn +∥φn∥ = 1 and | log ρn| ∥hn∥p = o(1) as n → +∞. We will omit the subscript n in δi,n = δi. Recall that +δαi +i += di,nρβi +n . Now, define Φi,n(y) := φi,n(δiy) for y ∈ Ωi,n := δ−1 +i +Ωǫn. Thus, extending φn = 0 in +IR2 \ Ωǫn and arguing in the same way as in [17, Claim 1, section 4] we can prove the following fact. +We provide a sketch of the proof. +Claim 5.1. There holds that the sequence Φi,n converges (up to subsequence) to Φ∗ +i = aiY0i for +i = 1, . . . , m, weakly in Hαi(IR2) and strongly in Lαi(IR2) as n → +∞ for some constant ai ∈ IR, +i = 1, . . . , m. +Proof: First, notice that ∥Φi,n∥H1 +0 (Ωi,n) = 1, for i = 1, . . . , m. Then, we want to prove that there is a +constant M > 0 such for all n (up to a subsequence) ∥Φi,n∥2 +Lαi ≤ M. Notice that for any i ∈ {1, . . . , m} +we find that in Ωi,n +∆Φi,n + δ2 +i K(δiy)Φi,n = δ2 +i hn(δiy). +(5.4) +Furthermore, it follows that Φi,n → Φ∗ +i weakly in H1 +0(Ωi,n) and strongly in Lp(K) for any K compact +sets in R2. Now, we multiply (5.4) by Φi,n for any i ∈ {1, . . . , m}, integrate by parts and we obtain +that +m +� +i=1 +2α2 +i ∥Φi,n∥2 +Lαi = 1 + o(1) +(5.5) +since +δ2 +i K(δiy) = 2α2 +i |y|αi−2 +(1 + |y|αi)2 + O +� � +j 0, as a consequence of [13, +Lemma 4.1] with m = 1 and ξ1 = 0. Now, introduce the coefficient γ0 as +γ0 +� +− 1 +2π log ǫ + H(0, 0) +� += 2, +(5.7) +so that from (2.21), γ0 = − 4π(α1−2) +β1+1 +· +1 +log ρ + O +� +1 +| log ρ|2 +� +. +For simplicity, we shall denote Πi(y) = 2α2 +i |y|αi−2 +(1+|y|αi)2 , for i = 1, . . . , m. +Claim 5.2. For all j = 1, . . . , m, there exist the limit, up to subsequences, +µj = +lim +n→+∞ log ρn +� +Ωj,n +Πj(y)Φj,n(y) dy. +(5.8) +Furthermore, it holds +2 +m +� +i=1 +ai = +� +− +β1 + 1 +4π(α1 − 2) + +m +� +i=1 +(−1)i+1 βi +2παi +� +µ1, +(5.9) +and for all j = 1, . . . , m − 1. +µj = (−1)j−1µ1. +(5.10) +Proof: Consider tests functions γ−1 +0 PZ0j’s and the assumption on hn, | log ρn| ∥hn∥∗ = o(1). Hence, +multiplying equation (5.4) by PǫZ0j and integrating by parts we obtain that +� +Ωǫ +hPZ0j = − +� +Ωǫ +φ|x|αj−2ewj Z0j + +� +Ωǫ +KφPǫZ0j = +� +Ωǫ +� +K − |x|αj−2ewj� +PǫZ0jφ ++ +� +Ωǫ +|x|αj−2ewj (PZ0j − Z0j) φ = +� +Ωǫ +φ +m +� +i=1 +i̸=j +|x|αi−2ewiPZ0j + +� +Ωǫ +|x|αj−2ewj � +1 − γ0G(x, 0) + O(ρ˜σ) +� +φ +in view of PǫZ0j = 0 on ∂Ωǫn and +� +Ωǫ ∆φPǫZ0j = +� +Ωǫ φ ∆PǫZ0j = +� +Ωǫ φ ∆Z0j. Now, multiplying by +γ−1 +0 +and estimating every integral term we find that γ−1 +0 +� +Ωǫ hPZ0j = O ( | log ρ| ∥h∥p) = o (1) , in view +of PZ0j = O(1) and γ−1 +0 += − +β1+1 +4π(α1−2) log ρ + O(1) = O(| log ρ|) in (5.7). Next, scaling we obtain that +γ−1 +0 +� +Ωǫ +φ|x|αj−2ewj = − +β1 + 1 +4π(α1 − 2) log ρ +� +Ωj,n +2α2 +j|y|αj−2 +(1 + |y|αj)2 Φj,n dy + o(1), +in view of +lim +� +Ωj,n +2α2 +j|y|αj−2 +(1 + |y|αj)2 Φj,n dy = aj +� +IR2 +2α2 +j|y|αj−2 +(1 + |y|αj)2 Y0,j dy = 0 + +SIGN-CHANGING BUBBLE TOWER SOLUTIONS ON PIERCED DOMAINS +23 +Also, by using (5.7) and expansion of PǫZ0j, we get that for i < j +γ−1 +0 +� +Ωǫ +|x|αi−2ewiφPZ0j = +� +Ωi,n +2α2 +i |y|αi−2 +(1 + |y|αi)2 Φi,n + + +2γ−1 +0 +1 + ( δi|y| +δj )αj + 1 +2π log(δi|y|) − H(δi|y|, 0) + + dy ++ O(ρ˜σ| log ρ|) = 2γ−1 +0 +� +Ωi,n +ΠiΦi,n(y) dy + 1 +2π log δi +� +Ωi,n +ΠiΦi,n(y) dy ++ 1 +2π +� +Ωi,n +ΠiΦi,n(y) log |y| dy + o(1), +and for i > j we have that +γ−1 +0 +� +Ωǫ +|x|αi−2ewiφPZ0j = +� +Ωi,n +ΠiΦi,n +��δj +δi +�αj +2γ−1 +0 +( δj +δi )αj + |y|αj + 1 +2π log(δi|y|) − H(δi|y|, 0) +� +dy ++ O(ρ˜σ| log ρ|) = 1 +2π log δi +� +Ωi,n +ΠiΦi,n(y) dy + 1 +2π +� +Ωi,n +ΠiΦi,n(y) log |y| dy + o(1), +in view of +� δj +δi +�αjγ−1 +0 += o(1). Furthermore, it follows that +� +Ωǫ +|x|αj−2ewjG(x, 0)φ = − 1 +2π log δj +� +Ωj,n +Πj(y)Φj,n(y) dy − 1 +2π +� +Ωj,n +Πj(y)Φj,n(y) log |y| dy + o(1). +From the choice of δj’s in (2.2), it follows that log δj = +1 +αj +� +log dj+βj log ρ +� +and from previous expansions +we deduce that for all j = 2, . . . , m − 1 +γ−1 +0 +� +Ωǫ +hPZ0j = +� +ij +� βi +2παi log ρ +� +Ωi,n +ΠiΦi,n(y) dy + 1 +2π +� +Ωi,n +ΠiΦi,n(y) log |y| dy +� +− +β1 + 1 +4π(α1 − 2) log ρ +� +Ωj,n +ΠjΦj,n dy ++ +βj +2παj log ρ +� +Ωj,n +Πj(y)Φj,n dy + 1 +2π +� +Ωj,n +Πj(y)Φj,n(y) log |y| dy + o(1), +Since for all i = 1, . . . , m +lim +� +Ωi,n +ΠiΦi,n(y) log |y| dy = ai +� +IR2 ΠiY0,i(y) log |y| dy = −4πai +(5.11) +we obtain that for all i = 2, . . . , m − 1 +2 +m +� +i=1 +ai = +� +ij +� βi +2παi log ρ +� +Ωi,n +ΠiΦi,n(y) dy +� ++ +� +− +β1 + 1 +4π(α1 − 2) + +βj +2παj +� +log ρ +� +Ωj,n +Πj(y)Φj,n dy + o(1). +(5.12) +Choosing j ∈ {2, . . ., m − 2} we rewrite previous expression for j + 1 as +2 +m +� +i=1 +ai = +� +ij+1 +� βi +2παi log ρ +� +Ωi,n +ΠiΦi,n +� ++ +� +− +β1 + 1 +4π(α1 − 2) + +βj+1 +2παj+1 +� +log ρ +� +Ωj+1,n +Πj+1Φj+1,n + o(1). +(5.13) +Hence, by subtracting (5.12) from (5.13) we obtain that +0 = − +β1 + 1 +4π(α1 − 2) log ρ +�� +Ωj,n +Πj(y)Φj,n dy + +� +Ωj+1,n +Πj+1(y)Φj+1,n dy +� ++ o(1), + +24 +P. FIGUEROA +so that, for all j = 2, . . . , m − 2 +lim log ρn +�� +Ωj,n +Πj(y)Φj,n dy + +� +Ωj+1,n +Πj+1(y)Φj+1,n dy +� += 0. +(5.14) +Similarly as above it extends to j = 1 +2 +m +� +i=1 +ai = +� +i>1 +� βi +2παi +log ρ +� +Ωi,n +ΠiΦi,ndy +� ++ +� +− +β1 + 1 +4π(α1 − 2) + +β1 +2πα1 +� +log ρ +� +Ω1,n +Π1Φ1,ndy + o(1) +and +lim log ρn +�� +Ω1,n +Π1(y)Φ1,n dy + +� +Ω2,n +Π2(y)Φ2,n dy +� += 0; +and to j = m +2 +m +� +i=1 +ai = +� +i j we obtain that +� +Ωǫ +|x|αi−2ewiφPǫwj = +� +Ωi,n +ΠiΦi +� +− 2αj log δi − 2 log +��δj +δi +�αj + |y|αj � ++ (4παj − γj)H(δiy, 0) ++ γj +2π log |δiy| + O(ρη) +� += −2αj log δi +� +Ωi,n +ΠiΦi − 2 +� +Ωi,n +ΠiΦi log +��δj +δi +�αj + |y|αj � ++ (4παj − γj) +� +Ωi,n +ΠiΦiH(δiy, 0) + γj log δi +2π +� +Ωi,n +ΠiΦi + γj +2π +� +Ωi,n +ΠiΦi log |y| + o(1), +and for i < j +� +Ωǫ +|x|αi−2ewiφPǫwj = +� +Ωi,n +ΠiΦi +� +− 2αj log δj − 2 log +� +1 + +�δi|y| +δj +�αj � ++ (4παj − γj)H(δiy, 0) + γj +2π log |δiy| ++ O(ρη) +� += −2αj log δj +� +Ωi,n +ΠiΦi − 2 +� +Ωi,n +ΠiΦi log +� +1 + +�δi|y| +δj +�αj� ++ (4παj − γj) +� +Ωi,n +ΠiΦiH(δiy, 0) ++ γj log δi +2π +� +Ωi,n +ΠiΦi + γj +2π +� +Ωi,n +ΠiΦi log |y| + o(1), +Now, by using the definition of γj’s in (2.20) and Lemma 2.1 we find that γj +2π = 2βj α1−2 +β1+1 + O +� +1 +| log ρ| +� +so that from the choice of δj’s we obtain that γj +2π log δi = 2βj α1−2 +β1+1 +βi +αi log ρ + O(1). Hence, from previous + +26 +P. FIGUEROA +expansions we deduce that for all j = 2, . . . , m − 1 +o(1) = +� +ij +�� +− 2αj log δi + γj log δi +2π +� � +Ωi,n +ΠiΦi + γj +2π +� +Ωi,n +ΠiΦi log |y| +− 2 +� +Ωi,n +ΠiΦi log +��δj +δi +�αj + |y|αj�� += −2 +� +Ωj,n +ΠjΦj log(1 + |y|αj) ++ +� +i≤j +� +− 2βj + 2βj α1 − 2 +β1 + 1 +βi +αi +� +log ρ +� +Ωi,n +ΠiΦi + +m +� +i=1 +2βj α1 − 2 +β1 + 1 +� +Ωi,n +ΠiΦi log |y| + o(1) ++ +� +i>j +�� +− 2αj βi +αi + 2βj α1 − 2 +β1 + 1 +βi +αi +� +log ρ +� +Ωi,n +ΠiΦi − 2 +� +Ωi,n +ΠiΦi log +��δj +δi +�αj + |y|αj �� +. +(5.15) +Since for all j = 1, . . . , m +lim +� +Ωj,n +ΠjΦj log(1 + |y|αj) = aj +� +IR2 ΠjY0j log(1 + |y|αj) = −2παjaj, +(5.16) +by using (5.8) +lim +� +Ωi,n +ΠiΦi log +��δj +δi +�αj ++ |y|αj� += αjai +� +IR2 ΠiY0i log |y| = −4παjai, +in view of δj +δi = o(1), and (5.11), as n → +∞ we get that +0 =4παjaj + +� +i≤j +� +− 2βj + 2βj α1 − 2 +β1 + 1 +βi +αi +� +µi + +m +� +i=1 +2βj α1 − 2 +β1 + 1 (−4πai) ++ +� +i>j +�� +− 2αj βi +αi + 2βj α1 − 2 +β1 + 1 +βi +αi +� +µi − 2(−4παjai) +� +so that +0 = 4παjaj + +� +i≤j +� +− 2βj + 2βj α1 − 2 +β1 + 1 +βi +αi +� +µi − 8πβj α1 − 2 +β1 + 1 +m +� +i=1 +ai ++ +� +i>j +� +− 2αj βi +αi + 2βj α1 − 2 +β1 + 1 +βi +αi +� +µi + 8παj +� +i>j +ai. +It is readily checked that 8παj +� +i>j ai = 4παj +� +i>j ai + 4παj +� +i≥j+1 ai, so we rewrite +0 = 4παj +� +i≥j +ai + 4παj +� +i≥j+1 +ai − 8πβj α1 − 2 +β1 + 1 +m +� +i=1 +ai − 2βjµ1 +� +i≤j +(−1)i+1 − 2αjµ1 +� +i>j +(−1)i+1 βi +αi ++ 2βj α1 − 2 +β1 + 1 µ1 +m +� +i=1 +(−1)i+1 βi +αi = 4παj +� +i≥j +ai + 4παj +� +i≥j+1 +ai − 8πβj α1 − 2 +β1 + 1 +m +� +i=1 +ai ++ µ1 +� +− 2βj 1 + (−1)j+1 +2 +− 2αj +� +i>j +(−1)i+1 βi +αi + 2βj α1 − 2 +β1 + 1 +m +� +i=1 +(−1)i+1 βi +αi +� +, +in view of (5.10). For simplicity we shall denote for j = 1, . . . , m +xj = +m +� +i=j +ai, +Aj = − βj +2π +1 + (−1)j+1 +2 +− αj +2π +� +i>j +(−1)i+1 βi +αi ++ βj +2π +α1 − 2 +β1 + 1 B, + +SIGN-CHANGING BUBBLE TOWER SOLUTIONS ON PIERCED DOMAINS +27 +and +Am+1 = − 1 +2π +� +− β1 + 1 +α1 − 2 + B +� +, +with +B = +m +� +i=1 +(−1)i+1 βi +αi +. +Hence, taking into account (5.9) and dividing by 4π, we have the following linear system in m + 1 +variables X = [x2 x3 . . . xm x1 µ1]T +0 = αjxj + αjxj+1 − 2βj α1 − 2 +β1 + 1 x1 + Ajµ1, +j = 2, . . . , m − 1 +0 = αmxm − 2βj α1 − 2 +β1 + 1 x1 + Amµm +0 = α1x2 + +� +α1 − 2β1 α1 − 2 +β1 + 1 +� +x1 + A1µ1 +0 = 2x1 + Am+1µ1 +which in matrix form becomes AX = ⃗0 with +A = + + +α2 +α2 +0 +· · · +0 +−2β2 +α1−2 +β1+1 +A2 +0 +α3 +α3 +· · · +0 +−2β3 +α1−2 +β1+1 +A3 +0 +0 +α4 +· · · +0 +−2β4 +α1−2 +β1+1 +A4 +... +... +... +... +... +... +... +0 +0 +0 +· · · +αm +−2βm +α1−2 +β1+1 +Am +α1 +0 +0 +· · · +0 +α1 − 2β1 +α1−2 +β1+1 +A1 +0 +0 +0 +· · · +0 +2 +Am+1 + + += + + +f1 +f2 +... +fm +fm+1 + + +Operating with rows fj’s of A in the following form +fm → fm + (−1)j α1 +αj+1 +fj, +j = 1, . . . , m − 1 +we are lead to consider the matrix + + +α2 +α2 +0 +· · · +0 +−2β2 α1−2 +β1+1 +A2 +0 +α3 +α3 +· · · +0 +−2β3 α1−2 +β1+1 +A3 +0 +0 +α4 +· · · +0 +−2β4 α1−2 +β1+1 +A4 +... +... +... +... +... +... +... +0 +0 +0 +· · · +αm +−2βm α1−2 +β1+1 +Am +0 +0 +0 +· · · +0 +α1 − 2α1B α1−2 +β1+1 +α1 +2π +� +− 1 + α1−2 +β1+1 B +� +0 +0 +0 +· · · +0 +2 +Am+1 + + +since +α1 − 2β1 +α1 − 2 +β1 + 1 + +m−1 +� +j=1 +(−1)j α1 +αj+1 +(−2βj+1)α1 − 2 +β1 + 1 = α1 − 2α1B α1 − 2 +β1 + 1 +and +A1 + +m−1 +� +j=1 +(−1)j α1 +αj+1 Aj+1 = − α1 +2π +� β1 +α1 + +m +� +i=2 +(−1)i+1 βi +αi +� +�� +� +=B +� ++ β1 +2π +α1 − 2 +β1 + 1 B + α1 +2π +α1 − 2 +β1 + 1 B +m−1 +� +j=1 +(−1)j βj+1 +αj+1 +� +�� +� += α1 +2π +α1−2 +β1+1 B2 ++ α1 +2π +m−1 +� +j=1 +(−1)j +� +− βj+1 +αj+1 +1 + (−1)j +2 ++ +� +i>j+1 +(−1)i βi +αi +� +� +�� +� +=0 += α1 +2π B +� +− 1 + α1 − 2 +β1 + 1 B +� +. +Notice that A is invertible, since +det +� +α1 − 2α1B α1−2 +β1+1 +α1 +2π +� +− 1 + α1−2 +β1+1 B +� +2 +Am+1 +� += α1 +4π · β1 + 1 +α1 − 2 > 0. +Therefore, x2 = x3 = · · · = xm = x1 = µ1 = 0, which implies our claim. +□ + +28 +P. FIGUEROA +Now, by using Claims 5.1-5.3 we deduce that Φj,n converges to zero weakly in Hαj(IR2) and strongly +in Lαj(IR2) as n → +∞. Thus, we arrives at a contradiction with (5.5) and it follows the priori estimate +∥φ∥ ≤ C| log ρ| ∥h∥p. It only remains to prove the solvability assertion. As usual, expressing (3.17) in +weak form, with the aid of Riesz’s representation theorem and Fredholm’s alternative, the existence +of a unique solution follows from the a priori estimate (3.18). This finishes the proof of Proposition +3.1. +□ +Proof of the Proposition 4.1. Arguing as above, it is enough to prove the a-priori estimate (4.18). +Let us assume by contradiction the existence of p > 1, sequences ρn → 0, functions hn ∈ Lp(Ωǫn), +φn ∈ W 2,2(Ωǫn) such that +L(φn) = hn in Ωǫn, +φn = 0 on ∂Ωǫn, +(5.17) +with ∥φn∥ = 1 and | log ρn| ∥hn∥p = o(1) as n → +∞. We will omit the subscript n in δi,n = δi. Recall +that δαi +i += di,nρβi +n . Now, define Φi,n(y) := φn(δiy) for y ∈ Ωi,n := δ−1 +i +Ωǫn, i = 1, . . . , m and extend +φn = 0 in IR2 \ Ωǫn. Arguing in the same way as in [13, Claim 1, section 4], we can prove the following +fact. We provide a sketch of the proof. +Claim 5.4. The sequence {Φi,n}n converges (up to a subsequence) to Φ∗ +i weakly in Hαi(IR2) and +strongly in Lαi(IR2). +Proof: First, we shall show that the sequence {Φi,n}n is bounded in Hαi(IR2). +Notice that +∥Φi,n∥H1 +0(Ωi,n) = 1 for i = 1, . . . , m. Thus, we want to prove that there is a constant M > 0 such +for all n (up to a subsequence) ∥Φi,n∥2 +Lαi ≤ M. Notice that for any i ∈ {1, . . . , m} we find that in Ωi,n +∆Φi,n + δ2 +i K0(δiy) (Φi,n + ˜c0,n) + δ2 +i K1(δiy) (Φi,n + ˜c1,n) = δ2 +i hn(δiy), +(5.18) +where for simplicity we denote ˜cj,n = ˜cj(φn), with ˜cj given by (5.1). Furthermore, it follows that +Φi,n → Φ∗ +i weakly in H1 +0(Ωi,n) and strongly in Lp(K) for any K compact sets in R2. Now, let χ a +smooth function with compact support in R2. We multiply (5.18) by χ, integrate by parts and we +obtain that +− +� +Ωi,n +∇Φi,n∇χ + +� +Ωi,n +� 2α2 +i |y|αi−2 +(1 + |y|αi)2 + o(1) +� +Φi,nχ + ˜c0,n +� +Ωi,n +� 2α2 +i |y|αi−2 +(1 + |y|αi)2 + o(1) +� +χ ++ +� +Ωi,n +o(1)Φi,nχ + ˜c1,n +� +Ωi,n +o(1)χ = +� +Ωi,n +δ2 +i hn(δiy)χ +for i odd and +− +� +Ωi,n +∇Φi,n∇χ + +� +Ωi,n +o(1)Φi,nχ + ˜c0,n +� +Ωi,n +o(1)χ + +� +Ωi,n +� 2α2 +i |y|αi−2 +(1 + |y|αi)2 + o(1) +� +Φi,nχ ++ ˜c1,n +� +Ωi,n +� 2α2 +i |y|αi−2 +(1 + |y|αi)2 + o(1) +� +χ = +� +Ωi,n +δ2 +i hn(δiy)χ +for i even, in view of +δ2 +i K0(δiy) = + + + + + + + + + + + + + + + +2α2 +i |y|αi−2 +(1 + |y|αi)2 + O +� � +j 0, so that for ψ0 and ψ1 we have that +� +Ωǫ +Kiψi = +� +1 − +1 +λiτ 2i +� +Ωǫ +Ki +� � +Ωǫ +Kiφ = O(ρ˜σ) +� +Ωǫ +Kiφ = O(ρ˜σ). +Also, we get that +� +Ωǫ +K0ψ0 = +m +� +j=1 +j odd +� +Ωǫ +|x|αj−2ewjψi = +m1 +� +j=1 +j odd +� +Ωj,n +2α2 +j|y|αj−2 +(1 + |y|αj)2 Ψ0,j,n(y) dy +and +� +Ωǫ +K1ψ1 = +m +� +j=1 +j even +� +Ωj,n +2α2 +j|y|αj−2 +(1 + |y|αj)2 Ψ1,j,n(y) dy. + +SIGN-CHANGING BUBBLE TOWER SOLUTIONS ON PIERCED DOMAINS +31 +Hence, it follows that +lim +n→+∞ log ρn +� +Ωǫn +K0ψ0 = 0 = +m +� +j=1 +j odd +µj = µ1 +m +� +j=1 +j odd +1, +so that, µj = 0 for all j = 1, . . . , m. +□ +Claim 5.7. There hold that +0 = ˜cl + aj + 2 +� +i>j +ai, +(5.26) +where l = 0 for j odd and l = 1 for j even. +Proof: To this aim we will use as tests functions Pǫwj. Hence, multiplying equation (5.4) by Pǫwj +and integrating by parts as in the previous Claim we obtain that +� +Ωǫ +hPwj = − +� +Ωǫ +[ψl − ˜cl,n]|x|αj−2ewj + +� +Ωǫ +[K0ψ0 + K1ψj]Pǫwj += − +� +Ωǫ +ψl|x|αj−2ewj + ˜cl,n +� +Ωǫ +|x|αj−2ewj + +� +Ωǫ +[K0ψ0 + K1ψ1]Pǫwj +in view of Pǫwj = 0 on ∂Ωǫn and +� +Ωǫ ∆ψlPǫwj = +� +Ωǫ ψl ∆Pǫwj. Arguing in the same way as in Claim +5.3 (replacing Ψl,i,n by Φi,n with l = 0 for j odd and l = 1 for j even in (5.15) and (5.16)) we find that +0 = 4παj˜cl + 4παjaj + +� +i≤j +� +− 2βj + 2βj +α1 − 2 +β1 + 1 +βi +αi +� +µi − 8πβj +α1 − 2 +β1 + 1 +m +� +i=1 +ai ++ +� +i>j +� +− 2αj +βi +αi ++ 2βj +α1 − 2 +β1 + 1 +βi +αi +� +µi + 8παj +� +i>j +ai, +in view of +� +Ωǫ +hPwj = O ( | log ρ| ∥h∥p) = o (1) , +� +Ωǫ +ψl|x|αj−2ewj = +� +Ωj,n +2α2 +j|y|αj−2 +(1 + |y|αj)2 Ψl,j,n = aj +� +IR2 +2α2 +j|y|αj−2 +(1 + |y|αj)2 Y0j dy + o(1) = o(1) +and +� +Ωǫ +|x|αj−2ewj = +� +Ωj,n +2α2 +j|y|αj−2 +(1 + |y|αj)2 dy = +� +IR2 +2α2 +j|y|αj−2 +(1 + |y|αj)2 dy + o(1) = 4παj + o(1). +By using (5.24) and (5.25) we obtain that (5.26) for l = 0 with j odd and l = 1 with j even. +□ +For the next step, consider the function ηj(x) = 4 +3 log(δαj +j ++ |x|αj) +δ +αj +j +−|x|αj +δ +αj +j ++|x|αj + 8 +3 +δ +αj +j +δ +αj +j ++|x|αj , for any +j ∈ {1, . . . , m} so that ∆ηj + |x|αj−2ewjηj = |x|αj−2ewjZ0j. Notice that +Pǫηj = ηj + 8π +3 αjH(x, 0) − ˜γjG(x, 0) + O(ρ˜σ) +uniformly in Ωǫ for some ˜σ > 0, +by using similar arguments as to obtain expansion in Lemma 2.1, as shown in [13, Lemma 4.1] with +m = 1 and ξ = 0, where the coefficients ˜γi’s, i = 1, . . . , m, are given by +˜γi +� +− 1 +2π log ǫ + H(0, 0) +� += 4 +3αi log δi + 8 +3 + 8π +3 αiH(0, 0) +(5.27) +From (2.2) it follows that ˜γi = − 8π(α1−2)βi +3(β1+1) ++ O +� +1 +| log ρ| +� +,. +Claim 5.8. There hold that aj + 4 � +i>j ai + 2˜cl = 0 with l = 0 for all j odd and l = 1 for all j even. +Consequently, combining with (5.26) it follows that ai = 0 for all i = 1, . . . , m. + +32 +P. FIGUEROA +Proof: We use the following test function Pǫηj. Thus, from the assumption on hn, | log ρn| ∥hn∥∗ = +o(1), we get the above relation between aj and ˜ci either for l = 0 and all j odd or for l = 1 and all j +even. Assume that l = 0 for all j odd or l = 1 for all j even. Multiplying equation (5.17) by Pǫηj and +again integrating by parts we obtain that +� +Ωǫ +hPηj = +� +Ωǫ +[ψl − ˜cl,n] +� +|x|αj −2ewj Z0j − |x|αj−2ewjηj +� ++ +� +Ωǫ +[K0ψ0 + K1ψ1] Pǫηj = +� +Ωǫ +ψi|x|αj−2ewjZ0j ++ +� +Ωǫ +|x|αj−2ewj ψl (Pηj − ηj) + +� +Ωǫ +� +K0ψ0 + K1ψ1 − |x|αj−2ewjψl +� +Pǫηj − ˜cl,n +� +Ωǫ +|x|αj −2ewj (Z0j − ηj) , +Now, estimating every integral term we find that +� +Ωǫ hPηj = O ( | log ρ| ∥h∥p) = o (1) for all j = +1, . . . , m, in view of Pηj = O(| log ρ|) and G(x, 0) = O(| log ρ|). Next, by scaling we obtain that either +for l = 0 and all j odd or l = 1 and all j even, it holds +� +Ωǫ +ψl|x|αj−2ewjZ0j = +� +Ωj,n +2α2 +j|y|αj−2 +(1 + |y|αj)2 Ψl,j,nY0j dy = aj +� +IR2 +2α2 +j|y|αj−2 +(1 + |y|αj)2 Y 2 +0j dy + o(1). +Note that +� +IR2 +2α2 +j|y|αj−2 +(1 + |y|αj)2 Y 2 +0j = +� +IR2 +2α2 +j|y|αj−2 +(1 + |y|αj)2 +�1 − |y|αj +1 + |y|αj +�2 +dy = 4π +3 αj +and +� +IR2 +2α2 +j|y|αj−2 +(1 + |y|αj)2 Y0j log |y| = +� +IR2 +2α2 +j|y|αj−2 +(1 + |y|αj)2 +1 − |y|αj +1 + |y|αj log |y| dy = −4π. +Also, by using (5.27) we get that +� +Ωǫ +|x|αj−2ewjψl (Pǫηj − ηj) = +� +Ωǫ +|x|αj−2ewjψl +� �8π +3 αj − ˜γj +� +H(x, 0) + 1 +2π ˜γj log |x| + O(ρ˜σ) +� += +�8π +3 αj − ˜γj +� � +Ωj,n +2α2 +j|y|αj−2 +(1 + |y|αj)2 Ψl,j,nH(δjy, 0) dy + ˜γj +2π log δj +� +Ωj,n +2α2 +j|y|αj−2 +(1 + |y|αj)2 Ψl,j,n dy ++ ˜γj +2π +� +Ωj,n +2α2 +j|y|αj−2 +(1 + |y|αj)2 Ψl,j,n log |y|dy + O(ρ˜σ) = −4(α1 − 2)βj +3(β! + 1) aj +� +IR2 +2α2 +j|y|αj−2 +(1 + |y|αj)2 Y0j log |y|dy+o(1), +in view of (5.24). Furthermore, using (4.15) we have that +Notice that +� +Ωj,n +2α2 +j|y|αj−2 +(1 + |y|αj)2 Ψi,j,n(y) dy = aj +� +IR2 +2α2 +j|y|αj−2 +(1 + |y|αj)2 Y0j(y) dy + o(1) = o(1), +since +� +IR2 +2α2 +j|y|αj−2 +(1 + |y|αj)2 Y0j(y) dy = +� +IR2 +2α2 +j|y|αj−2 +(1 + |y|αj)2 · 1 − |y|αj +1 + |y|αj dy = 0. +If i < j then +Pǫηj(δiy) = +�4 +3αj log δj + 4 +3 log +� +1 + +�δi|y| +δj +�αj��1 − +� δi|y| +δj +�αj +1 + +� δi|y| +δj +�αj + 8 +3 · +1 +1 + +� δi|y| +δj +�αj ++ ˜γj +2π log(δi|y|) + +�8π +3 αj − ˜γj +� +H(δiy, 0) + O(ρ˜σ) +and +� +Ωǫ +|x|αi−2ewiψlPǫηj = +� +Ωi,n +ΠiΨl,i,nPǫηj(δiy) = ˜γj +2π +� +Ωi,n +ΠiΨl,i,n(y) log |y| + o(1) += 16π(α1 − 2)βj +3(β1 + 1) +ai + o(1) + +SIGN-CHANGING BUBBLE TOWER SOLUTIONS ON PIERCED DOMAINS +33 +in view of δi|y| +δj += o(1) uniformly for y on compact subsets, (5.24) and dominated convergence. Similarly, +if i > j then +Pǫηj(δiy) = +�4 +3αj log δj + 4 +3 log +��δj +δi +�αj ++ |y|αj��� δj +δi +�αj − |y|αj +� δj +δi +�αj + |y|αj + 8 +3 +�δj +δi +�αj +· +1 +� δj +δi +�αj + |y|αj ++ ˜γj +2π log(δi|y|) + +�8π +3 αj − ˜γj +� +H(δiy, 0) + O(ρ˜σ) +and +� +Ωǫ +|x|αi−2ewiψlPǫηj = +� +Ωi,n +ΠiΨl,i,nPǫηj(δiy) = 4 +3 +� +Ωi,n +ΠiΨl,i,n log +��δj +δi +�αj + |y|αj �� δj +δi +�αj − |y|αj +� δj +δi +�αj + |y|αj ++ ˜γj +2π +� +Ωi,n +ΠiΨl,i,n(y) log |y| + o(1) = −4 +3αjai(−4π) + 16π(α1 − 2)βj +3(β1 + 1) +ai + o(1) +in view of δj +δi = o(1), +4 +3αj log δj +� +Ωi,n +ΠiΨl,i,n +� δj +δi +�αj − |y|αj +� δj +δi +�αj + |y|αj = 4 +3 log dj +� +Ωi,n +ΠiΨl,i,n +� δj +δi +�αj − |y|αj +� δj +δi +�αj + |y|αj − 4 +3βj log ρ +� +Ωi,n +ΠiΨl,i,n ++ 8 +3βj +�δj +δi +�αj log ρ +� +Ωi,n +ΠiΨl,i,n +1 +� δj +δi +�αj + |y|αj = o(1), +� +Ωi,n +ΠiΨl,i,n log +��δj +δi +�αj ++ |y|αj�� δj +δi +�αj − |y|αj +� δj +δi +�αj + |y|αj = − αjai +� +IR2 ΠiY0i log |y| + o(1), +(5.24) and dominated convergence. If l = 0 for j odd and l = 1 for j even, we get that +� +Ωǫ +� +K0ψ0 + K1ψ1 − |x|αj−2ewjψl +� +Pǫηj = +� +ij +�16π(α1 − 2)βj +3(β1 + 1) +ai + 16π +3 αjai +� ++ o(1) = 16π(α1 − 2)βj +3(β1 + 1) +m +� +i=1 +i̸=j +ai + 16π +3 αj +� +i>j +ai + o(1). +Besides, similarly as above we obtain that +� +Ωǫ +|x|αj−2ewj (Z0j − ηj) = +� +Ωǫ +|x|αj−2ewjZ0j − +� +Ωǫ +|x|αj−2ewj ηj = +� +B r +δj +(0)\B ǫj +δj +(0) +2α2 +j|y|αj −2 +(1 + |y|αj )2 +1 − |y|αj +1 + |y|αj +− +� +Ωj,n +2α2 +j|y|αj −2 +(1 + |y|αj )2 +�4 +3 log +� +δ +αj +j ++ δ +αj +j |y|αj� +Y0j(y) + 8 +3 +1 +1 + |y|αj +� +dy + O(δ +αj +j ) += O(ρ˜σ| log ρ|) − 4 +3αj log δj +� +Ωj,n +2α2 +j|y|αj−2 +(1 + |y|αj )2 Y0j(y) dy − 4 +3 +� +Ωj,n +2α2 +j|y|αj−2 +(1 + |y|αj)2 Y0j(y) log (1 + |y|αj ) dy +− 8 +3 +� +Ωj,n +2α2 +j|y|αj −2 +(1 + |y|αj )2 +1 +1 + |y|αj dy = −8π +3 αj + o(1), +in view of +� +IR2 +2α2 +j|y|αj−2 +(1 + |y|αj)2 Y0j(y) log (1 + |y|αj) dy = −2παj +and +� +IR2 +2α2 +j|y|αj−2 +(1 + |y|αj)2 +1 +1 + |y|αj dy = 2παj. +Therefore, we conclude that +o(1) = aj +�4π +3 αj + o(1) +� ++ 16π(α1 − 2)βj +3(β1 + 1) +aj + 16π(α1 − 2)βj +3(β1 + 1) +m +� +i=1 +i̸=j +ai + 16π +3 αj +� +i>j +ai − ˜cl,n +� +−8π +3 αj + o(1) +� +, +and the conclusion follows. +□ + +34 +P. FIGUEROA +Now, by using Claims 5.4-5.8 and arguing similarly to the proof of Proposition 3.1, we deduce the +a-priori estimate (4.18). This finishes the proof. +□ +Appendix A. +Lemma A.1. There exist p0 > 1 close to 1 such that all the following integrals are of order O (ρpηp) +for any 1 < p ≤ p0 and for some ηp > 0: (i) δ2−p +j +� +Aj +δj +�� +|y|αj−1 +(1+|y|αj )2 +��pdy; (ii) ρηδ2−2p +j +� +Aj +δj +�� +|y|αj−2 +(1+|y|αj )2 +��pdy; +(iii) δ2−2p +j +� δi +δj +�αip � +Aj +δj +�� +1 +|y|αj+2 +��pdy, for i < j; (iv) δ2−2p +j +� δj +δi +�αip � +Aj +δj +��|y|αj−2��pdy, for j < i; +(v) ρ(1+τ)pδ2+2τp +j +� +Aj +δj +�� (1+|y|αj )2τ +|y|(αj−2)τ +��pdy, for j odd and (vi) ρ(1+1/τ)pδ2+2p/τ +j +� +Aj +δj +�� (1+|y|αj )2/τ +|y|(αj−2)/τ +��pdy for j +even. +Proof: It is readily checked that for any 1 < p and any j = 1, . . . , m +δ2−p +j +� +Aj +δj +���� +|y|αj−1 +(1 + |y|αj)2 +���� +p +dy = O(δ2−p +j +) = O(ρ(2−p)βj/αj) +and +ρηδ2−2p +j +� +Aj +δj +���� +|y|αj−2 +(1 + |y|αj)2 +���� +p +dy = O(ρηδ2−2p +j +) = O(ρη+(2−2p)βj/αj). +Furthermore, fixing j = 1, . . . , m for any i < j we have that +� +Aj +δj +���� +1 +|y|αj+2 +���� +p +dy = O +��δj−1 +δj +�1−p− αip +2 � +. +Hence, there exist ηp > 0 such that δ2−2p +j +� δi +δj +�αip� δj−1 +δj +�1−p− αi +2 = O(ρpηp), since for p = 1 we have that +� δi +δj +�αi�δj−1 +δj +�− αi +2 = +� δi +δj−1 +�αi�δj−1 +δj +� αi +2 = O +� +ρ +αi( βi +αi − +βj−1 +αj−1 )+ αi +2 ( +βj−1 +αj−1 − +βj +αj )� +and αi +� +βi +αi − βj−1 +αj−1 +� ++ αi +2 +� +βj−1 +αj−1 − βj +αj +� +> 0, in view of i ≤ j − 1 < j and βl +αl is decreasing in l. Similarly, +for j < i we have that +δ2−2p +j +�δj +δi +�αip � +Aj +δj +��|y|αj−2��p dy = O +� +δ2−2p +j +�δj +δi +�αip�δj+1 +δj +�1−p+ αi +2 � +and for p = 1 we find that +� δj +δi +�αi� δj+1 +δj +� αi +2 = +� δj+1 +δi +�αi� δj +δj+1 +� αi +2 = O +� +ρ +αi( +βj+1 +αj+1 − βi +αi )+ αi +2 ( +βj +αj − +βj+1 +αj+1 )� +. Now, +if j odd we have that +� +Aj +δj +���� +(1 + |y|αj)2τ +|y|(αj−2)τ +���� +p +dy = O +��δj−1 +δj +�1+pτ− +αj pτ +2 ++ +�δj+1 +δj +�1+pτ+ +αj pτ +2 +� +. +Then, we get that ρ(1+τ)pδ2+2τp +j +� +δj−1 +δj +�1+pτ− +αj pτ +2 += O +� +ρ +(1+τ)p+ +βj +αj (1+τp+ +αj pτ +2 +)+ +βj−1 +αj−1 (1+pτ− +αj pτ +2 +)� +, and +for p = 1 we find that +1 + τ + βj +αj +� +1 + τ + αjτ +2 +� ++ βj−1 +αj−1 +� +1 + τ − αjτ +2 +� +> 0. +(A.1) +Indeed, αj += +αj−1+2 +τ ++ 2 implies +αjβj−1τ +2αj−1 += +βj−1 +2 ++ βj−1 +αj−1 (1 + τ). Also, βj−1 = +τβj+τ+1 +τ +and +1 +2(1 + τ) + βj +αj (1 + τ) > 0 implies that +1 + τ + βj +αj +� +1 + τ + αjτ +2 +� += 1 + τ + βj +αj +(1 + τ) + βjτ +2 +> βjτ +2 ++ 1 +2(1 + τ) +≥ βj−1 +2 += αjβj−1τ +2αj−1 +− βj−1 +αj−1 +(1 + τ) = − βj−1 +αj−1 +� +1 + τ − αjτ +2 +� + +SIGN-CHANGING BUBBLE TOWER SOLUTIONS ON PIERCED DOMAINS +35 +and (A.1) follows. Similarly, we get that +ρ(1+τ)pδ2+2τp +j +�δj+1 +δj +�1+pτ+ +αj pτ +2 += O +� +ρ +(1+τ)p+ +βj +αj (1+τp− +αjpτ +2 +)+ +βj−1 +αj−1 (1+pτ+ +αj pτ +2 +)� +, +and for p = 1, 1 + τ + βj +αj +� +1 + τ − αjτ +2 +� ++ βj−1 +αj−1 +� +1 + τ + αjτ +2 +� +> 0, by using that αj = αj+1−2 +τ +− 2 and +βj+1 = τβj − τ − 1. Therefore, there exist ηp > 0 such that +ρ(1+τ)pδ2+2pτ +j +� +Aj +δj +���� +(1 + |y|αj)2τ +|y|(αj−2)τ +���� +p +dy = O +� +ρpηp� +. +Next, similar arguments as above lead us to obtain that for j even, +1 + 1 +τ + βj +αj +� +1 + 1 +τ + αj +2τ +� ++ βj−1 +αj−1 +� +1 + 1 +τ − αj +2τ +� +> 0, +by using αj = (αj−1 + 2)τ + 2 and βj−1 = βj+τ+1 +τ +. Also, by using that αj = (αj+1 − 2)τ − 2 and +βj = τβj+1 + τ + 1 it follows that 1 + 1 +τ + βj +αj +� +1 + 1 +τ − αj +2τ +� ++ βj+1 +αj+1 +� +1 + 1 +τ + αj +2τ +� +> 0. Thus, there exist +ηp > 0 such that +ρ(1+1/τ)pδ2+2p/τ +j +� +Aj +δj +���� +(1 + |y|αj)2/τ +|y|(αj−2)/τ +���� +p +dy = O +� +ρpηp� +in view of +� +Aj +δj +���� +(1 + |y|αj)2/τ +|y|(αj−2)/τ +���� +p +dy = O +��δj−1 +δj +�1+ p +τ − +αj p +2τ + +�δj+1 +δj +�1+ p +τ + +αj p +2τ � +. +This completes the proof. +□ +Acknowledgements +The author would like to thank to Professor Angela Pistoia (U. Roma “La Sapienza”, Italy) for propose +him problem (1.2). Moreover, the author would like to express his gratitude to Professors Angela Pistoia +and Pierpaolo Esposito (U. Roma Tre, Italy) for many stimulating discussions about this problem and +related ones. This work has been supported by grant Fondecyt Regular No 1201884, Chile. +References +[1] M. Ahmedou, A. Pistoia, On the supercritical mean field equation on pierced domains, Proc. Amer. Math. Soc. 143 +(9) (2015), 3969–3984 +[2] D. Bartolucci, C.S. Lin, Existence and uniqueness for mean field equations on multiply connected domains at the +critical parameter, Math. Ann. 359 (2014), 1–44. +[3] D. Bartolucci, A. Pistoia, Existence and qualitative properties of concentrating solutions for the sinh-Poisson equa- +tion, IMA J. Appl. Math. 72 (2007), no. 6, 706-729. +[4] T. Bartsch, A. Pistoia, T. Weth, N-vortex equilibria for ideal fluids in bounded planar domains and new nodal +solutions of the sinh-Poisson and the Lane-Emden-Fowler equations. Comm. Math. 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Musso, Linearized theory for entire solutions of a singular Liouville equation, Proc. +Amer. Math. Soc. 140 (2) (2012), 581–588. +[11] M. del Pino, M. Kowalczyk, M. Musso, Singular limits in Liouville-type equations, Cal. Var. P.D.E., 24 (2005), +47–81. + +36 +P. FIGUEROA +[12] Z. Djadli, Existence result for the mean field problem on Riemann surfaces of all genuses, Commun. Contemp. +Math. 10 (2008), 205-220. +[13] P. Esposito, P. Figueroa, A. Pistoia, On the mean field equation with variable intensities on pierced domains, +Nonlinear Analysis 190 (2020) 111597. +[14] P. Esposito, M. Grossi,A. Pistoia, On the existence of blowing-up solutions for a mean field equation. Ann. Inst. H. +Poincar´e Anal. Non Lin´eaire 22 (2005), no. 2, 227–257. +[15] P. Esposito, J. Wei, Non-simple blow-up solutions for the Neumann two-dimensional sinh-Gordon equation. Calc. +Var. Partial Differential Equations 34 (2009), no. 3, 341–375. +[16] P. Figueroa, Singular limits for Liouville-type equations on the flat two-torus, Calc. Var. P.D.E 49 (2014), no- 1–2, +613–647. +[17] P. Figueroa, A note on sinh-Poisson equation with variable intensities on pierced domains, Asymptotic Analysis, +122 (2021) 327–348. +[18] P. Figueroa, Bubbling solutions for mean field equations with variable intensities on compact Riemann surfaces, +arXiv:2203.09731, accepted for publication in Journal D’Analyse Mathematique. +[19] P. Figueroa, L. Iturriaga, E. Topp, Sign-changing solutions for the sinh–Poisson equation with Robin Boundary +condition, in preparation. +[20] M. Grossi, A. Pistoia, Multiple Blow-Up Phenomena for the Sinh-Poisson Equation, Arch. Rational Mech. Anal. +209 (2013) 287–320. +[21] A. Jevnikar, An existence result for the mean field equation on compact surfaces in a doubly supercritical regime, +Proc. Roy. Soc. Edinburgh Sect A 143 (2013), 1021–1045. +[22] A. Jevnikar, Blow-up analysis and existence results in the supercritical case for an asymmetric mean field equation +with variable intensities, J. Diff. Eq. 263 (2017), no. 2, 972-1008 +[23] A. Jevnikar, J. Wei, W. Yang, Classification of blow-up limits for the sinh-Gordon equation, Differential Integral +Equations 31 (2018), 657–684. +[24] A. Jevnikar, J. Wei, W. Yang, On the topological degree of the mean field equation with two parameters, Indiana +Univ. Math. J. 67 (2018), no. 1, 29–88. +[25] J. Jost, G. Wang, D. Ye, C. Zhou, The blow up of solutions of the elliptic sinh-Gordon equation, Calc. Var. Partial +Differential Equations 31 (2008) no. 2, 263–276. +[26] A. Malchiodi, Morse theory and a scalar field equation on compact surfaces, Adv. Differential Equations 13 (2008), +1109–1129. +[27] H. Ohtsuka, T. Suzuki, Mean field equation for the equilibrium turbulence and a related functional inequality, Adv. +Differential Equations 11 (2006), 281–304. +[28] L. Onsager, Statistical hydrodynamics. Nuovo Cimento (9) 6 (1949), 279–287. +[29] A. Pistoia, T. Ricciardi, Concentrating solutions for a Liouville type equation with variable intensities in 2D- +turbulence, Nonlinearity 29 (2016), no. 2, 271–297. +[30] A. Pistoia, T. Ricciardi, Sign-changing tower of bubbles for a sinh-Poisson equation with asymmetric exponents, +Discrete Contin. Dyn. Syst. 37 (2017), 5651–5692. +[31] T. Ricciardi, Mountain pass solutions for a mean field equation from two-dimensional turbulence, Diff. Int. Eqs. 20 +(2007), 561–575. +[32] T. Ricciardi, R. Takahashi, Blow-up behavior for a degenerate elliptic sinh-Poisson equation with variable intensities, +Calc. Var. Partial Differential Equations 55 (2016), Paper No. 152, 25 pp. +[33] T. Ricciardi, R. Takahashi, G. Zecca, X. Zhang, On the existence and blow-up of solutions for a mean field equation +with variable intensities, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 27 (2016), 413–429. +[34] T. Ricciardi, G. Zecca, Minimal blow-up masses and existence of solutions for an asymmetric sinh- Poisson equation, +Math. Nachr. 290 (2017), no. 14-15, 2375-2387 +[35] K. Sawada, T. Suzuki, Derivation of the equilibrium mean field equations of point vortex and vortex filament system, +Theoret. Appl. Mech. Japan 56 (2008), 285–290. +Pablo Figueroa +Instituto de Ciencias F´ısicas y Matem´aticas +Facultad de Ciencias +Universidad Austral de Chile +Campus Isla Teja s/n, Valdivia, Chile +Email address: pablo.figueroa@uach.cl + diff --git a/hdAzT4oBgHgl3EQf4f6J/content/tmp_files/load_file.txt b/hdAzT4oBgHgl3EQf4f6J/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..6579894a10be069af31cb48c7b58d50d9dd3eed5 --- /dev/null +++ b/hdAzT4oBgHgl3EQf4f6J/content/tmp_files/load_file.txt @@ -0,0 +1,1765 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf,len=1764 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='01845v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='AP] 4 Jan 2023 SIGN-CHANGING BUBBLE TOWER SOLUTIONS FOR SINH-POISSON TYPE EQUATIONS ON PIERCED DOMAINS PABLO FIGUEROA Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' For asymmetric sinh-Poisson type problems with Dirichlet boundary condition arising as a mean field equation of equilibrium turbulence vortices with variable intensities of interest in hydrodynamic turbulence, we address the existence of sign-changing bubble tower solutions on a pierced domain Ωǫ := Ω \\ B(ξ, ǫ), where Ω is a smooth bounded domain in IR2 and B(ξ, ǫ) is a ball centered at ξ ∈ Ω with radius ǫ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Precisely, given a small parameter ρ > 0 and any integer m ≥ 2, there exist a radius ǫ = ǫ(ρ) > 0 small enough such that each sinh-Poisson type equation, either in Liouville form or mean field form, has a solution uρ with an asymptotic profile as a sign-changing tower of m singular Liouville bubbles centered at the same ξ and with ǫ(ρ) → 0+ as ρ approaches to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Introduction Let Ω be a smooth bounded domain in IR2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Given ǫ > 0 and ξ ∈ Ω, define Ωǫ := Ω\\B(ξ, ǫ), a pierced domain, where B(ξ, ǫ) is a ball centered at ξ with radius ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Inspired by results in [1, 13, 17, 20, 30], we are interested in constructing sign-changing bubble tower solutions to sinh-Poisson type equations, either in Liouville form or mean field form, with variable intensities and Dirichlet boundary conditions on a pierced domain Ωǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Precisely, on one hand, we consider the following problem in Liouville form � −∆u = ρ(V0(x)eu − νV1(x)e−τu) in Ωǫ u = 0 on ∂Ωǫ , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='1) and, on the other hand, we also study the problem in mean field form \uf8f1 \uf8f2 \uf8f3 −∆u = λ0V0(x)eu � Ωǫ V0eu − λ1τV1(x)e−τu � Ωǫ V1e−τu in Ωǫ u = 0 on ∂Ωǫ , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='2) where ρ > 0 is small, λ0, λ1 > 0, V0, V1 > 0 are smooth potentials in Ω, τ > 0, ǫ > 0 is a small number and ν ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Our aim is to construct for each problem a family of solutions uρ for a suitable choice of ǫ = ǫ(ρ), with an asymptotic profile as a sum of positive and negative singular Liouville bubbles centered at the same point ξ as ρ → 0, on the line of [20, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' These equations and related ones have attracted a lot of attention in recent years due to its relevance in the statistical mechanics description of 2D-turbulence, as initiated by Onsager [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Precisely, in this context Caglioti, Lions, Marchioro, Pulvirenti [6] and Sawada, Suzuki [35] derive the following equation: − ∆u = λ � [−1,1] αeαu � Ω eαudxdP(α) in Ω, u = 0 on ∂Ω, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='3) where Ω is a bounded domain in R2, u is the stream function of the flow, λ > 0 is a constant related to the inverse temperature and P is a Borel probability measure in [−1, 1] describing the point-vortex intensities distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' We observe that (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='3) is obtained under a deterministic assumption on the distribution of the vortex circulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Date: January 6, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' 2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' 35B44, 35J25, 35J60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' sinh-Poisson type equation, pierced domain, tower of bubbles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' 1 2 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' FIGUEROA Equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='3) includes several well-known problems depending on a suitable choice of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' For instance, if P = δ1 is concentrated at 1, then (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='3) corresponds to the classical mean field equation − ∆u = λ eu � Ω eudx in Ω, u = 0 on ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='4) Since there are plenty of results in the literature concerning (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='4), let us just quote [2, 7, 8, 9, 12, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' When P = σδ1 + (1 − σ)δ−τ with τ ∈ [−1, 1] and σ ∈ [0, 1], equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='3) becomes − ∆u = λ � σ eu � Ω eudx − (1 − σ)τ e−τu � Ω e−τudx � in Ω, u = 0 on ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='5) Setting λ0 = λσ, λ1 = λ(1−σ) and V0 = V1 = 1 problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='5) can be rewritten as (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='2) replacing Ωǫ by Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' If τ = 1 and V0 = V1 ≡ 1 problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='2) reduces to mean field equation of the equilibrium turbulence, see [5, 21, 24, 27, 31] or its related sinh-Poisson version, see [3, 4, 20, 23, 25], which have received a considerable interest in recent years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Recently, sign-changing solutions have been constructed in Ω for the sinh-Poisson equation with Robin boundary condition in [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Concerning results for τ > 0, Pistoia and Ricciardi built in [29] sequences of blowing-up solutions to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='2) (in Ω instead Ωǫ) when λ0, λ1τ 2 are close to 8π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Ricciardi and Takahashi in [32] provided a complete blow-up picture for solution sequences of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='2) and successively in [33] Ricciardi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' constructed min-max solutions when λ0 → 8π+ and λ1 → 0 on a multiply connected domain (in this case the nonlinearity e−τu may be treated as a lower-order term with respect to the main term eu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' A blow-up analysis and some existence results are obtained when τ > 0 in a compact Riemann surface in [22, 34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Bubbling solutions in a compact Riemann surface has been recently constructed in [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' On the other hand, on pierced domains, Ahmedou and Pistoia in [1] proved that on Ωǫ, there exists a solution to the classical mean field equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='4) which blows-up at ξ as ǫ → 0 for any λ > 8π (extra symmetric conditions are required when λ ∈ 8πN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' In [13] the authors constructed a family of solutions to the mean field equation with variable intensities (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='2) blowing-up positively and negatively at ξ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' , ξm1 and ξm1+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' , ξm, respectively, as ǫ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' , ǫm → 0 on a pierced domain with several holes (Ωǫ is replaced by in Ω \\ ∪m i=1B(ξi, ǫi) ), in the super-critical regime λ0 > 8πm1 and λ1τ 2 > 8π(m − m1) with m1 ∈ {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=', m}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Recently, in the same spirit of [13], the author in [17] addressed the sinh-Poisson type equation with variable intensities (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='1) on a pierced domain with several holes Ω \\ ∪m i=1B(ξi, ǫi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='1) is related, but not equivalent, to problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='2) by using the change ρ = λ0 � Ωǫ V0eu and ρν = λ1τ � Ωǫ V1e−τu .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' To the extent of our knowledge, there are by now just few results concerning non-simple blow-up for sinh-Poisson type problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Precisely, sign-changing solutions with non-simple blow-up has been built in [15] for the Neumann sinh-Poisson equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Grossi and Pistoia built in [20] a sign-changing bubble tower solutions for the sinh-Poisson version (τ = 1) in a symmetric domain Ω with respect to a fixed point ξ ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' After that, Pistoia and Ricciardi in [30] extend this construction to a sinh-Poisson type equation with asymmetric exponents (τ ̸= 1) under a symmetric assumption on Ω depending on either τ ∈ Q or τ /∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' In both situations [20, 30] the number of bubbles can be arbitrary large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' A matter of interest to us is whether do there exist sign-changing bubble tower solutions to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='1) for small values of ρ or to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='2) for some values of the parameters λ0, λ1, τ > 0 on a pierced domain Ωǫ, with bubbles centered at ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Our first result in this direction without symmetry assumptions reads as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Let m ≥ 2 be a positive integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' There exists ρ0 > 0 such that for all 0 < ρ < ρ0 there is ǫ = ǫ(ρ) small enough such that problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='1) has a sign-changing solution uρ in Ωǫ blowing-up at ξ in the sense that uρ = (−1)m+1 2π τ ν(m) (αm + 2)G(·, ξ) + o(1) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='6) SIGN-CHANGING BUBBLE TOWER SOLUTIONS ON PIERCED DOMAINS 3 locally uniformly in ¯Ω \\ {ξ} as ρ → 0+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Here, for simplicity, we denote ν(i) = 1 + (−1)i 2 = � 0 if i is odd 1 if i is even and σ(i) = 1 − (−1)i 2 = � 1 if i is odd 0 if i is even , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='7) αm is given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='4) with i = m and G(x, y) = − 1 2π log |x − y| + H(x, y) is the Green’s function of −∆ in Ω, where the regular part H is a harmonic function in Ω so that H(x, y) = 1 2π log |x − y| on ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Nevertheless, the latter result may not tell us whether (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='2) has sign-changing bubble tower solutions for some values of the parameters λ0, λ1, τ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Therefore, we perform directly to problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='2) a similar procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Conversely, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='2 below may not tell us whether (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='1) has sign-changing bubble tower solutions for all small ρ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Assume that λ0 and λ1 decompose for some α1 > 2 (see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='3)), as either λ0 = 2πm � α1 + (m − 2) � 1 + 1 τ �� , λ1τ 2 = 2πm [α1τ + m(1 + τ)] , if m even (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='8) or λ0 = 2π(m+ 1) � α1 + (m − 1) � 1 + 1 τ �� , λ1τ 2 = 2π(m− 1) [α1τ + (m − 1)(1 + τ)] , if m odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='9) In particular, we choose α1 > 2 and α1 /∈ 2IN if τ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Our second result (without symmetry assumptions) is the following Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' If either (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='8) or (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='9) holds for a positive integer m ≥ 2, then there exists a radius ǫ > 0 small enough such that problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='2) has a sign-changing solution uǫ in Ωǫ blowing-up at ξ in the sense of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='6) locally uniformly in ¯Ω \\ {ξ} as ǫ → 0+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Our solutions correspond to a superposition of highly concentrated vortex configurations of alternating orientation around the hole B(ξ, ǫ) and they extend some known results [20, 30] for symmetric domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' We also point out that a delicate point in the paper concerns the linear theory developed in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' A bit more complicated analysis is necessary in comparison with linear theories developed in previous works [1, 11, 13, 14, 16, 20, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Without loss of generality, we shall assume in the rest of the paper that ξ = 0 ∈ Ω, so that Ωǫ = Ω \\ B(0, ǫ) where ǫ > 0 is small and ν = 1, since we can replace νV2 by V2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' However, we need the presence of ν when we compare (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='1) with equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Finally, we point out some comments about the proofs of the theorems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Following the ideas presented in [20, 30] about bubble tower solutions for sinh-Poisson type equations and in [1, 13, 16] about construction of solutions on pierced domains, we find a solution uρ using a perturbative approach, precisely, we look for a solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='1) as uρ = U + φ, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='10) where U is a suitable ansatz built using the projection operator Pǫ onto H1 0(Ωǫ)(see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='1)) and φ ∈ H1 0(Ωǫ) is a small remainder term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Letting wδ,α(x) = log 2α2δα (δα+|x|α)2 be a solution to the singular Liouville equation ∆u + |x|α−2eu = 0 in IR2, � IR2 |x|α−2eu < +∞, the ansatz U is defined as follows U(x) = � i odd Pǫwi(x) − 1 τ � i even Pǫwi(x) = m � i=1 (−1)i+1 τ ν(i) Pǫwi(x), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='11) where wj := wδj,αj for j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' , m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' A careful choice of the parameters δj’s, αi’s and the radius ǫ, depending on ρ > 0, is made in section 2 (see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='2), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='3)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='4) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='21)) in order to make U be a good approximated solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Indeed, the error term R for (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='1) given by R = ∆U + ρ(V0(x)eU − V1(x)e−τU) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='12) 4 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' FIGUEROA is small in Lp-norm for p > 1 close to 1 (see Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' A linearization procedure around U leads us to re-formulate (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='1) in terms of a nonlinear problem for φ (see equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='13)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' We will prove the existence of such a solution φ to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='13) by using a fixed point argument, thanks to some estimates in subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='2 (see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='20) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='21)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' The corresponding solution uρ in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='10) behaves as a sign- changing tower of m singular Liouville bubbles thanks to the asymptotic properties of its main order term U (see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='23) in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' In Section 5 we will prove the invertibility of the linear operator naturally associated to the problem (see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='14)) stated in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' To conclude Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='2, the same procedure with the same ansatz is performed to equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='2), assuming (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='8)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='9) and ǫ = ǫ(ρ), where the error term is given by R = ∆U + λ0 V0(x)eU � Ωǫ V0eU − λ1τ V1(x)e−τU � Ωǫ V1e−τU .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='13) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Approximation of the solution In this section we shall make a choice of the parameters αi’s, δj’s and ǫ = ǫ(ρ) in order to make U a good approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Introduce the function Pǫw as the unique solution of � ∆Pǫw = ∆w in Ωǫ Pǫw = 0, on ∂Ωǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='1) For simplicity, we will denote h0 = H(0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' We have the following asymptotic expansion of Pwδ,α as δ → 0, as shown in [1, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='1] (see also [13, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='1]): Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' The function Pǫwδ,α satisfies Pǫwδ,α(x) = wδ,α(x) − log(2α2δα) + 4παH(x, 0) − γα δ,ǫG(x, 0) + O � δα + �ǫ δ �α + � 1 + ���log δ log ǫ ��� � ǫ � , uniformly in Ωǫ, where γα δ,ǫ is given by γα δ,ǫ = −2α log δ+4παh0 − 1 2π log ǫ+h0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' In particular, there holds Pǫwδ,α(x) = [4πα − γα δ,ǫ]G(x, 0) + O � δα + � ǫ δ �α + � 1 + ���log δ log ǫ ��� � ǫ � as ǫ → 0 locally uniformly in Ω \\ {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Given m ∈ IN with m ≥ 2, consider δj > 0 and αj > 2, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' , m, so that our approximating solution U is defined by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='11), parametrized by δj’s and α′ is with δj = δj(ρ, α1) (it also depends on τ, h0, V0(0) and V1(0)), where ν(i) is defined in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='7) and Pǫ the projection operator defined by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='1) for a suitable choice of ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' In order to have a good approximation, for any i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' , m we will assume that δαi i = diρβi, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='2) for small ρ > 0, where αi’s, βi’s, and di’s will be specified below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' We choose α1 > 2, with α1 ∈ \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 (m−2)/2 � k=0 � 2IN − 4k τ �c � � m/2 � k=1 �2 τ IN − 4k + 2 �c� if m is even (m−1)/2 � k=0 � 2IN − 4k τ �c � � (m−1)/2 � k=1 �2 τ IN − 4k + 2 �c� if m is odd , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='3) and for i ≥ 2 αi = \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 α1 + 2(i − 1) + 2(i − 1) τ if i is odd α1τ + 2(i − 1)τ + 2(i − 1) if i is even .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='4) Note that αi = � α1 + 2i − 2 + 2i − 2 τ � τ ν(i), for i ≥ 2 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='5) and αi > 0 for all i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' , m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Furthermore, several identities and properties of αi’s, βi’s, di’s and ǫ will be proven in order to have a good approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' From de definition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='4), it is readily checked SIGN-CHANGING BUBBLE TOWER SOLUTIONS ON PIERCED DOMAINS 5 that {α2k+1}k and {α2k}k are increasing in k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Since α1 > 2 and α2 = α1τ + 2τ + 2 > 2, it follows that αi > 2 for all i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' , m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Notice that all these sets are countable : 2IN − 4k τ for k = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' , m−2 2 and 2 τ IN − 4k + 2 for k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' , m 2 with m even;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' 2IN − 4k τ for k = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' , m−1 2 and 2 τ IN − 4k + 2 for k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' , m−1 2 with m odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Hence, its union is also countable set and the complement of its union is dense in IR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Therefore, there exist α1 ∈ IR satisfying (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' If αi’s are given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='4) then αj+1 = (αj + 2)τ(−1)j+1 + 2 = \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 (αj + 2)τ + 2 if j is odd αj + 2 τ + 2 if j is even (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='6) and j � i=1 (−1)i+1 τ ν(i) αi = � −j − j τ if j is even α1 + j − 1 + j−1 τ if j is odd (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='7) holds for all j ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' If α1 satisfies (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='3) then αi /∈ 2IN for all i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' , m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Proof: Assume first that i is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Then, i + 1 is even, from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='4) for i odd, αi = α1 + 2i − 2 + 2i−2 τ and direct computations lead us to obtain that (αi + 2)τ + 2 = � α1 + 2i + 2i − 2 τ � τ + 2 = (α1 + 2i)τ + 2i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' On the other hand, from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='4) for i + 1 even, we find that αi+1 = α1τ + 2iτ + 2i, so that αi+1 = (αi + 2)τ + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Direct computations in case i is even allows us to conclude (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' From the choice of αi’s and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='5), it follows that j � i=1 (−1)i+1 τ ν(i) αi = j � i=1 (−1)i+1 � α1 + 2i − 2 + 2i − 2 τ � = α1 j � i=1 (−1)i+1 + 2 j � i=1 (−1)i+1(i − 1) + 2 τ j � i=1 (−1)i+1(i − 1) and we conclude (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='7), in view of j � i=1 (−1)i+1(i − 1) = j � i=1 (−1)i+1i + j � i=1 (−1)i+1 = � − j 2 + 0 if j is even j+1 2 − 1 if j is odd .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Finally, assume that m is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' If α1 satisfies (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='3) then we have that α1 ∈ � 2IN − 4k τ �c, k = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=', m−2 2 and α1 ∈ � 2 τ IN−4k+2 �c, k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' , m 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' That is to say, α1 + 4k τ /∈ 2IN, k = 0, 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' , m−2 2 and τ(α1 + 4k − 2) /∈ 2IN, k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' , m 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Therefore, α2k+1 = α1 + 4k + 4k τ /∈ 2IN, k = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' , m−2 2 and α2k = α1τ +(4k−2)τ +4k−2 /∈ 2IN, k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' , m 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Similar argument lead us to conclude that αi /∈ 2IN for all i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' , m if m is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' □ In particular, we have that α1 > 2, α2 = (α1 + 2)τ + 2, α3 = α2+2 τ + 2, α4 = (α3 + 2)τ + 2, α5 = α4+2 τ + 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Furthermore, we have that 4π m � i=1 (−1)i+1 τ ν(i) αi − 2π(α1 − 2) = (−1)m+1 2π τ ν(m) (αm + 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='8) Now, we define βi as follows, if m is even then βl = τ ν(l) � m − l + m − l + 1 τ � = \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 (m − l)τ + m − l + 1, if l is even m − l + m − l + 1 τ if l is odd , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='9) 6 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' FIGUEROA for l = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' , m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' If m is odd then βl = τ ν(l) � m − l + 1 + m − l τ � = \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 (m − l + 1)τ + m − l, if l is even m − l + 1 + m − l τ if l is odd (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='10) for l = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' , m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' In any case, either m is odd or even, it is readily checked that βm = 1 and βi > 0 for all i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' , m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Note that we can rewrite βl = τ ν(l) �m − l + 1 τ ν(m) + m − l τ ν(m+1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='11) Hence, we shall prove the following useful properties from the definition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='9) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Note that 1 − σ(i) = ν(i) = σ(i + 1) and σ(i) = 1 − ν(i) = ν(i + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' If βi’s are given by either (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='9) or (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='10) then for any l = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' , m βl = \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 τβl−1 − τ − 1, if l is even βl−1 τ − 1 − 1 τ if l is odd (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='12) and for any l = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' , m − 1 βl − 1 2τ ν(l) = m � j=l+1 (−1)j+1 τ ν(j) βj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='13) Furthermore, it holds that βl αl < βl−1 αl−1 for any l = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' , m, so that, δi δj → 0 as ρ → 0 for any i < j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Proof: First, assume that m is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' So, βl’s are given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' If l is even, then we have that l − 1 is odd, so that, βl−1 = m − l + 1 + m − l + 2 τ and τβl−1 − τ − 1 = (m − l)τ + τ + m − l + 2 − τ − 1 = βl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Next, assume that l is odd (still m is even).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Similarly as above, it follows that βl−1 τ − 1 − 1 τ = βl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Arguing in the same way for βl’s given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='10) when m is odd, we conclude (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Now, for any m, we shall prove that βl = \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 1 + 2 m−l � i=1 (−1)i+1τ σ(i)βl+i, if l is even 1 + 2 m−l � i=1 (−1)i+1 τ σ(i) βl+i if l is odd .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='14) Assume that m is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' If l is even then ν(i + l) = 1 − σ(i) and from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='11) we get that m−l � i=1 (−1)i+1τ σ(i)βl+i = m−l � i=1 (−1)i+1τ �βl τ − � 1 + 1 τ � i � = βl m−l � i=1 (−1)i+1 − (τ + 1) m−l � i=1 (−1)i+1i and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='14) follows, in view of m − l is even, �m−l i=1 (−1)i+1 = 0 and �m−l i=1 (−1)i+1i = − m−l 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Next, assume that l is odd (still m is even) so that ν(i + l) = σ(i) and similarly as above m−l � i=1 (−1)i+1 τ σ(i) βl+i = m−l � i=1 (−1)i+1 � βl − � 1 + 1 τ � i � = βl m−l � i=1 (−1)i+1 − � 1 + 1 τ � m−l � i=1 (−1)i+1i = βl − 1 2 , in view of m − l is odd, �m−l i=1 (−1)i+1 = 1 and �m−l i=1 (−1)i+1i = m−l+1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Arguing in the same way for βl’s given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='10) when m is odd we conclude (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Now, from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='14) taking j = l + i we have that SIGN-CHANGING BUBBLE TOWER SOLUTIONS ON PIERCED DOMAINS 7 if l is even then βl = 1 + 2 m−l � i=1 (−1)i+1τ σ(i)βl+i = 1 + 2 m � j=l+1 (−1)j+1 τ τ ν(j) βi in view of σ(j − l) = σ(j) = 1 − ν(j), and if l is odd then βl = 1 + 2 m−l � i=1 (−1)i+1 τ σ(i) βl+i = 1 + 2 m � j=l+1 (−1)j τ ν(j) βi, in view of σ(j − l) = ν(j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Thus, we deduce (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Now, we know that 0 < αl−1 + αl−1τ + 2βl−1 + 2βl−1τ for any l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Hence, for l even we have that βl = τβl−1 − τ − 1 and αl = (αl−1 + 2)τ + 2 by using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='12) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='6) respectively, so that αl−1[τβl−1 − τ − 1] < [(αl−1 + 2)τ + 2]βl−1 ⇐⇒ αl−1βl < αlβl−1 By using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='6) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='12), for l odd we have that βl = βl−1 τ − 1 τ − 1 and αl = αl−1+2 τ + 2, and it is readily checked that αl−1βl < αlβl−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Thus, we deduce that δi δj → 0 as ρ → 0 for any i < j, in view of δi δj = d1/αi i ρβi/αi d1/αj j ρβj/αj = d1/αi i d1/αj j ρ βi αi − βj αj and βj αj < βi αi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' This finishes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' □ Now, we define di’s by log dm = am and log dl = al + 2τ ν(l) m � i=l+1 ai τ ν(i) = \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 al + 2 m � i=l+1 aiτ σ(i), if l is even al + 2 m � i=l+1 ai τ ν(i) if l is odd (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='15) for l = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' , m − 1, where al = log �τ ν(l)Vν(l)(0) 2α2 l � + (−1)l τ σ(l) 2π(αm + 2)h0 = \uf8f1 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f3 log �τV1(0) 2α2 l � + 2π(αm + 2)h0, if l is even log �V0(0) 2α2 l � − 2π τ (αm + 2)h0 if l is odd (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='16) when m even, while when m is odd al = log �τ ν(l)Vν(l)(0) 2α2 l � + (−1)l+1τ ν(l)2π(αm + 2)h0 = \uf8f1 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f3 log �τV1(0) 2α2 l � − 2π(αm + 2)τh0, if l is even log �V0(0) 2α2 l � + 2π(αm + 2)h0 if l is odd .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='17) Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' If di’s are given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='15) then for any l = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' , m − 1 log dl = \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 al + 2 m−l � i=1 (−1)i+1τ σ(i) log dl+i, if l is even al + 2 m−l � i=1 (−1)i+1 τ σ(i) log dl+i if l is odd (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='18) 8 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' FIGUEROA and consequently, log dl − al 2τ ν(l) = m � j=l+1 (−1)j+1 τ ν(j) log dj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='19) Proof: First, assume that m is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' So, al’s are given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' If l is even, then we have that ν(i + l) = 1 − σ(i), τ τ ν(j) = τσ(j) and m − l is even, so that, σ(m − l) = 0 and m−l � i=1 (−1)i+1τ σ(i) log dl+i = m−l−1 � i=1 (−1)i+1τ σ(i) � al+i + 2τ ν(l+i) m � j=l+i+1 aj τ ν(j) � − am = m−l−1 � i=1 (−1)i+1τ σ(i)al+i + 2τ m−l−1 � i=1 m � j=l+i+1 (−1)i+1 aj τ ν(j) − am = m−1 � j=l+1 (−1)j+1τ σ(j)aj + m � j=l+2 [1 + (−1)j]ajτ σ(j) − am = m � j=l+1 ajτ σ(j) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='18) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Similarly as above, if l is odd, then we have that ν(i + l) = σ(i), τ τ ν(j) = τσ(j) and m − l + 1 is even, so that, σ(m − l + 1) = 0 and m−l � i=1 (−1)i+1 τ σ(i) log dl+i = m−l−1 � i=1 (−1)i+1 τ σ(i) al+i + 2 m−l−1 � i=1 m � j=l+i+1 (−1)i+1 aj τ ν(j) + am τ = m−1 � j=l+1 (−1)j τ ν(j) aj + m � j=l+2 [1 + (−1)j+1] aj τ ν(j) + am τ = m � j=l+1 aj τ ν(j) and we conclude (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Similar arguments work out in case m is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' We deduce (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='19) by using the change j = l + i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' This conclude the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' □ Now, ǫ = ǫ(ρ) is chosen so that m � i=1 (−1)i+1 τ ν(i) γi = 2π(α1 − 2), where γj = γαj δj,ǫ, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' , m (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='20) and γα δ,ǫ is given in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Precisely, ǫ = ǫ(ρ) is given by ǫα1−2 = \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 [V0(0)]m[τV1(0)] m τ �m i=1 α4/τ ν(i) i e 2π τ (αm+2)h0�ρ 2 �m+ m τ if m is even [V0(0)]m+1[τV1(0)] m−1 τ �m i=1 α4/τ ν(i) i e2π(αm+2)h0�ρ 2 �m+1+ m−1 τ if m is odd (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='21) It is readily checked that ǫ(ρ) → 0+ as ρ → 0+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Moreover, ǫα1−2 ∼ ρβ1+1, in view of β1 = � m − 1 + m τ if m is even m + m−1 τ if m is odd (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='22) Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' If ǫ is given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='21) then it holds (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='20) and ǫ δ1 → 0 as ρ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Proof: First, assume that m is even so that, from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='21) it follows that (α1 − 2) log ǫ = m log V0(0) + m τ log[τV1(0)] + 2π τ (αm + 2)h0 − 4 m � i=1 log αi τ ν(i) + � m + m τ � log �ρ 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' On the other hand, from the definition of δi and γi it follows that � − 1 2π log ǫ + h0 � γi = −2 log di − 2βi log ρ + 4παih0 SIGN-CHANGING BUBBLE TOWER SOLUTIONS ON PIERCED DOMAINS 9 so that, � − 1 2π log ǫ + h0 � m � i=1 (−1)i+1 τ ν(i) γi = −2 m � i=1 (−1)i+1 τ ν(i) log di − 2 log ρ m � i=1 (−1)i+1 τ ν(i) βi + 4πh0 m � i=1 (−1)i+1 τ ν(i) αi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Hence, we compute the sums involved as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' From Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='4 we have that for any m, either odd or even, log d1 = a1 + 2 m � j=2 (−1)j τ ν(j) log dj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Then by using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='15) and m = 2k for some k ∈ IN we get that m � j=1 (−1)j τ ν(j) log dj = a1 + log d1 2 = m � i=1 ai τ ν(i) = k � i=1 1 τ � log[τV1(0)] + 2π(αm + 2)h0 − log(2α2 2i) � + k � i=1 � log V0(0) − 2π τ (αm + 2)h0 − log(2α2 2i−1) � = m 2 log V0(0) + m 2τ log[τV1(0)] − � m + m τ �log 2 2 − 2 m � i=1 log αi τ ν(i) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Also, from the choice of βi’s in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='9), it is readily checked that m � i=1 (−1)i+1 τ ν(i) βi = m � i=1 (−1)i+1 � m + m + 1 τ − � 1 + 1 τ � i � = m 2 + m 2τ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Therefore, by using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='7) we obtain that � − 1 2π log ǫ + h0 � m � i=1 (−1)i+1 τ ν(i) γi = −m log V0(0) − m τ log[τV1(0)] + � m + m τ � log 2 + 4 m � i=1 log αi τ ν(i) − � m + m τ � log ρ + 4πh0 � − m − m τ � = −(α1 − 2) log ǫ + 2π τ (αm + 2)h0 − 4πh0 � m + m τ � = −(α1 − 2) log ǫ + 2π(α1 − 2)h0, in view of αm+2 τ − 2m − 2m τ = α1 − 2 and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='20) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Similarly, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='20) is proven if m is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Finally, taking into account (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='22), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='21) and δα1 1 = d1ρβ1, we deduce that ǫ δ1 → 0 ⇐⇒ β1+1 α1−2 > β1 α1 , in view of ǫα1−2 ∼ ρβ1+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Indeed, it is readily checked that β1+1 α1−2 > β1+1 α1 > β1 α1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' □ Let us stress that the behavior of ǫ = ǫ(ρ) and δj = δj(ρ)’s with respect to ρ is given by ǫ δj , δi δj → 0 as ρ → 0 for i < j and ǫ → 0 as ρ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Assume that δi’s are given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='2) with αi’s, βi’s and di’s defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='4), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='9)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='10) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='15), respectively, and ǫ is given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Notice that log δj log ǫ = O(1) as ρ → 0, for all j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' , m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Now, define the shrinking annuli Ai = {x ∈ Ω | � δi−1δi < |x| ≤ � δiδi+1}, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' , m, where for simplicity we denote δ0 = ǫ2 δ1 and δm+1 = M2 0 δm with M0 = sup{|x| : x ∈ Ω}, so that, Ωǫ = ∪m j=1Aj, ∩m j=1Aj = ∅ and Aj δj approaches to IR2 as ρ → 0 for each j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' , m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' 10 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' FIGUEROA Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' There exist η > 0 such that the following expansions hold U(δjy) = (−1)j+1 τ ν(j) � −2 log δj − aj − log ρ + log |y|αj−2 (1 + |y|αj)2 � + (−1)m+1 2π τ ν(m) (αm + 2)H(δjy, 0) + O (ρη) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='23) for any j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' , m, uniformly for δjy ∈ Aj, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' , m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Proof: From the expansions of Pǫwj, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' , m and the definition of γj in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='1, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='8) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='20) we obtain that U(x) = m � i=1 (−1)i+1 τ ν(i) � wi − log(2α2 i δαi i ) � + 1 2π m � i=1 (−1)i+1 τ ν(i) γi log |x| + m � i=1 (−1)i+1 τ ν(i) [4παi − γi] H(x, 0) + O \uf8eb \uf8ed m � j=1 � δαj j + � ǫ δj �αj� + ǫ \uf8f6 \uf8f8 = m � i=1 (−1)i+1 τ ν(i) log 1 (δαi i + |x|αi)2 + (α1 − 2) log |x| + (−1)m+1 2π τ ν(m) (αm + 2)H(x, 0) + O (ρη) , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='24) where we choose 0 < η ≤ min � 1, β1 + 1 α1 − 2, min � αj �β1 + 1 α1 − 2 − βj αj � : j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' , m �� , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='25) in view βj are decreasing so that δαj j = O(ρ) for all j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' , m, � ǫ δj �αj = O � ρ αj( β1+1 α1−2 − βj αj )� and ǫ = O � ρ β1+1 α1−2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Now, from the choice of δi’s in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='2), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='13) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='19) we have that for j odd 2 m � i=j+1 (−1)i τ (ν(i) αi log δi = 2 m � i=j+1 (−1)i τ (ν(i) log di + 2 log ρ m � i=j+1 (−1)i τ (ν(i) βi = 2log dj − aj 2 + 2βj − 1 2 log ρ = αj log δj − aj − log ρ and similarly, for j even 2 �m i=j+1 (−1)i τ (ν(i) αi log δi = − 1 τ � αj log δj − aj − log ρ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Thus, we rewrite as 2 m � i=j+1 (−1)i τ (ν(i) αi log δi = (−1)j+1 τ ν(j) � αj log δj − aj − log ρ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='26) SIGN-CHANGING BUBBLE TOWER SOLUTIONS ON PIERCED DOMAINS 11 On the other hand,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' for any y ∈ A1 δ1 it holds that U(δ1y) = − 2α1 log δ1 + log 1 (1 + |y|α1)2 + (α1 − 2) log(δ1|y|) + m � i=2 (−1)i+1 τ ν(i) log 1 (δαi i + δαi 1 |y|αi)2 + (−1)m+1 2π τ ν(m) (αm + 2)H(δ1y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' 0) + O (ρη) = − (α1 + 2) log δ1 + m � i=2 (−1)i 2αi τ ν(i) log δi + log |y|α1−2 (1 + |y|α1)2 + m � i=2 (−1)i τ ν(i) 2 log � 1 + �δ1|y| δi �αi� + (−1)m+1 2π τ ν(m) (αm + 2)H(δ1y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' 0) + O (ρη) = − 2 log δ1 − a1 − log ρ + log |y|α1−2 (1 + |y|α1)2 + (−1)m+1 2π τ ν(m) (αm + 2)H(δ1y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' 0) + O \uf8eb \uf8ed m � j=2 �δ1 δj � αj 2 + ρη \uf8f6 \uf8f8 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='27) in view of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='26) and log � 1 + �δ1|y| δi �αi� = O ��δ1|y| δi �αi� = O ��δ1 δi � αi 2 � , i = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' , m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Choosing η > 0 satisfying (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='25) and 0 < η ≤ 1 2 min � αi � β1 α1 − βi αi � : 1 < i � (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='28) we get that � δ1 δi � αi 2 = O (ρη), i = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' , m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Similarly,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' for y ∈ Aj δj ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' 1 < j < m we find that U(δjy) = j−1 � i=1 (−1)i+1 τ ν(i) � − 2αi log(δj|y|) − 2 log � 1 + � δi δj|y| �αi� � + (α1 − 2) log(δj|y|) + (−1)j+1 τ ν(j) � − 2αj log δj + log 1 (1 + |y|αj)2 � + m � i=j+1 (−1)i τ ν(i) 2αi log δi + m � i=j+1 (−1)i τ ν(i) 2 log � 1 + �δj|y| δi �αi� + (−1)m+1 2π τ ν(m) (αm + 2)H(δjy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' 0) + O (ρη) = −2 log δj j � i=1 (−1)i+1 τ ν(i) αi + 2 m � i=j+1 (−1)i τ ν(i) αi log δi + (α1 − 2) log δj + (α1 − 2) log |y| − 2 log |y| j−1 � i=1 (−1)i+1 τ ν(i) αi + (−1)j+1 τ ν(j) log 1 (1 + |y|αj)2 + (−1)m+1 2π τ ν(m) (αm + 2)H(δjy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' 0) + O � j−1 � i=1 � δi δj � αi 2 + m � i=j+1 �δj δi � αi 2 + ρη � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='29) in view of log � 1 + � δi δj|y| �αi� = O �� δi δj|y| �αi� = O �� δi δj � αi 2 � , i < j, log � 1 + �δj|y| δi �αi� = O ��δj|y| δi �αi� = O ��δj δi � αi 2 � , j < i, 12 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' FIGUEROA and by using again (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Now, we choose η satisfying (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='25), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='28) and smaller, if necessary, so that 0 < η ≤ 1 2 min � αi � βi αi − βj αj � : i < j � , j = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' , m and 0 < η ≤ 1 2 min � αi �βj αj − βi αi � : j < i � , j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' , m − 1 thus, � δi δj � αi 2 = O (ρη), i < j and � δj δi � αi 2 = O (ρη), j < i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Hence, by using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='7) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='26) for j even we get that U(δjy) = −2 log δj � − j − j τ � − 1 τ [αj log δj − aj − log ρ] + (α1 − 2) log δj + (α1 − 2) log |y| − 2 log |y| � α1 + j − 2 + j − 2 τ � − 1 τ log 1 (1 + |y|αj)2 + (−1)m+1 2π τ ν(m) (αm + 2)H(δjy, 0) + O (ρη) and we conclude (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Similarly, for j odd we get that U(δjy) = −2 log δj � α1 + j − 1 + j − 1 τ � + αj log δj − aj − log ρ + (α1 − 2) log δj + (α1 − 2) log |y| − 2 log |y| � −j + 1 − j − 1 τ � + log 1 (1 + |y|αj)2 + (−1)m+1 2π τ ν(m) (αm + 2)H(δjy, 0) + O(ρη) and we obtain (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Also, we have for any y ∈ Am δm U(δmy) = m−1 � i=1 (−1)i+1 τ ν(i) � −2αi log(δm|y|) − 2 log � 1 + � δi δm|y| �αi�� + (α1 − 2) log(δm|y|) + (−1)m+1 τ ν(m) � −2αm log δm + log 1 (1 + |y|αm)2 � + (−1)m+1 2π τ ν(m) (αm + 2)H(δjy, 0) + O(ρη) = (−1)m+1 τ ν(m) � −2 log δm − am − log ρ + log |y|αm−2 (1 + |y|αm)2 + 2π(αm + 2)H(δmy, 0) � + O(ρη).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='30) This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Problem in Liouville form (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='1) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Error estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' In order to evaluate how well the approximation U satisfies the equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='1) (and get the invertibility of the linearized operator), we will use the norms ∥h∥p = �� Ωǫ |h(x)|p dx �1/p and ∥h∥ = �� Ωǫ |∇h(x)|2 dx �1/2 , the usual norms in the Banach spaces Lp(Ωǫ) and H1 0(Ωǫ), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Assume that δi’s are given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='2) with αi’s, βi’s and di’s defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='4), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='9)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='10) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='15), respectively, and ǫ is given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Let us evaluate the approximation rate of U in ∥ · ∥p, encoded in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='12) Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' There exist ρ0 > 0, a constant C > 0 and 1 < p0 < 2, such that for any ρ ∈ (0, ρ0) and p ∈ (1, p0) it holds ∥R∥p ≤ Cρηp (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='1) for some ηp > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Proof: First, note that ∆U = �m i=1 (−1)i τ ν(i) |x|αi−2ewi so that, for any 1 < j < m and y ∈ Aj δj we have that ∆U(δjy) = (−1)j τ ν(j) 2α2 j|y|αj−2 δ2 j (1 + |y|αj)2 + O � j−1 � i=1 � δi δj �αi 1 δ2 j |y|αi+2 + m � i=j+1 �δj δi �αi |y|αi−2 δ2 j � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='2) SIGN-CHANGING BUBBLE TOWER SOLUTIONS ON PIERCED DOMAINS 13 Similarly, we obtain that ∆U(δ1y) = − 2α2 j|y|αj−2 δ2 j (1 + |y|αj)2 + O � m � i=2 �δj δi �αi |y|αi−2 δ2 j � , y ∈ A1 δ1 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='3) and ∆U(δmy) = (−1)m τ ν(m) 2α2 j|y|αj−2 δ2 j (1 + |y|αj)2 + O � m−1 � i=1 � δi δj �αi 1 δ2 j |y|αi+2 � , y ∈ Am δm (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='4) By using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='16) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='23) (from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='6) for y ∈ Aj δj and any j odd we have that ρV0(δjy)eU(δjy) = 2α2 j|y|αj−2 δ2 j (1 + |y|αj)2 V0(δjy) V0(0) exp � (−1)m+1 2π τ ν(m) (αm + 2)[H(δjy, 0) − h0] + O(ρη) � = 2α2 j|y|αj−2 δ2 j (1 + |y|αj)2 [1 + O(δj|y| + ρη)] (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='5) and ρV1(δjy)e−τU(δjy) = ρV1(δjy) exp � − τ � − 2 log δj − aj − log ρ + log |y|αj−2 δ2 j (1 + |y|αj)2 + (−1)m+1 2π τ ν(m) (αm + 2)H(δjy, 0) � + O(ρη) � = O � ρ1+τδ2τ j (1 + |y|αj)2τ |y|(αj−2)τ � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='6) Similarly, by using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='17) and 1 − ν(m) = σ(m) for y ∈ Aj δj and j even ρV1(δjy)e−τU(δjy) = 2α2 j|y|αj−2 δ2 j τ(1 + |y|αj)2 V1(δjy) V1(0) exp � (−1)m2πτ σ(m)(αm + 2)[H(δjy, 0) − h0] + O(ρη) � = 2α2 j|y|αj−2 δ2 j τ(1 + |y|αj)2 [1 + O(δj|y| + ρη)] (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='7) and ρV0(δjy)eU(δjy) = O � ρ1+1/τδ2/τ j (1 + |y|αj)2/τ |y|(αj−2)/τ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='8) Hence, by using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='3), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='5) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='6) for δ1y ∈ A1 we get that R(δ1y) = 1 δ2 1 2α2 1|y|α1−2 (1 + |y|α1)2 O(δ1|y| + ρη) + O � ρ1+τδ2τ 1 (1 + |y|α1)2τ |y|(α1−2)τ + m � i=2 �δ1 δi �αi |y|αi−2 δ2 1 � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='9) by using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='2), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='5) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='6) for 1 < j < m, j odd and δjy ∈ Aj R(δjy) = 1 δ2 j 2α2 j|y|αj−2 (1 + |y|αj)2 O(δj|y| + ρη) + O � ρ1+τδ2τ j (1 + |y|αj)2τ |y|(αj−2)τ + j−1 � i=1 � δi δj �αi 1 δ2 j |y|αi+2 + m � i=j+1 �δj δi �αi |y|αi−2 δ2 j � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='10) by using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='2), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='7) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='8) for 1 < j < m, j even and δjy ∈ Aj R(δjy) = 1 δ2 j τ 2α2 j|y|αj−2 (1 + |y|αj)2 O(δj|y| + ρη) + O � ρ1+1/τδ2/τ j (1 + |y|αj)2/τ |y|(αj−2)/τ + j−1 � i=1 � δi δj �αi 1 δ2 j |y|αi+2 + m � i=j+1 �δj δi �αi |y|αi−2 δ2 j � (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='11) 14 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' FIGUEROA and by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='4) for δmy ∈ Am R(δmy) = 1 δ2mτ ν(m) 2α2 m|y|αm−2 (1 + |y|αm)2 O(δm|y| + ρη) + O � ρ1+τ (−1)m+1 δ2τ (−1)m+1 m (1 + |y|αm)2τ (−1)m+1 |y|(αm−2)τ (−1)m+1 + m−1 � i=1 � δi δm �αi 1 δ2m|y|αi+2 � (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='12) By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='9)-(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='12), we finally get that there exist ρ0 > 0 small enough and p0 > 1 close to 1, so that for all 0 < ρ ≤ ρ0 and 1 < p ≤ p0 � Ωǫ |R(x)|p dx = m � j=1 � Aj |R(x)|p dx = m � j=1 δ2 j � Aj |R(δjy)|p dy = O (ρpηp) , for some ηp > 0, see Appendix, Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='1 for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' The nonlinear problem and proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' In this subsection, we will look for a solution u of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='1) in the form u = U + φ, for some small remainder term φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' In terms of φ, the problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='1) is equivalent to find φ ∈ H1 0(Ωǫ) so that � L(φ) = −[R + Λ(φ) + N(φ)], in Ωǫ, φ = 0, on ∂Ωǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='13) where the linear operators L and Λ are defined as L(φ) = ∆φ + K(x)φ, K(x) = m � i=1 |x|αi−2ewi (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='14) and Λ(φ) = ρ[V0(x)eU + τV1(x)e−τU]φ − Kφ (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='15) The nonlinear part N is given by N(φ) = ρV0(x)eU� eφ − φ − 1 � − ρV1(x)e−τU� e−τφ + τφ − 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='16) Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' For any p > 1 there exists ρ0 > 0 so that for all 0 < ρ ≤ ρ0, h ∈ Lp(Ωǫ) there is a unique solution φ ∈ H1 0(Ωǫ) of � L(φ) = h in Ωǫ φ = 0 on ∂Ωǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='17) Moreover, there is a constant C > 0 independent of ρ such that ∥φ∥ ≤ C| log ρ|∥h∥p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='18) The latter proposition implies that the unique solution φ = T (h) of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='17) defines a continuous linear map from Lp(Ωǫ) into H1 0(Ωǫ), with norm bounded by C| log ρ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' We are now in position to study the nonlinear problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' There exist p0 > 1 and ρ0 > 0 so that for any 1 < p < p0 and all 0 < ρ ≤ ρ0, the problem � L(φ) = −[R + Λ(φ) + N(φ)], in Ωǫ φ = 0, on ∂Ωǫ (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='19) admits a unique solution φ(ρ) ∈ H1 0(Ωǫ), where N, Λ and R are given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='16), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='15) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='12), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Moreover, there is a constant C > 0 such that for some ηp > 0 ∥φ∥∞ ≤ Cρηp| log ρ| Here, ηp is the same as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' We shall use the following estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' There exist p0 > 1 and ρ0 > 0 so that for any 1 < p < p0 and all 0 < ρ ≤ ρ0 it holds ∥Λ(φ)∥p ≤ Cρη′ p∥φ∥, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='20) for all φ ∈ H1 0(Ωǫ) with ∥φ∥ ≤ Mρηp| log ρ|, for some η′ p > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' SIGN-CHANGING BUBBLE TOWER SOLUTIONS ON PIERCED DOMAINS 15 Proof: Arguing in the same way as in [13, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='3], for simplicity, denote W = ρ[V0eU +τV1e−τU], so that the linear operator Λ is re-written as Λ(φ) = (W − K)φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' By using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='5)-(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='6) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='7)-(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='8), we find that for i odd δ2 i W(δiy) = 2α2 i |y|αi−2 (1 + |y|α i )2 [1 + O(|δiy| + ρη)] + O � ρ1+τδ2+2τ i (1 + |y|αi)2τ |y|(αi−2)τ � uniformly for δiy ∈ Ai and for i even δ2 i W(δiy) = 2α2 i |y|αi−2 (1 + |y|α i )2 [1 + O(|δiy| + ρη)] + O � ρ1+1/τδ2+2/τ i (1 + |y|αi)2/τ |y|(αi−2)/τ � uniformly for x ∈ Ai and i even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Hence, for any q ≥ 1 and for i odd there holds ∥W − K∥q Lq(Ai) ≤ C � δ2−q i � Ai δi ���� 2α2 i |y|αi−1 (1 + |y|αi)2 ���� q dy + ρηqδ2−2q i � Ai δi ���� 2α2 i |y|αi−2 (1 + |y|αi)2 ���� q dy + ρ(1+τ)qδ2+2τq i � Ai δi ���� (1 + |y|αi)2τ |y|(αi−2)τ ���� q dy + � ji δ2−2q i � δi δj �αjq � Ai δi |y|(αj−2)qdy � ≤ Cρqη′ 1,q for some η′ 1,q > 0, see Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Similarly, there holds ∥W − K∥q Lq(Ai) ≤ Cρqη′ 2,q for any q ≥ 1 and for i even for some η′ 2,q > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Hence, we get that ∥Λ(φ)∥p ≤ ∥(W − K) φ∥p ≤ ∥W − K∥pr ∥φ∥ps ≤ Cρη′ p∥φ∥, where η′ p = min � η′ 1,pr, η′ 2,ps � with r, s satisfying 1 r + 1 s = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Furthermore, we have used the H¨older’s inequality ∥uv∥q ≤ ∥u∥qr∥v∥qs with 1 r + 1 s = 1 and the inclusions Lp(Ωǫ) ֒→ Lpr(Ωǫ) for any r > 1 and H1 0(Ωǫ) ֒→ Lq(Ωǫ) for any q > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Let us stress that we can choose p, r and s close enough to 1 such that η′ p > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' □ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' There exist p0 > 1 and ρ0 > 0 so that for any 1 < p < p0 and all 0 < ρ ≤ ρ0 it holds ∥N(φ1) − N(φ2)∥p ≤ Cρη′′ p ∥φ1 − φ2∥ (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='21) for all φi ∈ H1 0(Ωǫ) with ∥φi∥ ≤ Mρηp| log ρ|, i = 1, 2, and for some η′′ p > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' In particular, we have that ∥N(φ)∥p ≤ Cρη′′ p ∥φ∥ (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='22) for all φ ∈ H1 0(Ωǫ) with ∥φ∥ ≤ Mρηp| log ρ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Proof: We will argue in the same way as in [1, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='1], see also [13, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' First, denoting fi(φ) = Vi(x)e(−τ)i(U+φ) we point out that N(φ) = 1 � i=0 ρ {fi(φ) − fi(0) − f ′ i(0)[φ]} and N(φ1) − N(φ2) = 1 � i=0 ρf ′′ i (˜φµi)[φθi, φ1 − φ2], (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='23) by the mean value theorem, where φθi = θiφ1 + (1 − θi)φ2, ˜φµi = µiφθi for some θi, µi ∈ [0, 1], i = 0, 1, and f ′′ i (φ)[ψ, v] = τ 2iVi(x)e(−τ)i(U+φ)ψv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Using H¨older’s inequalities we get that ∥f ′′ i (φ)[ψ, v]∥p ≤ |τ|2i∥Vie(−τ)i(U+φ)∥pri∥ψ∥psi∥v∥pti (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='24) with 1 ri + 1 si + 1 ti = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' We have used the H¨older’s inequality and ∥uvw∥q ≤ ∥u∥qr∥v∥qs∥w∥qt with 1 r + 1 s + 1 t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Now, let us estimate ∥Vie(−τ)i(U+φ)∥pri with φ = ˜φµi, i = 0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='5) and the change of variable x = δiy let us estimate � Ai ��V0eU��q dx = O � ρ−qδ2−2q i � Ai δi ���� |y|αi−2 (1 + |y|αi)2 ���� q � 1 + O � ρη + δi|y| ��q � = O � ρ βi αi (2−2q)−q� (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='25) 16 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' FIGUEROA for any i odd and similarly, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='7) we get that � Ai ��V1e−τU��q dx = O � ρ βi αi (2−2q)−q� (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='26) for any i even, in view of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='6) we get the estimate � Ai ��V0eU��q dx = O � ρ q τ δ 2+ 2q τ i � Ai δi � |y|αi−2 (1 + |y|αi)2 �− q τ dy � = O � ρ−q+qηq� for all i even, in view of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Similarly, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='8) we deduce that � Ai ��V1e−τU��q dx = O(ρ−q+qηq) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='27) for i odd, again in view of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Therefore, by using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='25) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='27) and that βi αi are decreasing, we deduce that ��V0eU��q q = m � i=1 i odd O � ρ βi αi (2−2q)−q� + O � ρ−q+qηq� = O � ρ β1 α1 (2−2q)−q� for any q ≥ 1 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='28) and, by using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='26) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='27) we obtain that ��V1e−τU��q q = O � ρ β1 α1 (2−2q)−q� for any q ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='29) On the other hand, using the estimate |ea − 1| ≤ C|a| uniformly for any a in compact subsets of IR, H¨older’s inequality with 1 s′ i + 1 t′ i = 1, i = 0, 1, ∥˜φµi∥ ≤ Mρηp| log ρ| ≤ C, i = 0, 1 and triangle inequality we find that ��Vie(−τ)i(U+ ˜φµi )�� q = O � ρ η0,qs′ i −1+ηp| log ρ| + ρη0,q−1� , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='30) where for q > 1 we denote η0,q = β1(2−2q) α1q (on the line of [17, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='14) proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Note that η0,q < 0 for any q > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Also, choosing q and s′ i, i = 0, 1, close enough to 1, we get that 0 < ηp + η0,qs′ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Now, we can conclude the estimate by using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='23)-(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='30) to get ∥N(φ1) − N(φ2)∥p ≤ C 1 � i=0 ρ∥Vie(−τ)i(U+ ˜φµi )∥pri∥φθi∥∥φ1 − φ2∥ ≤ C 1 � i=0 ρηp+η0,pri| log ρ|∥φ1 − φ2∥ and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='21) follows, where η′′ p = 1 2 min{ηp + η0,pri : i = 0, 1} > 0 choosing ri close to 1 so that ηp + η0,pri > 0 for i = 0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Let us stress that p > 1 is chosen so that ηp > 0 and η′ p > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' □ Proof of the Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Taking into account Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='1, estimates (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='1), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='20), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='21), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='22) and standard arguments it turns out that for all ρ > 0 sufficiently small A is a contraction mapping of FM (for M large enough), and therefore a unique fixed point of A exists in FM, where A(φ) := −T (R + Λ(φ) + N(φ)) and FM = {φ ∈ H1 0(Ωǫ) : ∥φ∥ ≤ Mρηp| log ρ|}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' See the proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='2 in [13] for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' □ Proof of the Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Taking into account (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='10) and the definition of U, the existence of a solution to equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='1) follows directly by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' The asymptotic behavior of uρ as ρ → 0+ follows from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='23) in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='6 and estimate for φ in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Furthermore, we have that uρ has the desired concentration properties (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='6) as ρ → 0+ locally uniformly in ¯Ω \\ {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Problem in mean field form (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='2) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Error estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Assume that δi’s are given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='2) with αi’s, βi’s and di’s defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='4), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='9)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='10) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='15), respectively, and ǫ is given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Let us evaluate the error term in ∥ · ∥p, encoded in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' SIGN-CHANGING BUBBLE TOWER SOLUTIONS ON PIERCED DOMAINS 17 Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' There exist ρ0 > 0, a constant C > 0 and 1 < p0 < 2, such that for any ρ ∈ (0, ρ0) and p ∈ (1, p0) it holds ∥R∥p ≤ Cρηp (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='1) for some ηp > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Proof: By using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='5) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='8) we have that � Ωǫ V0(x)eU = m � j=1 � Aj δj δ2 j V0(δjy)eU(δjy)dy = m � j=1 j odd 1 ρ � Aj δj 2α2 j|y|αj−2 δ2 j (1 + |y|αj)2 [1 + O(δj|y| + ρη)] dy + m � j=1 j even O � ρ1/τδ2+2/τ j � Aj δj (1 + |y|αj)2/τ |y|(αj−2)/τ dy � = 1 ρ � 4π m � j=1 j odd αj + O(ρη) � (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='2) and similarly, from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='6) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='7) we get that � Ωǫ V1(x)e−τU = 1 ρτ � 4π m � j=1 j even αj + O(ρη) � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='3) in view of � R2 |y|αi−2 (1+|y|αi)2 dy = 2π αi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' From assumptions (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='8)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='9) it follows that λ0 = 4π �m j=1 j odd αj and λ1τ 2 = 4π �m j=1 j even αj so that, by using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='2)-(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='3) we find that for y ∈ Aj δj and any j odd λ0 V0(δjy)eU(δjy) � Ωǫ V0(x)eU = 2α2 j|y|αj−2 δ2 j (1 + |y|αj)2 [1 + O(δj|y| + ρη)] (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='4) and λ1τ V1(δjy)e−τU(δjy) � Ωǫ V1(x)e−τU = O � ρ1+τδ2τ j (1 + |y|αj)2τ |y|(αj−2)τ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='5) Similarly, for any y ∈ Aj δj and j even λ0 V0(δjy)eU(δjy) � Ωǫ V0(x)eU = O � ρ1+1/τδ2/τ j (1 + |y|αj)2/τ |y|(αj−2)/τ � (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='6) and λ1τ V1(δjy)e−τU(δjy) � Ωǫ V1(x)e−τU = 2α2 j|y|αj−2 δ2 j τ(1 + |y|αj)2 [1 + O(δj|y| + ρη)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='7) Hence, similar to the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='1 by using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='2)-(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='4) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='4)-(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='7) for δ1y ∈ A1 we get that R(δ1y) = 1 δ2 1 2α2 1|y|α1−2 (1 + |y|α1)2 O(δ1|y| + ρη) + O � ρ1+τδ2τ 1 (1 + |y|α1)2τ |y|(α1−2)τ + m � i=2 �δ1 δi �αi |y|αi−2 δ2 1 � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='8) for 1 < j < m, j odd and δjy ∈ Aj R(δjy) = 1 δ2 j 2α2 j|y|αj−2 (1 + |y|αj)2 O(δj|y| + ρη) + O � ρ1+τδ2τ j (1 + |y|αj)2τ |y|(αj−2)τ + j−1 � i=1 � δi δj �αi 1 δ2 j |y|αi+2 + m � i=j+1 �δj δi �αi |y|αi−2 δ2 j � (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='9) for 1 < j < m, j even and δjy ∈ Aj R(δjy) = 1 δ2 j τ 2α2 j|y|αj−2 (1 + |y|αj)2 O(δj|y| + ρη) + O � ρ1+1/τδ2/τ j (1 + |y|αj)2/τ |y|(αj−2)/τ + j−1 � i=1 � δi δj �αi 1 δ2 j |y|αi+2 + m � i=j+1 �δj δi �αi |y|αi−2 δ2 j � (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='10) 18 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' FIGUEROA and for δmy ∈ Am R(δmy) = 1 δ2mτ ν(m) 2α2 m|y|αm−2 (1 + |y|αm)2 O(δm|y| + ρη) + O � ρ1+τ (−1)m+1 δ2τ (−1)m+1 m (1 + |y|αm)2τ (−1)m+1 |y|(αm−2)τ (−1)m+1 + m−1 � i=1 � δi δm �αi 1 δ2m|y|αi+2 � (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='11) By (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='8)-(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='11) and similar ideas to get (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='1) we conclude the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' The nonlinear problem and proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' In this section we shall study the following nonlinear problem: � L(φ) = −[R + Λ0(φ) + N(φ)] in Ωǫ φ = 0, on ∂Ωǫ, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='12) where the linear operators L, Λ0 are defined as L(φ) = ∆φ + K0 � φ − 1 λ0 � Ωǫ K0φdx � + K1 � φ − 1 λ1τ 2 � Ωǫ K1φdx � (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='13) and Λ0(φ) = λ0 V0eU � Ωǫ V0eUdx � φ − � Ωǫ V0eUφdx � Ωǫ V0eUdx � + λ1τ 2 V1e−τU � Ωǫ V1e−τUdx � φ − � Ωǫ V1e−τUφdx � Ωǫ V1(x)e−τUdx � − K0 � φ − 1 λ0 � Ωǫ K0φdx � − K1 � φ − 1 λ1τ 2 � Ωǫ K1φdx � (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='14) with K0 = m � k=1 k odd |x|αk−2ewk, K1 = m � k=1 k even |x|αk−2ewk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='15) The nonlinear term N(φ) is given by N(φ) =λ0 � V0eU+φ � Ωǫ V0eU+φdx − V0eU � Ωǫ V0eUdx − V0eU � Ωǫ V0eUdx � φ − � Ωǫ V0eUφdx � Ωǫ V0eUdx �� − λ1τ � V1e−τ(U+φ)dx � Ωǫ V1e−τ(U+φ)dx − V1e−τU � Ωǫ V1e−τUdx + τ V1e−τU � Ωǫ V1e−τUdx � φ − � Ωǫ V1e−τUφdx � Ωǫ V1e−τUdx �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='16) It is readily checked that φ is a solution to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='13) if and only if uǫ given by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='11) is a solution to (?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' In Section 5 we will prove the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' For any p > 1, there exists ρ0 > 0 and C > 0 such that for any ρ ∈ (0, ρ0) and h ∈ Lp(Ωǫ) there exists a unique φ ∈ H1 0(Ωǫ) solution of L(φ) = h in Ωǫ, φ = 0 on ∂Ωǫ, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='17) which satisfies ∥φ∥ ≤ C| log ρ| ∥h∥p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='18) Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' There exist p0 > 1 and ρ0 > 0 so that for any 1 < p < p0 and all 0 < ρ ≤ ρ0 it holds ∥Λ0(φ)∥p ≤ Cρσ′ p∥φ∥, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='19) for all φ ∈ H1 0(Ωǫ) with ∥φ∥ ≤ Mρσp| log ρ|, for some σ′ p > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Proof: Arguing in the same way as in [13, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='3], denote Wi = λiτ 2iVie(−τ)iU � Ωǫ Vie(−τ)iU dx for i = 0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' By us- ing (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='4), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='6), Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='1 and similar computations to prove (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='20), we find that ∥W0 − K0∥q Lq(Ai) ≤ Cρqσ′ 0,q for any i for some σ′ 0,q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Similarly, we find that ∥W1 − K1∥q Lq(Ai) ≤ Cρqσ′ 1,q for any i for some σ′ 1,q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' It is possible to see that taking q > 1 close enough to 1, we get that σ′ i,q > 0 for i = 0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' SIGN-CHANGING BUBBLE TOWER SOLUTIONS ON PIERCED DOMAINS 19 Notice that Λ0 is a linear operator and we re-write Λ0(φ) as Λ0(φ) = 1 � i=0 � (Wi − Ki) φ − 1 λiτ 2i (Wi − Ki) � Ωǫ Wiφ + 1 λiτ 2i Ki � Ωǫ (Ki − Wi) φ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Hence, we get that ∥Λ0(φ)∥p ≤ 1 � i=0 � ∥Wi − Ki∥pri0 ∥φ∥psi0 + ∥Wi∥ri1 λiτ 2i ∥Wi − Ki∥p ∥φ∥si1 + ∥Ki∥p λiτ 2(i−1) ∥Ki − Wi∥ri2 ∥φ∥si2 � ≤ C 1 � i=0 � ρσ′ i,pri0 ∥φ∥ + ρσ′ i,p+σ3,ri1 ∥φ∥ + ρσ′ i,ri2 +σ3,p∥φ∥ � and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='19) follows, where σ′ p = min � σ′ i,pri0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' σ′ i,p + σ3,ri1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' σ′ i,ri2 + σ3,p | i = 0, 1 � with rij, sij, i = 0, 1, j = 0, 1, 2 satisfying 1 rij + 1 sij = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' We have used that ∥W0∥r01 r01 ≤ C � m � j=1 j odd δ2−2r01 j � Aj δj ����� 2α2 j|y|αj−2 (1 + |y|αi)2 ����� r01 + m � j=1 j even ρ(1+1/τ)r01δ 2+2 r01 τ j � Aj δj ���� (1 + |y|αj)2/τ |y|(αj−2)/τ ���� r01� ≤ Cρσ3,r01 and similarly, ∥W1∥r11 r11 ≤ Cρσ3,r11 , where σ3,q = β1 α1 (2−2q) and similarly that ∥Ki∥p p ≤ Cρσ3,p, i = 0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Note that 2−2q αjq < 1 for any j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' , m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Furthermore, we have used the H¨older’s inequality ∥uv∥q ≤ ∥u∥qr∥v∥qs with 1 r + 1 s = 1 and the inclusions Lp(Ωǫ) ֒→ Lpr(Ωǫ) for any r > 1 and H1 0(Ωǫ) ֒→ Lq(Ωǫ) for any q > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Let us stress that we can choose p, rij and sij, i = 1, 2, j = 0, 1, 2, close enough to 1 such that σ′ p > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' □ Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' There exist p0 > 1 and ρ0 > 0 so that for any 1 < p < p0 and all 0 < ρ ≤ ρ0 it holds ∥N(φ1) − N(φ2)∥p ≤ Cρσ′′ p ∥φ1 − φ2∥ (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='20) for all φi ∈ H1 0(Ωǫ) with ∥φi∥ ≤ Mρσp| log ρ|, i = 1, 2, and for some σ′′ p > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' In particular, we have that ∥N(φ)∥p ≤ Cρσ′′ p ∥φ∥ (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='21) for all φ ∈ H1 0(Ωǫ) with ∥φ∥ ≤ Mρσp| log ρ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Proof: Arguing in the same way as in [1, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='1], we denote gi(φ) = Vi(x)e(−τ)i(U+φ) � Ωǫ Vi(x)e(−τ)i(U+φ) and point out that N(φ) = 1 � i=0 λi(−τ)i {gi(φ) − gi(0) − g′ i(0)[φ]} and by the mean value theorem we get that N(φ1) − N(φ2) = 2 � i=1 λi(−τ)ig′′ i (˜φµi)[φθi, φ1 − φ2], (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='22) where φθi = θiφ1 + (1 − θi)φ2, ˜φµi = µiφθi for some θi, µi ∈ [0, 1], i = 0, 1, and g′′ i (φ)[ψ, v] = τ 2i � Vi(x)euiψv � Ωǫ Vi(x)eui − Vi(x)euiv � Ωǫ Vi(x)euiψ � � Ωǫ Vi(x)eui�2 − Vi(x)euiψ � Ωǫ Vi(x)euiv � � Ωǫ Vi(x)eui�2 − Vi(x)eui � Ωǫ Vi(x)euiψv � � Ωǫ Vi(x)eui�2 + 2 Vi(x)eui � Ωǫ Vi(x)euiv � Ωǫ Vi(x)euiψ � � Ωǫ Vi(x)eui�3 � , 20 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' FIGUEROA where for simplicity we denote ui = (−τ)i(U + φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Using H¨older’s inequalities we get that ∥g′′ i (φ)[ψ, v]∥p ≤ C � ∥Vieui∥pri ∥Vieui∥1 + ∥Vieui∥2 pri ∥Vieui∥2 1 + ∥Vieui∥3 pri ∥Vieui∥3 1 � ∥ψ∥∥v∥, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='23) with 1 ri + 1 si + 1 ti = 1, 1 ri + 1 qi = 1, 1 pri + 1 ˜ri = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' We have used the H¨older’s inequality, the inclusions presented in the previous Lemma and ∥uvw∥q ≤ ∥u∥qr∥v∥qs∥w∥qt with 1 r + 1 s + 1 t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Now, let us estimate ∥Vieui ∥pri ∥Vieui ∥1 with φ = ˜φµi, i = 0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Taking into account (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='28)-(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='29) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='2)-(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='3), we obtain that for i = 0, 1 ���Vie(−τ)iU��� q q = O � ρ β1 α1 (2−2q)−q� for any q ≥ 1 and ���Vie(−τ)iU��� 1 ≥ Ci ρ for some Ci > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Note that σ3,q − 1 ≤ 2−(αi+2)q αiq for any i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' , m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' By the previous estimates we find that ��Vie(−τ)iU+ ˜φµi �� q = O � ρ η0,qs′ i −1+ηp| log ρ| + ρη0,q−1� (on the line of [17, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='14) proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Also, choosing si, i = 0, 1, close enough to 1, we get that σp + η0,si > 0 and ���Vie(−τ)i(U+ ˜φµi )��� 1 ≥ Ci ρ − Cρη0,si −1+σp| log ρ| ≥ 1 ρ � Ci − Cρη0,si +σp | log ρ| � ≥ Ci 2ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Taking q = pri, we obtain the estimate for i = 0, 1 ∥Vie(−τ)i(U+ ˜φµi )∥pri ∥Vie(−τ)i(U+ ˜φµi )∥1 = O � ρσ3,pri � ρσp+σ3,prisi −σpri| log ρ| + 1 �� = O (ρσ3,pri ) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='24) choosing si > 1 close enough to 1 so that σp + σ3,prisi − σpri > 0, i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Now, we can conclude the estimate by using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='22)-(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='24) to get ∥N(φ1) − N(φ2)∥p ≤ C 1 � i=0 λiρσp+3σ3,pri | log ρ|∥φ1 − φ2∥ ≤ Cρσ′′ p ∥φ1 − φ2∥, where σ′′ p = 1 2 min{σp + 3σ3,pri : i = 0, 1} > 0 choosing ri close to 1 so that σp + 3σ3,pri > 0 for i = 0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Let us stress that p > 1 is chosen so that σp > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' □ Taking into account previous estimates (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='1), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='19), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='20) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='21), and by using the same argument as in the proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='2 we conclude the following Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' There exist p0 > 1 and ρ0 > 0 so that for any 1 < p < p0 and all 0 < ρ ≤ ρ0, the problem (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='12) admits a unique solution φ(ρ) ∈ H1 0(Ωǫ), where L, R, Λ0(φ) and N are given by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='13), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='13), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='14) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='16), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Moreover, there is a constant C > 0 such that ∥φ∥∞ ≤ Cρσp| log ρ|, for some σp > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Proof of the Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Arguing in the same way as in the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='1, the existence of a solution (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='10) to equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='2) follows directly by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='2 and the definition of U in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' The linear theory In this section we present the invertibility of the linear operators L and L defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='14) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='13) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Roughly speaking in the scale annulus Aj δj the operator L apporaches to the following linear operator in IR2 Lj(φ) = ∆φ + 2α2 j|y|αj−2 (1 + |y|αj)2 φ, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' , m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' SIGN-CHANGING BUBBLE TOWER SOLUTIONS ON PIERCED DOMAINS 21 It is well known that, in case αj ∈ 2IN, the bounded solutions of Lj(φ) = 0 in IR2 are precisely linear combinations of the functions Y1j(y) = |y| αj 2 1 + |y|αj cos �αj 2 θ � , Y2j(y) = |y| αj 2 1 + |y|αj sin �αj 2 θ � and Y0j(y) = 1 − |y|αj 1 + |y|αj , which are written in polar coordinates for j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' , m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' See [10] for a proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' In our case, we will consider solutions of Lj(φ) = 0 with αj /∈ 2IN for all j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' , m such that � IR2 |∇φ(y)|2 dy < +∞, which are multiples of Y0j, see [1, Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='1] for a proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Another key element in the study of L, which shows technical details, is to get rid of the presence of ˜cj(φ) = − 1 λjτ 2j � Ωǫ Kjφ j = 0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='1) Let us introduce the following Banach spaces for j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' , m Lαj(IR2) = � u ∈ W 1,2 loc (IR2) : � IR2 |y|αj−2 (1 + |y|αj)2 |u(y)|2 dy < +∞ � (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='2) and Hαj(IR2) = � u ∈ W 1,2 loc (IR2) : � IR2 |∇u(y)|2 dy + � IR2 |y|αj−2 (1 + |y|αj)2 |u(y)|2 dy < +∞ � (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='3) endowed with the norms ∥u∥Lαj := �� IR2 |y|αj−2 (1 + |y|αj)2 |u(y)|2 dy �1/2 and ∥u∥Hαj := �� IR2 |∇u(y)|2 dy + � IR2 |y|αj−2 (1 + |y|αj)2 |u(y)|2 dy �1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' We point out the compactness of the embedding iαj : Hαj(IR2) → Lαj(IR2), (see for example [20]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Let us assume the existence of p > 1, sequences ρn → 0, ǫn = ǫ(ρn) → 0, functions hn ∈ Lp(Ωǫn), φn ∈ H1 0(Ωǫn) such that L(φn) = hn in Ωǫn, φn = 0 on ∂Ωǫn ∥φn∥ = 1 and | log ρn| ∥hn∥p = o(1) as n → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' We will omit the subscript n in δi,n = δi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Recall that δαi i = di,nρβi n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Now, define Φi,n(y) := φi,n(δiy) for y ∈ Ωi,n := δ−1 i Ωǫn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Thus, extending φn = 0 in IR2 \\ Ωǫn and arguing in the same way as in [17, Claim 1, section 4] we can prove the following fact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' We provide a sketch of the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Claim 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' There holds that the sequence Φi,n converges (up to subsequence) to Φ∗ i = aiY0i for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' , m, weakly in Hαi(IR2) and strongly in Lαi(IR2) as n → +∞ for some constant ai ∈ IR, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' , m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Proof: First, notice that ∥Φi,n∥H1 0 (Ωi,n) = 1, for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' , m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Then, we want to prove that there is a constant M > 0 such for all n (up to a subsequence) ∥Φi,n∥2 Lαi ≤ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Notice that for any i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' , m} we find that in Ωi,n ∆Φi,n + δ2 i K(δiy)Φi,n = δ2 i hn(δiy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='4) Furthermore, it follows that Φi,n → Φ∗ i weakly in H1 0(Ωi,n) and strongly in Lp(K) for any K compact sets in R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' Now, we multiply (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='4) by Φi,n for any i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content=' , m}, integrate by parts and we obtain that m � i=1 2α2 i ∥Φi,n∥2 Lαi = 1 + o(1) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQf4f6J/content/2301.01845v1.pdf'} +page_content='5) since δ2 i K(δiy) = 2α2 i |y|αi−2 (1 + |y|αi)2 + O � � j> +a>0.6eelee0.71.55 +nI.C09 +1551.55 +n080.91.6 +mm1.6I.C1.6retiectancebsaerptance2+hGe absorptarice(a)(b(c)I.C1.555a +1.61.5551.51 +55OSG0.510.5- +e0.1nS +b0.21七 +e +CICe1+1+0.C(a))03angle 0, degangle 0. deq(f)angle 0, de0.4. +6030 +0 +15 +3060 +300 +15 +30I. +6030 +0 +10.5 +5 +305W553 +where θBIC is the angle corresponding to the off-Γ BIC +in the spectrum while α and β are fitting parameters +which can be obtained from numerical simulations. The +absorption coefficient can be written as +A = +4γ1(θ)γ2 +[ω − ¯ω(θ)]2 + [γ1(θ) + γ2]2 +(3) +The critical coupling points are topologically protected +Figure 3. +Absorptance spectra of the PhC-based structure +calculated by (a) FDTD and (b) TCMT theory. Comparison +of the absorptance spectra of the structure calculated by two +different approach at Γ-point. The period p = 620 nm and +d = 26.5 nm. The other parameters are the same as for Fig. 2. +objects associated with phase singularities of the reflec- +tion amplitude36. For this reason, the effects of critical +coupling and perfect light absorption survive under varia- +tion of the system’s parameters preserving all symmetries +of the structure. Thus, by changing the parameters one +can achieve the critical coupling in the Γ-point if +γ2 = βθ4 +BIC. +(4) +By varying the grating period we achieve the in-Γ crit- +ical coupling in the Ge-based structure with period +p = 620 nm which led to the following values of the +TCMT parameters α ≈ 4.3 · 10−6 µm−1deg−2, β ≈ +1.27 · 10−8 µm−1deg−4, λBIC = 1.55 µm. In Fig. 3 (a- +b) we compare the numerical data against Eq. (3). The +absorption spectrum at the normal incidence is shown in +Fig. 3 (c). One can see a good coincidence between the +full-wave simulations and the TCMT approximation of +the spectrum. +In summary, we have demonstrated that application +of a high-Q Tamm state makes in possible to engineer a +perfect light absorber which provides a 100% absorption +in a semiconductor grating. The proposed design may be +used at both normal and oblique incidence at the telecom +wavelength. We believe that the reported wavelength and +angle selectivity can be of use in small LiDAR detectors. +This research was funded by the Russian Science Foun- +dation (project no. 22-42-08003). This work was sup- +ported by the Higher Education Sprout Project of the +National Yang Ming Chiao Tung University and Ministry +of Education and the National Science and Technology +Council (NSTC 109-2628-E-007 -003 -MY3; 111-2923-E- +007 -008 -MY3; 111-2628-E-007-021 ). +The authors declare no conflicts of interest. +1 Y. Akahane, T. Asano, B.-S. Song, and S. Noda, Nature +425, 944 (2003). +2 B.-S. Song, S. Noda, T. Asano, and Y. Akahane, Nature +Materials 4, 207 (2005). +3 T. Asano, Y. Ochi, Y. Takahashi, K. Kishimoto, +and +S. Noda, Optics Express 25, 1769 (2017). +4 Y.-C. Hsiao, in Photonic Crystals - A Glimpse of the Cur- +rent Research Trends (IntechOpen, 2019). +5 M. SoljaˇCi´C and J. D. Joannopoulos, Nature Materials 3, +211 (2004). +6 S. G. Moiseev, I. A. Glukhov, Y. S. 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Maksimov, Physical Review Let- +ters 118, 267401 (2017). + diff --git a/ktE3T4oBgHgl3EQfJwko/content/tmp_files/load_file.txt b/ktE3T4oBgHgl3EQfJwko/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..471c4b80c626e27d23c4fb80812fac4102cd2667 --- /dev/null +++ b/ktE3T4oBgHgl3EQfJwko/content/tmp_files/load_file.txt @@ -0,0 +1,462 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf,len=461 +page_content='Enhanced light absorption in Tamm metasurface with a bound state in the continuum Rashid G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content=' Bikbaev1,2,∗, Dmitrii N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content=' Maksimov1,2,∗, Pavel S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content=' Pankin1,2, Ming-Jyun Ye3, Kuo-Ping Chen3,4, and Ivan V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content=' Timofeev1,2 1Kirensky Institute of Physics, Federal Research Center KSC SB RAS, 660036, Krasnoyarsk, Russia 2Siberian Federal University, Krasnoyarsk 660041, Russia 3College of Photonics, National Yang Ming Chiao Tung University, Tainan 711, Taiwan and 4Institute of Photonics Technologies, National Tsing Hua University, Hsinchu 300, Taiwan (Dated: January 12, 2023) We consider light absorption in a germanium grating placed on top of photonic-crystalline sub- strate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content=' Such a system supports an optical Tamm state decoupled from the continuous spectrum with its frequency within the photonic band gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content=' We have demonstrated that application of the Tamm state makes in possible to engineer extremely narrow absorber which provides a 100% absorption in a semiconductor grating in the critical coupling regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content=' The proposed design may be used at both normal and oblique incidence at the telecom wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content=' Localized modes in periodic structures are of great in- terest due to record high Q-factors1–3 as well as the op- portunity to design tunable devices by using liquid crys- tals4, nonlinear media5 or resonant materials6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content=' Along with studies on volume electromagnetic waves localized in structural defects, attention is paid to surface waves localized at the boundary with negative dielectric per- mittivity media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content=' In this case, the light is trapped at the boundary between plasmonic and dielectric struc- tures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content=' These localized states are called Tamm plas- mon polaritons7 (TPPs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content=' The TPPs are applied for engineering optical devices, such as photoelectrochemi- cal cells8, sensors9, lasers10, beam steerers11 and solar cells12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content=' Coupling of TPPs with other types of localized modes, for example, with a surface plasmon-polariton13, leads to hybrid modes which are widely applied in opti- cal sensors14,15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content=' Tailoring the parameters of TPP sup- porting structures makes it possible to set-up the critical coupling at which all radiation incident on the structure is absorbed at the TPP wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content=' This resonant ab- sorption mechanism is used in absorbers16,17 and pho- todetectors18,19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content=' Such surface-localized status can be ex- cited at the boundary between a photonic crystal and plasmonic20,21 or dielectric metasurfaces22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content=' In the latter case, such localized states are called optical Tamm states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content=' Lately, we have seen a surge of interest to photonic non-radiation states, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content=' optical bound states in the continuum (BICs)23–25 which have become an impor- tant instrument for engineering optical devices with en- hanced light-matter interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content=' In the presence of ma- terial absorption the BIC is shown to acquire finite-life, albeit remain localized and decoupled from the outgo- ing channels26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content=' Quasi-BIC in lossy periodic structures is found to be instrumental for enhancement of light ab- sorption27,28 in the critical coupling regime even in low loss dielectrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content=' Therefor, BIC concept opens novel op- portunities for highly efficient light absorbers29–33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content=' It has been demonstrated in27 that application of per- fect mirror in the substrate of a BIC supporting struc- tures makes it possible to set-up a perfect light absorber in the so-called critical coupling regime when the radia- tive and non-radiative Q-factors are equal to one another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content=' Schematic representation of the Au and PhC-based structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content=' H and L are height and width of nanostripes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content=' p is period of nanostripes along x-axis and d is Ge substrate thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content=' The PhC consists of alternating layers of silicon dioxide and titanium dioxide with refractive indices 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content='44 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content='45, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content=' The thicknesses of layers are 268.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content='3 nm and 156.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content='8 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content=' The thickness of the Au substrate is 200 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content=' In practice the mirror substrate for a dielectric structure can be implemented as an opaque metal film.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content=' All metals, however, exhibit significant material losses so the radia- tion is absorbed not only in the dielectric or semiconduc- tor but in the mirror itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content=' Thus, the electromagnetic energy is wasted on heating the substrate rather than produce the desirable photoelectric effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content=' In this letter we propose a set-up with a non-absorbing mirror which leads to 100% of incident radiation absorbed in the ger- manium (Ge) metasurface34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content=' In Table 1 of paper35, the calculated total Q-factor (1/Qtotal = 1/Qrad + 1/Qmat) of Ge is quite high at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content='5µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content=' For designing high-Q meta- surfaces at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content='5µm, it is more suitable than silicon in group IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content=' The system is depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content=' The structure con- sists of a periodic Ge grating placed on top of 1D photonic crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content=' Later on the set-up will be referred to as the PhC-based structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content=' For comparison we will consider a similar Ge grating but placed on an Au substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content=' The latter set-up will be referred to as the Au-based structure and is also shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content=' The Tamm metasurface structure has been proposed in arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content='04346v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content='optics] 11 Jan 2023 Ge TiO2 SiO2 PhC-basedstructure2 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content=' (a) Reflectance, (b) absorptance spectra of the Au-based and PhC-based structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content=' (c) Absortance of the Ge in Au- based and PhC-based structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content=' For Au-based structure H = 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content='5 nm and p = 300 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content=' For PhC-based structure H = 200 nm and p = 675 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content=' For both setups L = 350 nm and d = 25 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content=' (d-e) Reflectance and absorptance spectra of the Au-based and PhC-based structures at 27 deg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content=' and 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content='5 deg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content=', respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content=' (f) Absorptance spectra of the Ge in both setups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content=' our previous work36 where we demonstrated that an off-Γ BIC leads to emergence of critical coupling (CC) points in the parameter space of incident frequency ω and incident angle θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content=' In36 the critical coupling effect occurred as zero reflection due to tunnelling across the band-gap in the PhC substrate with a finite number of bilayers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content=' Here we suppose that the substrate is thick enough to suppress the tunneling and expect that the reflectance zeros are associated with the perfect light absorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content=' First, let us compare light absorption by the PhC- and Au-based structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content=' To carry out the comparison we defined the parameters of the structures such that both support an in-Γ BIC at frequency λBIC = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content='545 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content=' Cor- responding parameters of the structure are presented in caption of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content=' The BICs do not couple with the inci- dent light at the normal incidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content=' Therefore, to obtain the critical coupling one has to vary the angle of inci- dence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content=' The increase of the angle of incidence results in a drop of the radiative Q-factor of the leaky band host- ing the BIC until the radiation and material loss rates are equal to one another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content=' This effect is visible in the re- flectance spectra of the structures calculated by the finite difference time domain method shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content=' 2 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content=' In the PhC-based structure the critical coupling is achieved at θ = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content='5 deg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content=', thereas in the Au-based structure the critical coupling angle is θ = 27 deg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content=' The Au-based struc- ture is more lossy, hence the critical coupling is observed at a larger angle of incidence with a larger width of the resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content=' One can see in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content=' 2 (b) that in both struc- tures ≈ 100% of incident radiation is absorbed at the critical coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content=' However, as seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content=' 2 (c) less than 30% of energy is absorbed in Ge in the Au-based struc- ture while in the PhC-based structure the absorptance in Ge is A ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content=' Such a large difference in the absorption of the Ge is due to the fact that most of the radiation incident on the Au-based structure is absorbed in the plasmonic substrate, while in the PhC-based structure the substrate is all-dielectric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content=' For the reader’s conve- nience, the frequency dependencies of the absorption and reflection coefficients at the critical coupling are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content=' 2 (d-f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content=' Notice that with an in-Γ BIC the critical coupling is never obtained at the normal incidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content=' To engineer the perfect absorption in Ge at the normal incidence we re- turn to an off-Γ BIC reported in36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content=' Following36 we de- scribe the scattering spectrum of the Ge-based structure in the framework of the temporal coupled mode theory37 which yields the following solution for the reflection am- plitude r = � −1 + 2γ1(θ) i[ω − ¯ω(θ)] + γ1(θ) + γ2 � , (1) where ω is the frequency of the incident wave, ¯ω - the res- onant frequency, θ - the angle of incidence, γ1 is the loss rate due to coupling to radiation to the upper half-space, and γ2 is the loss rate due to absorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content=' According to38 the dispersion of γ1 and ¯ω are given by ¯ω = ω0 − αθ2 + O(θ4), γ1 = β(θ2 − θ2 BIC)2 + O(θ6), (2) I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content='C>> a>0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content='6eelee0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content='71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content='55 nI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content='C09 1551.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content='55 n080.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content='91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content='6 mm1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content='6I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content='C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content='6retiectancebsaerptance2+hGe absorptarice(a)(b(c)I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content='C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content='555a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content='61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content='5551.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content='51 55OSG0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content='510.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content='5- e0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content='1nS b0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content='21七 e CICe1+1+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content='C(a))03angle 0, degangle 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content=' deq(f)angle 0, de0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content=' 6030 0 15 3060 300 15 30I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content=' 6030 0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content='5 5 305W553 where θBIC is the angle corresponding to the off-Γ BIC in the spectrum while α and β are fitting parameters which can be obtained from numerical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content=' The absorption coefficient can be written as A = 4γ1(θ)γ2 [ω − ¯ω(θ)]2 + [γ1(θ) + γ2]2 (3) The critical coupling points are topologically protected Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content=' Absorptance spectra of the PhC-based structure calculated by (a) FDTD and (b) TCMT theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content=' Comparison of the absorptance spectra of the structure calculated by two different approach at Γ-point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content=' The period p = 620 nm and d = 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content='5 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content=' The other parameters are the same as for Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content=' objects associated with phase singularities of the reflec- tion amplitude36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content=' For this reason, the effects of critical coupling and perfect light absorption survive under varia- tion of the system’s parameters preserving all symmetries of the structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content=' Thus, by changing the parameters one can achieve the critical coupling in the Γ-point if γ2 = βθ4 BIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content=' (4) By varying the grating period we achieve the in-Γ crit- ical coupling in the Ge-based structure with period p = 620 nm which led to the following values of the TCMT parameters α ≈ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content='3 · 10−6 µm−1deg−2, β ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content='27 · 10−8 µm−1deg−4, λBIC = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content='55 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content=' 3 (a- b) we compare the numerical data against Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content=' The absorption spectrum at the normal incidence is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content=' 3 (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content=' One can see a good coincidence between the full-wave simulations and the TCMT approximation of the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content=' In summary, we have demonstrated that application of a high-Q Tamm state makes in possible to engineer a perfect light absorber which provides a 100% absorption in a semiconductor grating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content=' The proposed design may be used at both normal and oblique incidence at the telecom wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content=' We believe that the reported wavelength and angle selectivity can be of use in small LiDAR detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content=' This research was funded by the Russian Science Foun- dation (project no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content=' 22-42-08003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content=' This work was sup- ported by the Higher Education Sprout Project of the National Yang Ming Chiao Tung University and Ministry of Education and the National Science and Technology Council (NSTC 109-2628-E-007 -003 -MY3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content=' 111-2923-E- 007 -008 -MY3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content=' 111-2628-E-007-021 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content=' The authors declare no conflicts of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content=' 1 Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content=' Akahane, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content=' Asano, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content=' Song, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content=' Noda, Nature 425, 944 (2003).' metadata={'source': 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deg080.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content='915wavelen1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content='55 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content='540-2 ar0 ngle0,de20 eg40wavelen155 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content='5-40-2 a0 ngle0,de20 eg40FD TC(a 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content='64 vicius, Optics Express 28, 29033 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content=' 16 S.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content=' Kim, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content=' Lee, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content=' Ishii, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} +page_content=' Song, Advanced Optical Materials 10, 2102388 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE3T4oBgHgl3EQfJwko/content/2301.04346v1.pdf'} 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Storage in Magnesium and Its Alloys + + +Kaveh Edalati1,*, Etsuo Akiba2, Walter J. Botta3, Yuri Estrin4,5, Ricardo Floriano6, Daniel +Fruchart7,8, Thierry Grosdidier9,10, Zenji Horita1,11-13, Jacques Huot14, Hai-Wen Li15, Huai-Jun +Lin16, Ádám Révész17 and Michael J. Zehetbauer18 + +1 WPI, International Institute for Carbon-Neutral Energy Research (WPI-I2CNER), Kyushu +University, Fukuoka, Japan +2 International Research Center for Hydrogen Energy, Kyushu University, Fukuoka, Japan +3 Departamento de Engenharia de Materiais, Universidade Federal de São Carlos, Sao Carlos-SP, +Brazil +4 Department of Materials Science and Engineering, Monash University, Clayton, VIC 3800, +Australia +5 Department of Mechanical Engineering, The University of Western Australia, Crawley, WA +6009, Australia +6 School of Applied Sciences, University of Campinas (UNICAMP), Limeira, São Paulo, Brazil +7 Institut Néel, CNRS & UGA, 38042 Grenoble, France +8 JOMI-LEMAN SA, 74890 Fessy, France +9 Université de Lorraine, Laboratory of Excellence on Design of Alloy Metals for low-mass +Structures (DAMAS), Metz, F-57070, France +10 Université de Lorraine, Laboratoire d’Etude des Microstructures et de Mécanique des Matériaux +(LEM3 UMR 7239), Metz, F-57070, France +11 Graduate School of Engineering, Kyushu Institute of Technology, Kitakyushu, Japan +12 Magnesium Research Center, Kumamoto University, Kumamoto, Japan +13 Synchrotron Light Application Center, Saga University, Saga, Japan +14 Hydrogen Research Institute, Université du Québec à Trois-Rivières, 3351 des Forges, Trois- +Rivières, QC G9A 5H7, Canada +15 Hefei General Machinery Research Institute, Hefei 230031, China +16 Institute of Advanced Wear & Corrosion Resistance and Functional Materials, Jinan University, +Guangzhou 510632, China +17 Department of Materials Physics, Eötvös University, Budapest, H-1518, P.O.B. 32, Budapest, +Hungary +18 Faculty of Physics, University of Vienna, Boltzmanngasse 5, A-1090 Wien, Austria + +*Corresponding author (E-mail: kaveh.edalati@kyudai.jp; Tel/Fax: +81-92-802-6744) + + + +2 + +Abstract +Magnesium and its alloys are the most investigated materials for solid-state hydrogen storage in +the form of metal hydrides, but there are still unresolved problems with the kinetics and +thermodynamics of hydrogenation and dehydrogenation of this group of materials. Severe plastic +deformation (SPD) methods, such as equal-channel angular pressing (ECAP), high-pressure +torsion (HPT), intensive rolling and fast forging, have been widely used to enhance the activation, +air resistance, and hydrogenation/dehydrogenation kinetics of Mg-based hydrogen storage +materials by introducing ultrafine/nanoscale grains and crystal lattice defects. These severely +deformed materials, particularly in the presence of alloying additives or second-phase +nanoparticles, can show not only fast hydrogen absorption/desorption kinetics but also good +cycling stability. It was shown that some materials that are apparently inert to hydrogen can absorb +hydrogen after SPD processing. Moreover, the SPD methods were effectively used for hydrogen +binding-energy engineering and synthesizing new magnesium alloys with low thermodynamic +stability for reversible low/room-temperature hydrogen storage, such as nanoglasses, high-entropy +alloys, and metastable phases including the high-pressure γ-MgH2 polymorph. This article reviews +recent advances in the development of Mg-based hydrogen storage materials by SPD processing +and discusses their potential in future applications. +Keywords: severe plastic deformation (SPD); nanostructured materials; ultrafine-grained (UFG) +materials; magnesium hydride (MgH2); magnesium-based alloys; hydrogen absorption; +hydrogenation kinetics; hydrogen storage thermodynamics + + + + + + + + +Severe Plastic Deformation +Kinetics +Thermodynamics +7 +ZKBOECAF +6 +Hydrogen (wt%) +Plunger +ZK60 after ECAP [Krystian et al., 2010] +ripnyuk et al. (2004) +Load +Magnesium +0 +5 +10 +15 +20 +25 +30 +Discharging Time (min) +100 +Rotation +MgaNiPd +N=1500 +Sample +Die +HPT ZK60 +Pressure (MPa) +Powders +10 +wt% +[%1m] +T=623K.P=1.1MPa +17 +Pyrometer +Piston +Window +Hydrogen ( +54 +4 +TH2=305K +2 +Rolling +0.1 +o1stCycle +5thCycle +Cylinders +Plate +% +0.01 +HF Heating +0 +0.2 +0.4 +0.6 +0.8 +1.0 +Sample +Coil +Time (h) +HydrogenContent (wt.%)3 + +Table of Contents +1. Introduction +2. Processing Methods of Mg-Based Materials +2.1. Equal-Channel Angular Pressing +2.2. High-Pressure Torsion +2.3. Intensive Rolling +2.4. Fast Forging +3. Kinetic Features of Severely Deformed Mg-based Materials +3.1. Activation +3.2. Air Resistance and Hydrogenation Kinetics +3.3. Cycling Stability +3.4. Hydrogenation of Inert Materials +4. Synthesis of Mg-based Materials with Desirable Thermodynamics +4.1. Binding-Energy Engineering for Room-Temperature Hydrogen Storage +4.2. Nanoglasses +4.3. High-Entropy Alloys +4.4. Metastable Phases +5. Concluding Remarks and Outlooks +Acknowledgments +References + + + +4 + +1. Introduction +Hydrogen, the lightest element in the periodic table, is considered to be the cleanest fuel of +the 21st century. The burning of hydrogen produces only water, making it an appropriate fuel +without any negative effects on global warming by CO2 emission. Moreover, the electrochemical +reaction of hydrogen with oxygen in fuel cells has higher energy efficiency compared to the +combustion of fossil fuels which suggests another advantage of using hydrogen as a fuel [1]. +Despite the advantages of hydrogen, there are still drawbacks concerning hydrogen production, +utilization and storage that need to be addressed [2]. +Hydrogen is mainly produced from gas reforming and a large amount of CO2 is generated +during this process. There are significant attempts to produce hydrogen by water splitting using +renewable energies via electrolysis and photocatalysis [2], although the efficiency of +photocatalysis is still quite low [3]. Fuel cells have high efficiency for the utilization of hydrogen, +but reducing their working temperature, developing feasible catalysts, and reducing their price are +some major challenges in using fuel cells [4]. Hydrogen can be stored in the form of high-pressure +gas, liquid, or solid, but these methods have some limitations [5]. Storage of hydrogen in the form +of high-pressure gas needs special tanks due to safety issues and these tanks occupy large space +[2,5]. Storage of hydrogen in the liquid form is a compact technology, but liquifying hydrogen +consumes energy and liquid hydrogen can evaporate over time [2,5]. Solid-state hydrogen storage, +particularly in the form of metal hydrides and complex hydrides, provides the most compact and +safest technology for hydrogen storage, but most hydrides suffer either from high thermodynamic +stability (i.e. high working temperature), slow kinetics, difficult activation, or low gravimetric +storage capacity [5,6]. +Magnesium and its hydride MgH2 are the first hydrogen storage materials that were +introduced in 1951 [7]. Magnesium is an abundant element in the Earth’s crust, it is rather cheap, +and it can store 7.6 wt% of hydrogen [8]. However, magnesium exhibits strong Mg-H bonding, +which results in great thermodynamic stability of the hydride requiring high hydrogen desorption +temperatures well above 573 K [9]. Moreover, it suffers from poor activation and slow +hydrogenation/dehydrogenation kinetics mainly due to the presence of an oxide layer on its surface +and the slow diffusion of hydrogen in the bulk [8,9]. Alloying of magnesium, development of +composites and intermetallic compounds, the addition of catalysts, and microstructural +modifications are some strategies used to solve the thermodynamic and kinetic problems of Mg- +based hydrogen storage materials [8,9]. Processing by severe plastic deformation (SPD), first +proposed by Skripnyuk et al. [10], is an effective strategy used within the past two decades to +address both the kinetics and the thermodynamics of hydrogen storage in Mg-based materials. +In SPD, a large plastic strain is induced in a piece of material to produce a final product in +a bulk form with ultrafine grains or nanograins and large fractions of crystal lattice defects [11]. +Equal-channel angular pressing (ECAP) [12], high-pressure torsion (HPT) [13], and accumulative +roll-bonding (ARB) [14] are currently the most popular SPD processes, but there are some trends +to induce severe strain in hydrogen storage materials by conventional methods such as intensive +rolling [15], fast forging [16] and shot peening [17]. It was shown that the presence of a high + +5 + +density of grain boundaries and crystal lattice defects can provide pathways for hydrogen transport +to improve hydrogenation kinetics [18]. Unlike the ball milling technique which is traditionally +used to produce hydrogen storage materials in the form of powders with a large surface area [8,9], +the SPD processes produce bulk samples with a lower surface area. This results in better activation +and enhanced air resistance [18]. Furthermore, the SPD processes, particularly those with +extremely large shear strains, which are known as ultra-SPD, can be used to synthesize a wide +range of Mg-based hydrogen storage materials even from immiscible systems [19]. +In the present article, recent advances in the application of SPD to Mg-based hydrogen +storage materials are reviewed with a focus on processing, kinetic features, and thermodynamic +modification via the synthesis of novel materials. + +2. Processing Methods of Mg-Based Materials +Although numerous SPD methods have been developed to process structural materials with +advanced mechanical properties [11,12], only a limited range of these methods have been used for +processing hydrogen storage materials. The main impact of these methods is the refinement of +microstructure and the introduction of crystal lattice defects such as vacancies and dislocations. +Among the various SPD methods, HPT is capable of continuously inducing large strains under +high pressure and is the most efficient one for microstructure refinement and defect generation in +Mg-based compounds [13]. Since repetitive cycles and discontinuous straining are applied in +ECAP and intensive rolling, the processing strain and the concomitant microstructure refinement +are not as significant as HPT. The advantage of the former techniques is their suitability for +processing large pieces of hydrogen storage materials, which is beneficial for commercial +applications [11]. Fast forging can also induce a large strain within one cycle, but the amount of +strain and the degree of grain refinement by this method are lower compared with those enabled +by HPT processing [16]. In this section, the major studies conducted on SPD processing of Mg- +based hydrogen storage materials using ECAP, HPT, intensive rolling and fast forging are +reviewed. + +2.1. Equal-Channel Angular Pressing +In relation to the efficacy of the processing of magnesium alloys aiming at improving their +hydrogen storage properties, ECAP takes a special place among the techniques of SPD. Not only +is ECAP arguably the most popular SPD method [20-25], but it is also historically the first one +whose potency as a tool for accelerating the hydrogenation kinetics of magnesium alloys by SPD +was discovered [10]. In the ECAP method, a billet in the form of a rod or bar is repeatedly pressed +through a channel with a bending angle to introduce simple shear strain, as shown in Fig. 1a [20- +22]. The ability of ECAP to produce extreme grain refinement, down to submicron scale, in +magnesium and its alloys is well documented [22,26-29]. The ECAP-induced acceleration of +hydrogen absorption/desorption was first demonstrated for Mg-based alloy ZK60, Mg-4.95Zn- +0.71Zr (wt%), which is widely used for structural applications [10]. The effect of ECAP on the +dehydrogenation kinetics is shown in Fig. 1b. The data obtained for ZK60 by employing a different +ECAP facility, which represent a further improvement over the original results [10], are also shown +in Fig. 1b [30]. A striking drop in the particle size down to the submicron range for the material + +6 + +processed by ECAP and subjected to hydrogenation and dehydrogenation is shown in Fig. 1c. +Variants of the ZK60 alloy were the object of studies where a combination of ECAP with cold +rolling [31] or ARB [32] was used, albeit with lesser success. + + +Figure 1. (a) Schematic illustration of ECAP. (b) Dehydrogenation kinetics of alloy ZK60 after +different kinds of processing by ECAP (triangles) [10], ECAP followed by high-energy ball +milling (closed circles) [10] and ECAP with a different route (red curve) [30]. (c) Scanning +electron microscopy images of alloy ZK60 processed by ECAP after hydrogen +absorption/desorption cycle [10]. + +(a) +Plunger +Sample +Die +(b) +7 +ZK60ECAR +9 +(%1m) +5 +Hydrogen +3 +2 +ZK60afterECAP [Krystianetal.,2010] +coarsegrainedZK60 +ZK60after ECAPor HEBMSkripnyuk et al. (2004) +0 +5 +10 +15 +20 +25 +30 +DischargingTime (min) +(c) +500nm7 + + +Research conducted so far demonstrated that ECAP is on par or can even outperform the +more conventional ball milling technique, both methods improving the hydrogenation kinetics. +Indeed, the hydrogenation curve for the ball-milled ZK60 alloy was shown to be very close to, yet +lying beneath, the curve measured for the ECAP-processed one [10]. The great benefit of ECAP +is the possibility to obtain a promising hydrogen storage material in bulk, with an additional +advantage of avoiding potentially hazardous ball milling of powders. Several studies utilizing +ECAP as the processing route [33-36] and further SPD techniques were also applied to a range of +Mg-based alloys [37,38], notably the eutectic Mg89Ni11 alloy with a fine lamellar structure [39]. +A promising approach to enhancing the hydrogenation/dehydrogenation kinetics of +magnesium is the use of composites containing metal hydrides [40] or particles of carbonaceous +materials [41]. Specifically, a study of a Mg-based composite with 2 wt% of multiwall carbon +nanotubes processed by ECAP showed that the addition of nanotubes to magnesium leads to a +substantial acceleration of hydrogen desorption rate, a further advantage being the disappearance +of pressure hysteresis [42]. Viewed as a progenitor of SPD-based tools for improving the +hydrogenation kinetics of Mg-based alloys, ECAP processing can be regarded as a viable avenue +to large-scale hydrogen storage facilities [43]. + +2.2. High-Pressure Torsion +Among different bulk SPD methods, HPT [13] has successfully been applied to +manufacture a large variety of different hydrogen storage materials [43-47] due to the +exceptionally high shear strain that can be achieved in bulk sample volume [21,48]. During HPT +deformation, as schematically shown in Fig. 2a, a disc-shaped specimen is inserted between two +anvils and subjected to concurrent uniaxial pressure of several Gigapascals and torsional straining +by several revolutions [49]. There has been a wide range of applications of HPT to Mg-based +materials, as will be discussed here. +HPT deformation of MgH2 powders results in a significant grain refinement [50] and also +induces a strong (002) texture and the formation of the metastable γ-MgH2 phase [51] which +positively influences the overall hydrogen storage performance. The particle size of magnesium +powders has a strong effect on the hydrogen absorption kinetics, but the grain/crystal size is also +known to affect the kinetics significantly [52]. It was reported that a bimodal microstructure +develops when bulk magnesium is subjected to HPT, including the emergence of nanocrystals and +large recrystallized grains, resulting in a substantial improvement of the hydrogenation kinetics +[53]. +The average density of dislocations in commercial magnesium processed by HPT reaches +a very large value of 8x1015 m−2, which can act as hydrogen transport sites [54]. A ZK60 Mg- +based alloy processed by HPT exhibits a stable storage capacity of up to 100 hydrogenation cycles, +as shown in Fig. 2b and 2c [55]. The maximum hydrogen capacity of ball-milled Mg-Ni +nanopowder increases by 50% after HPT and reaches the theoretical value, due to the creation of +new hydrogen transport sites in the vicinity of dislocations [56], while the formation of Mg2NiH0.3 +hexagonal solid solution and the monoclinic Mg2NiH4 takes place [57-58]. Other lattice defects, +notably stacking faults, can also improve the hydrogenation kinetics of Mg2Ni processed by HPT +[59]. A large fraction of cracks in ultrafine Mg + 2 wt% Ni powder can act as pathways for +hydrogen transport from the surface of the HPT-processed disc [60]. As will be discussed later, +the extreme shear deformation during torsion can reach such a high magnitude that it is capable to +promote hydrogen uptake even in the non-absorbing MgNi2 phase [61]. + +8 + + + +Figure 2. (a) Schematic representation of HPT apparatus with uniaxial compression and +simultaneous torsion [49]. (b) Dehydrogenation and (c) hydrogenation kinetic curves obtained at +623 K and 1 MPa absorption pressure, and at 10 Pa desorption pressure for HPT-deformed ZK60 +alloy [55]. (d) Hydrogenation curves at different temperatures for Mg + 5 wt% Ni + 2 wt.% Nb2O5 +powder composite processed by HPT [62]. (e) Desorption kinetic curves obtained at 573 K and 1 +kPa for as-milled magnesium powders catalyzed by Nb2O5 and/or carbon nanotubes and +corresponding HPT-processed discs. (f) High-resolution lattice image of cycled HPT-processed +Mg + Nb2O5 + carbon nanotube (inset: selected area electron diffraction pattern) [65]. + + +(a) +(d) +STAGEII +STAGEI +HPT sample +Absorbed Hydrogen (wt%), +Load cell +5 +4 +UpperAnvil +3 +Sample +2 +623K +473 K +Sample +573 K +423 K +1 +523K +373K +623 K +LowerAnvil +Load +Load +Ldad +0 +0. +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +1.6 +1.8 +Rotation +2.0 +Time (h) +(b) +(e) +0.0 +(%1m) +Relative Hydrogen Content +ZK60HPT - cycle 40 +Desorbed Hydrogen (wt%) +0 +0.9 +-0.5 +cycle 60 +cycle 80 +0.8 +Hydrogen +-1.0 +cycle100 +0.7 +2 +Mg-NbO_HEBM +-1.5 +0.6 +Mg-NbO_HEBM+HPT +3 +Mg-CNT_HEBM +-2.0 +0.5 +Mg-CNT_HEBM+HPT +Mg-NbO-CNT_HEBM +0.4 +4. +Desorbed +Mg-NbO-CNT_HEBM+HPT +-2.5 +0.3 +5 +0.2 +-3.5 +0.1 +6 +0 +0 +1 +2 +3 +5 +6 +7 +8 +0 +1000 +2000 +3000 +4000 +5000 +6000 +Time (min) +Time (s) +(c) +(f) +1.38 +1 +Relative Hydrogen Content +Absorbed Hydrogen (wt%) +3.5 +1.5A +0.9 +Nbosgrains +2.45A +1A +3.0 +0.8 +2.6A +2.8A + 0.7 +2.5 + 0.6 +2.0 +0.5 +1.5 + 0.4 +0.3 +1.0 +ZK60HPT-cycle40 +0.2 +0.5 +cycle 60 +cycle80 +0.1 +cycle100 +0.0 +0 +Amorphous +0 +2 +4 +6 +10 +12 +14 +16 +CNT +carbon +Time (min) +5 nm9 + +The HPT process can be used effectively to fabricate new Mg-based composites or alloys +for hydrogen storage. A reasonable hydrogenation capacity can be obtained for a powder mixture +of Mg + 5 wt% Ni + 2 wt% Nb2O5 subjected to HPT at a temperature as low as 423 K, as shown +in Fig. 2d [62]. It was revealed recently that metastable phases can be developed even in +immiscible systems (such as Mg-V-Sn [19], Mg-V-Ni [19], Mg-Ti [63] and Mg-Zr [64]) during +ultra-SPD, thus new potential hydrogen storing materials can be manufactured. It was found in a +recent study that the combination of ball milling and the HPT process can further improve the +desorption kinetics of nanocrystalline magnesium catalyzed by Nb2O5 and/or carbon nanotubes +[65,66]. As confirmed by transmission electron microscopy, the carbon nanotubes acting as +diffusion channels for hydrogen are preserved during plastic deformation by ball milling and HPT +as well as during sorption cycling, as shown in Fig. 2e and 2f [65]. The combined catalytic effect +of metal-oxide particles and carbon nanotubes can be substituted by applying only metal-oxide +nanotubes [67]. +HPT can also be utilized for the synthesis of high-entropy materials, such as MgVTiCrFe +for hydrogen storage [68]. Moreover, the hydrogenation performance of fully disordered systems, +like glassy alloys, can be improved significantly when the materials are subjected to severe shear +deformation, including a reduced hydrogenation temperature and improved hydrogen sorption +kinetics for Mg65Ni20Cu5Ce10 [69] and Mg65Ni20Cu5Y10 [70] metallic glasses. At the same time, +the hydride formation enthalpy increases noticeably which is especially more pronounced in the +most deformed perimeter region of the HPT-processed discs [71]. Despite the potential of HPT to +be applied to a wide range of materials, its main limitation is currently the small size of the sample +which is an obstacle to its development for commercial applications. + +2.3. Intensive Rolling +Despite the successful use of high-energy ball milling methods to produce nanostructured +Mg-based alloys, two processing techniques of ECAP schematically shown in Fig. 1a [10] and +intensive rolling schematically shown in Fig. 3a-c [72], were reported almost at the same time, as +alternatives enabling the production of “bulk” samples with refined and more air-resistant +microstructures. Metal processing based on cold rolling, such as co-lamination via repetitive +rolling, was first used in hydride-forming Mg-based alloys to produce Mg-Ni composite structures +[72]. Heat treatment of a deformed sandwich of Mg and Ni foils resulted in the formation of an +intermetallic Mg2Ni compound. A similar result was observed for Mg-Al composite alloys with +the final heat treatment forming the Mg17Al12 compound that also absorbed hydrogen reversibly +[73]. Cold rolling of a stack of Mg/Pd foils produced an air-resistant laminated compound with a +shorter activation time compared to a ball-milled sample of the same composition as well as to +pure magnesium, as shown in Fig. 3d [74]. Repetitive rolling was also used to prepare laminate +composites of Mg/Cu [75,76] and Mg/Pd [75] with the micrometer-order layered structure +containing a high density of dislocations and vacancies. A similar route to co-lamination, but +involving a bonding process during intensive rolling, designated as ARB shown schematically in +Fig. 3a [77], was used to prepare Mg/Ti multilayers and Mg/stainless steel composites [78]. +Activation for hydrogen absorption improved with an increased number of fold and roll operations, +which caused increasing refinement of the hard second-phase particles. +Intensive cold rolling was effectively used to refine the microstructure and disperse the +particles of catalysts. Intensive rolling was used to process mixtures of MgH2-Fe as an alternative +to reach the grain size reduction typically obtained by ball milling [79,80]. Cold rolling of +commercial MgH2 was considered equivalent to ball milling in terms of microstructure refinement + +10 + +[81], as also observed for Mg-LaNi5 composites [82]. Figs. 3b and 3c show the schematics of a +vertical and horizontal rolling machine used to process MgH2 powders, respectively [34]. The +presence of additives in Mg-based alloys processed by cold rolling proved to be as efficient as +ball-milled materials despite the distribution of the additive particles being much less +homogeneous. A uniform distribution of additive particles was also observed in MgH2 powders +containing different types of additives [83]. It was shown that the uniformity of FeF3 distribution +in MgH2 powders can be improved by a combination of short-time ball milling followed by +intensive rolling [84]. +There have been some improvements in the rolling process for treating hydrogen storage +materials. The use of a protective atmosphere during rolling allows further refinement of the +microstructure in commercial magnesium [85] and MgH2 [86] since a greater number of rolling +passes can be applied without surface contamination. Fig. 3e shows the refined and textured +microstructure achieved in MgH2 after 35 rolling passes under an argon atmosphere which can +result in fast hydrogenation and dehydrogenation kinetics. Figs. 3f and 3g show the positive effect +of the number of rolling passes under a protective atmosphere on the kinetics of hydrogen +absorption and desorption in MgH2 [86]. The same route was used to process MgH2-LaNi5 +mixtures under a protective atmosphere, resulting in compacted composite flakes [87], which +exhibited faster hydrogen absorption kinetics and reduced desorption temperatures in comparison +with single-phase MgH2. Low-temperature rolling in AZ91 alloy can improve the hydrogen +storage properties compared with rolling at ambient temperature due to the larger density of +microcracks and consequently high density of exposed interfaces and formation of stronger (002) +texture [88]. +Cold rolling has also been used as a further processing step to improve the activation and +hydrogen absorption kinetics of Mg-based samples initially prepared by a different technique. It +was shown that cold rolling following other modes of SPD processing of magnesium and its alloys +results in a rolling texture that can enhance the kinetics in the first hydrogenation cycle by exposing +less densely packed atomic planes to the hydrogen atmosphere [24]. Studies of other processing +combinations, such as cold rolling of melt-spun ribbons [89], cold rolling of HPT-processed discs +[90], and ARB processing of ECAP-treated ZK60 alloy [32] demonstrated that in addition to the +refined microstructure, the presence of free surfaces (or interfaces) and texture have positive +effects on the hydrogen storage properties of Mg-based alloys. Intensive rolling is likely to have +the greatest potential for large-scale processing of hydrogen storage materials, although the +magnitude of plastic deformation in this method is smaller than in ball milling followed by HPT. + + +11 + + +Figure 3. (a) Schematic illustration of ARB process [77], and sketches of (b) vertical and (c) +horizontal rolling machines used to process MgH2 powders [34]. Hydrogen absorption curves +under a hydrogen pressure of 1.5 MPa at 623 K for pure magnesium and Mg - 2.5 at% Pd after +cold rolling (CR) and ball milling (BM) [74]. (e) Dark-field image and corresponding selected area +electron diffraction pattern, taken by transmission electron microscopy, for MgH2 processed by +cold rolling with 35 cycles. (f) Hydrogen absorption and (g) desorption curves for MgH2 processed +by different numbers of passes of cold rolling under a protective atmosphere [86]. + + +(a) +(b) +(c) +Rolling Cylinders +Cutting +Stacking +Powders +Rolling +Cylinders +Plate +Powders +Roll Bonding +StainlessSteelPlates +(d) +(e) +(%1m) +8 +7 +5 +Mg-Pd2.5at.%CR +4 +Mg-Pd2.5at.%BM +Mg-Pd2.5at.%CR.reactivated +口 +PureMgCR +2 +500 +1000 +1500 +2000 +2500 +100 nm +Time (min) +(f) +(g) +Cold Rolled MgH2 +--As-received +0 +-10-passes +330°C/0.1MPa +口-20-passes +5 +-O-35-passes +-1 +50-passes +4 +-2 +8- +2 +As-received +-4 +-10-passes +Cold Rolled MgH, +-20-passes +330°C/1.5MPa +1 +.-35-passes +-5 +-50-passes +0 +9- +0 +180 +360 +540 +720 +9001080126014401620 +0 +300 +600 +900 +1200 +15001800210024002700 +Time (s) +Time (s)12 + +2.4. Fast Forging +As discussed earlier, the kinetic performance of hydrogen storage Mg-based materials can +be improved by: (i) reducing the particle/crystallite sizes to the nanometer level combined with +including a high lattice strain and a large density of extended defects (dislocations, stacking faults, +twins, etc.) by plastic deformation (high-energy ball milling [91-96], HPT [24,35,44,50,51,97,98], +ECAP [10,33,99,100-102], intensive rolling [81,103,104]), and (ii) using various additives which +can act as catalysts to accelerate hydrogen sorption [104-108]. In any case, mass production of +MgH2 must satisfy practical requirements such as processing conditions (ease of manufacture, +efficiency, cost, etc.) and material performance (maximum hydrogen uptake, fast sorption, stability, +lifetime, etc.) [109]. These requirements can be achieved using fast forging, a less conventional +SPD method [16,110,111]. +A photograph of the fast-forging facility is shown in Fig. 4a. In this method, the height of +ingots of bulk magnesium, Mg-based alloys or their compacted composites is reduced by about a +factor of ten within about 5x10-3 s, while the initial temperature of the sample is set using an +induction high-frequency coil. Since the mechanical energy of the hammer is almost entirely +dissipated in the material without elastic rebound, it leads to refining the grain size plastically, +while simultaneously healing the sample frictionally [16,110,111]. Such grain refinement by fast +forging can improve the hydrogenation kinetics similarly to other SPD methods. +In addition to microstructural modification, fast forging can be used to synthesize +composites of Mg-based hydrogen storage materials. As a trial, homogeneous mixtures of strongly +compacted powders of Mg+Ni with initial particle sizes of 5-30 µm for magnesium and 30-40 µm +for nickel were fast forged. To optimize forging conditions and determine the deformation fields, +a two-dimensional calculation model was developed by considering the particle size, distribution +of nickel among magnesium particles and their relative hardness [112]. Numerical simulation of +the adiabatic compression processes quantifies a marked increase of sample temperature for +average strains of 80-90%, while the additional heat generated by an increase in strain was found +to drop [113,114]. Fig. 4b shows the conversion rate of Mg89/Ni11 powder mixtures to form Mg2Ni. +The examination of absorption traces at 614 K and 2 MPa of H2 indicate that fast forging of +magnesium in a brittle state at low temperature promotes the formation of defects and cracks +enhancing hydrogen diffusion in the bulk, while forging at high temperature and a ductile state of +the material allows the formation of Mg2Ni as a catalyst, as reported for various SPD routes +[39,114-118]. +In another study, compacts of 95 wt% Mg + 5 wt% MgH2 were processed by combining +fast forging with two passes of ECAP at room temperature. Pressure-composition-temperature +isotherm analyses showed that at 593 K and under a hydrogen pressure of 1 MPa, the fast forging +was effective to achieve up to 7 wt% of hydrogen uptake in ~140 min, as shown in Fig. 4c [119]. +The performance characteristics were found to be rather similar to those of the ECAP-processed +samples with 6 wt% hydrogen uptake at 593 K and 1 MPa in 60 min [120,121]. Such improvements +by fast forging are also comparable with those achieved using other SPD routes [122,123]. These +results confirm that the fast-forging process can be considered as an efficient SPD route enabling +mass production of MgH2 for hydrogen storage [124]. + + +13 + + +Figure 4. (a) Fast forging facility with controllable atmosphere and temperature having a 150 kg +hammer falling from up to 1.5 meters with a forging time of less than 0.02 s (left), schematic +illustration of working chamber (center) and the appearance of samples before and after forging +(right) [16]. (b) The conversion rate of Mg89/Ni11 powder mixture to Mg2Ni after fast forging at +different temperatures [114]. (c) First hydrogen absorption kinetic curves at 593 K and under a +hydrogen pressure of 1 MPa for 95 wt% Mg + 5% wt% MgH2 compact processed by fast forging +[119]. + +3. Kinetic Features of Severely Deformed Mg-based Materials +As +mentioned +above, +many +investigations +have +been +done +on +the +hydrogenation/dehydrogenation kinetics of metal hydrides. The popular ways to enhance kinetics +are the addition of a catalyst or change of stoichiometry [125-127]. Other efficient ways to enhance +kinetics discussed in several sections of the present review are mechanical deformation of powders +[128] or bulk samples [129-132]. However, for most practical applications, dehydrogenation can +be relatively slow, and thermodynamics (plateau pressure) is more crucial than kinetics. For the +hydrogenation part, because of the high enthalpy of the formation of the hydrides, the main rate- + +(a) +Piston: +Pyrometer +Window +SampleBeforeForging +Hammer +Workingchamber +HF Heating +Sample +SampleAfterForging +Coil +(b) +50 +1 +40 +6 +Hydrogen (wt%) +5 +30 +20 +3 +T +=506°C +eut +2 +10 +1 +0 +0 +0 +100 +200 +300 +400 +500 +600 +700 +800 +0 +50 +100 +150 +200 +EffectiveTemperature(oC) +Time (min)14 + +limiting step will be the heat transfer. For example, storing one kilogram of hydrogen in +magnesium hydride for six minutes will require a heat transfer of about 100 kW. Therefore, for +practical hydrogen tanks made of metal hydrides, the main challenge will be to manage a rapid +heat transfer. In addition, for commercial applications, the cost of the metal hydride should be as +low as possible. +Despite the significance of plateau pressure, heat transfer and cost, it is still essential to +find strategies to address the kinetic drawbacks of hydrogen storage materials. SPD processing +generates a high density of crystal lattice defects and refines the microstructure of magnesium and +its alloys and as a result improves its hydrogenation kinetics, while keeping its reversibility. In +contrast with other methods used to enhance hydrogenation kinetics, such as alloying, catalyst +addition, synthesis of composites, and amorphization [133,134], the SPD methods do not need any +extra additives. Furthermore, as opposed to ball milling which produces potentially hazardous fine +powders [133,134], the SPD-processed materials are in a bulk form with less contact with the air +atmosphere. In addition, they have a large density of crystal lattice defects, which can act as fast +hydrogen pathways for easy activation. These kinetic features of severely deformed materials can +sometimes result in hydrogen uptake in materials that are normally inert to hydrogen. These kinetic +features are briefly reviewed in this section. + +3.1. Activation +One issue that is often overlooked is the problem of the activation of hydrogen storage +materials. For metal hydrides, the term activation means the process by which the alloy is prepared +for reversible hydrogen sorption [135]. Typically, it involves submitting the alloy to hydrogen +pressure and/or temperature much higher than the ones predicated by thermodynamics. However, +this implies an additional cost for the manufacturer because the storage tank needs to be +overdesigned so that it can sustain not only the service conditions but also the activation conditions. +The ideal situation would be to perform the activation process under the same regime as the service +conditions of the storage tank. The problem of activation has been investigated for a long time, but +it is still not clear what mechanisms are at play [136]. What is known is that microstructure plays +a crucial role in the activation of metal hydrides. Mechanical deformation has been shown to +improve the activation kinetics of metal hydrides [99]. The most widely used method is ball milling +[137,138], but recently other mechanical methods or a combination of them have been shown to +improve the activation of metal hydrides. Some of them are cold rolling [74,89,139,140], ECAP +[30,31], and HPT [59]. As examples of the effectiveness of mechanical deformation, Fig. 5 shows +the effect of HPT on (a) microstructure and (b) the first hydrogenation of magnesium [53], and (c) +the effect of ECAP passes on the first hydrogenation of AZ61 magnesium alloy [141]. Clearly, +mechanical deformation introduces ultrafine grains and crystal lattice defects and accelerates the +first hydrogenation. + + +15 + + +Figure 5. (a) Microstructure and corresponding selected area electron diffraction of magnesium +processed by ten turns of HPT (N = 10) [53]. (b) First hydrogenation (activation) of magnesium at +423 K and under 3 MPa of hydrogen before and after processing by HPT for N = 0.25 and N = 10 +turns [53]. (c) First hydrogenation of AZ61 alloy at 643 K and a hydrogen pressure of 3.5 MPa +before and after processing by ECAP for N = 1, 4 and 8 passes [141]. + +3.2. Air Resistance and Hydrogenation Kinetics +The kinetics of hydrogen absorption and desorption in magnesium is slow, and thus, this +issue should be addressed by increasing the surface area, generation of grain boundaries, +generation of interphase boundaries, alloying with transition metals (Tm = Co, Fe, Mn, Ni, Nb, Ti, +V), and the addition of catalysts. The milling process in the form of high-energy ball milling, + +a +1 μm +(b) +10 +Hydrogenation: P = 3 MPa, T= 423 K +9 +(wt. +Mg (99.9%) +8 +HPT: P = 6 GPa, @ = 0.2 rpm, T = 298 K +Storage +6 +5 +N=10(=160) +4 +3 +N = 0.25 +2 +Anneal (=0) +(ε= 4) +0 +0 +20 +40 +60 +80 +100 +120 +ExposureTime (ks) +(c) +(%1M) +AZ61 +5 +N:ECAPPasses +4 +3 +2 +Absorbed +V-AZ61 +1 +0-AZ61N=1 +O—AZ61N=4 +←-AZ61N=8 +0 +100 +200 +300 +400 +500 +Time (s)16 + +mechanical alloying and reactive ball milling (i.e. milling under a hydrogen atmosphere) is +effective to induce all these effects [40,92,126,142,143]. However, in some cases, the milling +process does not necessarily improve the kinetics because the generated active surfaces can react +easily to be oxidized during processing or subsequent handling. In addition, because of this high +surface reactivity, the produced powders are pyrophoric which can create significant safety issues. +Thus, the ball-milled magnesium powders should be handled under a controlled atmosphere such +as in the glove box in the laboratory. Therefore, despite the efficiency of ball milling and +particularly reactive ball milling in mechanical alloying, uniform addition of catalysts, grain size +reduction, and generation of high densities of grain boundaries and defects [50,144,145], the +difficulties in handling the active powders under air atmosphere have greatly restricted its potential +applications. Several SPD techniques can produce similar benefits as ball milling in enhancing the +hydrogen sorption kinetics with an additional advantage of the bulk materials processed by these +techniques being more air resistant and less contaminated compared with ball-milled powders [98]. +The low surface area of bulk materials processed by SPD compared to powders allows for storing +them under an air atmosphere for a long time. The presence of crystal lattice defects, particularly +grain boundaries, provides numerous pathways for hydrogen transport from the partially oxidized +surface to the bulk to produce the hydride phase [18]. Moreover, it was shown that the +consolidation of powders to a compact material using SPD methods not only enhances the air +resistance but also accelerates the hydrogen storage kinetics [52,60,146]. These SPD methods for +additional powder consolidation include intensive extrusion [139,147], intensive rolling [58,139], +HPT [44,63,148,149], ECAP [36,150], accumulative fold-forging [151] and fast forging [118]. +HPT is often selected for powder consolidation of hydrogen storage materials because it +induces large shear strains combined with high pressure. A comparative study of HPT imparted +on (i) micro-sized atomized magnesium powder and (ii) nano-sized condensed magnesium powder +showed an additional impact of SPD on the kinetics of absorption/desorption [52]. The +microstructures of these samples before and after HPT processing are shown in Figs. 6a-d, +illustrating the size of the powder precursors and the SPD-induced consolidation. The examination +of the first hydrogenation kinetics at 673 K under a hydrogen pressure of 3.5 MPa, shown in Fig. +6e, indicates that the finer condensed magnesium powder has faster hydrogenation kinetics than +the atomized one [60,146]. For both types of powder precursors, HPT processing has the effect of +improving the hydrogenation kinetics through the introduction of structural defects, microstructure +refinement, and breaking of the powder surface oxide layer that was redistributed inside the +microstructure to act as a possible catalyst [52]. Another advantage of HPT consolidation is the +reduction of the hysteresis between the absorption and desorption plateau pressure during the +pressure-composition isotherm experiments [52]. It should be noted that the apparent lower +hydrogen uptake capacities after HPT processing in Fig. 6e are due to the large scale of the +diffusion path in the bulk HPT samples being rapidly surrounded by compact MgH2, which does +not allow the samples to react completely with hydrogen. This effect is reduced in subsequent +cycles due to hydrogen-induced fragmentation of samples and diminished diffusion path. +In summary, in addition to the enhancement of the kinetics of hydrogen storage, the +consolidation of powders by SPD can also have further benefits. First, the initial powder can be +mixed with catalysts almost without any limitations. Second, the produced bulk materials are air- +resistant and can be stored under atmospheric conditions [152-154]. Third, the absorption and +desorption processes occur usually much faster than in non-consolidated precursor powders +[52,139]. Fourth, the microstructural features such as the crystallite size, crystal lattice defects and + +17 + +texture, which were suggested to influence the hydrogenation kinetics [148,150,153], can be easily +tailored by controlling the imposed strain and the heat treatment [118]. + + +Figure 6. Microstructure of (a) atomized magnesium micro-sized powder, (b) condensed nano- +sized powder, (c) HPT-consolidated atomized powder and (d) HPT-consolidated condensed +powder, including (e) kinetics for their first hydrogenation [52]. + +3.3. Cycling Stability +Over the last two decades, numerous investigations of hydrogen storage in +mechanochemically treated magnesium and Mg-based alloys, including SPD-processed ones, have +been reported [130]. The top-down SPD methods have been successfully applied to increase + +(a) Atomized Mg Powder +(b) Condensed Mg Powder +40 μm +500 μm +(c) Atomized Mg Powder + HPT +(d) Condensed Mg Powder + HPT +HPT Disc +1 μm +1 μm +口 +(e) +Hydrogen Content (wt.%) +8 +Mg +Atomized Micro-powder +-HPTMicro-powder +Condensed Nano-powder +-HPT Nano-powder +0 +60 +120 +180 +240300360 +420 +Time (min)18 + +absorption/desorption kinetics with the additional advantage of replacing highly expensive and +contamination/oxidation-risky bottom-up methods such as ball milling [10]. Moreover, a +combination of SPD methods and ab initio calculations have been applied to decrease the +desorption temperature by substitutional mechanical alloying of magnesium with elements of +optimized type and concentration [129]. The kinetics of hydrogen storage is very sensitive to +numerous parameters such as the surface and morphology of the storage material (i.e. oxide layers, +porosity, texture, microstructure, and the density/type of SPD-induced crystal lattice defects), and +these parameters may change in different experimental procedures or during absorption/desorption +cycles [130]. An examination of publications on SPD processing of Mg-based hydrogen storage +materials indicates that the results of investigations do not always match unless all the parameters +mentioned above are carefully controlled to have a reliable understanding of the hydrogen +dissociation (see, e.g. [155]), diffusion and storage processes. Another issue with these +publications is that most of them only studied one or a few absorption/desorption cycles, while the +stability of the hydride during high-cycle hydrogen storage is a critical issue. +The hydride formation stability is intimately connected with the thermal stability of the +microstructure with regard to the temperature necessary for a minimum time to complete +absorption/desorption. While in systems with high melting temperatures, the microstructure is +stable because of a much lower absorption/desorption temperature compared to the melting point +(e.g. TiFe [130]), this is not necessarily true for materials with medium and low melting +temperatures, like magnesium. Here, stability can be reached by creating alloys or by adding +nanoparticles of a second phase that can stabilize the microstructure beyond the hydrogenation +temperature and act as nucleation sites for hydride [156]. Krystian et al. [30] for the first time +reported this stability for up to 1,000 cycles for ECAP-processed Mg-based ZK60 alloy, but not +for ECAP-processed pure magnesium. Nevertheless, Chiu et al. [157] confirmed the stability of +another ECAP-processed Mg-based alloy, AZ31. Grill et al. [55] not only reported highly stable +storage characteristics for HPT-processed ZK60 alloy but also found that SPD-induced second- +phase particles can act as nuclei for heterogeneous hydride formation, according to a thorough +Johnson-Mehl-Avrami analysis [158] yielding an Avrami exponent of n = 1. Popilevsky et al. +[159] reported something similar in the case of composites of magnesium with multiwall carbon +nanotubes where they applied some mechanochemical treatment and achieved chains of carbon +nanoparticles that became nucleation sites for the hydrogenation. For SPD-processed Mg-based +alloys, through combined differential scanning calorimetry and X-ray diffraction analyses, Ojdanic +et al. [160], Cengeri et al. [161], and very recently Abbasi et al. [162] found thermally stable SPD- +induced vacancy agglomerates [160-162] which heterogeneously nucleate the hydride phase, +instead of second-phase precipitates. +Fig. 7a shows the hydrogen absorption rate in a Mg-0.5Zn-1Gd-0.62Y-0.67Nd (wt%) alloy +processed by HPT and ECAP after ten hydrogenation cycles. The hydrogenation kinetics in both +samples remain fast even after ten hydrogenation cycles, although the ECAP-processed sample +shows slightly better performance. Fig. 7 demonstrates the long-cycle hydrogenation (up to 100 +cycles) in ZK60 processed by (a) HPT and (b) friction stir processing (FSP), confirming that both +samples exhibit high cycling stability. From all these long-cycle storage characteristics shown in +Fig. 7, FSP gave rise to the best hydrogen storage kinetics and the best storage capacity, followed +by those of ECAP and HPT. Future investigations should be conducted to understand how the type +and/or density of the vacancy agglomerates depend on the applied SPD method and thus govern +the long-cycle hydrogenation in different Mg-based compounds. + + +19 + + +Figure 7. (a) Hydrogen absorption at T = 613 K after ten hydrogenation cycles in Mg-0.5Zn-1Gd- +0.62Y-0.67Nd (wt%) alloy processed by HPT and ECAP to von Mises equivalent strain of 4.3 +[162]. Cyclic hydrogen absorption kinetic curves at T = 623 K for Mg-based ZK60 alloy processed +by (b) HPT and (c) FSP to comparable von Mises equivalent strains of 7 [161]. + + +a) +76543210 +Hydrogen (wt%) +ECAP:4Passes +HPT:0.25Rotations +Mg-0.5Zn-1Gd-0.62Y-0.67Nd (wt%) +2 +0 +4 +6 +8 +10 +12 +Time (h) +(b) HPT ZK60 +wt% +Hydrogen (wt%) +86 +T= 623K,P= 1.1MPa +7654321 +４２ +10080 +0 +60 +53.0 +20 +2.02.5 +1.5 +1.0 +0 +0.5 +Time (h) +0 +(c) FSPZK60 +wt% +7654321 +Hydrogen (wt%) +8642 +T=623K,P=1.3MPa +0 +60 +20 +52.02.53.0 +1.5 +00 +0.51.0 +Time (h)20 + +3.4. Hydrogenation of Inert Materials +We reiterate that SPD is a useful group of processes to produce nano-sized grains decorated +with crystal lattice defects [20]. As reviewed in [48,163,164], among the SPD processes, HPT is a +unique method that is applicable to hard and brittle materials including intermetallics [165,166], +glasses [167,168], ceramics [169,170] and semiconductors [171-174]. In particular, by taking this +advantage, HPT was applied to an intermetallic Mg2Ni compound which gave rise to a significant +improvement of the hydrogenation kinetics of the material due to the grain refinement down to the +nanoscale [59]. The enhanced kinetics was observed even after annealing of the HPT-processed +Mg2Ni, which was associated with the formation of many stacking fault defects [59]. In addition +to these kinetic effects, which were discussed earlier in this article, defect generation by HPT +processing can lead to hydrogen storage in materials that are otherwise inert to hydrogen. This +positive effect in some materials such as TiFe [175], TiV [176] and Ti-Fe-Mn [177] is due to the +activation problem having been resolved by HPT processing. In addition, it was shown that the +formation of defects by HPT processing can influence the local metal-hydrogen interaction and +the hydride formation thermodynamics [178]. It is this thermodynamic advantage that enables the +partial absorption of hydrogen in some materials that are normally inert to hydrogen. For example, +HPT was successfully applied to store hydrogen in MgNi2 which is considered to be inert to +hydrogenation [61]. Some experimental pieces of evidence of the hydrogen absorption capability +of MgNi2 are presented below. +Fig. 8a displays X-ray diffraction profiles after HPT processing for N = 2, 5 and 10 turns, +including a profile before HPT (labelled by N = 0) [61]. Peak broadening occurred as the number +of HPT revolutions increased, indicating that strain was introduced in the sample. The results of +the quantitative analysis for hydrogen uptake before and after hydrogenation are plotted in Fig. 8b +with respect to the number of HPT turns. The hydrogen content increased with the number of HPT +revolutions but remained unchanged for the samples that were not subjected to hydrogenation +irrespective of the number of HPT turns. This comparison shows that hydrogen was stored in +MgNi2 after the HPT process and the amount increased with the number of HPT turns. The data +for crystallite size and anisotropic strain derived from the X-ray diffraction profiles indicated that +hydrogen likely stay at lattice defects such as dislocations and grain boundaries but not at the +tetragonal sites [61]. Despite the clear evidence that hydrogen could be stored in MgNi2, the +amount absorbed was as small as 0.1 wt%. The data presented in Figs. 8a and 8b were obtained +six months after the hydrogenation of HPT-processed samples. These low values might be +associated with the hydrogen trapped in dislocations or grain boundaries while most of the +hydrogen in the crystal lattice was desorbed over the six-month storage. In another study, the +amount of absorbed hydrogen was measured using thermo-gravimetric-differential analysis one +day after the hydrogenation of HPT-processed samples [179]. The measurements returned a high +value of ~1.5 wt% for the weight loss, as shown in Fig. 8c [179]. No change in the X-ray profiles +occurred six months after HPT (Fig. 8d), suggesting that a low storage capacity reported in [61] +might stem from the desorption of hydrogen while keeping the material for six months in the air. +The entirety of the data presented leads one to conclude that HPT has a great potential to activate +even inert Mg-based materials for hydrogen storage through modifications of both the +thermodynamics and the kinetics of hydrogen sorption. +. + + +21 + + +Figure 8. (a) X-ray diffraction profiles after HPT processing for N = 2, 5 and 10 revolutions +including profile measured before HPT (labelled N = 0) [61]. (b) Plots of hydrogen content against +the number of HPT revolutions, where hydrogenation was made at 373 K under 1 MPa for 24 +hours after HPT processing [61]. (c) Thermo-gravimetric differential thermal analysis one day (A) +and six months (B) after HPT processing for ten revolutions and hydrogenation [179]. (d) X-ray +diffraction profiles measured one day (A) and six months (B) after HPT processing for ten +revolutions and hydrogenation including the intensity difference between profiles of A and B +(labeled A-B) [179]. + +4. Synthesis of Mg-based Materials with Desirable Thermodynamics +Despite significant progress in the improvement of the kinetics of hydrogenation in +magnesium and its alloys by using catalysts or by application of SPD, there has been rather limited +success in the development of Mg-based alloys with appropriate thermodynamics for +dehydrogenation at room temperature. The main reason for such limited advancements is the large +binding energy between the magnesium and hydrogen atoms. Compositional modification is + +(a) +(b) +(a.u.) +0.14 +MgNi2 +MgNi2 +0.12 +HPT:2.8GPa +HPT:2.8GPa +Intensity +N=10 +0.10 +Hydrogenation +0.08 +373K,1MPa,10h +0.06 +after +before +0.04 +N=2 +0.02 +0 +N=0 +2110 +13 +110 += +2 +4 +6 +8 +10 +40 +60 +80 +100 +120 +20 (degree) +Number ofRevolution +(c) +(d) +MgNi2 +MgNi2 +HPT:2.8GPa +HPT:2.8GPa +Differential Thermal Analysis /a.u. +/mass% +Normalized Intensity /a.u. +B(6monthsafterHPT+HG) +B (6monthsafterHPT+HG) +A (1.dayafter HPT+HG) +Hydrogenation(HG) +373K,1MPa,10h +A (1dayafterHPT+HG) +A +A-B +B +400 +500 +600 +700 +40 +60 +80 +100 +120 +300 +Temperature /K +20 /degree22 + +generally considered the most effective strategy to modify the thermodynamics of hydrogen +storage in Mg-based alloys [180-183]. The SPD processing does not influence the overall +thermodynamics of hydrogen storage in magnesium unless a phase transformation occurs, +mechanical alloying is achieved, or small crystals with sizes of a few nanometers are formed. +However, the SPD methods and particularly HPT have been used in recent years to synthesize new +Mg-based alloys which can store hydrogen at low temperatures or even at room temperature. Some +major studies are discussed in this section which include the concept of binding-energy +engineering and the developments of nanoglasses, high-entropy alloys and metastable phases. + +4.1. Binding-Energy Engineering for Room-Temperature Hydrogen Storage +Magnesium hydride MgH2 described above as a promising hydrogen storage material +suffers from high dehydrogenation enthalpy (75 kJ/mol H2) due to the strong binding between the +magnesium and hydrogen atoms [38,184]. Several major research activities to overcome the key +issue of the excessively high thermodynamic stability of MgH2 have been developed: (i) +nanoengineering, such as the production of thin films [185-188] and nanoparticles [189-192] +including confined [193], non-confined [194], core-shell [195] or composite [196] states; (ii) +fabrication of metastable hydrides such as γ-MgH2 [51,98]; and (iii) alloying to form intermetallic +compounds, notably Mg2Ni [197], or solid solutions including Mg-In alloys [198,199]. Among +them, alloying is the most feasible way to develop new materials by tuning the chemical +composition. However, the thermodynamic immiscibility of Mg in many systems limits the +fabrication of Mg-based alloys comprised of three or more component elements, especially by the +melting process. +HPT was recently found to be a powerful tool to fabricate new metastable Mg-based alloys +with superior hydrogen storage properties at room temperature [19,129]. A successful example is +Mg4NiPd, which is expected to have moderate hydrogen binding energy within the Mg2Ni and +Mg2Pd ones, as predicted by the first-principles calculations (Fig. 9a) [129]. The nominal Mg4NiPd +fabricated by the melting process contained three phases, Mg8Ni3Pd, MgNi2 and Mg5Pd2, with +heterogeneous elemental distributions (Fig. 9b). The homogeneity of the elemental distribution +significantly improved to the atomic scale by 1,500 turns of HPT (Fig. 9b), and as a result, a new +phase of Mg4NiPd with a partly ordered CsCl-type structure was obtained (Fig. 9c). More +importantly, at room temperature, the HPT-processed Mg4NiPd could reversibly absorb and desorb +0.7 wt% of hydrogen (Fig. 9d), which was suggested to occupy the 4Mg-1Ni-1Pd site with a +hydrogen binding energy of -0.12 eV (Fig. 9e). These results demonstrated the success of +destabilizing the dehydrogenation thermodynamics of Mg-based alloys to room temperature by +applying binding energy tailoring principles which could be realized by SPD processing. This +strategy can also be applied more universally to design new materials with superior functional +properties beyond the scope of the phase equilibria. + + +23 + + +Figure 9. (a) Concept of binding-energy engineering used to design Mg4NiPd. (b) Elemental +mapping of Mg4NiPd after casting (N = 0) and after HPT processing for N = 20, 100, 500 and +1,500 turns. (c) X-ray diffraction profile and (d) pressure-composition isotherm of Mg4NiPd after +1,500 HPT turns. (e) Crystal structure of homogeneous Mg4NiPd simulated using first-principles +calculations [129]. + +4.2. Nanoglasses +Owing to the thermodynamic stability of MgH2, its hydrogen storage temperature is higher +than 573 K [9,184]. Although, as discussed above, nanostructuring is an efficient strategy to + +(a) +TARGET +(b) +NE0 +460K +iNo Hydrogenation +Mg:NiPd +TDeh: 630 K +Mg +Mg2Ni +Mg2Pd +MgNi2 +1.1 +■Mg5Pd2 +0 +N:Numberof Turns +Hydrogen Binding Energy (eV) +(c) +Mg4NiPd +N=1500 +40 +um +Intensity +口 +OrderedBCC +N=500 +N=1500 +100 +40μm +40um +20 +30 +40 +50 +60 +70 +80 +90 +Bragg Angle, 20 (deg.) +(d) +H Atom per Mg+Ni+Pd Atom +(e) +0 +0.1 +0.2 +0.3 +0.4 +100 +Mg4NiPd +N=1500 +(MPa) +10 +Pressure +TH2 = 305 K +0 1st Cycle +H2 +0.1 +5th Cycle +0.01 +0 +0.2 +0.4 +0.6 +0.8 +1.0 +Mg +Ni +Pd +Hvdrogen Content (wt.%24 + +improve the hydrogen storage kinetics of MgH2, the thermodynamic stability of nanostructured or +nanoconfined MgH2 can hardly be reduced [200-202]. Hydrogen storage properties of Mg-based +materials are expected to be varied significantly in their amorphous form compared with their +crystalline counterparts. As distinct from the defined chemical compositions in crystalline alloys, +amorphous alloys usually have a wider range of chemical compositions which is related to the +glass-forming ability of the amorphous alloys [203]. Therefore, the tunability of hydrogen storage +properties in amorphous alloys is supposed to be much wider than that in crystalline alloys. +In the past two decades, various Mg-based amorphous alloys have been prepared by melt +spinning or ball milling and studied as hydrogen storage materials. These includes Mg-Ni [204], +Mg-Ni-Y [205], Mg-Ni-RE [206], Mg-Ni-Fe [207], Mg-Ni-La [208], Mg-Mm-Ni [209,210], Mg- +Y-Ni [211], Mg65Ni20Cu5Y10 [70], Mg90Ni8RE2 (RE = Y, Nd, Gd) [212], Mg85Ni15−xMx (M = Y or +La, x = 0 or 5) [213], Mg100−xNix alloys (x = 0.5, 1, 2, 5) [214], LaMg11Ni [215], Mg-Ce-Ni [216], +RMg2Ni (R = La, Ce, Pr, Nd) [217], Mg11Y2Ni2 [218] and Mg80Ce10Ni10 [219]. However, most +studies showed that the amorphous alloys would decompose into crystalline alloys or hydrides +upon hydrogenation/dehydrogenation [204-219], and thus the hydrogen storage reversibility of +amorphous alloys is usually not satisfactory. It was reported that the Mg-Ce-Ni amorphous alloys +could reversibly absorb and desorb 0.3 wt.% of hydrogen at room temperature [220]; however, the +reversible hydrogen storage of this alloy could not be improved by increasing the temperature up +to 423 K [221]. Therefore, improving the reversible hydrogen storage capacity of Mg-based +amorphous alloys at low temperatures remains a challenging issue. +SPD was shown to be beneficial for improving the hydrogen storage properties of Mg- +based amorphous materials, particularly for enhancing hydrogen storage kinetics and reducing +hydrogen storage temperature. For example, high-pressure calorimetry measurements revealed +that hydrogen uptake in the fully amorphous Mg65Ni20Cu5Y10 alloy, prepared by melt spinning +followed by HPT processing, occurs at a significantly lower temperature compared to the fully +crystallized state [70]. In another trial, the HPT technique was used to process the melt-spun +Mg65Ce10Ni20Cu5 amorphous alloy to produce a nanoglass alloy [69]. Fig. 10a shows that the X- +ray diffraction pattern of the melt-spun Mg65Ce10Ni20Cu5 alloy is typically amorphous, while the +HPT-treated samples contain a small fraction of crystalline phases. Fig. 10b demonstrates that +superior hydrogen storage could be achieved in these HPT-treated alloys at a low temperature of +393 K, while the melt-spun sample was not active at this temperature. It was suggested that the +HPT-induced nanoglass formation (i.e. amorphous phase with a large fraction of interfaces at the +nanometer scale) leads to a large number of hydrogen diffusion channels and hydrogen storage +sites for the improvement of hydrogenation properties (Fig. 10c). Such hydrogen storage +performance in the HPT-processed sample is comparable with the performance of thin films of +Mg65Ce10Ni20Cu5. As shown in Figs. 10d and 10e, the amorphous thin films with thicknesses of +50-300 nm could reversibly absorb and desorb hydrogen with almost unchanged desorption +kinetics at a low temperature of 393 K [222]. Therefore, SPD-processed nanoglasses and nanosized +amorphous thin films are promising candidates for hydrogen storage at moderate temperatures, but +nanoglasses can be produced by SPD in the bulk form with a larger size which is beneficial for +practical applications. + + +25 + + +Figure 10. Hydrogen storage in Mg65Ce10Ni20Cu5 nanoglass and amorphous thin films. (a) X-ray +diffraction patterns of the melt-spun alloy before and after HPT processing for 1, 5 and 10 turns. +Inset: an image of a disc-shaped specimen after the HPT process [69]. (b) Hydrogenation kinetics +of melt-spun and HPT-treated samples [69]. (c) High-resolution image of sample processed by one +HPT turn taken by transmission electron microscopy [69]. (d) Cross-sectional images of films with +thicknesses of 300, 250 and 50 nm taken by scanning electron microscopy [222]. (e) Hydrogen +desorption kinetics of amorphous thin films [222]. + +4.3. High-Entropy Alloys +In a quest for new materials for hydrogen storage, multicomponent and high-entropy alloys +and their corresponding hydrides have attracted wide attention due to the huge compositional field +that can be accessed and the unexpected potentialities of hydrogen storage properties they promise +[223,224]. The vast compositional field offers virtually endless opportunities for designing and/or +discovering new alloys with superior hydrogen storage properties [225]. The concept of high- +entropy materials was first proposed in 2004 by independent studies by Yeh et al. [226] and Cantor +et al. [227]. They introduced a new family of metallic alloys that was based on a mixture of at least +five principal elements, each having an atomic percentage in the range of 5 at% to 35 at%. The +presence of multiprincipal elements leads to the stabilization of single-phase solid solutions with +main crystal structures of body-centered cubic (BCC), face-centered cubic (FCC) or hexagonal +close-packed (HCP). High-entropy hydrides also consist of at least five principal elements and +hydrogen, as shown in Fig. 11a [228]. +Several high-entropy alloys for hydrogen storage have been proposed mainly including +refractory elements such as TiVZrNb [229], TiZrNbTa [230], TiVZrNbHf [231,232], TiZrNbHfTa + +(a) +(c) +-Mg.Cu +1.6 +Melt-spun +a-CeMg +HPT-IN +Intensity (a.u.) +(%1m) +HPT-5N +1.2 +HPT-10N +HPT-10 +len +0.8 +Hydroge +HPT +0.4 +Melt-spun +0.0 +20.nm +20 +30 +40 +50 +60 +3 +4 +5 +6 +2 (degree) +Time (h) +(d) +300nm +2*250nm +10*50nm +1μm +1μm +1μum +(e +0.0 +0.0 +0.0 +Hydrogen (wt%) +1st +(%m) +(wt%) +1st +2nd +1st +2nd +0.5 +3rd +-0.5 +2nd +3rd +4th +3rd +en +len +4th +5th +-1.0 +4th +1.0 +Hydroge +0.5 +5th +1.5 +-1.5 +120°C +120°C +120°℃ +-2.9 +4 +12 +8 +12 +-1.0 +8 +8 +12 +Time (h) +Time (h) +Time (h)26 + +[233,234], TiZrNbFeNi [235] and TiZrNbCrFe [236]. Despite the remarkable results observed in +hydrogen storage properties so far, the gravimetric storage capacities of these alloys are relatively +low due to the high atomic weight of refractory elements (zirconium, hafnium, niobium, tantalum). +Accordingly, one of the strategies to solve this limitation is to proceed with the substitution of +refractory elements with lightweight elements with a high affinity to hydrogen such as magnesium. +To that end, Mg-containing high-entropy alloys have been explored for hydrogen storage such as +MgZrTiFe0.5Co0.5Ni0.5 +[237], +MgTiNbCr0.5Mn0.5Ni0.5 +[238], +MgVAlCrNi +[239], +Mg12Al11Ti33Mn11Nb33 [240], Mg35Al15Ti25V10Zn15 [241] and MgVTiCrFe [242]. These Mg- +containing high-entropy alloys show the presence of single solid-solution phases and are mainly +prepared by high-energy ball milling. The SPD techniques such as HPT have also been +successfully used to synthesize new Mg-containing multicomponent alloys [243] and high-entropy +alloys [68], although the hydrogen storage performance of these alloys is not still comparable with +the Ti-Zr-containing high-entropy alloys with the Laves phase structure [228,243]. +Among various SPD methods, HPT is currently the most popular one to synthesize +multicomponent and high-entropy alloys [244]. MgVCr is a Mg-based multicomponent hydrogen +storage alloy with the BCC structure that was synthesized by HPT [242]. As shown in Fig. 11b, +the application of HPT to a powder mixture of magnesium, vanadium and chromium, could lead +to the formation of a new alloy with a single BCC phase. This multicomponent alloy stored up to +0.9 wt% of hydrogen at ambient temperature, but the reversibility of the storage was rather poor. +The MgTiVCrFe alloy is a Mg-containing high-entropy alloy that was synthesized by ball +milling followed by HPT processing [68]. The alloy showed a BCC solid solution phase together +with a high level of amorphization. The hydrogen uptake capacity of alloy MgTiVCrFe at 623 K +was found to be 0.3 wt%, as seen in Fig. 11c. The outcomes of the investigations conducted so far +indicate that SPD processing is a possible pathway to synthesize Mg-containing high-entropy alloy. +However, this research needs to be empowered by developing theoretical concepts for material +design, as attempted earlier for Ti-containing high-entropy hydrogen storage materials [228,243]. + + +27 + + +Figure 11. (a) Typical structure of high-entropy hydrides [243]. (b) X-ray diffraction profiles of a +mixture of magnesium, vanadium and chromium powders before and after HPT processing for N += 10, 100 and 1,200 revolutions [242]. (c) Hydrogen pressure-composition isotherms at 623 K for +MgTiVCrFe synthesized by ball milling followed by HPT processing [68]. + + +(a) +Zr +OCr +OMn +OFe +ONi.H +(b) +MgvCr +HPT:P=3GPa,0=1rpm,T=300K +vMgH2 +oCr +BCC-Ma-V-Cr +口 +口 +口 +口 +VW +N=1200 +N=100 +N=10 +N=0 +2030405060708090100110120 +Bragg Angle, 20(deg.) +C +Pressure (MPa) +-1stAbsorption +0.1 +O-1stDesorption +2ndAbsorption +O-2ndDesorption +0.01 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Hydrogen Content (wt%)28 + +4.4. Metastable Phases +Following a publication in 1988 on the application of HPT to an aluminum alloy, the main +goal of SPD processing has become the production of ultrafine-grained materials [245]. However, +a survey in the classic publications indicates that the first SPD methods were invented to control +polymorphic phase transformations under high pressure and large shear strain [246]. Nowadays, +the SPD methods and particularly HPT are still used to control polymorphic phase transformations +[247-249] and solid-state reactions [250,251]. The main reason for a significant interest in using +HPT to control phase transformation is the high applicable pressure in this method which can lead +to the synthesis of high-pressure phases [247,249]. Moreover, the shear strain in this method can +be increased almost without any limitations, a fact that was used to employ ultra-SPD for +synthesizing new materials by mechanical alloying through HPT [46]. Moreover, a combination +of high pressure and high strain in this method permits for synthesizing hard and brittle materials +such as TiFe hydrogen storage material [252]. This capability of HPT was successfully used to +synthesize various Mg-X alloys and Mg2X intermetallics (X: 21 different elements) [253], while +the synthesis of Mg-based compounds by melting techniques is usually hard. It is remarkable that +HPT was recently used to synthesize new metastable hydrogen storage materials even from +immiscible systems such as Mg-Ti [63,98], Mg-Zr [64], Mg-Hf [254], Mg-V-Cr [242] and Mg- +Ni-Pd [129]. As discussed earlier, some of these metastable alloys such as Mg4NiPd exhibit +thermodynamics suitable for reversible room-temperature hydrogen storage, while some others +such as MgHf show poor reactivity with hydrogen [254]. Here, the impact of HPT on the +production of the high-pressure γ-MgH2 phase with low thermodynamic stability is discussed [51]. +As shown in Fig. 12a, MgH2 has an α phase with the tetragonal structure at ambient +pressure and shows transitions to a γ phase with the orthorhombic structure at 0.39-5.5 GPa and to +a β phase with the cubic structure at 3.9-9.7 GPa [255]. Some studies suggested that the partial +formation of γ phase after mechanical milling [256], electrochemical process [257] and plasma +sputtering [258] can result in the destabilization of hydride for easy hydrogen desorption. To +examine this issue, HPT was recently used to synthesize almost 100% of γ-MgH2 and examine its +dehydrogenation behavior [51]. As shown in the X-ray diffraction profiles of Fig. 12b, a transition +from the α phase to the γ phase occurs by HPT processing under 5 GPa, while the fraction of the +high-pressure phase increases with increasing the number of HPT turns (i.e. increasing strain). +Examination of lattice images by high-resolution transmission electron microscopy, shown in Figs. +12c and 12d, also indicates the formation of large amounts of γ phase (almost 100%) after HPT +processing for N = 15 turns. Differential scanning calorimetry and thermogravimetry analyses, +shown in Figs. 12e and 12f, clearly confirm a reduction in the dehydrogenation temperature by +increasing the fraction of the γ phase. These results confirm that the production of metastable γ- +MgH2 is a strategy to reduce thermodynamic stability, as suggested by first-principles calculations +in [51]. However, the reversibility of this phase is another issue that should be addressed by +adjusting the chemical composition, as attempted for some other metastable phases such as +Mg4NiPd [129]. +In conclusion, HPT appears as an effective SPD method to synthesize metastable alloys +and hydrides for hydrogen storage. However, the method has two drawbacks for practical +applications. First, the sample in HPT is in the form of a disc with small sizes. Second, strain is +not uniform along the disc radius resulting in the nonuniform distribution of metastable phases. +There are now some signs of progress in upscaling the sample size in HPT [163], which can open +a path for future applications of the method. Moreover, it was suggested that using ring samples +rather than disc samples can diminish the heterogeneity in the final product [259]. + +29 + + + +Figure 12. (a) Pressure-temperature phase diagram of magnesium hydride. (b) X-ray diffraction +profiles of MgH2 processed by HPT for various turns (N). High-resolution lattice images of MgH2 +after HPT processing for (c) 1 and (d) 15 turns. (e) Heat flow in differential scanning calorimetry +and (f) mass loss in thermogravimetry for MgH2 processed by HPT for various turns [51]. + +5. Concluding Remarks and Outlook +Severe plastic deformation methods are receiving significant attention in the context of the +processing of hydrogen storage materials. The initial motivation to apply these processes to +magnesium and its alloys was the improvement of the kinetics of hydrogen storage, but currently, + +(a) +(b) +10 +Normalized Intensity +MgH2 +βCubic +HPT: T = 300 K. P = 5 GPa, = 1 rpm +(GPa) +8 +6 +yOrthorhombic +Pressure +N=15 +N=4 +4 +N=1 +N=0.5 +2 +α Tetragonal +O=N +Powder +0! +0 +200 +400 +600 +800 +1000 +22 +26 +30 +34 +38 +Temperature (K) +Bragg Angle, 20(deg.) +(c) N= 1 +[210] [111] +(d) N= 15 +[111][111] +[020] +101 +α[121] +[101] +2 nm +2 nm +(e) +(f) +MgH2 +MgH2 +Heat Flow (W/g) +HPT:T=300K,P=5GPa,@=1rpm +(%M) +HPT: T=300 K, P=5GPa, の=1 rpm +N=15 +N= 15 +N= 4 +N=4 +N= 1 +N= 1 +N=0.5 +Mass +N= 0.5 +N=0 +N=0 +15W/g +Powder +5wt.% +Powder +550 +600 +650 +700 +750 +550 +600 +650 +700 +750 +Temperature (K) +Temperature (K)30 + +there are numerous reports on the application of these methods to produce novel materials with +appropriate thermodynamics and low-temperature hydrogen storage capability. Currently, a wide +range of processes including equal-channel angular pressing, high-pressure torsion, intensive +rolling and fast forging are employed for processing Mg-based hydrogen storage materials, as +discussed in Section 2. While high-pressure torsion is the most powerful technique for fundamental +studies, the three other methods have a greater potential for commercial applications. The +application of severe plastic deformation introduces ultrafine grains and high densities of crystal +lattice defects in bulk samples, and this results in better activation, high air resistance, fast +hydrogen absorption/desorption kinetics, and appropriate cycling stability, as discussed in Section +3. It was shown that some materials which are apparently inert to hydrogen become active after +processing and can store hydrogen. The large shear strain and high pressure involved in severe +plastic deformation enable the synthesis of new materials such as nanoglasses, high-entropy alloys +and metastable phases with thermodynamics amenable for hydrogen storage, as discussed in +Section 4. A combination of theoretical binding-energy engineering and severe plastic deformation +can successfully lead to achieving reversible hydrogen storage in Mg-based alloys even at room +temperature. +The findings reviewed in this article introduce severe plastic deformation as a strong tool +to produce advanced hydrogen storage materials. However, there are several issues that need to be +addressed in the future. (i) Scaling up the processes is still a key issue for commercialization. (ii) +Heat transfer is of paramount significance in hydrogen storage materials, but so far this aspect has +not been sufficiently examined for severely deformed magnesium. (iii) Cycling stability of +severely deformed Mg-based hydrogen storage materials requires further investigation. (iv) +Addition of catalysts in conjunction with SPD processing may be a potential pathway to enhancing +the hydrogenation/dehydrogenation kinetics at low temperatures. (v) More importantly, further +studies on the combination of thermodynamic calculations and experiments are needed to discover +new materials that can satisfy the desired requirements for stationary and mobile hydrogen storage +applications. By considering the worldwide trend to realize carbon-neutral energy and utilize +hydrogen as a zero CO2 emission fuel, it is expected that in the future, severe plastic deformation +technologies will contribute to the hydrogen economy more prominently. + +Acknowledgments +K.E. was supported in part by the Light Metals Educational Foundation of Japan, and in +part by the MEXT, Japan through Grants-in-Aid for Scientific Research on Innovative Areas +(JP19H05176 & JP21H00150) and Challenging Research Exploratory (JP22K18737). +W.J.B. is grateful to the Brazilian agencies FAPESP (grant number 2013/05987-8) and +CNPq (grant number 421181-2018-4 and 307397-2019-0) for the financial support and to the +Laboratory of Structural Characterization (LCE-DEMa-UFSCar) for general electron microscopy +facilities. +R.F. thanks for the financial support from FAPESP (grant number 2022/01351-0). +T.G. thanks the support from the French State through the ANR-21-CE08-0034-01 project +as well as the program “Investment in the future” operated by the National Research Agency +(ANR) and referenced under No. ANR-11-LABX-0008-01 (Labex DAMAS). +H.L. appreciates the financial support from the National Natural Science Foundation of +China (grant number 52171205). +H.J.L. thanks the financial support from the National Natural Science Foundation of China +(grant number 52071157). + +31 + +References +[1] +M. 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Botta3, Yuri Estrin4,5, Ricardo Floriano6, Daniel Fruchart7,8, Thierry Grosdidier9,10, Zenji Horita1,11-13, Jacques Huot14, Hai-Wen Li15, Huai-Jun Lin16, Ádám Révész17 and Michael J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Zehetbauer18 1 WPI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' International Institute for Carbon-Neutral Energy Research (WPI-I2CNER),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Kyushu University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Fukuoka,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Japan 2 International 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Materials Science and Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Monash University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Clayton,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' VIC 3800,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Australia 5 Department of Mechanical Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' The University of Western Australia,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Crawley,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' WA 6009,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Australia 6 School of Applied Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' University of Campinas (UNICAMP),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Limeira,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' São Paulo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Brazil 7 Institut Néel,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' CNRS & UGA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' 38042 Grenoble,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' France 8 JOMI-LEMAN SA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' 74890 Fessy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' France 9 Université de Lorraine,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Laboratory of Excellence on Design of Alloy Metals for low-mass Structures (DAMAS),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Metz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' F-57070,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' France 10 Université de Lorraine,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Laboratoire d’Etude des Microstructures et de Mécanique des Matériaux (LEM3 UMR 7239),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Metz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' F-57070,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' France 11 Graduate School of Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Kyushu Institute of Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Kitakyushu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Japan 12 Magnesium Research Center,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Kumamoto University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Kumamoto,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Japan 13 Synchrotron Light Application Center,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Saga University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Saga,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Japan 14 Hydrogen Research Institute,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Université du Québec à Trois-Rivières,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' 3351 des Forges,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Trois- Rivières,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' QC G9A 5H7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Canada 15 Hefei General Machinery Research Institute,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Hefei 230031,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' China 16 Institute of Advanced Wear & Corrosion Resistance and Functional Materials,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Jinan University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Guangzhou 510632,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' China 17 Department of Materials Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Eötvös University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Budapest,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' H-1518,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' 32, Budapest, Hungary 18 Faculty of Physics, University of Vienna, Boltzmanngasse 5, A-1090 Wien, Austria Corresponding author (E-mail: kaveh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='edalati@kyudai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='jp;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Tel/Fax: +81-92-802-6744) 2 Abstract Magnesium and its alloys are the most investigated materials for solid-state hydrogen storage in the form of metal hydrides, but there are still unresolved problems with the kinetics and thermodynamics of hydrogenation and dehydrogenation of this group of materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Severe plastic deformation (SPD) methods, such as equal-channel angular pressing (ECAP), high-pressure torsion (HPT), intensive rolling and fast forging, have been widely used to enhance the activation, air resistance, and hydrogenation/dehydrogenation kinetics of Mg-based hydrogen storage materials by introducing ultrafine/nanoscale grains and crystal lattice defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' These severely deformed materials, particularly in the presence of alloying additives or second-phase nanoparticles, can show not only fast hydrogen absorption/desorption kinetics but also good cycling stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' It was shown that some materials that are apparently inert to hydrogen can absorb hydrogen after SPD processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Moreover, the SPD methods were effectively used for hydrogen binding-energy engineering and synthesizing new magnesium alloys with low thermodynamic stability for reversible low/room-temperature hydrogen storage, such as nanoglasses, high-entropy alloys, and metastable phases including the high-pressure γ-MgH2 polymorph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' This article reviews recent advances in the development of Mg-based hydrogen storage materials by SPD processing and discusses their potential in future applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Keywords: severe plastic deformation (SPD);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' nanostructured materials;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' ultrafine-grained (UFG) materials;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' magnesium hydride (MgH2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' magnesium-based alloys;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' hydrogen absorption;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' hydrogenation kinetics;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' hydrogen storage thermodynamics Severe Plastic Deformation Kinetics Thermodynamics 7 ZKBOECAF 6 Hydrogen (wt%) Plunger ZK60 after ECAP [Krystian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=', 2010] ripnyuk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' (2004) Load Magnesium 0 5 10 15 20 25 30 Discharging Time (min) 100 Rotation MgaNiPd N=1500 Sample Die HPT ZK60 Pressure (MPa) Powders 10 wt% [%1m] T=623K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='P=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='1MPa 17 Pyrometer Piston Window Hydrogen ( 54 4 TH2=305K 2 Rolling 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='1 o1stCycle 5thCycle Cylinders Plate % 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='01 HF Heating 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='0 Sample Coil Time (h) HydrogenContent (wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='%)3 Table of Contents 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Introduction 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Processing Methods of Mg-Based Materials 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Equal-Channel Angular Pressing 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' High-Pressure Torsion 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Intensive Rolling 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Fast Forging 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Kinetic Features of Severely Deformed Mg-based Materials 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Activation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Air Resistance and Hydrogenation Kinetics 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Cycling Stability 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Hydrogenation of Inert Materials 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Synthesis of Mg-based Materials with Desirable Thermodynamics 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Binding-Energy Engineering for Room-Temperature Hydrogen Storage 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Nanoglasses 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' High-Entropy Alloys 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Metastable Phases 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Concluding Remarks and Outlooks Acknowledgments References 4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Introduction Hydrogen, the lightest element in the periodic table, is considered to be the cleanest fuel of the 21st century.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' The burning of hydrogen produces only water, making it an appropriate fuel without any negative effects on global warming by CO2 emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Moreover, the electrochemical reaction of hydrogen with oxygen in fuel cells has higher energy efficiency compared to the combustion of fossil fuels which suggests another advantage of using hydrogen as a fuel [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Despite the advantages of hydrogen, there are still drawbacks concerning hydrogen production, utilization and storage that need to be addressed [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Hydrogen is mainly produced from gas reforming and a large amount of CO2 is generated during this process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' There are significant attempts to produce hydrogen by water splitting using renewable energies via electrolysis and photocatalysis [2], although the efficiency of photocatalysis is still quite low [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Fuel cells have high efficiency for the utilization of hydrogen, but reducing their working temperature, developing feasible catalysts, and reducing their price are some major challenges in using fuel cells [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Hydrogen can be stored in the form of high-pressure gas, liquid, or solid, but these methods have some limitations [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Storage of hydrogen in the form of high-pressure gas needs special tanks due to safety issues and these tanks occupy large space [2,5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Storage of hydrogen in the liquid form is a compact technology, but liquifying hydrogen consumes energy and liquid hydrogen can evaporate over time [2,5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Solid-state hydrogen storage, particularly in the form of metal hydrides and complex hydrides, provides the most compact and safest technology for hydrogen storage, but most hydrides suffer either from high thermodynamic stability (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' high working temperature), slow kinetics, difficult activation, or low gravimetric storage capacity [5,6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Magnesium and its hydride MgH2 are the first hydrogen storage materials that were introduced in 1951 [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Magnesium is an abundant element in the Earth’s crust, it is rather cheap, and it can store 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='6 wt% of hydrogen [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' However, magnesium exhibits strong Mg-H bonding, which results in great thermodynamic stability of the hydride requiring high hydrogen desorption temperatures well above 573 K [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Moreover, it suffers from poor activation and slow hydrogenation/dehydrogenation kinetics mainly due to the presence of an oxide layer on its surface and the slow diffusion of hydrogen in the bulk [8,9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Alloying of magnesium, development of composites and intermetallic compounds, the addition of catalysts, and microstructural modifications are some strategies used to solve the thermodynamic and kinetic problems of Mg- based hydrogen storage materials [8,9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Processing by severe plastic deformation (SPD), first proposed by Skripnyuk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' [10], is an effective strategy used within the past two decades to address both the kinetics and the thermodynamics of hydrogen storage in Mg-based materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' In SPD, a large plastic strain is induced in a piece of material to produce a final product in a bulk form with ultrafine grains or nanograins and large fractions of crystal lattice defects [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Equal-channel angular pressing (ECAP) [12], high-pressure torsion (HPT) [13], and accumulative roll-bonding (ARB) [14] are currently the most popular SPD processes, but there are some trends to induce severe strain in hydrogen storage materials by conventional methods such as intensive rolling [15], fast forging [16] and shot peening [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' It was shown that the presence of a high 5 density of grain boundaries and crystal lattice defects can provide pathways for hydrogen transport to improve hydrogenation kinetics [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Unlike the ball milling technique which is traditionally used to produce hydrogen storage materials in the form of powders with a large surface area [8,9], the SPD processes produce bulk samples with a lower surface area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' This results in better activation and enhanced air resistance [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Furthermore, the SPD processes, particularly those with extremely large shear strains, which are known as ultra-SPD, can be used to synthesize a wide range of Mg-based hydrogen storage materials even from immiscible systems [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' In the present article, recent advances in the application of SPD to Mg-based hydrogen storage materials are reviewed with a focus on processing, kinetic features, and thermodynamic modification via the synthesis of novel materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Processing Methods of Mg-Based Materials Although numerous SPD methods have been developed to process structural materials with advanced mechanical properties [11,12], only a limited range of these methods have been used for processing hydrogen storage materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' The main impact of these methods is the refinement of microstructure and the introduction of crystal lattice defects such as vacancies and dislocations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Among the various SPD methods, HPT is capable of continuously inducing large strains under high pressure and is the most efficient one for microstructure refinement and defect generation in Mg-based compounds [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Since repetitive cycles and discontinuous straining are applied in ECAP and intensive rolling, the processing strain and the concomitant microstructure refinement are not as significant as HPT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' The advantage of the former techniques is their suitability for processing large pieces of hydrogen storage materials, which is beneficial for commercial applications [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Fast forging can also induce a large strain within one cycle, but the amount of strain and the degree of grain refinement by this method are lower compared with those enabled by HPT processing [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' In this section, the major studies conducted on SPD processing of Mg- based hydrogen storage materials using ECAP, HPT, intensive rolling and fast forging are reviewed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Equal-Channel Angular Pressing In relation to the efficacy of the processing of magnesium alloys aiming at improving their hydrogen storage properties, ECAP takes a special place among the techniques of SPD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Not only is ECAP arguably the most popular SPD method [20-25], but it is also historically the first one whose potency as a tool for accelerating the hydrogenation kinetics of magnesium alloys by SPD was discovered [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' In the ECAP method, a billet in the form of a rod or bar is repeatedly pressed through a channel with a bending angle to introduce simple shear strain, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' 1a [20- 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' The ability of ECAP to produce extreme grain refinement, down to submicron scale, in magnesium and its alloys is well documented [22,26-29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' The ECAP-induced acceleration of hydrogen absorption/desorption was first demonstrated for Mg-based alloy ZK60, Mg-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='95Zn- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='71Zr (wt%), which is widely used for structural applications [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' The effect of ECAP on the dehydrogenation kinetics is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' 1b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' The data obtained for ZK60 by employing a different ECAP facility, which represent a further improvement over the original results [10], are also shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' 1b [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' A striking drop in the particle size down to the submicron range for the material 6 processed by ECAP and subjected to hydrogenation and dehydrogenation is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' 1c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Variants of the ZK60 alloy were the object of studies where a combination of ECAP with cold rolling [31] or ARB [32] was used, albeit with lesser success.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' (a) Schematic illustration of ECAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' (b) Dehydrogenation kinetics of alloy ZK60 after different kinds of processing by ECAP (triangles) [10], ECAP followed by high-energy ball milling (closed circles) [10] and ECAP with a different route (red curve) [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' (c) Scanning electron microscopy images of alloy ZK60 processed by ECAP after hydrogen absorption/desorption cycle [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' (a) Plunger Sample Die (b) 7 ZK60ECAR 9 (%1m) 5 Hydrogen 3 2 ZK60afterECAP [Krystianetal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=',2010] coarsegrainedZK60 ZK60after ECAPor HEBMSkripnyuk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' (2004) 0 5 10 15 20 25 30 DischargingTime (min) (c) 500nm7 Research conducted so far demonstrated that ECAP is on par or can even outperform the more conventional ball milling technique, both methods improving the hydrogenation kinetics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Indeed, the hydrogenation curve for the ball-milled ZK60 alloy was shown to be very close to, yet lying beneath, the curve measured for the ECAP-processed one [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' The great benefit of ECAP is the possibility to obtain a promising hydrogen storage material in bulk, with an additional advantage of avoiding potentially hazardous ball milling of powders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Several studies utilizing ECAP as the processing route [33-36] and further SPD techniques were also applied to a range of Mg-based alloys [37,38], notably the eutectic Mg89Ni11 alloy with a fine lamellar structure [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' A promising approach to enhancing the hydrogenation/dehydrogenation kinetics of magnesium is the use of composites containing metal hydrides [40] or particles of carbonaceous materials [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Specifically, a study of a Mg-based composite with 2 wt% of multiwall carbon nanotubes processed by ECAP showed that the addition of nanotubes to magnesium leads to a substantial acceleration of hydrogen desorption rate, a further advantage being the disappearance of pressure hysteresis [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Viewed as a progenitor of SPD-based tools for improving the hydrogenation kinetics of Mg-based alloys, ECAP processing can be regarded as a viable avenue to large-scale hydrogen storage facilities [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' High-Pressure Torsion Among different bulk SPD methods, HPT [13] has successfully been applied to manufacture a large variety of different hydrogen storage materials [43-47] due to the exceptionally high shear strain that can be achieved in bulk sample volume [21,48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' During HPT deformation, as schematically shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' 2a, a disc-shaped specimen is inserted between two anvils and subjected to concurrent uniaxial pressure of several Gigapascals and torsional straining by several revolutions [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' There has been a wide range of applications of HPT to Mg-based materials, as will be discussed here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' HPT deformation of MgH2 powders results in a significant grain refinement [50] and also induces a strong (002) texture and the formation of the metastable γ-MgH2 phase [51] which positively influences the overall hydrogen storage performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' The particle size of magnesium powders has a strong effect on the hydrogen absorption kinetics, but the grain/crystal size is also known to affect the kinetics significantly [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' It was reported that a bimodal microstructure develops when bulk magnesium is subjected to HPT, including the emergence of nanocrystals and large recrystallized grains, resulting in a substantial improvement of the hydrogenation kinetics [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' The average density of dislocations in commercial magnesium processed by HPT reaches a very large value of 8x1015 m−2, which can act as hydrogen transport sites [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' A ZK60 Mg- based alloy processed by HPT exhibits a stable storage capacity of up to 100 hydrogenation cycles, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' 2b and 2c [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' The maximum hydrogen capacity of ball-milled Mg-Ni nanopowder increases by 50% after HPT and reaches the theoretical value, due to the creation of new hydrogen transport sites in the vicinity of dislocations [56], while the formation of Mg2NiH0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='3 hexagonal solid solution and the monoclinic Mg2NiH4 takes place [57-58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Other lattice defects, notably stacking faults, can also improve the hydrogenation kinetics of Mg2Ni processed by HPT [59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' A large fraction of cracks in ultrafine Mg + 2 wt% Ni powder can act as pathways for hydrogen transport from the surface of the HPT-processed disc [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' As will be discussed later, the extreme shear deformation during torsion can reach such a high magnitude that it is capable to promote hydrogen uptake even in the non-absorbing MgNi2 phase [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' 8 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' (a) Schematic representation of HPT apparatus with uniaxial compression and simultaneous torsion [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' (b) Dehydrogenation and (c) hydrogenation kinetic curves obtained at 623 K and 1 MPa absorption pressure, and at 10 Pa desorption pressure for HPT-deformed ZK60 alloy [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' (d) Hydrogenation curves at different temperatures for Mg + 5 wt% Ni + 2 wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='% Nb2O5 powder composite processed by HPT [62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' (e) Desorption kinetic curves obtained at 573 K and 1 kPa for as-milled magnesium powders catalyzed by Nb2O5 and/or carbon nanotubes and corresponding HPT-processed discs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' (f) High-resolution lattice image of cycled HPT-processed Mg + Nb2O5 + carbon nanotube (inset: selected area electron diffraction pattern) [65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' (a) (d) STAGEII STAGEI HPT sample Absorbed Hydrogen (wt%), Load cell 5 4 UpperAnvil 3 Sample 2 623K 473 K Sample 573 K 423 K 1 523K 373K 623 K LowerAnvil Load Load Ldad 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='8 Rotation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='0 Time (h) (b) (e) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='0 (%1m) Relative Hydrogen Content ZK60HPT - cycle 40 Desorbed Hydrogen (wt%) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='5 cycle 60 cycle 80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='8 Hydrogen 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='0 cycle100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='7 2 Mg-NbO_HEBM 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='6 Mg-NbO_HEBM+HPT 3 Mg-CNT_HEBM 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='5 Mg-CNT_HEBM+HPT Mg-NbO-CNT_HEBM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Desorbed Mg-NbO-CNT_HEBM+HPT 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='3 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='1 6 0 0 1 2 3 5 6 7 8 0 1000 2000 3000 4000 5000 6000 Time (min) Time (s) (c) (f) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='38 1 Relative Hydrogen Content Absorbed Hydrogen (wt%) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='5A 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='9 Nbosgrains 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='45A 1A 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='6A 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='8A 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='0 ZK60HPT-cycle40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='5 cycle 60 cycle80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='1 cycle100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='0 0 Amorphous 0 2 4 6 10 12 14 16 CNT carbon Time (min) 5 nm9 The HPT process can be used effectively to fabricate new Mg-based composites or alloys for hydrogen storage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' A reasonable hydrogenation capacity can be obtained for a powder mixture of Mg + 5 wt% Ni + 2 wt% Nb2O5 subjected to HPT at a temperature as low as 423 K, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' 2d [62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' It was revealed recently that metastable phases can be developed even in immiscible systems (such as Mg-V-Sn [19], Mg-V-Ni [19], Mg-Ti [63] and Mg-Zr [64]) during ultra-SPD, thus new potential hydrogen storing materials can be manufactured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' It was found in a recent study that the combination of ball milling and the HPT process can further improve the desorption kinetics of nanocrystalline magnesium catalyzed by Nb2O5 and/or carbon nanotubes [65,66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' As confirmed by transmission electron microscopy, the carbon nanotubes acting as diffusion channels for hydrogen are preserved during plastic deformation by ball milling and HPT as well as during sorption cycling, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' 2e and 2f [65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' The combined catalytic effect of metal-oxide particles and carbon nanotubes can be substituted by applying only metal-oxide nanotubes [67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' HPT can also be utilized for the synthesis of high-entropy materials, such as MgVTiCrFe for hydrogen storage [68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Moreover, the hydrogenation performance of fully disordered systems, like glassy alloys, can be improved significantly when the materials are subjected to severe shear deformation, including a reduced hydrogenation temperature and improved hydrogen sorption kinetics for Mg65Ni20Cu5Ce10 [69] and Mg65Ni20Cu5Y10 [70] metallic glasses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' At the same time, the hydride formation enthalpy increases noticeably which is especially more pronounced in the most deformed perimeter region of the HPT-processed discs [71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Despite the potential of HPT to be applied to a wide range of materials, its main limitation is currently the small size of the sample which is an obstacle to its development for commercial applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Intensive Rolling Despite the successful use of high-energy ball milling methods to produce nanostructured Mg-based alloys, two processing techniques of ECAP schematically shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' 1a [10] and intensive rolling schematically shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' 3a-c [72], were reported almost at the same time, as alternatives enabling the production of “bulk” samples with refined and more air-resistant microstructures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Metal processing based on cold rolling, such as co-lamination via repetitive rolling, was first used in hydride-forming Mg-based alloys to produce Mg-Ni composite structures [72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Heat treatment of a deformed sandwich of Mg and Ni foils resulted in the formation of an intermetallic Mg2Ni compound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' A similar result was observed for Mg-Al composite alloys with the final heat treatment forming the Mg17Al12 compound that also absorbed hydrogen reversibly [73].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Cold rolling of a stack of Mg/Pd foils produced an air-resistant laminated compound with a shorter activation time compared to a ball-milled sample of the same composition as well as to pure magnesium, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' 3d [74].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Repetitive rolling was also used to prepare laminate composites of Mg/Cu [75,76] and Mg/Pd [75] with the micrometer-order layered structure containing a high density of dislocations and vacancies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' A similar route to co-lamination, but involving a bonding process during intensive rolling, designated as ARB shown schematically in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' 3a [77], was used to prepare Mg/Ti multilayers and Mg/stainless steel composites [78].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Activation for hydrogen absorption improved with an increased number of fold and roll operations, which caused increasing refinement of the hard second-phase particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Intensive cold rolling was effectively used to refine the microstructure and disperse the particles of catalysts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Intensive rolling was used to process mixtures of MgH2-Fe as an alternative to reach the grain size reduction typically obtained by ball milling [79,80].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Cold rolling of commercial MgH2 was considered equivalent to ball milling in terms of microstructure refinement 10 [81], as also observed for Mg-LaNi5 composites [82].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' 3b and 3c show the schematics of a vertical and horizontal rolling machine used to process MgH2 powders, respectively [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' The presence of additives in Mg-based alloys processed by cold rolling proved to be as efficient as ball-milled materials despite the distribution of the additive particles being much less homogeneous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' A uniform distribution of additive particles was also observed in MgH2 powders containing different types of additives [83].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' It was shown that the uniformity of FeF3 distribution in MgH2 powders can be improved by a combination of short-time ball milling followed by intensive rolling [84].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' There have been some improvements in the rolling process for treating hydrogen storage materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' The use of a protective atmosphere during rolling allows further refinement of the microstructure in commercial magnesium [85] and MgH2 [86] since a greater number of rolling passes can be applied without surface contamination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' 3e shows the refined and textured microstructure achieved in MgH2 after 35 rolling passes under an argon atmosphere which can result in fast hydrogenation and dehydrogenation kinetics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' 3f and 3g show the positive effect of the number of rolling passes under a protective atmosphere on the kinetics of hydrogen absorption and desorption in MgH2 [86].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' The same route was used to process MgH2-LaNi5 mixtures under a protective atmosphere, resulting in compacted composite flakes [87], which exhibited faster hydrogen absorption kinetics and reduced desorption temperatures in comparison with single-phase MgH2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Low-temperature rolling in AZ91 alloy can improve the hydrogen storage properties compared with rolling at ambient temperature due to the larger density of microcracks and consequently high density of exposed interfaces and formation of stronger (002) texture [88].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Cold rolling has also been used as a further processing step to improve the activation and hydrogen absorption kinetics of Mg-based samples initially prepared by a different technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' It was shown that cold rolling following other modes of SPD processing of magnesium and its alloys results in a rolling texture that can enhance the kinetics in the first hydrogenation cycle by exposing less densely packed atomic planes to the hydrogen atmosphere [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Studies of other processing combinations, such as cold rolling of melt-spun ribbons [89], cold rolling of HPT-processed discs [90], and ARB processing of ECAP-treated ZK60 alloy [32] demonstrated that in addition to the refined microstructure, the presence of free surfaces (or interfaces) and texture have positive effects on the hydrogen storage properties of Mg-based alloys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Intensive rolling is likely to have the greatest potential for large-scale processing of hydrogen storage materials, although the magnitude of plastic deformation in this method is smaller than in ball milling followed by HPT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' 11 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' (a) Schematic illustration of ARB process [77], and sketches of (b) vertical and (c) horizontal rolling machines used to process MgH2 powders [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Hydrogen absorption curves under a hydrogen pressure of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='5 MPa at 623 K for pure magnesium and Mg - 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='5 at% Pd after cold rolling (CR) and ball milling (BM) [74].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' (e) Dark-field image and corresponding selected area electron diffraction pattern, taken by transmission electron microscopy, for MgH2 processed by cold rolling with 35 cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' (f) Hydrogen absorption and (g) desorption curves for MgH2 processed by different numbers of passes of cold rolling under a protective atmosphere [86].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' (a) (b) (c) Rolling Cylinders Cutting Stacking Powders Rolling Cylinders Plate Powders Roll Bonding StainlessSteelPlates (d) (e) (%1m) 8 7 5 Mg-Pd2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='5at.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='%CR 4 Mg-Pd2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='5at.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='%BM Mg-Pd2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='5at.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='%CR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='reactivated 口 PureMgCR 2 500 1000 1500 2000 2500 100 nm Time (min) (f) (g) Cold Rolled MgH2 --As-received 0 10-passes 330°C/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='1MPa 口-20-passes 5 O-35-passes 1 50-passes 4 2 8- 2 As-received 4 10-passes Cold Rolled MgH, 20-passes 330°C/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='5MPa 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='-35-passes 5 50-passes 0 9- 0 180 360 540 720 9001080126014401620 0 300 600 900 1200 15001800210024002700 Time (s) Time (s)12 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Fast Forging As discussed earlier, the kinetic performance of hydrogen storage Mg-based materials can be improved by: (i) reducing the particle/crystallite sizes to the nanometer level combined with including a high lattice strain and a large density of extended defects (dislocations, stacking faults, twins, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=') by plastic deformation (high-energy ball milling [91-96], HPT [24,35,44,50,51,97,98], ECAP [10,33,99,100-102], intensive rolling [81,103,104]), and (ii) using various additives which can act as catalysts to accelerate hydrogen sorption [104-108].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' In any case, mass production of MgH2 must satisfy practical requirements such as processing conditions (ease of manufacture, efficiency, cost, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=') and material performance (maximum hydrogen uptake, fast sorption, stability, lifetime, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=') [109].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' These requirements can be achieved using fast forging, a less conventional SPD method [16,110,111].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' A photograph of the fast-forging facility is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' 4a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' In this method, the height of ingots of bulk magnesium, Mg-based alloys or their compacted composites is reduced by about a factor of ten within about 5x10-3 s, while the initial temperature of the sample is set using an induction high-frequency coil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Since the mechanical energy of the hammer is almost entirely dissipated in the material without elastic rebound, it leads to refining the grain size plastically, while simultaneously healing the sample frictionally [16,110,111].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Such grain refinement by fast forging can improve the hydrogenation kinetics similarly to other SPD methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' In addition to microstructural modification, fast forging can be used to synthesize composites of Mg-based hydrogen storage materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' As a trial, homogeneous mixtures of strongly compacted powders of Mg+Ni with initial particle sizes of 5-30 µm for magnesium and 30-40 µm for nickel were fast forged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' To optimize forging conditions and determine the deformation fields, a two-dimensional calculation model was developed by considering the particle size, distribution of nickel among magnesium particles and their relative hardness [112].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Numerical simulation of the adiabatic compression processes quantifies a marked increase of sample temperature for average strains of 80-90%, while the additional heat generated by an increase in strain was found to drop [113,114].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' 4b shows the conversion rate of Mg89/Ni11 powder mixtures to form Mg2Ni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' The examination of absorption traces at 614 K and 2 MPa of H2 indicate that fast forging of magnesium in a brittle state at low temperature promotes the formation of defects and cracks enhancing hydrogen diffusion in the bulk, while forging at high temperature and a ductile state of the material allows the formation of Mg2Ni as a catalyst, as reported for various SPD routes [39,114-118].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' In another study, compacts of 95 wt% Mg + 5 wt% MgH2 were processed by combining fast forging with two passes of ECAP at room temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Pressure-composition-temperature isotherm analyses showed that at 593 K and under a hydrogen pressure of 1 MPa, the fast forging was effective to achieve up to 7 wt% of hydrogen uptake in ~140 min, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' 4c [119].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' The performance characteristics were found to be rather similar to those of the ECAP-processed samples with 6 wt% hydrogen uptake at 593 K and 1 MPa in 60 min [120,121].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Such improvements by fast forging are also comparable with those achieved using other SPD routes [122,123].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' These results confirm that the fast-forging process can be considered as an efficient SPD route enabling mass production of MgH2 for hydrogen storage [124].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' 13 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' (a) Fast forging facility with controllable atmosphere and temperature having a 150 kg hammer falling from up to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='5 meters with a forging time of less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='02 s (left), schematic illustration of working chamber (center) and the appearance of samples before and after forging (right) [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' (b) The conversion rate of Mg89/Ni11 powder mixture to Mg2Ni after fast forging at different temperatures [114].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' (c) First hydrogen absorption kinetic curves at 593 K and under a hydrogen pressure of 1 MPa for 95 wt% Mg + 5% wt% MgH2 compact processed by fast forging [119].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Kinetic Features of Severely Deformed Mg-based Materials As mentioned above, many investigations have been done on the hydrogenation/dehydrogenation kinetics of metal hydrides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' The popular ways to enhance kinetics are the addition of a catalyst or change of stoichiometry [125-127].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Other efficient ways to enhance kinetics discussed in several sections of the present review are mechanical deformation of powders [128] or bulk samples [129-132].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' However, for most practical applications, dehydrogenation can be relatively slow, and thermodynamics (plateau pressure) is more crucial than kinetics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' For the hydrogenation part, because of the high enthalpy of the formation of the hydrides, the main rate- (a) Piston: Pyrometer Window SampleBeforeForging Hammer Workingchamber HF Heating Sample SampleAfterForging Coil (b) 50 1 40 6 Hydrogen (wt%) 5 30 20 3 T =506°C eut 2 10 1 0 0 0 100 200 300 400 500 600 700 800 0 50 100 150 200 EffectiveTemperature(oC) Time (min)14 limiting step will be the heat transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' For example, storing one kilogram of hydrogen in magnesium hydride for six minutes will require a heat transfer of about 100 kW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Therefore, for practical hydrogen tanks made of metal hydrides, the main challenge will be to manage a rapid heat transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' In addition, for commercial applications, the cost of the metal hydride should be as low as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Despite the significance of plateau pressure, heat transfer and cost, it is still essential to find strategies to address the kinetic drawbacks of hydrogen storage materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' SPD processing generates a high density of crystal lattice defects and refines the microstructure of magnesium and its alloys and as a result improves its hydrogenation kinetics, while keeping its reversibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' In contrast with other methods used to enhance hydrogenation kinetics, such as alloying, catalyst addition, synthesis of composites, and amorphization [133,134], the SPD methods do not need any extra additives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Furthermore, as opposed to ball milling which produces potentially hazardous fine powders [133,134], the SPD-processed materials are in a bulk form with less contact with the air atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' In addition, they have a large density of crystal lattice defects, which can act as fast hydrogen pathways for easy activation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' These kinetic features of severely deformed materials can sometimes result in hydrogen uptake in materials that are normally inert to hydrogen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' These kinetic features are briefly reviewed in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Activation One issue that is often overlooked is the problem of the activation of hydrogen storage materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' For metal hydrides, the term activation means the process by which the alloy is prepared for reversible hydrogen sorption [135].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Typically, it involves submitting the alloy to hydrogen pressure and/or temperature much higher than the ones predicated by thermodynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' However, this implies an additional cost for the manufacturer because the storage tank needs to be overdesigned so that it can sustain not only the service conditions but also the activation conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' The ideal situation would be to perform the activation process under the same regime as the service conditions of the storage tank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' The problem of activation has been investigated for a long time, but it is still not clear what mechanisms are at play [136].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' What is known is that microstructure plays a crucial role in the activation of metal hydrides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Mechanical deformation has been shown to improve the activation kinetics of metal hydrides [99].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' The most widely used method is ball milling [137,138], but recently other mechanical methods or a combination of them have been shown to improve the activation of metal hydrides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Some of them are cold rolling [74,89,139,140], ECAP [30,31], and HPT [59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' As examples of the effectiveness of mechanical deformation, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' 5 shows the effect of HPT on (a) microstructure and (b) the first hydrogenation of magnesium [53], and (c) the effect of ECAP passes on the first hydrogenation of AZ61 magnesium alloy [141].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Clearly, mechanical deformation introduces ultrafine grains and crystal lattice defects and accelerates the first hydrogenation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' 15 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' (a) Microstructure and corresponding selected area electron diffraction of magnesium processed by ten turns of HPT (N = 10) [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' (b) First hydrogenation (activation) of magnesium at 423 K and under 3 MPa of hydrogen before and after processing by HPT for N = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='25 and N = 10 turns [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' (c) First hydrogenation of AZ61 alloy at 643 K and a hydrogen pressure of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='5 MPa before and after processing by ECAP for N = 1, 4 and 8 passes [141].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Air Resistance and Hydrogenation Kinetics The kinetics of hydrogen absorption and desorption in magnesium is slow, and thus, this issue should be addressed by increasing the surface area, generation of grain boundaries, generation of interphase boundaries, alloying with transition metals (Tm = Co, Fe, Mn, Ni, Nb, Ti, V), and the addition of catalysts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' The milling process in the form of high-energy ball milling, a 1 μm (b) 10 Hydrogenation: P = 3 MPa, T= 423 K 9 (wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Mg (99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='9%) 8 HPT: P = 6 GPa, @ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='2 rpm, T = 298 K Storage 6 5 N=10(=160) 4 3 N = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='25 2 Anneal (=0) (ε= 4) 0 0 20 40 60 80 100 120 ExposureTime (ks) (c) (%1M) AZ61 5 N:ECAPPasses 4 3 2 Absorbed V-AZ61 1 0-AZ61N=1 O—AZ61N=4 ←-AZ61N=8 0 100 200 300 400 500 Time (s)16 mechanical alloying and reactive ball milling (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' milling under a hydrogen atmosphere) is effective to induce all these effects [40,92,126,142,143].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' However, in some cases, the milling process does not necessarily improve the kinetics because the generated active surfaces can react easily to be oxidized during processing or subsequent handling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' In addition, because of this high surface reactivity, the produced powders are pyrophoric which can create significant safety issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Thus, the ball-milled magnesium powders should be handled under a controlled atmosphere such as in the glove box in the laboratory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Therefore, despite the efficiency of ball milling and particularly reactive ball milling in mechanical alloying, uniform addition of catalysts, grain size reduction, and generation of high densities of grain boundaries and defects [50,144,145], the difficulties in handling the active powders under air atmosphere have greatly restricted its potential applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Several SPD techniques can produce similar benefits as ball milling in enhancing the hydrogen sorption kinetics with an additional advantage of the bulk materials processed by these techniques being more air resistant and less contaminated compared with ball-milled powders [98].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' The low surface area of bulk materials processed by SPD compared to powders allows for storing them under an air atmosphere for a long time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' The presence of crystal lattice defects, particularly grain boundaries, provides numerous pathways for hydrogen transport from the partially oxidized surface to the bulk to produce the hydride phase [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Moreover, it was shown that the consolidation of powders to a compact material using SPD methods not only enhances the air resistance but also accelerates the hydrogen storage kinetics [52,60,146].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' These SPD methods for additional powder consolidation include intensive extrusion [139,147], intensive rolling [58,139], HPT [44,63,148,149], ECAP [36,150], accumulative fold-forging [151] and fast forging [118].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' HPT is often selected for powder consolidation of hydrogen storage materials because it induces large shear strains combined with high pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' A comparative study of HPT imparted on (i) micro-sized atomized magnesium powder and (ii) nano-sized condensed magnesium powder showed an additional impact of SPD on the kinetics of absorption/desorption [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' The microstructures of these samples before and after HPT processing are shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' 6a-d, illustrating the size of the powder precursors and the SPD-induced consolidation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' The examination of the first hydrogenation kinetics at 673 K under a hydrogen pressure of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='5 MPa, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' 6e, indicates that the finer condensed magnesium powder has faster hydrogenation kinetics than the atomized one [60,146].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' For both types of powder precursors, HPT processing has the effect of improving the hydrogenation kinetics through the introduction of structural defects, microstructure refinement, and breaking of the powder surface oxide layer that was redistributed inside the microstructure to act as a possible catalyst [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Another advantage of HPT consolidation is the reduction of the hysteresis between the absorption and desorption plateau pressure during the pressure-composition isotherm experiments [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' It should be noted that the apparent lower hydrogen uptake capacities after HPT processing in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' 6e are due to the large scale of the diffusion path in the bulk HPT samples being rapidly surrounded by compact MgH2, which does not allow the samples to react completely with hydrogen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' This effect is reduced in subsequent cycles due to hydrogen-induced fragmentation of samples and diminished diffusion path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' In summary, in addition to the enhancement of the kinetics of hydrogen storage, the consolidation of powders by SPD can also have further benefits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' First, the initial powder can be mixed with catalysts almost without any limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Second, the produced bulk materials are air- resistant and can be stored under atmospheric conditions [152-154].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Third, the absorption and desorption processes occur usually much faster than in non-consolidated precursor powders [52,139].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Fourth, the microstructural features such as the crystallite size, crystal lattice defects and 17 texture, which were suggested to influence the hydrogenation kinetics [148,150,153], can be easily tailored by controlling the imposed strain and the heat treatment [118].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Microstructure of (a) atomized magnesium micro-sized powder, (b) condensed nano- sized powder, (c) HPT-consolidated atomized powder and (d) HPT-consolidated condensed powder, including (e) kinetics for their first hydrogenation [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Cycling Stability Over the last two decades, numerous investigations of hydrogen storage in mechanochemically treated magnesium and Mg-based alloys, including SPD-processed ones, have been reported [130].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' The top-down SPD methods have been successfully applied to increase (a) Atomized Mg Powder (b) Condensed Mg Powder 40 μm 500 μm (c) Atomized Mg Powder + HPT (d) Condensed Mg Powder + HPT HPT Disc 1 μm 1 μm 口 (e) Hydrogen Content (wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='%) 8 Mg Atomized Micro-powder HPTMicro-powder Condensed Nano-powder HPT Nano-powder 0 60 120 180 240300360 420 Time (min)18 absorption/desorption kinetics with the additional advantage of replacing highly expensive and contamination/oxidation-risky bottom-up methods such as ball milling [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Moreover, a combination of SPD methods and ab initio calculations have been applied to decrease the desorption temperature by substitutional mechanical alloying of magnesium with elements of optimized type and concentration [129].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' The kinetics of hydrogen storage is very sensitive to numerous parameters such as the surface and morphology of the storage material (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' oxide layers, porosity, texture, microstructure, and the density/type of SPD-induced crystal lattice defects), and these parameters may change in different experimental procedures or during absorption/desorption cycles [130].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' An examination of publications on SPD processing of Mg-based hydrogen storage materials indicates that the results of investigations do not always match unless all the parameters mentioned above are carefully controlled to have a reliable understanding of the hydrogen dissociation (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' [155]), diffusion and storage processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Another issue with these publications is that most of them only studied one or a few absorption/desorption cycles, while the stability of the hydride during high-cycle hydrogen storage is a critical issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' The hydride formation stability is intimately connected with the thermal stability of the microstructure with regard to the temperature necessary for a minimum time to complete absorption/desorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' While in systems with high melting temperatures, the microstructure is stable because of a much lower absorption/desorption temperature compared to the melting point (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' TiFe [130]), this is not necessarily true for materials with medium and low melting temperatures, like magnesium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Here, stability can be reached by creating alloys or by adding nanoparticles of a second phase that can stabilize the microstructure beyond the hydrogenation temperature and act as nucleation sites for hydride [156].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Krystian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' [30] for the first time reported this stability for up to 1,000 cycles for ECAP-processed Mg-based ZK60 alloy, but not for ECAP-processed pure magnesium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Nevertheless, Chiu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' [157] confirmed the stability of another ECAP-processed Mg-based alloy, AZ31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Grill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' [55] not only reported highly stable storage characteristics for HPT-processed ZK60 alloy but also found that SPD-induced second- phase particles can act as nuclei for heterogeneous hydride formation, according to a thorough Johnson-Mehl-Avrami analysis [158] yielding an Avrami exponent of n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Popilevsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' [159] reported something similar in the case of composites of magnesium with multiwall carbon nanotubes where they applied some mechanochemical treatment and achieved chains of carbon nanoparticles that became nucleation sites for the hydrogenation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' For SPD-processed Mg-based alloys, through combined differential scanning calorimetry and X-ray diffraction analyses, Ojdanic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' [160], Cengeri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' [161], and very recently Abbasi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' [162] found thermally stable SPD- induced vacancy agglomerates [160-162] which heterogeneously nucleate the hydride phase, instead of second-phase precipitates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' 7a shows the hydrogen absorption rate in a Mg-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='5Zn-1Gd-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='62Y-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='67Nd (wt%) alloy processed by HPT and ECAP after ten hydrogenation cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' The hydrogenation kinetics in both samples remain fast even after ten hydrogenation cycles, although the ECAP-processed sample shows slightly better performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' 7 demonstrates the long-cycle hydrogenation (up to 100 cycles) in ZK60 processed by (a) HPT and (b) friction stir processing (FSP), confirming that both samples exhibit high cycling stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' From all these long-cycle storage characteristics shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' 7, FSP gave rise to the best hydrogen storage kinetics and the best storage capacity, followed by those of ECAP and HPT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Future investigations should be conducted to understand how the type and/or density of the vacancy agglomerates depend on the applied SPD method and thus govern the long-cycle hydrogenation in different Mg-based compounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' 19 Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' (a) Hydrogen absorption at T = 613 K after ten hydrogenation cycles in Mg-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='5Zn-1Gd- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='62Y-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='67Nd (wt%) alloy processed by HPT and ECAP to von Mises equivalent strain of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='3 [162].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Cyclic hydrogen absorption kinetic curves at T = 623 K for Mg-based ZK60 alloy processed by (b) HPT and (c) FSP to comparable von Mises equivalent strains of \uf07e7 [161].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' a) 76543210 Hydrogen (wt%) ECAP:4Passes HPT:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='25Rotations Mg-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='5Zn-1Gd-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='62Y-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='67Nd (wt%) 2 0 4 6 8 10 12 Time (h) (b) HPT ZK60 wt% Hydrogen (wt%) 86 T= 623K,P= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='1MPa 7654321 ４２ 10080 0 60 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='0 20 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='5 Time (h) 0 (c) FSPZK60 wt% 7654321 Hydrogen (wt%) 8642 T=623K,P=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='3MPa 0 60 20 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='5 00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='0 Time (h)20 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Hydrogenation of Inert Materials We reiterate that SPD is a useful group of processes to produce nano-sized grains decorated with crystal lattice defects [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' As reviewed in [48,163,164], among the SPD processes, HPT is a unique method that is applicable to hard and brittle materials including intermetallics [165,166], glasses [167,168], ceramics [169,170] and semiconductors [171-174].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' In particular, by taking this advantage, HPT was applied to an intermetallic Mg2Ni compound which gave rise to a significant improvement of the hydrogenation kinetics of the material due to the grain refinement down to the nanoscale [59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' The enhanced kinetics was observed even after annealing of the HPT-processed Mg2Ni, which was associated with the formation of many stacking fault defects [59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' In addition to these kinetic effects, which were discussed earlier in this article, defect generation by HPT processing can lead to hydrogen storage in materials that are otherwise inert to hydrogen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' This positive effect in some materials such as TiFe [175], TiV [176] and Ti-Fe-Mn [177] is due to the activation problem having been resolved by HPT processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' In addition, it was shown that the formation of defects by HPT processing can influence the local metal-hydrogen interaction and the hydride formation thermodynamics [178].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' It is this thermodynamic advantage that enables the partial absorption of hydrogen in some materials that are normally inert to hydrogen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' For example, HPT was successfully applied to store hydrogen in MgNi2 which is considered to be inert to hydrogenation [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Some experimental pieces of evidence of the hydrogen absorption capability of MgNi2 are presented below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' 8a displays X-ray diffraction profiles after HPT processing for N = 2, 5 and 10 turns, including a profile before HPT (labelled by N = 0) [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Peak broadening occurred as the number of HPT revolutions increased, indicating that strain was introduced in the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' The results of the quantitative analysis for hydrogen uptake before and after hydrogenation are plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' 8b with respect to the number of HPT turns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' The hydrogen content increased with the number of HPT revolutions but remained unchanged for the samples that were not subjected to hydrogenation irrespective of the number of HPT turns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' This comparison shows that hydrogen was stored in MgNi2 after the HPT process and the amount increased with the number of HPT turns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' The data for crystallite size and anisotropic strain derived from the X-ray diffraction profiles indicated that hydrogen likely stay at lattice defects such as dislocations and grain boundaries but not at the tetragonal sites [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Despite the clear evidence that hydrogen could be stored in MgNi2, the amount absorbed was as small as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='1 wt%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' The data presented in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' 8a and 8b were obtained six months after the hydrogenation of HPT-processed samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' These low values might be associated with the hydrogen trapped in dislocations or grain boundaries while most of the hydrogen in the crystal lattice was desorbed over the six-month storage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' In another study, the amount of absorbed hydrogen was measured using thermo-gravimetric-differential analysis one day after the hydrogenation of HPT-processed samples [179].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' The measurements returned a high value of ~1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='5 wt% for the weight loss, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' 8c [179].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' No change in the X-ray profiles occurred six months after HPT (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' 8d), suggesting that a low storage capacity reported in [61] might stem from the desorption of hydrogen while keeping the material for six months in the air.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' The entirety of the data presented leads one to conclude that HPT has a great potential to activate even inert Mg-based materials for hydrogen storage through modifications of both the thermodynamics and the kinetics of hydrogen sorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' 21 Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' (a) X-ray diffraction profiles after HPT processing for N = 2, 5 and 10 revolutions including profile measured before HPT (labelled N = 0) [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' (b) Plots of hydrogen content against the number of HPT revolutions, where hydrogenation was made at 373 K under 1 MPa for 24 hours after HPT processing [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' (c) Thermo-gravimetric differential thermal analysis one day (A) and six months (B) after HPT processing for ten revolutions and hydrogenation [179].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' (d) X-ray diffraction profiles measured one day (A) and six months (B) after HPT processing for ten revolutions and hydrogenation including the intensity difference between profiles of A and B (labeled A-B) [179].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Synthesis of Mg-based Materials with Desirable Thermodynamics Despite significant progress in the improvement of the kinetics of hydrogenation in magnesium and its alloys by using catalysts or by application of SPD, there has been rather limited success in the development of Mg-based alloys with appropriate thermodynamics for dehydrogenation at room temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' The main reason for such limited advancements is the large binding energy between the magnesium and hydrogen atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Compositional modification is (a) (b) (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=') 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='14 MgNi2 MgNi2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='12 HPT:2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='8GPa HPT:2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='8GPa Intensity N=10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='10 Hydrogenation 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='08 373K,1MPa,10h 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='06 after before 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='04 N=2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='02 0 N=0 2110 13 110 = 2 4 6 8 10 40 60 80 100 120 20 (degree) Number ofRevolution (c) (d) MgNi2 MgNi2 HPT:2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='8GPa HPT:2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='8GPa Differential Thermal Analysis /a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' /mass% Normalized Intensity /a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' B(6monthsafterHPT+HG) B (6monthsafterHPT+HG) A (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='dayafter HPT+HG) Hydrogenation(HG) 373K,1MPa,10h A (1dayafterHPT+HG) A A-B B 400 500 600 700 40 60 80 100 120 300 Temperature /K 20 /degree22 generally considered the most effective strategy to modify the thermodynamics of hydrogen storage in Mg-based alloys [180-183].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' The SPD processing does not influence the overall thermodynamics of hydrogen storage in magnesium unless a phase transformation occurs, mechanical alloying is achieved, or small crystals with sizes of a few nanometers are formed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' However, the SPD methods and particularly HPT have been used in recent years to synthesize new Mg-based alloys which can store hydrogen at low temperatures or even at room temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Some major studies are discussed in this section which include the concept of binding-energy engineering and the developments of nanoglasses, high-entropy alloys and metastable phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Binding-Energy Engineering for Room-Temperature Hydrogen Storage Magnesium hydride MgH2 described above as a promising hydrogen storage material suffers from high dehydrogenation enthalpy (75 kJ/mol H2) due to the strong binding between the magnesium and hydrogen atoms [38,184].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Several major research activities to overcome the key issue of the excessively high thermodynamic stability of MgH2 have been developed: (i) nanoengineering, such as the production of thin films [185-188] and nanoparticles [189-192] including confined [193], non-confined [194], core-shell [195] or composite [196] states;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' (ii) fabrication of metastable hydrides such as γ-MgH2 [51,98];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' and (iii) alloying to form intermetallic compounds, notably Mg2Ni [197], or solid solutions including Mg-In alloys [198,199].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Among them, alloying is the most feasible way to develop new materials by tuning the chemical composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' However, the thermodynamic immiscibility of Mg in many systems limits the fabrication of Mg-based alloys comprised of three or more component elements, especially by the melting process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' HPT was recently found to be a powerful tool to fabricate new metastable Mg-based alloys with superior hydrogen storage properties at room temperature [19,129].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' A successful example is Mg4NiPd, which is expected to have moderate hydrogen binding energy within the Mg2Ni and Mg2Pd ones, as predicted by the first-principles calculations (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' 9a) [129].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' The nominal Mg4NiPd fabricated by the melting process contained three phases, Mg8Ni3Pd, MgNi2 and Mg5Pd2, with heterogeneous elemental distributions (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' 9b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' The homogeneity of the elemental distribution significantly improved to the atomic scale by 1,500 turns of HPT (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' 9b), and as a result, a new phase of Mg4NiPd with a partly ordered CsCl-type structure was obtained (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' 9c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' More importantly, at room temperature, the HPT-processed Mg4NiPd could reversibly absorb and desorb 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='7 wt% of hydrogen (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' 9d), which was suggested to occupy the 4Mg-1Ni-1Pd site with a hydrogen binding energy of -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='12 eV (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' 9e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' These results demonstrated the success of destabilizing the dehydrogenation thermodynamics of Mg-based alloys to room temperature by applying binding energy tailoring principles which could be realized by SPD processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' This strategy can also be applied more universally to design new materials with superior functional properties beyond the scope of the phase equilibria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' 23 Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' (a) Concept of binding-energy engineering used to design Mg4NiPd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' (b) Elemental mapping of Mg4NiPd after casting (N = 0) and after HPT processing for N = 20, 100, 500 and 1,500 turns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' (c) X-ray diffraction profile and (d) pressure-composition isotherm of Mg4NiPd after 1,500 HPT turns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' (e) Crystal structure of homogeneous Mg4NiPd simulated using first-principles calculations [129].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Nanoglasses Owing to the thermodynamic stability of MgH2, its hydrogen storage temperature is higher than 573 K [9,184].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Although, as discussed above, nanostructuring is an efficient strategy to (a) TARGET (b) NE0 460K iNo Hydrogenation Mg:NiPd TDeh: 630 K Mg Mg2Ni Mg2Pd MgNi2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='1 ■Mg5Pd2 0 N:Numberof Turns Hydrogen Binding Energy (eV) (c) Mg4NiPd N=1500 40 um Intensity 口 OrderedBCC N=500 N=1500 100 40μm 40um 20 30 40 50 60 70 80 90 Bragg Angle, 20 (deg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=') (d) H Atom per Mg+Ni+Pd Atom (e) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='4 100 Mg4NiPd N=1500 (MPa) 10 Pressure TH2 = 305 K 0 1st Cycle H2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='1 5th Cycle 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='01 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='0 Mg Ni Pd Hvdrogen Content (wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='%24 improve the hydrogen storage kinetics of MgH2, the thermodynamic stability of nanostructured or nanoconfined MgH2 can hardly be reduced [200-202].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Hydrogen storage properties of Mg-based materials are expected to be varied significantly in their amorphous form compared with their crystalline counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' As distinct from the defined chemical compositions in crystalline alloys, amorphous alloys usually have a wider range of chemical compositions which is related to the glass-forming ability of the amorphous alloys [203].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Therefore, the tunability of hydrogen storage properties in amorphous alloys is supposed to be much wider than that in crystalline alloys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' In the past two decades, various Mg-based amorphous alloys have been prepared by melt spinning or ball milling and studied as hydrogen storage materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' These includes Mg-Ni [204], Mg-Ni-Y [205], Mg-Ni-RE [206], Mg-Ni-Fe [207], Mg-Ni-La [208], Mg-Mm-Ni [209,210], Mg- Y-Ni [211], Mg65Ni20Cu5Y10 [70], Mg90Ni8RE2 (RE = Y, Nd, Gd) [212], Mg85Ni15−xMx (M = Y or La, x = 0 or 5) [213], Mg100−xNix alloys (x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='5, 1, 2, 5) [214], LaMg11Ni [215], Mg-Ce-Ni [216], RMg2Ni (R = La, Ce, Pr, Nd) [217], Mg11Y2Ni2 [218] and Mg80Ce10Ni10 [219].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' However, most studies showed that the amorphous alloys would decompose into crystalline alloys or hydrides upon hydrogenation/dehydrogenation [204-219], and thus the hydrogen storage reversibility of amorphous alloys is usually not satisfactory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' It was reported that the Mg-Ce-Ni amorphous alloys could reversibly absorb and desorb 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='3 wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='% of hydrogen at room temperature [220];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' however, the reversible hydrogen storage of this alloy could not be improved by increasing the temperature up to 423 K [221].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Therefore, improving the reversible hydrogen storage capacity of Mg-based amorphous alloys at low temperatures remains a challenging issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' SPD was shown to be beneficial for improving the hydrogen storage properties of Mg- based amorphous materials, particularly for enhancing hydrogen storage kinetics and reducing hydrogen storage temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' For example, high-pressure calorimetry measurements revealed that hydrogen uptake in the fully amorphous Mg65Ni20Cu5Y10 alloy, prepared by melt spinning followed by HPT processing, occurs at a significantly lower temperature compared to the fully crystallized state [70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' In another trial, the HPT technique was used to process the melt-spun Mg65Ce10Ni20Cu5 amorphous alloy to produce a nanoglass alloy [69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' 10a shows that the X- ray diffraction pattern of the melt-spun Mg65Ce10Ni20Cu5 alloy is typically amorphous, while the HPT-treated samples contain a small fraction of crystalline phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' 10b demonstrates that superior hydrogen storage could be achieved in these HPT-treated alloys at a low temperature of 393 K, while the melt-spun sample was not active at this temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' It was suggested that the HPT-induced nanoglass formation (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' amorphous phase with a large fraction of interfaces at the nanometer scale) leads to a large number of hydrogen diffusion channels and hydrogen storage sites for the improvement of hydrogenation properties (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' 10c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Such hydrogen storage performance in the HPT-processed sample is comparable with the performance of thin films of Mg65Ce10Ni20Cu5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' As shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' 10d and 10e, the amorphous thin films with thicknesses of 50-300 nm could reversibly absorb and desorb hydrogen with almost unchanged desorption kinetics at a low temperature of 393 K [222].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Therefore, SPD-processed nanoglasses and nanosized amorphous thin films are promising candidates for hydrogen storage at moderate temperatures, but nanoglasses can be produced by SPD in the bulk form with a larger size which is beneficial for practical applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' 25 Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Hydrogen storage in Mg65Ce10Ni20Cu5 nanoglass and amorphous thin films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' (a) X-ray diffraction patterns of the melt-spun alloy before and after HPT processing for 1, 5 and 10 turns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Inset: an image of a disc-shaped specimen after the HPT process [69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' (b) Hydrogenation kinetics of melt-spun and HPT-treated samples [69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' (c) High-resolution image of sample processed by one HPT turn taken by transmission electron microscopy [69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' (d) Cross-sectional images of films with thicknesses of 300, 250 and 50 nm taken by scanning electron microscopy [222].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' (e) Hydrogen desorption kinetics of amorphous thin films [222].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' High-Entropy Alloys In a quest for new materials for hydrogen storage, multicomponent and high-entropy alloys and their corresponding hydrides have attracted wide attention due to the huge compositional field that can be accessed and the unexpected potentialities of hydrogen storage properties they promise [223,224].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' The vast compositional field offers virtually endless opportunities for designing and/or discovering new alloys with superior hydrogen storage properties [225].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' The concept of high- entropy materials was first proposed in 2004 by independent studies by Yeh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' [226] and Cantor et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' [227].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' They introduced a new family of metallic alloys that was based on a mixture of at least five principal elements, each having an atomic percentage in the range of 5 at% to 35 at%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' The presence of multiprincipal elements leads to the stabilization of single-phase solid solutions with main crystal structures of body-centered cubic (BCC), face-centered cubic (FCC) or hexagonal close-packed (HCP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' High-entropy hydrides also consist of at least five principal elements and hydrogen, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' 11a [228].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Several high-entropy alloys for hydrogen storage have been proposed mainly including refractory elements such as TiVZrNb [229], TiZrNbTa [230], TiVZrNbHf [231,232], TiZrNbHfTa (a) (c) Mg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='Cu 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='6 Melt-spun a-CeMg HPT-IN Intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=') (%1m) HPT-5N 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='2 HPT-10N HPT-10 len 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='8 Hydroge HPT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='4 Melt-spun 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='0 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='nm 20 30 40 50 60 3 4 5 6 2 (degree) Time (h) (d) 300nm 2*250nm 10*50nm 1μm 1μm 1μum (e 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='0 Hydrogen (wt%) 1st (%m) (wt%) 1st 2nd 1st 2nd 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='5 3rd 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='5 2nd 3rd 4th 3rd en len 4th 5th 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='0 4th 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='0 Hydroge 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='5 5th 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='5 120°C 120°C 120°℃ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='9 4 12 8 12 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='0 8 8 12 Time (h) Time (h) Time (h)26 [233,234], TiZrNbFeNi [235] and TiZrNbCrFe [236].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Despite the remarkable results observed in hydrogen storage properties so far, the gravimetric storage capacities of these alloys are relatively low due to the high atomic weight of refractory elements (zirconium, hafnium, niobium, tantalum).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Accordingly, one of the strategies to solve this limitation is to proceed with the substitution of refractory elements with lightweight elements with a high affinity to hydrogen such as magnesium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' To that end, Mg-containing high-entropy alloys have been explored for hydrogen storage such as MgZrTiFe0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='5Co0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='5Ni0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='5 [237], MgTiNbCr0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='5Mn0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='5Ni0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='5 [238], MgVAlCrNi [239], Mg12Al11Ti33Mn11Nb33 [240], Mg35Al15Ti25V10Zn15 [241] and MgVTiCrFe [242].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' These Mg- containing high-entropy alloys show the presence of single solid-solution phases and are mainly prepared by high-energy ball milling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' The SPD techniques such as HPT have also been successfully used to synthesize new Mg-containing multicomponent alloys [243] and high-entropy alloys [68], although the hydrogen storage performance of these alloys is not still comparable with the Ti-Zr-containing high-entropy alloys with the Laves phase structure [228,243].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Among various SPD methods, HPT is currently the most popular one to synthesize multicomponent and high-entropy alloys [244].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' MgVCr is a Mg-based multicomponent hydrogen storage alloy with the BCC structure that was synthesized by HPT [242].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' 11b, the application of HPT to a powder mixture of magnesium, vanadium and chromium, could lead to the formation of a new alloy with a single BCC phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' This multicomponent alloy stored up to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='9 wt% of hydrogen at ambient temperature, but the reversibility of the storage was rather poor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' The MgTiVCrFe alloy is a Mg-containing high-entropy alloy that was synthesized by ball milling followed by HPT processing [68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' The alloy showed a BCC solid solution phase together with a high level of amorphization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' The hydrogen uptake capacity of alloy MgTiVCrFe at 623 K was found to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='3 wt%, as seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' 11c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' The outcomes of the investigations conducted so far indicate that SPD processing is a possible pathway to synthesize Mg-containing high-entropy alloy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' However, this research needs to be empowered by developing theoretical concepts for material design, as attempted earlier for Ti-containing high-entropy hydrogen storage materials [228,243].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' 27 Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' (a) Typical structure of high-entropy hydrides [243].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' (b) X-ray diffraction profiles of a mixture of magnesium, vanadium and chromium powders before and after HPT processing for N = 10, 100 and 1,200 revolutions [242].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' (c) Hydrogen pressure-composition isotherms at 623 K for MgTiVCrFe synthesized by ball milling followed by HPT processing [68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' (a) Zr OCr OMn OFe ONi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='H (b) MgvCr HPT:P=3GPa,0=1rpm,T=300K vMgH2 oCr BCC-Ma-V-Cr 口 口 口 口 VW N=1200 N=100 N=10 N=0 2030405060708090100110120 Bragg Angle, 20(deg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=') C Pressure (MPa) 1stAbsorption 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='1 O-1stDesorption 2ndAbsorption O-2ndDesorption 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='0 Hydrogen Content (wt%)28 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Metastable Phases Following a publication in 1988 on the application of HPT to an aluminum alloy, the main goal of SPD processing has become the production of ultrafine-grained materials [245].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' However, a survey in the classic publications indicates that the first SPD methods were invented to control polymorphic phase transformations under high pressure and large shear strain [246].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Nowadays, the SPD methods and particularly HPT are still used to control polymorphic phase transformations [247-249] and solid-state reactions [250,251].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' The main reason for a significant interest in using HPT to control phase transformation is the high applicable pressure in this method which can lead to the synthesis of high-pressure phases [247,249].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Moreover, the shear strain in this method can be increased almost without any limitations, a fact that was used to employ ultra-SPD for synthesizing new materials by mechanical alloying through HPT [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Moreover, a combination of high pressure and high strain in this method permits for synthesizing hard and brittle materials such as TiFe hydrogen storage material [252].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' This capability of HPT was successfully used to synthesize various Mg-X alloys and Mg2X intermetallics (X: 21 different elements) [253], while the synthesis of Mg-based compounds by melting techniques is usually hard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' It is remarkable that HPT was recently used to synthesize new metastable hydrogen storage materials even from immiscible systems such as Mg-Ti [63,98], Mg-Zr [64], Mg-Hf [254], Mg-V-Cr [242] and Mg- Ni-Pd [129].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' As discussed earlier, some of these metastable alloys such as Mg4NiPd exhibit thermodynamics suitable for reversible room-temperature hydrogen storage, while some others such as MgHf show poor reactivity with hydrogen [254].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Here, the impact of HPT on the production of the high-pressure γ-MgH2 phase with low thermodynamic stability is discussed [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' 12a, MgH2 has an α phase with the tetragonal structure at ambient pressure and shows transitions to a γ phase with the orthorhombic structure at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='39-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='5 GPa and to a β phase with the cubic structure at 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='9-9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='7 GPa [255].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Some studies suggested that the partial formation of γ phase after mechanical milling [256], electrochemical process [257] and plasma sputtering [258] can result in the destabilization of hydride for easy hydrogen desorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' To examine this issue, HPT was recently used to synthesize almost 100% of γ-MgH2 and examine its dehydrogenation behavior [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' As shown in the X-ray diffraction profiles of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' 12b, a transition from the α phase to the γ phase occurs by HPT processing under 5 GPa, while the fraction of the high-pressure phase increases with increasing the number of HPT turns (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' increasing strain).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Examination of lattice images by high-resolution transmission electron microscopy, shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' 12c and 12d, also indicates the formation of large amounts of γ phase (almost 100%) after HPT processing for N = 15 turns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Differential scanning calorimetry and thermogravimetry analyses, shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' 12e and 12f, clearly confirm a reduction in the dehydrogenation temperature by increasing the fraction of the γ phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' These results confirm that the production of metastable γ- MgH2 is a strategy to reduce thermodynamic stability, as suggested by first-principles calculations in [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' However, the reversibility of this phase is another issue that should be addressed by adjusting the chemical composition, as attempted for some other metastable phases such as Mg4NiPd [129].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' In conclusion, HPT appears as an effective SPD method to synthesize metastable alloys and hydrides for hydrogen storage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' However, the method has two drawbacks for practical applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' First, the sample in HPT is in the form of a disc with small sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Second, strain is not uniform along the disc radius resulting in the nonuniform distribution of metastable phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' There are now some signs of progress in upscaling the sample size in HPT [163], which can open a path for future applications of the method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Moreover, it was suggested that using ring samples rather than disc samples can diminish the heterogeneity in the final product [259].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' 29 Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' (a) Pressure-temperature phase diagram of magnesium hydride.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' (b) X-ray diffraction profiles of MgH2 processed by HPT for various turns (N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' High-resolution lattice images of MgH2 after HPT processing for (c) 1 and (d) 15 turns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' (e) Heat flow in differential scanning calorimetry and (f) mass loss in thermogravimetry for MgH2 processed by HPT for various turns [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Concluding Remarks and Outlook Severe plastic deformation methods are receiving significant attention in the context of the processing of hydrogen storage materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' The initial motivation to apply these processes to magnesium and its alloys was the improvement of the kinetics of hydrogen storage, but currently, (a) (b) 10 Normalized Intensity MgH2 βCubic HPT: T = 300 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' P = 5 GPa, = 1 rpm (GPa) 8 6 yOrthorhombic Pressure N=15 N=4 4 N=1 N=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='5 2 α Tetragonal O=N Powder 0!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' 0 200 400 600 800 1000 22 26 30 34 38 Temperature (K) Bragg Angle, 20(deg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=') (c) N= 1 [210] [111] (d) N= 15 [111][111] [020] 101 α[121] [101] 2 nm 2 nm (e) (f) MgH2 MgH2 Heat Flow (W/g) HPT:T=300K,P=5GPa,@=1rpm (%M) HPT: T=300 K, P=5GPa, の=1 rpm N=15 N= 15 N= 4 N=4 N= 1 N= 1 N=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='5 Mass N= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='5 N=0 N=0 15W/g Powder 5wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='% Powder 550 600 650 700 750 550 600 650 700 750 Temperature (K) Temperature (K)30 there are numerous reports on the application of these methods to produce novel materials with appropriate thermodynamics and low-temperature hydrogen storage capability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Currently, a wide range of processes including equal-channel angular pressing, high-pressure torsion, intensive rolling and fast forging are employed for processing Mg-based hydrogen storage materials, as discussed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' While high-pressure torsion is the most powerful technique for fundamental studies, the three other methods have a greater potential for commercial applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' The application of severe plastic deformation introduces ultrafine grains and high densities of crystal lattice defects in bulk samples, and this results in better activation, high air resistance, fast hydrogen absorption/desorption kinetics, and appropriate cycling stability, as discussed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' It was shown that some materials which are apparently inert to hydrogen become active after processing and can store hydrogen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' The large shear strain and high pressure involved in severe plastic deformation enable the synthesis of new materials such as nanoglasses, high-entropy alloys and metastable phases with thermodynamics amenable for hydrogen storage, as discussed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' A combination of theoretical binding-energy engineering and severe plastic deformation can successfully lead to achieving reversible hydrogen storage in Mg-based alloys even at room temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' The findings reviewed in this article introduce severe plastic deformation as a strong tool to produce advanced hydrogen storage materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' However, there are several issues that need to be addressed in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' (i) Scaling up the processes is still a key issue for commercialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' (ii) Heat transfer is of paramount significance in hydrogen storage materials, but so far this aspect has not been sufficiently examined for severely deformed magnesium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' (iii) Cycling stability of severely deformed Mg-based hydrogen storage materials requires further investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' (iv) Addition of catalysts in conjunction with SPD processing may be a potential pathway to enhancing the hydrogenation/dehydrogenation kinetics at low temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' (v) More importantly, further studies on the combination of thermodynamic calculations and experiments are needed to discover new materials that can satisfy the desired requirements for stationary and mobile hydrogen storage applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' By considering the worldwide trend to realize carbon-neutral energy and utilize hydrogen as a zero CO2 emission fuel, it is expected that in the future, severe plastic deformation technologies will contribute to the hydrogen economy more prominently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Acknowledgments K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' was supported in part by the Light Metals Educational Foundation of Japan, and in part by the MEXT, Japan through Grants-in-Aid for Scientific Research on Innovative Areas (JP19H05176 & JP21H00150) and Challenging Research Exploratory (JP22K18737).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' is grateful to the Brazilian agencies FAPESP (grant number 2013/05987-8) and CNPq (grant number 421181-2018-4 and 307397-2019-0) for the financial support and to the Laboratory of Structural Characterization (LCE-DEMa-UFSCar) for general electron microscopy facilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' thanks for the financial support from FAPESP (grant number 2022/01351-0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' thanks the support from the French State through the ANR-21-CE08-0034-01 project as well as the program “Investment in the future” operated by the National Research Agency (ANR) and referenced under No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' ANR-11-LABX-0008-01 (Labex DAMAS).' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Villela, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Miraglia, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' dos Santos, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='J.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Hauback, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Botta, Nanostructured MgH2 obtained by cold rolling combined with short-time high-energy ball milling, Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' 16 (2013) 158-163.' metadata={'source': 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B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Hauback, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Botta, Cold rolling of MgH2 powders containing different additives, Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Hydrog.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Hauback, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Botta, MgH2-based nanocomposites prepared by short-time high energy ball milling followed by cold rolling: a new processing route, Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Hydrog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Energy 39 (2014) 4404-4413.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' [85] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Floriano, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Leiva, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Carvalho, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Ishikawa, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Botta, Nanocrystalline Mg produced by cold rolling under inert atmosphere: a powerful tool for Mg activation, Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Hydrog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQfTwwC/content/2301.05009v1.pdf'} +page_content=' Energy 39 (2014) 4959-4965.' metadata={'source': 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generalizable way. This reflects in +empirical literature that either focuses on outputs instead of transformative processes, or preemptively +assumes the nature of technology. +Building on recent advances in information theory, we develop a +method to quantify technological sophistication. Our approach explicitly addresses the transformative +process where inputs interact to generate outputs; providing a more direct inference about the nature +of technology. +More specifically, we measure the degree to which an industry’s inputs interact in a +synergistic fashion. Synergies create novel outputs, so we conjecture that synergistic technologies are +more sophisticated. We test this conjecture by estimating synergy scores for industries across nearly 150 +countries using input-output datasets. We find that these scores predict popular export-based measures of +economic complexity that are commonly used as proxies for economic sophistication. The method yields +synergistic interaction networks that provide further insights on the structure of industrial processes. For +example, they reveal that industries from the tertiary sector tend to be disassortative sector-wise. To the +extent of our knowledge, our findings are the first data-driven inference of technological sophistication +within production processes (on an industrial scale). Thus, they provide the technological foundations +of economic complexity and represent an important step toward its empirical microfoundations. +1 +arXiv:2301.04579v1 [econ.GN] 11 Jan 2023 + +1 +Introduction +The transformative process of turning a set of inputs into an output is prevalent in various socioeconomic +and physical systems. In socioeconomic systems, it often takes the shape of a production process, and dif- +ferent disciplines analyze it through distinct theoretical frameworks and quantitative tools. For instance, +management sciences specialize in supply chains and global value chains, human and economic geography +utilize systems analysis, economists study input-output models and fit production functions, systems en- +gineering construct design structure matrices, and innovation scholars focus on combinatorial models and +network analysis. Independently of the domain of application, it is commonly agreed that technological +sophistication is a building block in the study of economic complexity. Thus, quantifying the degree and +structure of technological sophistication is critical to understand the evolution, performance, fragility, and +resilience of production systems. +By production process, we refer to the procedure through which a certain technology transforms a set of +inputs into an output. Due to the focus of this paper, we use the terms technological, industrial, productive, +and economic sophistication interchangeably, to refer to the capacity of a production process to generate +novel outputs from a set of inputs. Thus, an industry is considered more sophisticated if it is produces +more novelty than others when using the same inputs. In this paper, we develop a method to quantify +the technological sophistication of production processes. We test our method using two major input-output +datasets, and validate it through independent export data and well-established economic complexity in- +dices that are commonly used as proxies for technological sophistication at a national level. We find that +our synergy score predicts these indices, and that the inferred synergistic interaction networks have non- +trivial topologies that characterize production technologies. To the extent of our knowledge, this is the first +framework that quantifies technological sophistication by empirically inferring the nature of input-input and +input-output relationships (as opposed to assuming them). Our approach is non-parametric, so it does not +require the ex ante assumption of specific production functions or design structure matrices. Instead, by +exploiting the mutual information between input data in an output signal, it allows to infer the structure +of synergistic interactions. This is a major innovation as it facilitates both the overall quantification of +technological sophistication across firms, industries, sectors, and countries, as well as the estimation of the +interaction networks living at the heart of production processes; currently considered a black box by eco- +nomic complexity scholars. Furthermore, it provides an empirical basis to justify the specification of certain +production functions in input-output models. The method is of general purpose as it can adapted to other +contexts where transformative processes may be important (both in socioeconomic and physical systems). +In the rest of the paper we provide an overview of existing approaches, describe the methods and datasets +2 + +used, and present our results. +2 +Knowledge gap +In the theoretical literature of technological innovation, there is a consensus agreeing on the principle that +technological change can be described as a recombinant process through which novelty emerges from new +ways of coupling existing devices (or technologies) to fulfill a certain purpose[1–12]. +This consensus is +empirically supported by evidence on how new inventions recombine existing ones [13–18] (see [19] for a +comprehensive review). Arguably, technological sophistication is intimately related to this transformative +process, so producing empirical metrics to characterize it would provide generalizable foundations for its +quantification. Unfortunately, much of the empirical evidence in innovation studies remains domain-specific +(e.g., patents [20] and cities [21]), making it difficult to build generalizable empirical frameworks. In our +view, the lack of such frameworks obeys two limiting factors: (1) technology can be highly diverse across +industries and countries, and (2) generating reliable estimates for production models with numerous inputs +requires big data, something difficult to obtain from input-output (IO) tables (with the exception of recent +developments of inter-firm transaction data [22–24]). +IO scholars and analysts circumvent the lack of big data by modeling production networks that assume, +ex ante, the nature of the input-input and input-output interactions through production functions [24– +27]. This approach has the benefit of shifting the problem of modeling technological sophistication from +one of inferring structures to one of fitting parameters. This shift comes with problems that have been +recently discusses along the lines of misspecification [28], estimation biases [29], and aggregation artifacts +[30, 31]. Thus, assuming (parametric) production functions instead of inferring interaction structures limits +our capacity to quantify technological sophistication. +The aforementioned limitations have become such an important issue that supply-chain surveys have +been conducted in an attempt to determine the degree of dependence on certain inputs by certain industries +(see the IHS Markit survey in [32] and [33]). Moreover, such information, has been incorporated in the +new generation of IO models [24, 32]. Hence, a data-driven approach that would facilitate the inference of +interaction structures in production processes would help alleviating some of these problems. +Systems engineering, has created an alternative approach to production functions by developing the con- +cept of Design Structure Matrices (DSMs) [16, 34, 35]. DSMs describe networks of interactions between +the different components of a production processes. They provide a comprehensive tool to represent tech- +nological sophistication, but they require substantial knowledge about the process itself. Thus, DSMs build +on a top-down approach that demands substantial ex ante knowledge; making it difficult to scale up to +3 + +more aggregate levels such as industries or sectors. These aggregation levels are crucial for the design and +implementation of country- or industry-wide innovation strategies and policy intervention. A data-driven +framework to infer DSMs, or some of their components, would complement this literature in important ways. +In recent years, the literature of economic complexity has emerged with new proposals on how to quantify +technological sophistication [36–46] (see [47] for a comprehensive review). Building on existing literature on +export diversification [48–52], two dominant approaches have become the gold standard in this field: the +Economic Fitness Index (EFI) [40] and the Economic Complexity Index (ECI) [40]. While these frameworks +are built on different theoretical foundations (see [53] for a rigorous comparison), both of them try to map +export profiles into indices that capture different elements of economic sophistication.1 In spite of this grow- +ing literature, the technological foundations of production sophistication remain mostly theoretical. That is, +while these frameworks provide a proxy for technological sophistication by looking at exports, they do not +address production processes and hence, treat them like a black box where interactions remain obscure. This +very issue has been recently raised in a keynote speech by one of the creators of the ECI, Ricardo Hausmann. +Using a biological metaphor, Hausmann highlights the lack of a production account in the quantification of +technological sophistication and calls for scholars to move beyond the phenotypic view of economic sophis- +tication (the output/export side) to try to understand the genotypic one (the technological/transformative +side) [54]. +Evidently, the quantification of technological sophistication is a problem that pertains to multiple disci- +plines and has crucial implications in the design and implementation of economic and innovation strategies. +The lack of generalizable quantitative methods that address the production process explicitly poses a major +barrier to advance research in these fields. At the same time, it creates a void that maintains these various +disciplines, to a great extent, disconnected. Our contribution seeks to fill this gap and to provide a general +framework that facilitates the quantification of technological sophistication. Such contribution can help to +develop deeper insights on the microfoundations of economic complexity. +3 +Framework and data +Let us provide a succinct description of the methodology and the data employed in our analysis, and leave +more specific details for section 6. Broadly speaking, our method seeks to estimate the mutual information +between pairs of inputs in a given industry (from IO tables), and decompose their contribution to the +output into different modes of information sharing. We focus on a particular mode known as synergistic +information, i.e., the information cannot be obtained from any of the inputs alone but only exists as a virtue +1From these two indices, only the EFI has been applied at the industry level. +4 + +of the interaction between the inputs. Using this information, we measure the degree of synergy between +the inputs and estimate a synergy score. +A higher synergy score means that input interactions produce more novel information during the pro- +duction process. This interpretation aligns with the established notion of a recombinant process generating +novel outputs. Thus, we take an interpretative leap and claim that the synergy score quantifies technological +sophistication. In other words, an industry that exhibits higher synergy scores should have a more sophisti- +cated technology underlying its production process. This is our main conjecture, and we provide evidence to +support it in the later sections of this paper. Furthermore, we explore the estimated synergistic interaction +networks to learn new and more nuanced insights about the nature of technology. +3.1 +The synergy score +Consider an industry and three associated datasets. Data Y contain the changes in the output of the industry, +while X1 and X2 capture the changes in inputs 1 and 2 respectively. We are interested in quantifying how +much information do X1 and X2 provide about Y . It can be measured using the total mutual information +I(X1, X2; Y ) between the output and the inputs. In essence, the total mutual information is a measure of +the amount of uncertainty in Y , that is reduced by knowing X1 and X2. Furthermore, it is possible quantify +how much of this uncertainty reduction is a result of the interactions between the inputs. We can achieve this +by following the partial information decomposition (PID) proposed by [55], where the mutual information I +that X1 and X2 provide about Y can be described as +I(X1, X2; Y ) = Syn(X1, X2; Y ) + Oth(X1, X2; Y ). +(1) +In Equation 1, the mutual information is decomposed into the synergistic type and other ones (we explain +the others in section 6). Synergistic information can only be obtained from the interaction between X1 and +X2. It is an analogue for input complementarity in the IO literature. If either input is removed from the +production process, all the synergistic information would be lost from the output signal. +Arguably, more sophisticated production processes involve more synergistic information because they +generate more novel outputs by recombining the same inputs in innovative ways. Thus, we use this type of +information as a synergy score and compute it for all unique pairs of inputs in the data.2 Further details on +the estimation method can be found in section subsection 6.1. +2Equation 1 can be generalized for more than two inputs. However, producing reliable estimates for more than three inputs +can become too demanding data-wise. Nevertheless, we show in the SI that our main results hold when we estimate synergies +between triplets instead of pairs. +5 + +3.2 +Data +To empirically capture various degrees of technological sophistication, it is important to assemble a dataset +with substantial variation in terms of industrial development. Thus, having a large number of countries is +key, as most technological variation would be expected between nations with different levels of development. +We find such coverage in the Eora26 dataset [56, 57], a global set of input-output tables with harmonized +industries across a large number of countries. The subset extracted from Eora26 contains annual input- +output tables for 26 industries across 148 countries during the period 1995-2020. We use the time series of +each input and the corresponding total output to compute the synergy scores associated with a particular +country and industry. Thus, we exploit the temporal variation in the data to build the scores, and then +focus on comparing countries and industries. The original time series are transformed into log-fluctuations +(i.e., growth rates).3 +To further validate our results, we test our method using the OECD IO tables, covering 66 countries +across 45 unique industries. While these data are smaller in country coverage, they provide more industries +and are considered of a higher quality (because the data providers are subjected to harmonized reporting +standards). While an obvious implication of using OECD data is the loss of technological diversity (due to +the absence of lower-income countries), we show that our results are robust. +In search of evidence to validate our main conjecture, we test if our synergy scores contribute in a +significant way to the prediction of economic complexity indices. We focus on predicting the EFI, as it +has been shown to better capture economic complexity (the ECI correlates more with GDP) [20, 53] and +because, to the extent of our knowledge, it is the only index that has been used to study industry-level +economic sophistication. Nevertheless, in Figure 3 and in the SI, we provide evidence of robustness when +using the ECI and a third index, GENEPY [53], that generalizes both the EFI and the ECI. Since all the +economic complexity indices are export-based metrics, we construct them by using an independent dataset +on export flows [58] with the same country and temporal coverage as the Eora26 dataset. Due to the need +for collapsing time in the calculation of the synergy score, we compute inter-temporal average complexity +indices. Further details on these calculations are provided in the section 6. +3.3 +Descriptive features +Before proceeding to our results, let us show some interesting features of the synergy scores. In Figure 1, +we show an example of synergy networks estimated by clustering IO tables into four groups (see section 6 +for an explanation on the clustering procedure). The industry represented in each network ‘Textiles and +3This transformation yields normally-distributed growth rates at the industrial level, which makes the data compatible with +Gaussian estimators of mutual information. +6 + +Wearing Apparel’. Each panel represents the synergistic interaction structure between its inputs (‘Textiles +and Wearing Apparel’ appear as a node as well as it its output is also used as input). Hence, each node +represents an input and each weighted edge captures the synergy score between a pair of inputs. Overall, +these networks capture the average technological structure of a group of countries. +The main take-away of this example is that an industry that uses the same set of inputs in every country +group may exhibit very different interaction structures between these inputs. For instance, clusters 1 and +2, which have middle and high income countries, exhibit technologies with more synergistic interactions. +Likewise, the most central nodes, are not necessarily the same across country groups. +This highlights +the importance of quantifying and understanding the internal structure of production processes as part of +measuring technological sophistication. +Figure 1: Synergistic interaction networks of the ‘Textiles and Wearing Apparel’ industry +Cluster 1 +Median GNI per capita $35,220 +Cluster 2 +Median GNI per capita $7,215 +Cluster 3 +Median GNI per capita $1,855 +Cluster 4 +Median GNI per capita $895 +Industries +Agriculture +Mining and Quarrying +Fishing +Food & Beverages +Textiles and Wearing Apparel +Wood and Paper +Petroleum, Chemical and Non-Metallic Mineral Products +Metal Products +Electrical and Machinery +Transport Equipment +Other Manufacturing +Recycling +Electricity, Gas and Water +Construction +Maintenance and Repair +Wholesale Trade +Retail Trade +Hotels and Restraurants +Transport +Post and Telecommunications +Finacial Intermediation and Business Activities +Public Administration +Education, Health and Other Services +Private Households +Others +Re-export & Re-import +Sectors +Primary +Secondary +Tertiary +Other +Notes: The nodes represent industries. They are colored according to their sectors. The width of the edges is proportional to +the synergy score. +Sources: Authors’ own calculations using Eora26 data. +7 + +An insightful feature that can be drawn from a network representation of technology is the diversity +of synergies, quantified through network-variance [59]. When analyzing the relationship between network- +variance and the overall level of synergy in the network (through the first eigenvalue of the adjacency +matrix), we find that industry sophistication does not grow without increments in the diversity of synergistic +interactions. Figure 2a shows this finding using all the networks inferred from Eora26. It suggests that, +to quantify technological sophistication, it is not enough to measure the overall synergy level, but also +to understand how synergies are distributed within the production process, i.e., to infer the interaction +structure. +Figure 2: Synergy score features +(a) Synergy level and diversity of synergistic interactions +0 +10 +20 +30 +Synergy Variance +1e +5 +0 +2 +4 +6 +8 +10 +Largest Eigenvalue +(b) Distribution of synergy scores +0.0 +0.1 +0.2 +0.3 +Synergy +0 +5 +10 +15 +20 +25 +30 +Density +Notes: Figure 2a shows quantities for synergistic interaction networks (see 6 for details on how we obtain 100+ networks). +Figure 2b presents the distribution of pairwise synergy scores across all the edges of all the synergistic networks. +Sources: Authors’ own calculations with Eora26 data. +One last interesting feature of synergistic interactions is their scarcity, as suggested by Figure 2b. Put +it differently, highly synergistic interactions are much less abundant than low-synergy ones. This is con- +sistent with the literature on innovation dynamics [60, 61], where it is argued that innovative recombinant +processes happen sporadically. Together, these features link our measure of production sophistication to +well-established ideas from the innovation and technology literature. +8 + +4 +Results +4.1 +Predicting economic complexity +First, we investigate the relationship between the synergy scores and the complexity indices. Figure 3 shows +a clear positive correlation between the synergy score and the three indices. It suggests that synergistic +interactions may predict these proxies of economic sophistication, providing support to our conjecture on +being able to quantify technological sophistication. At this point, we would like to highlight the importance +of these results in view of the stark differences between our method and the EFI, GENEPY, and ECI. While +we infer technological sophistication by directly analyzing input-input and input-output interactions through +the mutual information in their growth rates, complexity indices take an approach that focuses on the co- +occurrence of outputs (so no input-output interactions are taken into account). We find it remarkable that +our method is able to capture the cross-country and cross-industry variation produced by the complexity +indices, especially when these two distinct approaches use independent datasets. +Figure 3: Synergy scores and their association to industry sophistication +(a) Economic fitness index +4 +3 +2 +1 +log (Synergy) +10 +8 +6 +4 +2 +0 +Industry log (Fitness) +Binned average +Linear fit on raw data +(b) GENEPY index +4 +3 +2 +1 +log (Synergy) +9 +8 +7 +6 +5 +4 +Industry log (GENEPY) +Binned average +Linear fit on raw data +(c) Economic complexity index +4 +3 +2 +1 +log (Synergy) +0.2 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Industry Complexity +Binned average +Linear fit on raw data +Notes: With the purpose of a clear visualization, we display data binned according to the synergy scores. Each dot corresponds +to the average value withing the corresponding bin. Since the synergy distribution is right-skewed, we analyze the logarithm of +the scores. The linear fit is estimated using the entire data (not the binned one). +Sources: Authors’ own calculations with Eora26 data. +Next, let us formally test whether the synergy scores contribute, in a significant way, to the prediction of +the EFI. We estimate linear regression models controlling for different factors that may contribute to tech- +nological sophistication.The intuition behind this exercise is that, if the synergy score displays a significant +coefficient in the prediction of output-based complexity indices (calculated through independent methods +and data), then our method is valid and quantifies the degree of technological sophistication across industries. +In Table 1, we present 7 linear regression models with 8 possible independent variables that contribute +to the prediction of the EFI.4 Our variable of interest is the logarithm of the synergy score (we take log +4Figure 3a suggests that a linear specification is the most adequate when the EFI is the dependent variable. +9 + +due to the skewed distribution shown in Figure 2). In the first model, we estimate the association shown +in Figure 3a; without any controls. +Model 2 includes log GDP per capita, and aims at controlling for +country-specific factors such as higher income, better public governance, and better infrastructure.Model 3 +adds dummy variables indicating if the industry belongs to the primary, secondary, or tertiary sector (with +the additional sector being ‘Other’). With this, we try to control for sector-specific factors like regulatory +frameworks (e.g., tax prerogatives and unionization practices). In model 4, we include industry-specific total +output. Finally, models 5 to 7 introduce industry-specific energy-related controls that are available in the +Eora26 data. The ratio of energy consumption to total output is a proxy of the technological efficiency of +the industry, which may relate to its level of sophistication. +In all regression models, the coefficient of log synergy remains positive and statistically significant. Fur- +thermore, its magnitude is relatively stable across the seven models. Interestingly, the coefficient is larger +than the one of GDP per capita.5 +This result validates our conjecture of more synergistic information +implying higher technological sophistication. In the SI, we show that these results are robust when using +GENEPY index and ECI as the dependent variable.6 +To further validate our regression results, we perform the same tests using the IO tables from the OECD. +Table 2 shows that our results hold with these data. In fact, the magnitude of the log synergy coefficients +nearly doubled with respect to the ones reported in Table 1. +4.2 +Synergistic interaction networks +Next, let us look at more nuanced results through the synergistic interaction networks. While the Eora26 +networks are relatively small (with 26 nodes), we have a more than 100 realizations as every industry employs +the same set of inputs at this level of aggregation. Thus, we can study the variation of network topology +and whether it relates to technological sophistication. More specifically, we are interested in the relationship +between the mean synergy score of a node and its topological features, namely, centrality, clustering, and +mixing.7 +5But one should be careful of not reading too much out of the controls’ coefficients, as they serve the only purpose of +accounting for potential confounders [62]. Expecting specific signs and magnitudes in the coefficients of control variables is a +common mistake known in epidemiology as the table 2 fallacy [63] and in econometrics as the interpretation of endogenous +controls [64]. +6Note that the low asjusted R2 does not invalidate our results, as this is quite common in cross-sectional regressions with +a large number of observations. If the data were aggregated, for example, by averaging the synergy scores of each industry +(instead of using the input-level observations), then the explained variance would increase substantially (see the SI for evidence +on this when testing the robustness to aggregated sophistication indices). This is consistent with Figure 3, where the data have +been binned. +7We employ the binary backbone version of the networks to perform this analysis. See section 6 for further details. +10 + +Table 1: Linear models predicting the economic fitness index of industries +Predictor +Model 1 +Model 2 +Model 3 +Model 4 +Model 5 +Model 6 +Model 7 +Log synergy +2.4570∗∗ +2.3528∗∗ +2.6970∗∗∗ +2.1796∗∗∗ +2.5636∗∗∗ +2.6875∗∗∗ +2.1795∗∗∗ +(1.0484) +(1.0053) +(1.0254) +(0.7582) +(0.9219) +(1.0247) +(0.7591) +Log GDP per. cap. +0.6523 +0.6403 +−0.4684 +0.3125 +0.6346 +−0.4682 +(0.5745) +(0.5685) +(0.5618) +(0.5481) +(0.5681) +(0.5642) +Primary sector +−5.8356∗∗ +−7.9426∗∗ +−6.2379∗∗ +−5.9475∗∗ +−7.9447∗∗ +(2.7029) +(3.2428) +(2.9617) +(2.6803) +(3.1856) +Secondary sector +−1.4582 +−5.7442∗∗ +−2.0697 +−1.5831 +−5.7460∗∗ +(1.1045) +(2.3370) +(1.2813) +(1.1066) +(2.2913) +Tertiary sector +−6.3603∗∗∗ +−8.9321∗∗∗ +−7.0664∗∗∗ +−6.4499∗∗∗ +−8.9336∗∗∗ +(2.0858) +(2.6226) +(2.0998) +(2.1006) +(2.6112) +Log output +1.4227∗∗ +1.4223∗∗ +(0.6162) +(0.6283) +Log energy +0.3827 +(0.4706) +Energy intensity +−1.6006 +−0.0405 +(1.4897) +(1.6730) +Adjusted R2 +0.0237 +0.0286 +0.0454 +0.0891 +0.0521 +0.0460 +0.0891 +No. of observations +695,500 +695,500 +695,500 +695,500 +695,175 +695,500 +695,500 +Notes: OLS regression coefficients. The model intercept is omitted. The dependent variable is the EFI, calculated for each +industry in the dataset, and averaged across the years in the sample period. The stars denote the level of significance. The +number in parenthesis is the clustered standard error. Confidence levels are indicated by * for 90%, ** for 95%, and *** for +99%. +Source: Authors’ own calculations using Eora26 data. +Table 2: Linear models predicting the economic fitness index of industries with OECD data +Predictor +Model 1 +Model 2 +Model 3 +Model 4 +Log synergy +4.3811∗∗∗ +4.2163∗∗∗ +4.4180∗∗∗ +4.5699∗∗∗ +(1.2116) +(1.1707) +(1.1405) +(1.1588) +Log GDP per. cap. +0.6275 +0.6192 +0.3155 +(0.5588) +(0.5560) +(0.5490) +Primary sector +1.4040 +0.4751 +(1.8543) +(2.0307) +Secondary sector +−2.5265 +−3.4749 +(2.5887) +(2.5931) +Tertiary sector +1.0486 +0.1175 +(1.8531) +(1.8746) +Log output +0.8703∗∗ +(0.3507) +Adjusted R2 +0.0297 +0.0332 +0.0406 +0.0598 +No. of observations +1,550,494 +1,550,494 +1,550,494 +1,550,494 +Notes: OLS regression coefficients. The model intercept is omitted. The dependent variable is the EFI, calculated for each +industry in the dataset, and averaged across the years in the sample period. The stars denote the level of significance. The +number in parenthesis is the clustered standard error. Confidence levels are indicated by * for 90%, ** for 95%, and *** for +99%. +Source: Authors’ own calculations using OECD data. +11 + +Figure 4a shows a positive association between the synergy score and eigenvector centrality.8 +This +association becomes negligible when tested under a null model (see section 6 for details on the null model). +It suggests that more synergistic inputs tend to interact with a larger set of different inputs in a production +process. Locally, a node with a higher synergy score tends to play a brokerage role between those inputs +with which it interacts. This can be seen in Figure 4b, where a negative relationship between synergy and +clustering coefficient is evident. Once more, this stylized fact dissipates when estimated under the null model. +Figure 4: Properties of synergistic interaction networks +(a) Synergy and centrality +0.0 +0.5 +1.0 +1.5 +2.0 +Total Synergy +0.18 +0.19 +0.20 +0.21 +0.22 +0.23 +0.24 +0.25 +Eigenvector Centrality +Empirical +Null model +(b) Synergy and clustering +0.0 +0.5 +1.0 +1.5 +2.0 +Total Synergy +0.20 +0.25 +0.30 +0.35 +0.40 +0.45 +Clustering coefficient +Empirical +Null model +Source: Authors’ own calculations with Eora26 data. +Finally, since synergy seems to correlate strongly to centrality and local structure, we consider whether +certain types of technology inherently contribute to the structure of synergistic interaction networks. While +fully answering this question is ambitious and beyond the scope of this paper, we take a first step by exploiting +the taxonomy of industries into sectors. It has been long argued that membership to a particular sector +conveys information about structural differences between industries and their underlying technologies, as +they correspond to the main stages of production [65, 66]. Accordingly, we are interested in learning if, +in synergistic interaction networks, sector membership conditions the probability of interacting with inputs +from the same sector and with inputs from different ones. +Using all the networks in the sample, we compute the mixing probability between each possible combina- +8Similar positive relationship is observed when using alternative measures of centrality like the betweenness, Katz, and +degree. +12 + +tion of sectors. In other words, we estimate how likely it is to find two input industries from specific sectors +interacting synergistically in a production process. Then, we perform a T-test of this probability, using the +one of the null model as the reference estimator. A positive T value of mixing probabilities would indicate +that two inputs from the corresponding two sectors tend to interact synergistically more than expected un- +der the null. Figure 5 shows the T values of every pair of sectors. Interestingly, the tertiary sector, when +interacting with other sectors, has the highest mixing probabilities. This suggests that, in terms of economic +complexity, tertiary-sector industries play a mediator role for industries in other sectors, enabling innovation +and more sophisticated production processes. +Figure 5: Sector mixing +Secondary_Primary +Other_Primary +Primary_Primary +Secondary_Secondary +Other_Secondary +Other_Other +Other_Tertiary +Tertiary_Primary +Tertiary_Secondary +Tertiary_Tertiary +Sector combination +6 +4 +2 +0 +2 +4 +6 +8 +Difference in mixing probabilities +T values (Empitical - Null) +*** +*** +*** +*** +* +*** +*** +*** +Notes: The T-tests are performed using the mixing probabilities calculated from the empirical networks and the ones produced +by the null model. Confidence levels are indicated by * for 90%, ** for 95%, and *** for 99%. +Source: Authors’ own calculations with Eora26 data. +5 +Discussion and conclusions +The quantification of technological sophistication in production processes is an elusive problem that is rele- +vant to several disciplines. This paper introduces the first method that explicitly addresses input-input and +13 + +input-output interactions, opening the black box of production processes underpinning economic complexity. +By estimating the amount of synergistic mutual information between the inputs of a production process, +we quantify the degree of technological sophistication across various industries and countries. Moreover, we +infer the structure of production processes by constructing synergistic interaction networks, revealing–never +before seen–features that characterize industrial sophistication. These networks provide empirical grounds +to select and justify production functions in IO models; something missing in this large body of literature +and a major limitation in IO empirical studies. They also reveal the structural role of industries with a high +degree of synergistic interactions. Altogether, our results represent a major step towards understanding the +microfoundations of economic complexity. +Some of the main limitations of this approach are that it requires larger data than what is typically +found in IO tables. This means that, in this study, we need to develop a clustering procedure, and that +our inferences are not for specific countries, but for groups. Another limitation is that we have to sacrifice +the temporal dimension as we need to exploit time variation to produce the estimations. Ideally one would +have high-frequency IO tables to compute the synergy scores by sub-periods. This would allow to study, +for example, the evolution of technological sophistication and of its networked structure across countries +and industries. Fortunately, new firm-transaction datasets are being generated as we write this manuscript, +so the future for using the proposed approach looks very promising. Furthermore, because our framework +works on the premise of a generic production process its applications could extend to other domains such as +the study physical/biological systems. +6 +Methods +Partial information decomposition (PID) requires a large number of observations per variable, typically in the +order of 200 for a process with two inputs and one output (the number of required observations scales with +the number of inputs). Such data do not exist for a single industry in a country’s IO tables. Nevertheless, we +can overcome this limitation by performing a data-clustering procedure based on the technological similarity +of countries in a given industry. Thus, we create a workflow that facilitates the pre-processing and inference +tasks. The entire pipeline should be repeated for each industry that one would like to include in the analysis. +The workflow consists of two major steps, and an illustrative sketch is provided in Figure 6. +1. Clustering: Grouping countries that exhibit similar production patterns in a target industry, prefer- +ably in roughly equally sized groups. This means that two countries that are in the same cluster in a +given industry A, may be in different ones when analyzing another industry B. +14 + +Figure 6: The two-step workflow +1. Clustering +1.1 Data Transformation +Input – output tables are +transformed to extract inputs to a +given target industry across all +countries and years. +Countries +Years +1.2 Feature extraction +Log Marginal Product of inputs. + +Labour Efficiency + Quality of +Infrastructure + Income per capita. +PCA 2 +PCA 1 +1.3 K-constrained clustering +Similar sized four clusters of +countries are identified with +similar production technology. +2. Pairwise synergies +Inputs +Output +Inputs +Output +2.1 Data Transformation +Cluster level log fluctuations are +extracted from the industry level +input – output table, for the inputs +to and corresponding output of +the target industry. +2.2 Partial Information Decomposition +Taking pair of inputs and corresponding +output log fluctuations, we use PID to +estimate total synergy among the +inputs. +2.3 Network of synergies +Iterating through all possible pairs of +input, we can generate a network of +synergy that exists in the production +process of the target industry. +𝑋! +𝑋" +𝐼(𝑋!𝑋";𝑌) +2. Estimation: Using PID to estimate synergy scores for each pair of inputs in the target industry, and +constructing its corresponding synergistic interaction network. +In the rest of this section, we explain the PID method and the clustering procedure in detail. Since the +economic complexity indices are standard metrics in the literature, we explain how we calculate them at the +industry level in the SI. +6.1 +Synergy score +Since the seminal paper of Claude Shannon in 1948 [67], information theory has evolved substantially, +spanning beyond the study of communication. It has become a powerful tool to quantify interdependencies +15 + +Agriculture +Fishing +Mining and Quarrying +0.30 +Food &Beverages +Textiles and Wearing Apparel +Woodand'Paper +Petroleum, Chemical and Non-Metallic Mineral Products +-0.25 +Metal Products +Electrical and Machinery +Transport Equipment +Other Manufacturing +0.20 +Recycling +Electricity, Gas and Water +Construction +Maintenance and Repair +0.15 +Wholesale Trade +Retail Trade +Hotels and Restraurants +Transport +0.10 +Post and Telecommunications +Finacial Intermediation and Business Activities +Public Administration +Education,Health and Other Services +-0.05 +Private Households +Others +Re-export & Re-import +0.00 +0 +5 +10 +15 +20 +25between variables in many complex systems like the ecology [68], financial markets [69], the brain [70] +and thermodynamics [71]. Shannon entropy measures the variability of a system in a given state-space. +Leveraging this concept, Shannon proposed a measure of dependence between two variables, commonly +known as mutual information. The ability to assess the variability of a system–and the interdependency +among its variables–made information theory ideal for the empirical study of complex systems. We adapt +these concepts and tools to quantify the interdependencies between an industry’s inputs through a synergy +score. +6.1.1 +Mutual information +Consider a random variable X, defined on a discrete state space X. The Shannon entropy is written as +H(x) = − +� +x∈X +p(x) log p(x), +(2) +where p(x) is the probability of the realization x of variable X on the state space X. Entropy is highest when +the variable explores all possible states with equal probability. Similarly, the shared uncertainty between +two random variables X and Y can be written as the mutual information +I(X; Y ) = +� +y∈Y +� +x∈X +p(x, y) log +� p(x, y) +p(x)p(y) +� +, +(3) +where p(x, y) represents the joint distribution of the two random variables. The measure I(X; Y ) becomes +0 if X and Y are independent, such that p(x, y) = p(x)p(y). +Next, let us assume that X and Y are Gaussian random variables. Then, the corresponding mutual +information for two Gaussian random variables can be written as (see[72] for a complete derivation), +I(X; Y ) = 1 +2 log +� detΣ(X) +detΣ(X|Y ) +� +, +(4) +where detΣ(X) represents the determinant of the covariance matrix of X, and Σ(X|Y ) represents the con- +ditional covariance, which can be written as +Σ(X|Y ) = Σ(X) − Σ(X, Y )Σ(Y )−1Σ(Y, X). +(5) +The definition in Equation 4 also exists for a multi-variate setting. For three or more variables, higher- +order effects such as synergistic interactions can be observed as well. Interestingly, these synergistic interac- +tions are present in the groups of variables (inputs) as a whole, so they absent when considered individually. +16 + +������������(������������, ������������; ������������) +������������(������������; ������������) +������������(������������; ������������) +Unique to Y +Unique to X +Redundant +Synergistic +Figure 7: Partial information decomposition +In a seminal paper, Williams and Beer [55] formalize the measurement of synergy for the case with two +inputs and one output using PID. Thus, we proceed to briefly explain the PID framework. +6.1.2 +Partial information decomposition +PID decomposes the mutual information between a pair of inputs and the output into Synergistic, Redundant +and Unique information. This decomposition was first introduced in [55], and has been instrumental to study +different types of interactions between random variables [73–75]. +Let us look at the case of mutual information between two input variables (X1, X2) and one output +variable (Y ). +Here, the total mutual information about the output provided by the two inputs can be +represented as the following Venn diagramFigure 7. +Redundant information could come from either X1 or X2, so this information would be preserved if one +of the inputs was removed. It is an analogue input substitutability in the IO literature. Unique information +comes from each input only, so it represents the unique contribution that each input makes to the output. +Both of these types of information are contained in the term Oth(X1, X2; Y ) from Equation 1. Thus, let us +rewrite it in its extended form +17 + +I(X1, X2; Y ) = Syn(X1, X2; Y ) + Red(X1, X2; Y ) + Unq(X1; Y ) + Unq(X2; Y ). +(6) +A general formulation of Equation 6 for n random variables is provided by [55].9 +The unique and the total joint information about the output provided by the inputs can be exactly +calculated. However, to estimate the synergy and redundancy between the inputs, we need to assume a +redundancy function. Here we employ the minimum mutual information redundancy estimator developed +for multivariate Gaussian systems[75]. +This redundancy estimator assumes the total redundancy to be +equivalent to the unique mutual information about the output provided by the weakest input. Formally, this +is +Red(X1, X2; Y ) = min(I(X1; Y ), I(X2; Y )). +Following [75], using this redundancy function yields the estimator of the synergistic interaction to be +Syn(X1, X2; Y ) = I(X1, X2; Y ) − max(I(X1; Y ), I(X2; Y )) +Note that Syn(X1, X2; Y ) is a distance measure because mutual information can be written in terms of +the Kullback-Leibler (KL) distance between two random variables. This means that the synergy score is +comparable across the industries and countries in our study because all the production processes have the +same number of random variables, providing the same bounds to their KL distances. In terms of units, +Syn(X1, X2; Y ) can be encoded in bits to obtain a more concrete measurement of how much synergistic +information represents from the total mutual information. +To remove potential numerical artefacts, we implement a bias correction procedure by performing a +randomized estimation where the inputs are shuffled. The synergy score calculated using the randomized +data is then subtracted from the true synergy score. This ensures that the statistical significance of the +estimated mutual information by removing the effect of spurious correlations. +6.1.3 +Synergistic interaction networks +The synergistic interaction network of a given industry is constructed using the synergy scores of every pair +of inputs. Each node in this network represents an industry (including itself) providing an input. The edges +are have no direction and are weighted according to the pairwise synergy score. Each network has 26 nodes. +It is possible to obtain fully connected graphs, which may trivialize certain network analysis. Thus, +9Performing the PID for more joint variables demands substantially more observations. To demonstrate the robustness of +our pairwise results, the SI shows that our findings hold when using triplets instead of pairs. +18 + +we extract the backbone of each network by using the method discussed in [76], which builds on the well- +known disparity filter [77] by automating the selection of a filtering threshold. To estimate the measures +of centrality, clustering, and mixing probabilities discussed in figure Figure 4, we use the binary version of +these backbone networks. +6.2 +Data and prepossessing +6.2.1 +Main datasets +Input-output tables: +The IO data are obtained from the Eora26, constructed by [56, 57]. These data +come from IO tables from organizations such as the UN, Eurostat, and the OECD among others. It is the +largest input-output dataset in terms of country coverage (181 economies). It consists of 26 harmonized +industries. Eora26 has become standard in the study of global value chains [78] and material footprint [79]. +It has also been shown to be consistent with similar, but smaller, global databases [80]. In order to achieve +its extensive country coverage, Eora26 inputs missing data–especially for low-income countries–through a +multi-region input-output model. Finally, in the SI, we provide the details of the complementary OECD +dataset. +International trade: +In order to measure industry sophistication, we calculate the economic complexity +index using the BACI international trade dataset [58], which is independent from Eora26. +These data +contain trade records between 200 countries for more than 5000 products during the 1995-2020 period. +These products are uniquely classified into 15 of the 26 industries of Eora26 through the HS92-to-ISIC +correspondence tables of the UN Statistics Division. While there are alternative trade datasets BACI has +been the only one so far used to measure economic complexity at the level of industries [81], we use it as our +primary source for estimating industry sophistication. +Development indicators: +We use three auxiliary development indicators to improve our clustering pro- +cedure (see subsection 6.3). The first is the World Bank’s gross national income (GNI) per capita indicator. +The second and third are the labor efficiency and infrastructure indicators from the World Economic Fo- +rum’s Global Competitive Index Report. The GNI covers the same period as Eora26, while the other two +indicators are available from 2007-2017.10 +10The SI shows that this auxiliary information is very useful to obtain coherent technology clusters without trivially becoming +the leading feature. +19 + +6.2.2 +Preprocessing input-output data +Using the Eora26 input-output tables, we construct time series for the total output of each industry (in a +specific country), as well as for the total inflow (combining both domestic and foreign) of each of its inputs. +Formally, let Ti(c′),j(c) denote a transaction where industry i in country c′ sells a total value T to industry j +in country c (in USD basic prices). Then, the total input inflow from industry i to j is +Xi,j(c) = +� +c′ +Ti(c′),j(c). +(7) +Similarly, the total output (Yj) of industry j in country c can be defined as +Yj(c) = +� +c′,i +Tj(c),i(c′) + +� +c′ +Fj(c),c′, +(8) +where Fj(c),c′ represents the final demand of goods and services of sector j of country c′ in any country c. +By using the multi-region transaction matrices T and F we identify the total inter-industry inputs Xi,j(c) +and the total output Yj(c) of industry j in country c. These time series may be affected by temporal trends +that prevent them from exhibiting Gaussian distributions. Thus, we convert these flows into log-fluctuations, +a common practice in the study of financial time series. The log-fluctuations for the input i to industry j +and country c (Xi,j(c)), and the corresponding output (Y t +j (c)) can be written as +ˆXt +i,j(c) = log Xt +i,j(c) +Xt−1 +i,j (c) +ˆY t +j (c) = log +Y t +j (c) +Y t−1 +j +(c). +(9) +For a given industry, a vector of output (or input) log-fluctuations can be collated across years and +countries for data-augmentation purposes. This is necessary in the application of PID since the country- +specific vectors ˆXt +i,j(c) and ˆY t +j (c) are not long enough to fulfill the requirements of the data estimation +method [75]. Ideally, such collation should consider clustering countries with similar technologies in a given +industry. Thus, in the next section, we explain how to achieve this. +6.3 +Clustering similar technologies +For a target industry, we cluster countries with similar technologies to augment the size of the data. We +measure technological similarity by using a popular concept in economics: the marginal product. +The +marginal product is the relative change in the output of an industry with respect to change in one of its +20 + +inputs. This measure enables us to dissect the effect of each input at a first level of approximation. The +marginal product of the input coming from industry j in industry i (in country c) is +MP t +i,j(c) = +Y t +j (c) − Y t−1 +j +(c) +Xt +i,j(c) − Xt−1 +i,j (c). +(10) +The median marginal product for all the available years, is taken to be a feature of the input to a +particular industry. This quantity usually exhibits a fat-tailed distribution across countries because of the +characteristic output differences among the economies. Thus, we log-transform it. +We use log MP as the key set of features for clustering countries with similar production technologies, +along with the auxiliary development indicators described in subsubsection 6.2.1. 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Constrained K-Means Clustering. Technical Report MSR-TR- +2000-65, Microsoft Research, 2000. +27 + diff --git a/mdE3T4oBgHgl3EQfiQrl/content/tmp_files/load_file.txt b/mdE3T4oBgHgl3EQfiQrl/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..768a9cb7cb67e1b76fccdb428168a37ff62aef0b --- /dev/null +++ b/mdE3T4oBgHgl3EQfiQrl/content/tmp_files/load_file.txt @@ -0,0 +1,1010 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf,len=1009 +page_content='Quantifying the Technological Foundations of Economic Complexity Hardik Rajpal1,2 and Omar A Guerrero1 1The Alan Turing Institute, London 2Imperial College London, London Abstract The problem of inferring the technological structure of production processes and, hence, quantifying technological sophistication, remains largely unsolved;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' at least in a generalizable way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' This reflects in empirical literature that either focuses on outputs instead of transformative processes, or preemptively assumes the nature of technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Building on recent advances in information theory, we develop a method to quantify technological sophistication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Our approach explicitly addresses the transformative process where inputs interact to generate outputs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' providing a more direct inference about the nature of technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' More specifically, we measure the degree to which an industry’s inputs interact in a synergistic fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Synergies create novel outputs, so we conjecture that synergistic technologies are more sophisticated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' We test this conjecture by estimating synergy scores for industries across nearly 150 countries using input-output datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' We find that these scores predict popular export-based measures of economic complexity that are commonly used as proxies for economic sophistication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' The method yields synergistic interaction networks that provide further insights on the structure of industrial processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' For example, they reveal that industries from the tertiary sector tend to be disassortative sector-wise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' To the extent of our knowledge, our findings are the first data-driven inference of technological sophistication within production processes (on an industrial scale).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Thus, they provide the technological foundations of economic complexity and represent an important step toward its empirical microfoundations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='04579v1 [econ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='GN] 11 Jan 2023 1 Introduction The transformative process of turning a set of inputs into an output is prevalent in various socioeconomic and physical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' In socioeconomic systems, it often takes the shape of a production process, and dif- ferent disciplines analyze it through distinct theoretical frameworks and quantitative tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' For instance, management sciences specialize in supply chains and global value chains, human and economic geography utilize systems analysis, economists study input-output models and fit production functions, systems en- gineering construct design structure matrices, and innovation scholars focus on combinatorial models and network analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Independently of the domain of application, it is commonly agreed that technological sophistication is a building block in the study of economic complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Thus, quantifying the degree and structure of technological sophistication is critical to understand the evolution, performance, fragility, and resilience of production systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' By production process, we refer to the procedure through which a certain technology transforms a set of inputs into an output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Due to the focus of this paper, we use the terms technological, industrial, productive, and economic sophistication interchangeably, to refer to the capacity of a production process to generate novel outputs from a set of inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Thus, an industry is considered more sophisticated if it is produces more novelty than others when using the same inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' In this paper, we develop a method to quantify the technological sophistication of production processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' We test our method using two major input-output datasets, and validate it through independent export data and well-established economic complexity in- dices that are commonly used as proxies for technological sophistication at a national level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' We find that our synergy score predicts these indices, and that the inferred synergistic interaction networks have non- trivial topologies that characterize production technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' To the extent of our knowledge, this is the first framework that quantifies technological sophistication by empirically inferring the nature of input-input and input-output relationships (as opposed to assuming them).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Our approach is non-parametric, so it does not require the ex ante assumption of specific production functions or design structure matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Instead, by exploiting the mutual information between input data in an output signal, it allows to infer the structure of synergistic interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' This is a major innovation as it facilitates both the overall quantification of technological sophistication across firms, industries, sectors, and countries, as well as the estimation of the interaction networks living at the heart of production processes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' currently considered a black box by eco- nomic complexity scholars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Furthermore, it provides an empirical basis to justify the specification of certain production functions in input-output models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' The method is of general purpose as it can adapted to other contexts where transformative processes may be important (both in socioeconomic and physical systems).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' In the rest of the paper we provide an overview of existing approaches, describe the methods and datasets 2 used, and present our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' 2 Knowledge gap In the theoretical literature of technological innovation, there is a consensus agreeing on the principle that technological change can be described as a recombinant process through which novelty emerges from new ways of coupling existing devices (or technologies) to fulfill a certain purpose[1–12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' This consensus is empirically supported by evidence on how new inventions recombine existing ones [13–18] (see [19] for a comprehensive review).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Arguably, technological sophistication is intimately related to this transformative process, so producing empirical metrics to characterize it would provide generalizable foundations for its quantification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Unfortunately, much of the empirical evidence in innovation studies remains domain-specific (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=', patents [20] and cities [21]), making it difficult to build generalizable empirical frameworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' In our view, the lack of such frameworks obeys two limiting factors: (1) technology can be highly diverse across industries and countries, and (2) generating reliable estimates for production models with numerous inputs requires big data, something difficult to obtain from input-output (IO) tables (with the exception of recent developments of inter-firm transaction data [22–24]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' IO scholars and analysts circumvent the lack of big data by modeling production networks that assume, ex ante, the nature of the input-input and input-output interactions through production functions [24– 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' This approach has the benefit of shifting the problem of modeling technological sophistication from one of inferring structures to one of fitting parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' This shift comes with problems that have been recently discusses along the lines of misspecification [28], estimation biases [29], and aggregation artifacts [30, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Thus, assuming (parametric) production functions instead of inferring interaction structures limits our capacity to quantify technological sophistication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' The aforementioned limitations have become such an important issue that supply-chain surveys have been conducted in an attempt to determine the degree of dependence on certain inputs by certain industries (see the IHS Markit survey in [32] and [33]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Moreover, such information, has been incorporated in the new generation of IO models [24, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Hence, a data-driven approach that would facilitate the inference of interaction structures in production processes would help alleviating some of these problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Systems engineering, has created an alternative approach to production functions by developing the con- cept of Design Structure Matrices (DSMs) [16, 34, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' DSMs describe networks of interactions between the different components of a production processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' They provide a comprehensive tool to represent tech- nological sophistication, but they require substantial knowledge about the process itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Thus, DSMs build on a top-down approach that demands substantial ex ante knowledge;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' making it difficult to scale up to 3 more aggregate levels such as industries or sectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' These aggregation levels are crucial for the design and implementation of country- or industry-wide innovation strategies and policy intervention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' A data-driven framework to infer DSMs, or some of their components, would complement this literature in important ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' In recent years, the literature of economic complexity has emerged with new proposals on how to quantify technological sophistication [36–46] (see [47] for a comprehensive review).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Building on existing literature on export diversification [48–52], two dominant approaches have become the gold standard in this field: the Economic Fitness Index (EFI) [40] and the Economic Complexity Index (ECI) [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' While these frameworks are built on different theoretical foundations (see [53] for a rigorous comparison), both of them try to map export profiles into indices that capture different elements of economic sophistication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='1 In spite of this grow- ing literature, the technological foundations of production sophistication remain mostly theoretical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' That is, while these frameworks provide a proxy for technological sophistication by looking at exports, they do not address production processes and hence, treat them like a black box where interactions remain obscure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' This very issue has been recently raised in a keynote speech by one of the creators of the ECI, Ricardo Hausmann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Using a biological metaphor, Hausmann highlights the lack of a production account in the quantification of technological sophistication and calls for scholars to move beyond the phenotypic view of economic sophis- tication (the output/export side) to try to understand the genotypic one (the technological/transformative side) [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Evidently, the quantification of technological sophistication is a problem that pertains to multiple disci- plines and has crucial implications in the design and implementation of economic and innovation strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' The lack of generalizable quantitative methods that address the production process explicitly poses a major barrier to advance research in these fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' At the same time, it creates a void that maintains these various disciplines, to a great extent, disconnected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Our contribution seeks to fill this gap and to provide a general framework that facilitates the quantification of technological sophistication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Such contribution can help to develop deeper insights on the microfoundations of economic complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' 3 Framework and data Let us provide a succinct description of the methodology and the data employed in our analysis, and leave more specific details for section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Broadly speaking, our method seeks to estimate the mutual information between pairs of inputs in a given industry (from IO tables), and decompose their contribution to the output into different modes of information sharing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' We focus on a particular mode known as synergistic information, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=', the information cannot be obtained from any of the inputs alone but only exists as a virtue 1From these two indices, only the EFI has been applied at the industry level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' 4 of the interaction between the inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Using this information, we measure the degree of synergy between the inputs and estimate a synergy score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' A higher synergy score means that input interactions produce more novel information during the pro- duction process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' This interpretation aligns with the established notion of a recombinant process generating novel outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Thus, we take an interpretative leap and claim that the synergy score quantifies technological sophistication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' In other words, an industry that exhibits higher synergy scores should have a more sophisti- cated technology underlying its production process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' This is our main conjecture, and we provide evidence to support it in the later sections of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Furthermore, we explore the estimated synergistic interaction networks to learn new and more nuanced insights about the nature of technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='1 The synergy score Consider an industry and three associated datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Data Y contain the changes in the output of the industry, while X1 and X2 capture the changes in inputs 1 and 2 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' We are interested in quantifying how much information do X1 and X2 provide about Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' It can be measured using the total mutual information I(X1, X2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Y ) between the output and the inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' In essence, the total mutual information is a measure of the amount of uncertainty in Y , that is reduced by knowing X1 and X2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Furthermore, it is possible quantify how much of this uncertainty reduction is a result of the interactions between the inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' We can achieve this by following the partial information decomposition (PID) proposed by [55], where the mutual information I that X1 and X2 provide about Y can be described as I(X1, X2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Y ) = Syn(X1, X2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Y ) + Oth(X1, X2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' (1) In Equation 1, the mutual information is decomposed into the synergistic type and other ones (we explain the others in section 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Synergistic information can only be obtained from the interaction between X1 and X2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' It is an analogue for input complementarity in the IO literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' If either input is removed from the production process, all the synergistic information would be lost from the output signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Arguably, more sophisticated production processes involve more synergistic information because they generate more novel outputs by recombining the same inputs in innovative ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Thus, we use this type of information as a synergy score and compute it for all unique pairs of inputs in the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='2 Further details on the estimation method can be found in section subsection 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' 2Equation 1 can be generalized for more than two inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' However, producing reliable estimates for more than three inputs can become too demanding data-wise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Nevertheless, we show in the SI that our main results hold when we estimate synergies between triplets instead of pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' 5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='2 Data To empirically capture various degrees of technological sophistication, it is important to assemble a dataset with substantial variation in terms of industrial development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Thus, having a large number of countries is key, as most technological variation would be expected between nations with different levels of development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' We find such coverage in the Eora26 dataset [56, 57], a global set of input-output tables with harmonized industries across a large number of countries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' The subset extracted from Eora26 contains annual input- output tables for 26 industries across 148 countries during the period 1995-2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' We use the time series of each input and the corresponding total output to compute the synergy scores associated with a particular country and industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Thus, we exploit the temporal variation in the data to build the scores, and then focus on comparing countries and industries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' The original time series are transformed into log-fluctuations (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=', growth rates).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='3 To further validate our results, we test our method using the OECD IO tables, covering 66 countries across 45 unique industries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' While these data are smaller in country coverage, they provide more industries and are considered of a higher quality (because the data providers are subjected to harmonized reporting standards).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' While an obvious implication of using OECD data is the loss of technological diversity (due to the absence of lower-income countries), we show that our results are robust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' In search of evidence to validate our main conjecture, we test if our synergy scores contribute in a significant way to the prediction of economic complexity indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' We focus on predicting the EFI, as it has been shown to better capture economic complexity (the ECI correlates more with GDP) [20, 53] and because, to the extent of our knowledge, it is the only index that has been used to study industry-level economic sophistication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Nevertheless, in Figure 3 and in the SI, we provide evidence of robustness when using the ECI and a third index, GENEPY [53], that generalizes both the EFI and the ECI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Since all the economic complexity indices are export-based metrics, we construct them by using an independent dataset on export flows [58] with the same country and temporal coverage as the Eora26 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Due to the need for collapsing time in the calculation of the synergy score, we compute inter-temporal average complexity indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Further details on these calculations are provided in the section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='3 Descriptive features Before proceeding to our results, let us show some interesting features of the synergy scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' In Figure 1, we show an example of synergy networks estimated by clustering IO tables into four groups (see section 6 for an explanation on the clustering procedure).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' The industry represented in each network ‘Textiles and 3This transformation yields normally-distributed growth rates at the industrial level, which makes the data compatible with Gaussian estimators of mutual information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' 6 Wearing Apparel’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Each panel represents the synergistic interaction structure between its inputs (‘Textiles and Wearing Apparel’ appear as a node as well as it its output is also used as input).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Hence, each node represents an input and each weighted edge captures the synergy score between a pair of inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Overall, these networks capture the average technological structure of a group of countries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' The main take-away of this example is that an industry that uses the same set of inputs in every country group may exhibit very different interaction structures between these inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' For instance, clusters 1 and 2, which have middle and high income countries, exhibit technologies with more synergistic interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Likewise, the most central nodes, are not necessarily the same across country groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' This highlights the importance of quantifying and understanding the internal structure of production processes as part of measuring technological sophistication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Figure 1: Synergistic interaction networks of the ‘Textiles and Wearing Apparel’ industry Cluster 1 Median GNI per capita $35,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='220 Cluster 2 Median GNI per capita $7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='215 Cluster 3 Median GNI per capita $1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='855 Cluster 4 Median GNI per capita $895 Industries Agriculture Mining and Quarrying Fishing Food & Beverages Textiles and Wearing Apparel Wood and Paper Petroleum,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Chemical and Non-Metallic Mineral Products Metal Products Electrical and Machinery Transport Equipment Other Manufacturing Recycling Electricity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Gas and Water Construction Maintenance and Repair Wholesale Trade Retail Trade Hotels and Restraurants Transport Post and Telecommunications Finacial Intermediation and Business Activities Public Administration Education,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Health and Other Services Private Households Others Re-export & Re-import Sectors Primary Secondary Tertiary Other Notes: The nodes represent industries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' They are colored according to their sectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' The width of the edges is proportional to the synergy score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Sources: Authors’ own calculations using Eora26 data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' 7 An insightful feature that can be drawn from a network representation of technology is the diversity of synergies, quantified through network-variance [59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' When analyzing the relationship between network- variance and the overall level of synergy in the network (through the first eigenvalue of the adjacency matrix), we find that industry sophistication does not grow without increments in the diversity of synergistic interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Figure 2a shows this finding using all the networks inferred from Eora26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' It suggests that, to quantify technological sophistication, it is not enough to measure the overall synergy level, but also to understand how synergies are distributed within the production process, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=', to infer the interaction structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Figure 2: Synergy score features (a) Synergy level and diversity of synergistic interactions 0 10 20 30 Synergy Variance 1e 5 0 2 4 6 8 10 Largest Eigenvalue (b) Distribution of synergy scores 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='3 Synergy 0 5 10 15 20 25 30 Density Notes: Figure 2a shows quantities for synergistic interaction networks (see 6 for details on how we obtain 100+ networks).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Figure 2b presents the distribution of pairwise synergy scores across all the edges of all the synergistic networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Sources: Authors’ own calculations with Eora26 data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' One last interesting feature of synergistic interactions is their scarcity, as suggested by Figure 2b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Put it differently, highly synergistic interactions are much less abundant than low-synergy ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' This is con- sistent with the literature on innovation dynamics [60, 61], where it is argued that innovative recombinant processes happen sporadically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Together, these features link our measure of production sophistication to well-established ideas from the innovation and technology literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' 8 4 Results 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='1 Predicting economic complexity First, we investigate the relationship between the synergy scores and the complexity indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Figure 3 shows a clear positive correlation between the synergy score and the three indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' It suggests that synergistic interactions may predict these proxies of economic sophistication, providing support to our conjecture on being able to quantify technological sophistication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' At this point, we would like to highlight the importance of these results in view of the stark differences between our method and the EFI, GENEPY, and ECI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' While we infer technological sophistication by directly analyzing input-input and input-output interactions through the mutual information in their growth rates, complexity indices take an approach that focuses on the co- occurrence of outputs (so no input-output interactions are taken into account).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' We find it remarkable that our method is able to capture the cross-country and cross-industry variation produced by the complexity indices, especially when these two distinct approaches use independent datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Figure 3: Synergy scores and their association to industry sophistication (a) Economic fitness index 4 3 2 1 log (Synergy) 10 8 6 4 2 0 Industry log (Fitness) Binned average Linear fit on raw data (b) GENEPY index 4 3 2 1 log (Synergy) 9 8 7 6 5 4 Industry log (GENEPY) Binned average Linear fit on raw data (c) Economic complexity index 4 3 2 1 log (Synergy) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='0 Industry Complexity Binned average Linear fit on raw data Notes: With the purpose of a clear visualization, we display data binned according to the synergy scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Each dot corresponds to the average value withing the corresponding bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Since the synergy distribution is right-skewed, we analyze the logarithm of the scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' The linear fit is estimated using the entire data (not the binned one).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Sources: Authors’ own calculations with Eora26 data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Next, let us formally test whether the synergy scores contribute, in a significant way, to the prediction of the EFI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' We estimate linear regression models controlling for different factors that may contribute to tech- nological sophistication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='The intuition behind this exercise is that, if the synergy score displays a significant coefficient in the prediction of output-based complexity indices (calculated through independent methods and data), then our method is valid and quantifies the degree of technological sophistication across industries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' In Table 1, we present 7 linear regression models with 8 possible independent variables that contribute to the prediction of the EFI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='4 Our variable of interest is the logarithm of the synergy score (we take log 4Figure 3a suggests that a linear specification is the most adequate when the EFI is the dependent variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' 9 due to the skewed distribution shown in Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' In the first model, we estimate the association shown in Figure 3a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' without any controls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Model 2 includes log GDP per capita, and aims at controlling for country-specific factors such as higher income, better public governance, and better infrastructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='Model 3 adds dummy variables indicating if the industry belongs to the primary, secondary, or tertiary sector (with the additional sector being ‘Other’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' With this, we try to control for sector-specific factors like regulatory frameworks (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=', tax prerogatives and unionization practices).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' In model 4, we include industry-specific total output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Finally, models 5 to 7 introduce industry-specific energy-related controls that are available in the Eora26 data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' The ratio of energy consumption to total output is a proxy of the technological efficiency of the industry, which may relate to its level of sophistication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' In all regression models, the coefficient of log synergy remains positive and statistically significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Fur- thermore, its magnitude is relatively stable across the seven models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Interestingly, the coefficient is larger than the one of GDP per capita.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='5 This result validates our conjecture of more synergistic information implying higher technological sophistication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' In the SI, we show that these results are robust when using GENEPY index and ECI as the dependent variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='6 To further validate our regression results, we perform the same tests using the IO tables from the OECD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Table 2 shows that our results hold with these data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' In fact, the magnitude of the log synergy coefficients nearly doubled with respect to the ones reported in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='2 Synergistic interaction networks Next, let us look at more nuanced results through the synergistic interaction networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' While the Eora26 networks are relatively small (with 26 nodes), we have a more than 100 realizations as every industry employs the same set of inputs at this level of aggregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Thus, we can study the variation of network topology and whether it relates to technological sophistication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' More specifically, we are interested in the relationship between the mean synergy score of a node and its topological features, namely, centrality, clustering, and mixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='7 5But one should be careful of not reading too much out of the controls’ coefficients, as they serve the only purpose of accounting for potential confounders [62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Expecting specific signs and magnitudes in the coefficients of control variables is a common mistake known in epidemiology as the table 2 fallacy [63] and in econometrics as the interpretation of endogenous controls [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' 6Note that the low asjusted R2 does not invalidate our results, as this is quite common in cross-sectional regressions with a large number of observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' If the data were aggregated, for example, by averaging the synergy scores of each industry (instead of using the input-level observations), then the explained variance would increase substantially (see the SI for evidence on this when testing the robustness to aggregated sophistication indices).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' This is consistent with Figure 3, where the data have been binned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' 7We employ the binary backbone version of the networks to perform this analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' See section 6 for further details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' 10 Table 1: Linear models predicting the economic fitness index of industries Predictor Model 1 Model 2 Model 3 Model 4 Model 5 Model 6 Model 7 Log synergy 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='4570∗∗ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='3528∗∗ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='6970∗∗∗ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='1796∗∗∗ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='5636∗∗∗ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='6875∗∗∗ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='1795∗∗∗ (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='0484) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='0053) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='0254) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='7582) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='9219) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='0247) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='7591) Log GDP per.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' cap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='6523 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='6403 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='4684 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='3125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='6346 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='4682 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='5745) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='5685) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='5618) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='5481) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='5681) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='5642) Primary sector −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='8356∗∗ −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='9426∗∗ −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='2379∗∗ −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='9475∗∗ −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='9447∗∗ (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='7029) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='2428) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='9617) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='6803) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='1856) Secondary sector −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='4582 −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='7442∗∗ −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='0697 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='5831 −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='7460∗∗ (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='1045) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='3370) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='2813) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='1066) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='2913) Tertiary sector −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='3603∗∗∗ −8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='9321∗∗∗ −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='0664∗∗∗ −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='4499∗∗∗ −8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='9336∗∗∗ (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='0858) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='6226) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='0998) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='1006) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='6112) Log output 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='4227∗∗ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='4223∗∗ (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='6162) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='6283) Log energy 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='3827 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='4706) Energy intensity −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='6006 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='0405 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='4897) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='6730) Adjusted R2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='0237 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='0286 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='0454 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='0891 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='0521 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='0460 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='0891 No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' of observations 695,500 695,500 695,500 695,500 695,175 695,500 695,500 Notes: OLS regression coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' The model intercept is omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' The dependent variable is the EFI, calculated for each industry in the dataset, and averaged across the years in the sample period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' The stars denote the level of significance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' The number in parenthesis is the clustered standard error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Confidence levels are indicated by * for 90%, ** for 95%, and *** for 99%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Source: Authors’ own calculations using Eora26 data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Table 2: Linear models predicting the economic fitness index of industries with OECD data Predictor Model 1 Model 2 Model 3 Model 4 Log synergy 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='3811∗∗∗ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='2163∗∗∗ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='4180∗∗∗ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='5699∗∗∗ (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='2116) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='1707) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='1405) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='1588) Log GDP per.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' cap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='6275 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='6192 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='3155 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='5588) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='5560) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='5490) Primary sector 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='4040 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='4751 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='8543) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='0307) Secondary sector −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='5265 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='4749 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='5887) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='5931) Tertiary sector 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='0486 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='1175 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='8531) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='8746) Log output 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='8703∗∗ (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='3507) Adjusted R2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='0297 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='0332 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='0406 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='0598 No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' of observations 1,550,494 1,550,494 1,550,494 1,550,494 Notes: OLS regression coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' The model intercept is omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' The dependent variable is the EFI, calculated for each industry in the dataset, and averaged across the years in the sample period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' The stars denote the level of significance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' The number in parenthesis is the clustered standard error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Confidence levels are indicated by * for 90%, ** for 95%, and *** for 99%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Source: Authors’ own calculations using OECD data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' 11 Figure 4a shows a positive association between the synergy score and eigenvector centrality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='8 This association becomes negligible when tested under a null model (see section 6 for details on the null model).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' It suggests that more synergistic inputs tend to interact with a larger set of different inputs in a production process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Locally, a node with a higher synergy score tends to play a brokerage role between those inputs with which it interacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' This can be seen in Figure 4b, where a negative relationship between synergy and clustering coefficient is evident.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Once more, this stylized fact dissipates when estimated under the null model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Figure 4: Properties of synergistic interaction networks (a) Synergy and centrality 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='0 Total Synergy 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='21 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='23 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='25 Eigenvector Centrality Empirical Null model (b) Synergy and clustering 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='0 Total Synergy 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='45 Clustering coefficient Empirical Null model Source: Authors’ own calculations with Eora26 data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Finally, since synergy seems to correlate strongly to centrality and local structure, we consider whether certain types of technology inherently contribute to the structure of synergistic interaction networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' While fully answering this question is ambitious and beyond the scope of this paper, we take a first step by exploiting the taxonomy of industries into sectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' It has been long argued that membership to a particular sector conveys information about structural differences between industries and their underlying technologies, as they correspond to the main stages of production [65, 66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Accordingly, we are interested in learning if, in synergistic interaction networks, sector membership conditions the probability of interacting with inputs from the same sector and with inputs from different ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Using all the networks in the sample, we compute the mixing probability between each possible combina- 8Similar positive relationship is observed when using alternative measures of centrality like the betweenness, Katz, and degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' 12 tion of sectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' In other words, we estimate how likely it is to find two input industries from specific sectors interacting synergistically in a production process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Then, we perform a T-test of this probability, using the one of the null model as the reference estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' A positive T value of mixing probabilities would indicate that two inputs from the corresponding two sectors tend to interact synergistically more than expected un- der the null.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Figure 5 shows the T values of every pair of sectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Interestingly, the tertiary sector, when interacting with other sectors, has the highest mixing probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' This suggests that, in terms of economic complexity, tertiary-sector industries play a mediator role for industries in other sectors, enabling innovation and more sophisticated production processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Figure 5: Sector mixing Secondary_Primary Other_Primary Primary_Primary Secondary_Secondary Other_Secondary Other_Other Other_Tertiary Tertiary_Primary Tertiary_Secondary Tertiary_Tertiary Sector combination 6 4 2 0 2 4 6 8 Difference in mixing probabilities T values (Empitical - Null) *** *** *** *** *** *** *** Notes: The T-tests are performed using the mixing probabilities calculated from the empirical networks and the ones produced by the null model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Confidence levels are indicated by * for 90%, ** for 95%, and *** for 99%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Source: Authors’ own calculations with Eora26 data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' 5 Discussion and conclusions The quantification of technological sophistication in production processes is an elusive problem that is rele- vant to several disciplines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' This paper introduces the first method that explicitly addresses input-input and 13 input-output interactions, opening the black box of production processes underpinning economic complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' By estimating the amount of synergistic mutual information between the inputs of a production process, we quantify the degree of technological sophistication across various industries and countries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Moreover, we infer the structure of production processes by constructing synergistic interaction networks, revealing–never before seen–features that characterize industrial sophistication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' These networks provide empirical grounds to select and justify production functions in IO models;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' something missing in this large body of literature and a major limitation in IO empirical studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' They also reveal the structural role of industries with a high degree of synergistic interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Altogether, our results represent a major step towards understanding the microfoundations of economic complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Some of the main limitations of this approach are that it requires larger data than what is typically found in IO tables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' This means that, in this study, we need to develop a clustering procedure, and that our inferences are not for specific countries, but for groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Another limitation is that we have to sacrifice the temporal dimension as we need to exploit time variation to produce the estimations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Ideally one would have high-frequency IO tables to compute the synergy scores by sub-periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' This would allow to study, for example, the evolution of technological sophistication and of its networked structure across countries and industries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Fortunately, new firm-transaction datasets are being generated as we write this manuscript, so the future for using the proposed approach looks very promising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Furthermore, because our framework works on the premise of a generic production process its applications could extend to other domains such as the study physical/biological systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' 6 Methods Partial information decomposition (PID) requires a large number of observations per variable, typically in the order of 200 for a process with two inputs and one output (the number of required observations scales with the number of inputs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Such data do not exist for a single industry in a country’s IO tables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Nevertheless, we can overcome this limitation by performing a data-clustering procedure based on the technological similarity of countries in a given industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Thus, we create a workflow that facilitates the pre-processing and inference tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' The entire pipeline should be repeated for each industry that one would like to include in the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' The workflow consists of two major steps, and an illustrative sketch is provided in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Clustering: Grouping countries that exhibit similar production patterns in a target industry, prefer- ably in roughly equally sized groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' This means that two countries that are in the same cluster in a given industry A, may be in different ones when analyzing another industry B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' 14 Figure 6: The two-step workflow 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Clustering 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='1 Data Transformation Input – output tables are transformed to extract inputs to a given target industry across all countries and years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Countries Years 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='2 Feature extraction Log Marginal Product of inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' + Labour Efficiency + Quality of Infrastructure + Income per capita.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' PCA 2 PCA 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='3 K-constrained clustering Similar sized four clusters of countries are identified with similar production technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Pairwise synergies Inputs Output Inputs Output 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='1 Data Transformation Cluster level log fluctuations are extracted from the industry level input – output table, for the inputs to and corresponding output of the target industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='2 Partial Information Decomposition Taking pair of inputs and corresponding output log fluctuations, we use PID to estimate total synergy among the inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='3 Network of synergies Iterating through all possible pairs of input, we can generate a network of synergy that exists in the production process of the target industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' 𝑋!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' 𝑋" 𝐼(𝑋!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='𝑋";' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='𝑌) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Estimation: Using PID to estimate synergy scores for each pair of inputs in the target industry, and constructing its corresponding synergistic interaction network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' In the rest of this section, we explain the PID method and the clustering procedure in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Since the economic complexity indices are standard metrics in the literature, we explain how we calculate them at the industry level in the SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='1 Synergy score Since the seminal paper of Claude Shannon in 1948 [67], information theory has evolved substantially, spanning beyond the study of communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' It has become a powerful tool to quantify interdependencies 15 Agriculture Fishing Mining and Quarrying 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content="30 Food &Beverages Textiles and Wearing Apparel Woodand'Paper Petroleum, Chemical and Non-Metallic Mineral Products 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='25 Metal Products Electrical and Machinery Transport Equipment Other Manufacturing 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='20 Recycling Electricity, Gas and Water Construction Maintenance and Repair 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='15 Wholesale Trade Retail Trade Hotels and Restraurants Transport 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='10 Post and Telecommunications Finacial Intermediation and Business Activities Public Administration Education,Health and Other Services 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='05 Private Households Others Re-export & Re-import 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='00 0 5 10 15 20 25between variables in many complex systems like the ecology [68], financial markets [69], the brain [70] and thermodynamics [71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Shannon entropy measures the variability of a system in a given state-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Leveraging this concept, Shannon proposed a measure of dependence between two variables, commonly known as mutual information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' The ability to assess the variability of a system–and the interdependency among its variables–made information theory ideal for the empirical study of complex systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' We adapt these concepts and tools to quantify the interdependencies between an industry’s inputs through a synergy score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='1 Mutual information Consider a random variable X, defined on a discrete state space X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' The Shannon entropy is written as H(x) = − � x∈X p(x) log p(x), (2) where p(x) is the probability of the realization x of variable X on the state space X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Entropy is highest when the variable explores all possible states with equal probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Similarly, the shared uncertainty between two random variables X and Y can be written as the mutual information I(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Y ) = � y∈Y � x∈X p(x, y) log � p(x, y) p(x)p(y) � , (3) where p(x, y) represents the joint distribution of the two random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' The measure I(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Y ) becomes 0 if X and Y are independent, such that p(x, y) = p(x)p(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Next, let us assume that X and Y are Gaussian random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Then, the corresponding mutual information for two Gaussian random variables can be written as (see[72] for a complete derivation), I(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Y ) = 1 2 log � detΣ(X) detΣ(X|Y ) � , (4) where detΣ(X) represents the determinant of the covariance matrix of X, and Σ(X|Y ) represents the con- ditional covariance, which can be written as Σ(X|Y ) = Σ(X) − Σ(X, Y )Σ(Y )−1Σ(Y, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' (5) The definition in Equation 4 also exists for a multi-variate setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' For three or more variables, higher- order effects such as synergistic interactions can be observed as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Interestingly, these synergistic interac- tions are present in the groups of variables (inputs) as a whole, so they absent when considered individually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' 16 ������������(������������, ������������;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' ������������) ������������(������������;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' ������������) ������������(������������;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' ������������) Unique to Y Unique to X Redundant Synergistic Figure 7: Partial information decomposition In a seminal paper, Williams and Beer [55] formalize the measurement of synergy for the case with two inputs and one output using PID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Thus, we proceed to briefly explain the PID framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='2 Partial information decomposition PID decomposes the mutual information between a pair of inputs and the output into Synergistic, Redundant and Unique information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' This decomposition was first introduced in [55], and has been instrumental to study different types of interactions between random variables [73–75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Let us look at the case of mutual information between two input variables (X1, X2) and one output variable (Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Here, the total mutual information about the output provided by the two inputs can be represented as the following Venn diagramFigure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Redundant information could come from either X1 or X2, so this information would be preserved if one of the inputs was removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' It is an analogue input substitutability in the IO literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Unique information comes from each input only, so it represents the unique contribution that each input makes to the output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Both of these types of information are contained in the term Oth(X1, X2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Y ) from Equation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Thus, let us rewrite it in its extended form 17 I(X1, X2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Y ) = Syn(X1, X2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Y ) + Red(X1, X2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Y ) + Unq(X1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Y ) + Unq(X2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' (6) A general formulation of Equation 6 for n random variables is provided by [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='9 The unique and the total joint information about the output provided by the inputs can be exactly calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' However, to estimate the synergy and redundancy between the inputs, we need to assume a redundancy function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Here we employ the minimum mutual information redundancy estimator developed for multivariate Gaussian systems[75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' This redundancy estimator assumes the total redundancy to be equivalent to the unique mutual information about the output provided by the weakest input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Formally, this is Red(X1, X2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Y ) = min(I(X1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Y ), I(X2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Y )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Following [75], using this redundancy function yields the estimator of the synergistic interaction to be Syn(X1, X2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Y ) = I(X1, X2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Y ) − max(I(X1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Y ), I(X2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Y )) Note that Syn(X1, X2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Y ) is a distance measure because mutual information can be written in terms of the Kullback-Leibler (KL) distance between two random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' This means that the synergy score is comparable across the industries and countries in our study because all the production processes have the same number of random variables, providing the same bounds to their KL distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' In terms of units, Syn(X1, X2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Y ) can be encoded in bits to obtain a more concrete measurement of how much synergistic information represents from the total mutual information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' To remove potential numerical artefacts, we implement a bias correction procedure by performing a randomized estimation where the inputs are shuffled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' The synergy score calculated using the randomized data is then subtracted from the true synergy score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' This ensures that the statistical significance of the estimated mutual information by removing the effect of spurious correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='3 Synergistic interaction networks The synergistic interaction network of a given industry is constructed using the synergy scores of every pair of inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Each node in this network represents an industry (including itself) providing an input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' The edges are have no direction and are weighted according to the pairwise synergy score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Each network has 26 nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' It is possible to obtain fully connected graphs, which may trivialize certain network analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Thus, 9Performing the PID for more joint variables demands substantially more observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' To demonstrate the robustness of our pairwise results, the SI shows that our findings hold when using triplets instead of pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' 18 we extract the backbone of each network by using the method discussed in [76], which builds on the well- known disparity filter [77] by automating the selection of a filtering threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' To estimate the measures of centrality, clustering, and mixing probabilities discussed in figure Figure 4, we use the binary version of these backbone networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='2 Data and prepossessing 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='1 Main datasets Input-output tables: The IO data are obtained from the Eora26, constructed by [56, 57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' These data come from IO tables from organizations such as the UN, Eurostat, and the OECD among others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' It is the largest input-output dataset in terms of country coverage (181 economies).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' It consists of 26 harmonized industries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Eora26 has become standard in the study of global value chains [78] and material footprint [79].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' It has also been shown to be consistent with similar, but smaller, global databases [80].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' In order to achieve its extensive country coverage, Eora26 inputs missing data–especially for low-income countries–through a multi-region input-output model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Finally, in the SI, we provide the details of the complementary OECD dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' International trade: In order to measure industry sophistication, we calculate the economic complexity index using the BACI international trade dataset [58], which is independent from Eora26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' These data contain trade records between 200 countries for more than 5000 products during the 1995-2020 period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' These products are uniquely classified into 15 of the 26 industries of Eora26 through the HS92-to-ISIC correspondence tables of the UN Statistics Division.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' While there are alternative trade datasets BACI has been the only one so far used to measure economic complexity at the level of industries [81], we use it as our primary source for estimating industry sophistication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Development indicators: We use three auxiliary development indicators to improve our clustering pro- cedure (see subsection 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' The first is the World Bank’s gross national income (GNI) per capita indicator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' The second and third are the labor efficiency and infrastructure indicators from the World Economic Fo- rum’s Global Competitive Index Report.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' The GNI covers the same period as Eora26, while the other two indicators are available from 2007-2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='10 10The SI shows that this auxiliary information is very useful to obtain coherent technology clusters without trivially becoming the leading feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' 19 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='2 Preprocessing input-output data Using the Eora26 input-output tables, we construct time series for the total output of each industry (in a specific country), as well as for the total inflow (combining both domestic and foreign) of each of its inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Formally, let Ti(c′),j(c) denote a transaction where industry i in country c′ sells a total value T to industry j in country c (in USD basic prices).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Then, the total input inflow from industry i to j is Xi,j(c) = � c′ Ti(c′),j(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' (7) Similarly, the total output (Yj) of industry j in country c can be defined as Yj(c) = � c′,i Tj(c),i(c′) + � c′ Fj(c),c′, (8) where Fj(c),c′ represents the final demand of goods and services of sector j of country c′ in any country c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' By using the multi-region transaction matrices T and F we identify the total inter-industry inputs Xi,j(c) and the total output Yj(c) of industry j in country c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' These time series may be affected by temporal trends that prevent them from exhibiting Gaussian distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Thus, we convert these flows into log-fluctuations, a common practice in the study of financial time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' The log-fluctuations for the input i to industry j and country c (Xi,j(c)), and the corresponding output (Y t j (c)) can be written as ˆXt i,j(c) = log Xt i,j(c) Xt−1 i,j (c) ˆY t j (c) = log Y t j (c) Y t−1 j (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' (9) For a given industry, a vector of output (or input) log-fluctuations can be collated across years and countries for data-augmentation purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' This is necessary in the application of PID since the country- specific vectors ˆXt i,j(c) and ˆY t j (c) are not long enough to fulfill the requirements of the data estimation method [75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Ideally, such collation should consider clustering countries with similar technologies in a given industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Thus, in the next section, we explain how to achieve this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='3 Clustering similar technologies For a target industry, we cluster countries with similar technologies to augment the size of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' We measure technological similarity by using a popular concept in economics: the marginal product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' The marginal product is the relative change in the output of an industry with respect to change in one of its 20 inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' This measure enables us to dissect the effect of each input at a first level of approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' The marginal product of the input coming from industry j in industry i (in country c) is MP t i,j(c) = Y t j (c) − Y t−1 j (c) Xt i,j(c) − Xt−1 i,j (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' (10) The median marginal product for all the available years, is taken to be a feature of the input to a particular industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' This quantity usually exhibits a fat-tailed distribution across countries because of the characteristic output differences among the economies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Thus, we log-transform it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' We use log MP as the key set of features for clustering countries with similar production technologies, along with the auxiliary development indicators described in subsubsection 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' The indicators on GNI, labor efficiency, and infrastructure help avoiding trivial clusters that could result from the coincidental similarity of marginal products due to non-technological reasons such as exogenous shocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Finally, we employ the k-means-constrained clustering algorithm [82] to define four country clusters in a target industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='11 We choose this constrained version of the k-means algorithm because it allows finding clusters that are balanced in size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='12 The hyperparameters of the algorithm, including the number of clusters, are optimized using consensus clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content='13 References [1] Joseph A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Schumpeter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfiQrl/content/2301.04579v1.pdf'} +page_content=' Business Cycles, volume 1.' metadata={'source': 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b/nNFST4oBgHgl3EQfLDje/content/tmp_files/2301.13739v1.pdf.txt @@ -0,0 +1,4269 @@ +EXACT SOLUTION OF TASEP AND VARIANTS WITH INHOMOGENEOUS SPEEDS +AND MEMORY LENGTHS +KONSTANTIN MATETSKI AND DANIEL REMENIK +ABSTRACT. In [MQR21; MR23] an explicit biorthogonalization method was developed that applies to a +class of determinantal measures which describe the evolution of several variants of classical interacting +particle systems in the KPZ universality class. The method leads to explicit Fredholm determinant +formulas for the multipoint distributions of these systems which are suitable for asymptotic analysis. In +this paper we extend the method to a broader class of determinantal measures which is applicable to +systems where particles have different jump speeds and different memory lengths. As an application of +our results we study three particular examples: some variants of TASEP with two blocks of particles +having different speeds, a version of discrete time TASEP which mixes particles with sequential and +parallel update, and a version of sequential TASEP where a single particle with long memory length +(equivalently, a long “caterpillar”) is added to the right of the system. In the last two cases we also include +a formal asymptotic analysis which shows convergence to the KPZ fixed point. +CONTENTS +1. +Introduction +1 +2. +Motivating examples: interacting particle systems +4 +2.1. +Continuous time TASEP +6 +2.2. +Discrete time TASEPs with right Bernoulli jumps +6 +2.3. +Caterpillars with right Bernoulli jumps +6 +2.4. +Other types of caterpillars +7 +2.5. +PushASEP +7 +3. +A biorthogonal ensemble formula for determinantal measures +8 +4. +An explicit biorthogonalization scheme +11 +4.1. +Setting +11 +4.2. +The biorthogonalization problem +12 +4.3. +The boundary value problem +15 +4.4. +Main result: representation in terms of random walk hitting times +16 +5. +Application to particle systems +24 +5.1. +Two-speed variants of TASEP +24 +5.2. +Mixed sequential and parallel TASEP and KPZ fixed point limit +28 +5.3. +One long caterpillar +33 +References +36 +1. INTRODUCTION +In the (continuous time, one-dimensional) totally asymmetric simple exclusion process (TASEP), +particles perform totally asymmetric nearest neighbour random walks on the integer lattice Z subject to +the exclusion rule: each particle independently attempts jumps to the neighbouring site to the right at +rate 1, the jumps being allowed only when the destination site is empty. Despite its simplicity, TASEP +1 +arXiv:2301.13739v1 [math.PR] 31 Jan 2023 + +EXACT SOLUTION OF TASEP AND VARIANTS WITH INHOMOGENEOUS SPEEDS AND MEMORY LENGTHS +2 +presents a very rich asymptotic behavior, and due to its tractability it has become a paradigmatic model +in out-of-equilibrium statistical physics. +Much of the interest in TASEP arises from the central role it plays as a member of the KPZ +universality class, a broad collection of physical and probabilistic models including particle systems, +one-dimensional random growth models, directed polymers, stochastic reaction-diffusion equations, +and random stirred fluids. Models in the KPZ class share a common asymptotic fluctuation behavior, +identified by their (in general, conjectural) convergence, under the characteristic 1:2:3 scaling, to a +universal, scale-invariant Markov process known as the KPZ fixed point, which was first constructed in +[MQR21] as the scaling limit of TASEP. This 1:2:3 scaling refers to the ratios between the exponents +used to rescale the fluctuations, space and time: for KPZ models, as t → ∞ one has fluctuations +growing like t1/3 with non-trivial spatial correlations arising at a scale of t2/3. +What makes TASEP special in this context is that its distribution can be expressed as a marginal of +an (in general, signed) determinantal point process. For general initial data, this was first discovered +in [Sas05; Bor+07] (building on exact determinantal formulas for the transition probabilities of the +system derived in [Sch97] using the coordinate Bethe ansatz), where it was used to study the special +case of periodic initial data, with particles initially occupying sites at 2Z. There the associated spatial +fluctuations in the long time 1:2:3 scaling limit were derived; they lead to the Airy1 process, whose +marginals are given by the Tracy-Widom GOE distribution from Random Matrix Theory [TW96]. For +another choice of special initial data known as step, where particles initially occupy sites at Z<0, there +is an even richer algebraic structure, and the analogous scaling limit had been known since the early +2000’s [Joh00; PS02; Joh03], leading to the Airy2 process and Tracy-Widom GUE marginals [TW94]. +The method employed in [Sas05; Bor+07] leads to an expression for the multi-point distribution +of TASEP as the Fredholm determinant of a kernel defined implicitly as the solution of a certain +biorthogonalization problem which depends on the initial data of the system. For step initial data, the +biorthogonalization turns out to be (in a certain, concrete sense) trivial, while for periodic initial data +the authors were able to solve it explicitly. The solution of the biorthogonalization problem for general +initial data was discovered in [MQR21], and leads to a kernel which can be expressed in terms of the +hitting time of a certain random walk to a curve defined by the initial data. In the 1:2:3 scaling limit, +this kernel naturally rescales to an analogous kernel defined in terms of Brownian hitting times, whose +Fredholm determinants yield the finite dimensional distributions of the KPZ fixed point. +TASEP is part of a family of exactly solvable models in the KPZ class for which a description in +terms of biorthogonal ensembles is available. Besides continuous time TASEP, this family includes +discrete time TASEP with both sequential and parallel update, with pushing and blocking dynamics, +and with Bernoulli and geometric jumps, as well several generalizations. In [MR23] we extended the +explicit biorthogonalization method of [MQR21] to a general class of determinantal measures which +includes these models and several others. The purpose of this paper is to develop a further generalization +of the method to cover extensions of these models to the case where particles have different speeds +and different memory lengths. By the speed of a particle we mean, in the context of continuous time +TASEP, simply its jump rate. The memory length of a particle, on the other hand, is easier to interpret in +the case of discrete time TASEP: it refers to the amount of time a site remains blocked after a particle +occupying it leaves. Memory lengths 0 and 1 translate respectively into the standard discrete time +TASEPs with sequential and parallel updates. For more general memory lengths, the system is no +longer Markovian, but it can be reinterpreted as a Markovian system of interacting caterpillars, which +occupy a variable number of sites in the lattice. The case of systems of caterpillars with equal lengths +associated to TASEP and its variants was studied in [MR23]. +The biorthogonal ensemble representation for TASEP-like systems in the case of inhomogeneous +speeds is well known [BF08; BFS08]. Those papers focus on two particle systems, PushASEP (a +combination of TASEP with blocking and pushing dynamics) and TASEP with parallel update, for +which they compute scaling limits in the case of periodic initial data. They actually obtain more general +multipoint distributions along “space-like paths” (i.e. the distribution of collections of particles at +different times, but subject to a certain ordering in space-time). As we will explain in the next section, +the case of TASEP with general memory lengths, or caterpillars, can be recovered by considering an + +EXACT SOLUTION OF TASEP AND VARIANTS WITH INHOMOGENEOUS SPEEDS AND MEMORY LENGTHS +3 +extension of this setting to one where particles are also allowed to start evolving at different times. +The biorthogonal ensemble representation in this setting was obtained in some generality in [MR23], +but the explicit solution of the biorthogonalization problem in that paper was restricted to the case +corresponding to equal caterpillar lengths and equal speeds. +Our goal in this paper is thus to complete this program in the general setting of inhomogeneous +speeds and memory lengths, by providing an explicit formula for the biorthogonal kernel appearing in +these formulas which is amenable to asymptotic analysis. As in [MR23], we will actually work in a +more general, abstract setting, which will cover all the examples we have mentioned so far, and several +more. +To illustrate the use of such formulas we include three applications. In the first one we will consider +the two-speed setting studied in [BFS09]. In that paper, the authors considered continuous time TASEP +with a leading block of particles with a different speed. They obtained explicit contour integral formulas +in the case of periodic initial data, for which they were able to perform the asymptotic analysis necessary +to describe its limiting behavior depending on the parameters of the model (the length and speed of +the leading block). Here we will obtain similar formulas for systems in a slightly more general setting +which includes continuous and discrete time TASEP with both sequential and parallel update. Our +result is of course applicable to more general initial data, and the resulting formulas can be used to +perform asymptotic analysis of these formulas in the general case, but we leave this for future work. +In the next two applications we consider discrete time TASEP with equal speeds but different lengths: +in the first one we study a system which mixes particles updating sequentially and in parallel, while in +the second one we consider the case where a single long caterpillar is placed at the right of the system. +In both cases we derive, formally, their limits under the proper KPZ 1:2:3 rescaling. +We finish this introduction by mentioning that in the particular case of discrete time TASEP with right +Bernoulli jumps, the explicit kernel for inhomogeneous speeds was obtained recently, and independently, +by Bisi, Liao, Saenz, and Zygouras [Bis+22]. In that paper they also provide a new derivation of +the biorthogonal ensemble representation of the system which uses combinatorial properties of the +Robinson-Schensted-Knuth correspondence together with intertwining relations to express the transition +kernel of the system in terms of an ensemble of non-intersecting lattice paths. +Outline. In Sec. 2 we describe several interacting particle systems (and some of their generalizations +to systems of caterpillars) in the KPZ universality class, whose distributions are particular cases of the +determinantal measure considered in Sec. 3. Under quite general assumptions, we prove in Thm. 3.3 +that a marginal of this measure can be written as a Fredholm determinant of a kernel described implicitly +through the solution of a biorthogonalization problem. Sec. 4 is devoted to the explicit solution of this +problem in an abstract setting, leading to an explicit formula for the kernel in Thm. 4.10. Finally, in +Sec. 5 we study this kernel and its KPZ scaling limit for the particular examples mentioned above. +Notation. We will use the same notation and conventions employed in [MR23]. We use the standard +notation N for the set of natural numbers {1, 2, . . . }, and we use N0 = N ∪ {0}. For n ∈ N we define +the set �n� = {1, . . . , n}. For N ≥ 2 the Weyl chamber is +ΩN = {⃗x ∈ ZN : xN < xN−1 < · · · < x1}. +Throughout the paper we consider various kernels K : Z × Z −→ R, which we identify with integral +operators acting on suitable families of functions f : Z → C as +Kf(x) = +� +y∈Z +K(x, y)f(y). +(1.1) +We prefer not to specify precisely the domains of such operators and always interpret them in terms +of absolutely convergent sums (1.1). The composition of two such operators K and L is defined as +KL(x, y) = � +z∈Z K(x, z)L(z, y), provided that the sum is absolutely convergent. Then we say that +K−1 is an inverse of an operator K if KK−1(x, y) = K−1K(x, y) = I(x, y), where I is the identity +operator I(x, y) = 1x=y. We use the standard notation K∗(x1, x2) = K(x2, x1) for the adjoint. + +EXACT SOLUTION OF TASEP AND VARIANTS WITH INHOMOGENEOUS SPEEDS AND MEMORY LENGTHS +4 +Our kernels will often be defined in terms of functions written as contour integrals. The contours in +these integrals will be usually γr, a circle in the complex plane with radius r and centered at the origin. +Whenever the contour is different, it will be specified explicitly. +For a closed subset U of C we say that a complex function f is analytic on U if it is analytic on +some open domain which contains U. A particular case of interest will be when U is the closed annulus +on the complex plane centered at the origin and with radii 0 < r < R, which we denote by Ar,R. +Finally, for a fixed vector a ∈ Rm and indices n1 < · · · < nm we let +χa(nj, x) = 1x>aj +and +¯χa(nj, x) = 1x≤aj, +(1.2) +which we also regard as multiplication operators acting on the space ℓ2({n1, . . . , nm} × Z). +2. MOTIVATING EXAMPLES: INTERACTING PARTICLE SYSTEMS +The main result of this paper will be stated in Sec. 4 in an abstract setting which, in general, does +not necessarily originate from determinantal measures connected to particle systems. In order to +motivate that setting and to provide some physical intuition, we begin by presenting in this section some +particular cases of that result, stated in the context of the variants of TASEP and systems of interacting +caterpillars with inhomogeneous jump speeds and lengths (equivalently, discrete time TASEPs with +inhomogeneous speeds and memory lengths) discussed in the introduction. We will begin by presenting +the general formula which we will obtain for the multipoint distribution of this type of systems. At this +stage we will not be precise about the details of and assumptions on the systems to which this result +will apply. Later on we will introduce particular cases corresponding to several particle systems and +systems of caterpillars to which the result will apply. The precise setting for our general result will be +provided in Secs. 3 and 4. +A (forward) caterpillar of length L ≥ 1 is an element X of the set +KL = +� +(X1, . . . , XL) ∈ ZL : Xi − Xi+1 ∈ {0, 1}, i ∈ �L − 1� +� +. +A caterpillar thus has L ordered sections X1 ≥ X2 ≥ · · · ≥ XL; we will call X1 the head of the +caterpillar. A system of N ≥ 2 interacting caterpillars of lengths ⃗L = (L1, . . . , LN) ≥ 1 takes values +in the set +ΩN,⃗L = +� +X = (X(1), . . . , X(N)): X(i) ∈ KLi : X1(i + 1) < XLi(i), i ∈ �N − 1� +� +, +i.e., the caterpillar X(i) has length Li and no two caterpillars overlap. For X ∈ ΩN,⃗L we define +Xhead = (X1(i) : i ∈ �N�) ∈ ΩN to be the vector of heads of the caterpillars, which can be thought of +as N particles located at the sites X1(i) for i ∈ �N�. +Now for fixed speeds vi > 0, i ∈ �N�, we will consider certain specific dynamics for caterpillars +Xt ∈ ΩN,⃗L in time t, which is either in R+ or in N0. The simplest example is the case of continuous +time TASEP, where all caterpillars have length 1 and the i-th one jumps to the right at rate vi except +that jumps onto already occupied sites are forbidden. We provide below other examples of dynamics +of caterpillars to which our results are applicable; those with lengths 2 or more all evolve in discrete +time (we remark that there is also a generalized version of continuous time TASEP which has the flavor +of a length-2 system, but its definition does not quite fit the setting of this section, although it can be +analyzed in the framework of Sec. 4, see [MR23, Sec. 3.3]). +We say that the system of caterpillars Xt has initial condition ⃗y ∈ ΩN if X0 ∈ ΩN,⃗L is given by +X1 +0(k) = · · · = XLk +0 (k) = yk for each k ∈ �N�; in words, the k-th caterpillar starts with all its +sections at yk. With a little ambiguity, we will write in this case X0 = ⃗y ∈ ΩN. Throughout the paper +we will be restricted to work in the case when the initial condition ⃗y is in the set +ΩN(⃗L) = {⃗x ∈ ZN : xi − xi+1 ≥ (Li − 1) ∨ 1 for i = 1, . . . , N − 1}. +The following holds for all of the systems of caterpillars considered in this paper: for fixed initial +condition ⃗y ∈ ΩN(⃗L), for any t ≥ 0 and 1 ≤ n1 < · · · < nm ≤ N, and for any real a1, . . . , am, the + +EXACT SOLUTION OF TASEP AND VARIANTS WITH INHOMOGENEOUS SPEEDS AND MEMORY LENGTHS +5 +distribution function of the heads of the system can be written in the form +P +� +Xhead +t +(i) > ai, i = 1, . . . , m +� += det +� +I − ¯χaKt ¯χa +� +ℓ2({n1,...,nm}×Z), +(2.1a) +where the kernel Kt is given implicitly in terms of the solution of a certain biorthogonalization problem +which involves the initial data ⃗y. The precise form of the biorthogonal kernel Kt is presented in Sec. 3. +We will see shortly one way to interpret the restriction to X0 = ⃗y with ⃗y ∈ ΩN(⃗L) in our setting. The +restriction actually arises as a requirement in the proof of this representation (which for systems of +caterpillars can be found in [MR23]), but it appears in general to be necessary for this representation +to hold. The same restriction will be crucial in the proof of our main abstract result, Thm. 4.10, from +which the results presented in this section will be corollaries. +Our main result will provide an explicit formula for the kernel Kt appearing in (2.1a). The formula +follows from solving explicitly the biorthogonalization problem defining the kernel for general initial +condition and representing the result in terms of a hitting problem for a certain random walk to a +curve defined by the initial data. This representation is such that the appropriate scaling limits can be +obtained naturally, by computing the limits of the kernels involved in the formula and recognizing that +the random walk hitting problem converges to a similar problem for a Brownian motion; we present +examples of this in Secs. 5.2 and 5.3 (see also [MQR21] where the scheme was implemented in detail +for continuous time TASEP). +In order to state our formula we first need to make several definitions. Consider a meromorphic +function ϕ : U −→ C, where the domain U ⊆ C contains 0 and all values vi, which is analytic and +non-zero in an annulus Ar,¯r with radii 0 < r < min vi and ¯r > max vi. Fix also a real parameter +θ ∈ (r, min vi). We introduce the kernels +Q(ℓ,n](x, y) = +1 +2πi +� +γr +dw +θx−y +wx−y−n+ℓ+1 +n +� +i=ℓ+1 +αiϕ(w)Li−1−1 +vi − w +with αi = vi−θ +θ ϕ(θ)1−Li−1, integer 0 ≤ ℓ < n and L0 = 1, and +Q+ +(ℓ,n](x, y) = +1 +2πi +� +γr +dw +θx−y +wx−y−n+ℓ+1 +n +� +i=ℓ+1 +α+ +i ϕ(w)Li−1 +vi − w +, +with α+ +i = vi−θ +θ ϕ(θ)1−Li. These two kernels are Markov. We let B+ +m be the time-inhomogeneous +random walk which has transitions from time m−1 to time m, m ≥ 1, with step distribution Q+ +(m−1,m]. +For a fixed initial condition ⃗y ∈ ΩN(⃗L) we define the stopping time +τ + = min{m = 0, . . . , N − 1 : B+ +m > ym+1}, +i.e., τ + is the hitting time of the strict epigraph of the “curve” (ym+1)m=0,...,n−1 by the random walk +(B+ +m)m≥0 (we set τ + = ∞ if the walk does not go above the curve by time N − 1). +Next for integer n ≥ 1 and 0 ≤ m < n and for a real t ≥ 0 we define the kernels +S−n(x, y) = +1 +2πi +� +γr +dw +θy−x +wy−x+n+1 ϕ(w)t +�n +i=1(vi − w) +�n−1 +i=1 α+ +i ϕ(w)Li−1 , +¯S(m,n](x, y) = − 1 +2πi +� +Γ⃗v +dw +θx−y +wx−y−n+m+1 ϕ(w)−t +�n−1 +i=m+1 α+ +i ϕ(w)Li−1 +�n +i=m+1(vi − w) +, +and +¯Sepi(⃗y) +n +(x, y) = EB+ +0 =x +� ¯S(τ +,n](B+ +τ +, y)1τ + ai, i = 1, . . . , m +� += det +� +I − ¯χaKt ¯χa +� +ℓ2({n1,...,nm}×Z) +for t ≥ 0, 1 ≤ n1 < · · · < nm ≤ N, and a1, . . . , am ∈ R. Our result, which is valid for all the systems +of caterpillars considered in this paper, is that the kernel Kt is given by +Kt(ni, xi; nj, xj) = −Q(ni,nj](xi, xj)1ni Xt(2) > · · · > Xt(N) evolving as follows: the i-th particle tries to make unit jumps +to the right at rate vi > 0, but attempted jumps are permitted only if the destination site is empty. Except +for the exclusion restriction, jumps by different particles occur independently. +Proposition 2.1. The distribution function of Xt = Xhead +t +for continuous time TASEP is given by (2.1) +with ϕ(w) = ew and Li = 1 for all i ∈ �N�. +2.2. Discrete time TASEPs with right Bernoulli jumps. Next we introduce discrete time TASEP +with right Bernoulli jumps and with inhomogeneous speeds. There are two natural variants of this +model: sequential and parallel update. Fix speed parameters pi ∈ (0, 1), i ∈ �N�. Again we have +particles occupying Z at locations Xt(1) > Xt(2) > · · · > Xt(N). Now to go from time t to time +t + 1, particles are updated one by one, from right to left in the sequential case and from left to right +in the parallel case, as follows: the i-th particle jumps to the right with probability pi and stays put +with probability qi = 1 − pi, but if a particle tries to jump on top of an occupied site, the transition is +blocked. Note that in the case of sequential update, a particle trying to jump at time t is blocked by the +position of its right neighbor at time t + 1, while in the case of parallel update the particle is blocked by +its neighbor at time t. +Proposition 2.2. The distribution function of Xt = Xhead +t +for discrete time TASEP with right Bernoulli +jumps is given by (2.1) with ϕ(w) = 1 + w and vi = pi/qi, and with Li = 1 for all i ∈ �N� in the case +of sequential update and Li = 2 for all i ∈ �N� in the parallel case. +2.3. Caterpillars with right Bernoulli jumps. Now for fixed parameters pi ∈ (0, 1), i ∈ �N�, we +define a dynamics for caterpillars Xt ∈ ΩN,⃗L in discrete time t ∈ N0. The transition from time t to time +t + 1 occurs in the following way, with the positions of the caterpillars being updated consecutively for +i ∈ �N� (i.e., from right to left): +• The head of the i-th caterpillar makes a unit step to the right with probability pi ∈ (0, 1) (i.e., +X1 +t+1(i) = X1 +t (i) + 1), provided that the destination site is empty. Otherwise it stays put (i.e., +X1 +t+1(i) = X1 +t (i)). +• The remaining sections of the i-th caterpillar move according to Xj +t+1(i) = Xj−1 +t +(i), j = 2, . . . , Li. +In words, the heads jump as in TASEP with right Bernoulli jumps, but are blocked by the whole +caterpillar to its right, while each of the remaining sections of each caterpillar follows the movement of +the section to its right in the previous time step. One sees directly that the new configuration Xt+1 is +again in ΩN,⃗L and that this choice of dynamics defines a Markov chain on ΩN,⃗L. +It is easy to see from the definition of its dynamics that the heads in this system of caterpillars evolve +as a version of discrete time TASEP, with right to left update, where particle i at time t is blocked by +particle i − 1 according to its location at time t − Li−1, which provides the interpretation of caterpillars +as encoding the memory lengths of the system. +Based on the last observation, it is natural to couple the model with a version of sequential TASEP +with different starting times. In this extension of TASEP we fix starting times 0 ≥ T1 ≥ T2 ≥ · · · ≥ TN +and an initial configuration of particles ⃗y ∈ ΩN, and run the process with particle i starting its evolution +at Xr-B +Ti (i) = yi at time Ti. In other words, from time TN to time TN−1 only the N-th particle moves +with the other particles staying put, then at time TN−1 particle N − 1 starts moving, and the two move +together up to time TN−2, when particle N −2 joins them, and so on. Throughout its evolution, particle +i jumps to the right with probability pi, provided that the target site is empty. The coupling between the +models is given in the following result (which for constant Lj appeared as Lem. 2.1 in [MR23]), and +follows directly from the definitions of the two models: + +EXACT SOLUTION OF TASEP AND VARIANTS WITH INHOMOGENEOUS SPEEDS AND MEMORY LENGTHS +7 +Lemma 2.3. Let the process Xr-B +t +start at initial times #—T = (− � +1≤j 0 for each i. +The following result, whose proof can be found in [MR23] (Lem. 5.6), will be often in this section +and the next one to compute compositions of the kernels of a certain form: +Lemma 3.1. Consider two kernels S1 and S2 given by +Si(x, y) = +1 +2πi +� +γ +dw +θx−y +wx−y+1 φi(w), +where φ1, φ2 are complex functions which are both analytic on an annulus Ar1,r2 for some r1 < r2 +and γ is any simple, positively oriented closed contour contained in Ar1,r2. Then the sum defining the +product S1S2 is absolutely convergent and +S1S2(x, y) = +1 +2πi +� +γ +dw +θx−y +wx−y+1 φ1(w)φ2(w). +Define the kernel +Vi(x1, x2) = +1 +2πi +� +γ¯ρ +dw (w − vi)−1 +wx2−x1 += vx1−x2 +i +1x1≥x2 +for i ∈ �N� and x1, x2 ∈ Z, where ¯ρ > maxi vi. The inverse of Vi is +V−1 +i +(x1, x2) = +1 +2πi +� +γρ +dw +w − vi +wx2−x1+2 = 1x1=x2 − vi1x1=x2+1, +where ρ > 0. For k ∈ �N� we set +V[k] = V1 V2 · · · Vk, +V[−k] = V−1 +k +· · · V−1 +2 +V−1 +1 +, +with the convention V[0] = I. The kernels of these operators can be written explicitly (using Lem. 3.1) +as +V[k](x1, x2) = +1 +2πi +� +γ¯ρ +dw +�k +i=1(w − vi)−1 +wx2−x1−k+1 +, +V[−k](x1, x2) = +1 +2πi +� +γρ +dw +�k +i=1(w − vi) +wx2−x1+k+1 . +We also introduce the (multiplication) kernels +ϑk(x1, x2) = v−x1 +k +1x1=x2, +ϑ−k(x1, x2) = vx2 +k 1x1=x2. +Next we introduce a kernel +Rt(x1, x2) = +1 +2πi +� +γρ +dw +ϕ(w)t +wx2−x1+1 , +which depends on a given complex function ϕ. We will assume that ϕ and the radii ρ and ¯ρ satisfy the +following: +Assumption 3.2. + +EXACT SOLUTION OF TASEP AND VARIANTS WITH INHOMOGENEOUS SPEEDS AND MEMORY LENGTHS +9 +(i) ϕ: U −→ C, where the domain U ⊆ C contains 0 and all values vi, and ϕ has at most a finite +number of singularities in U. +(ii) ϕ is analytic on an annulus Aρ,¯ρ ⊆ U with radii 0 < ρ < mini vi and ¯ρ > maxi vi. +(iii) ϕ(w) ̸= 0 for all w ∈ Aρ,¯ρ. +For k, ℓ ∈ �N� and t ∈ T we define the function +Fk,ℓ(x1, x2; t) = +� +ϑk V[k]Rt V[−ℓ]ϑ−ℓ +� +(x1, x2) += +1 +2πi +� +γ¯ρ +dw (w/vk)x1 +(w/vℓ)x2 +�ℓ +i=1(w − vi) +�k +i=1(w − vi) +ϕ(w)t +wℓ−k+1 . +Finally, for ⃗y, ⃗x ∈ ΩN and s ≤ t, we define +Gs,t(⃗y, ⃗x) = +� N +� +i=1 +ϕ(vi)s−t +� +det +� +Fk,ℓ(yk, xℓ; t − s) +� +k,ℓ∈�N�. +(3.1) +The function (3.1) defines, by convolution, an (in general, signed) measure on particle configurations +in a space-time domain. We are interested in the projections of this measure to special sets known as +space-like paths, which we introduce now. For (n1, t1), (n2, t2) ∈ �N�×T we write (n1, t1) ≺ (n2, t2) +if n1 ≤ n2, t1 ≥ t2 and (n1, t1) ̸= (n2, t2). We write n = (n, t) to denote elements of �N� × T. Then +we define the set of space-like paths as +SN = +� +m≥1 +� +(ni)i∈�m� : ni ∈ �N� × T, ni ≺ ni+1 +� +. +For a space-like path S = {(n1, t1), . . . , (nm, tm)} ∈ SN and for ⃗y ∈ ΩN and ⃗x ∈ Ωm, we set +G+ +S (⃗y, ⃗x) = +� +⃗x(ti)∈Ωni: +xni(ti)=xi,i∈�m� +G0,tm(⃗y, ⃗x(tm)) +m−1 +� +i=1 +Gti+1,ti(⃗x≤ni(ti+1), ⃗x(ti)). +(3.2) +We use ⃗x(ti) to parametrize vectors by time points. In particular, we postulate that ⃗x(ti) and ⃗x(ti+1) are +different vectors even if ti = ti+1 (this slight abuse of notation, which makes clear the correspondence +between vectors and the associated time points, will simplify the presentation later on). For TN ≤ +· · · ≤ T1 and for ⃗x ∈ ΩN and ⃗y ∈ ZN, we set +G−#— +T (⃗y, ⃗x) = +� N +� +i=1 +ϕ(vi)Ti +� +det +� +Fk,ℓ(yk, xℓ; −Tk) +� +k,ℓ∈�N�. +(3.3) +Convolving (3.2) and (3.3) in the case T1 ≤ tm, we define +G #— +T ,S(⃗y, ⃗x) = +� +⃗z∈ΩN +G−#— +T (⃗y, ⃗z)G+ +S (⃗z, ⃗x). +Our goal is to obtain a formula for the following integrated version of G #— +T ,S: for ⃗y ∈ ZN, ⃗a ∈ Zm, +M #— +T ,S(⃗y,⃗a) = +� +⃗x∈Ωm: +xi>ai,i∈�m� +G #— +T ,S(⃗y, ⃗x). +(3.4) +In words, one should think of a collection of N particles evolving in time, such that the i-th particle +starts at location yi at time Ti. Then for a fixed space-like path S, containing pairs (ni, ti), G #— +T ,S(⃗y, ⃗x) +defines a measure on ⃗x ∈ Ωm, with the i-th element of ⃗x intepreted as the position of the ni-th particle +at time ti. M #— +T ,S(⃗y,⃗a) is then the measure of the set of all particle configurations so that the ni-th +particle is located strictly to the right from ai at time ti. +Before stating the result we need to introduce a certain space of functions Vn(⃗v, θ). For fixed n ∈ N, +θ > 0 and given a vector ⃗v as above, let u1(n) < u2(n) < · · · < uν(n)(n) denote the distinct values + +EXACT SOLUTION OF TASEP AND VARIANTS WITH INHOMOGENEOUS SPEEDS AND MEMORY LENGTHS +10 +among the first n entries v1, . . . , vn of ⃗v and let βk(n) be the multiplicity of uk(n) among these entries +(in particular, �ν(n) +k=1 βk(n) = n). Then we define +Vn(⃗v, θ) = span +� +x ∈ Z �−→ xℓ(uk(n)/θ)x : 1 ≤ k ≤ ν(n), 0 ≤ ℓ < βk(n) +� +. +(3.5) +Finally, given a space like path S = {n1, . . . , nm} as above and a fixed vector a ∈ Rm we extend +the notation introduced in (1.2) to χa(nj, x) = 1 − ¯χa(nj, x) = 1x>aj. +Theorem 3.3. Let the function ϕ and the values vi satisfy Assum. 3.2, and fix TN ≤ · · · ≤ T1 and +a space-like path S, the time points of which are all greater than T1. Then the function (3.4) can be +written as +M #— +T ,S(⃗y,⃗a) = det +� +I − ¯χaK ¯χa +� +ℓ2(S×Z), +(3.6) +where det is the Fredholm determinant and: +(1) The kernel K : (S × Z)2 −→ R depends on #—T and ⃗y, and is given by +K(ni, xi; nj, xj) = −φ(ni,nj)(xi, xj)1ni≺nj + +nj +� +k=1 +Ψni +ni−k(xi)Φnj +nj−k(xj), +(3.7) +for ni = (ni, ti) and nj = (nj, tj) in S. +(2) For ni and nj as before, such that ni ≺ nj, the function φ(ni,nj) is defined as +φ(ni,nj)(xi, xj) = +1 +2πi +� +γρ +dw θxi−xjϕ(w)ti−tj +wxi−xj−nj+ni+1 +nj +� +k=ni+1 +(vk − w)−1. +(3) For n = (n, t) ∈ S and k ∈ �N�, the function Ψn +n−k is given by +Ψn +n−k(x) = +1 +2πi +� +γρ +dw θx−ykϕ(w)t−Tk +wx−yk+n−k+1 +�n +i=1(vi − w) +�k +i=1(vi − w) +. +(4) The functions Φn +n−k, for k ∈ �n� and n = (n, t), are uniquely characterized by: +(a) The biorthogonality relation � +x∈Z Ψn +ℓ(x)Φn +k(x) = 1k=ℓ, for each k, ℓ = 0, . . . , n − 1. +(b) span{x ∈ Z �−→ Φn +k(x) : 0 ≤ k < n} = Vn(⃗v, θ). +In applications to particle systems we are usually interested in the case S = {(i, t + Ti) : i ∈ �N�} +for some T1 ≥ · · · ≥ TN, corresponding to starting particle i at time t + Ti. In this case each point +n = (n, t) in S is determined by its first component n and the the kernel in (3.7) can be reexpressed as +a kernel K : (�N� × Z)2 −→ R given by +K(ni, xi; nj, xj) = −φ(ni,nj)(xi, xj)1ni0)N and ⃗y ∈ ZN, +which play the role of the particle speeds and initial positions1. +4.1. Setting. We consider a family a strictly positive measure (qℓ(i))i∈Z on Z, ℓ ∈ �N − 1�, which +satisfies: +Assumption 4.1. +(i) For each ℓ ∈ �N − 1� there is a κℓ ∈ N0 such that qℓ(i) = 1 for all i > κℓ, +(ii) There is a θ ∈ (0, minj∈�N� vj) such that or each ℓ ∈ �N − 1�, � +i∈Z qℓ(i)(θ/vℓ)i < ∞ and +� +i∈Z qℓ(i)(θ/vℓ+1)i < ∞. +Next we introduce a function aℓ(w), ℓ ∈ �N − 1�, which is constructed out of the measures qℓ +through the following Laurent series: +aℓ(w) = +� +i≤κℓ +(qℓ(i + 1) − qℓ(i))(w/vℓ)i. +(4.1) +For convenience we also set +q0(i) = 1i>0, +κ0 = 0 +and +a0(w) = 1. +(4.2) +We also consider a fixed complex function ψ. We assume that ψ and the aℓ’s satisfy: +Assumption 4.2. There are radii r and ¯r satisfying 0 < r < θ < min vi, and ¯r > max vi (with θ given +in Assum. 4.1) such that aℓ(w) is analytic on {w ∈ C: |w| ≥ r} while 1/aℓ(w), ψ(w) and 1/ψ(w) are +analytic and non-zero on the annulus Ar,¯r. +Using the functions aℓ we introduce the Markov kernels +Qℓ(x, y) = αℓ +2πi +� +γr +dw θx−y +wx−y +aℓ−1(w) +vℓ − w +(4.3) +for x, y ∈ Z and ℓ ∈ �N�, with +αℓ = +vℓ − θ +aℓ−1(θ)θ = +1 +� +i∈Z(θ/vℓ)iqℓ−1(i). +Due to Assump. 4.1(ii), the sum in this expression is finite. Note that since r < vℓ, the contour γr in +the integral in (4.3) includes only the pole at w = 0. Using (4.1) we can write explicitly, for ℓ ∈ �N�, +Qℓ(x, y) = αℓ(θ/vℓ)x−yqℓ−1(x − y). +(4.4) +which shows that Qℓ is indeed a Markov kernel (by the definition of αℓ; recall also that qℓ−1 is a positive +measure). In particular, since for x − y > κℓ we have qℓ(x − y) = 1, +Qℓ(x, y) = αℓ(θ/vℓ)x−y +∀ x − y > κℓ−1. +(4.5) +1In applications we usually consider systems with infinitely many particles but where the evolution of the first N particles +is independent of the remaining ones; since our formulas will yield the finite-dimensional distributions of the system, this +restriction to ⃗v and ⃗y of size N is not consequential. + +EXACT SOLUTION OF TASEP AND VARIANTS WITH INHOMOGENEOUS SPEEDS AND MEMORY LENGTHS +12 +Note in particular that Q1 is simply the transition kernel of a geometric random walk: +Q1(x, y) = v1 − θ +θ +(θ/v1)x−y1x>y. +Remark 4.3. Note that we have defined Qℓ using the function aℓ−1. It might seem more natural to use +aℓ in the definition, but in our setting this is not the case: thinking about the systems of caterpillars +from Sec. 2.3, the dynamics of the head of the ℓ-th caterpillar depends on its “speed” pℓ and on the +length Lℓ−1 of the caterpillar to its right, but not on its own length. This is also why we do not need to +introduce the measures qℓ and the functions aℓ for ℓ = N. +The inverse kernel of Qℓ is +Q−1 +ℓ (x, y) = α−1 +ℓ +2πi +� +γr +dw +θx−y +wx−y+2 +vℓ − w +aℓ−1(w). +(4.6) +Given integers 0 ≤ ℓ < n we denote +Q(ℓ,n](x, y) = Qℓ+1 · · · Qn(x, y) = +1 +2πi +� +γr +dw +θx−y +wx−y−n+ℓ+1 +n +� +i=ℓ+1 +αiai−1(w) +vi − w +, +(4.7) +whose inverse is +Q−1 +(ℓ,n](x, y) = Q−1 +n · · · Q−1 +ℓ+1(x, y) = +1 +2πi +� +γr +dw +θx−y +wx−y+n−ℓ+1 +n +� +i=ℓ+1 +vi − w +αiai−1(w), +where we used Lem. 3.1. We note that these formulas make sense also for ℓ = n if we postulate that +the (empty) products in this case are equal to 1: Q(n,n] = Q−1 +(n,n] = I. We also set Q[ℓ,n] = Q(ℓ−1,n] +for 1 ≤ ℓ ≤ n and Q[1,k) = Q[1,k−1] for k ≥ 2. +Similarly, using the function ψ we define a kernel R and its inverse R−1 as +R(x, y) = +1 +2πi +� +γr +dw +θx−y +wx−y+1 ψ(w), +R−1(x, y) = +1 +2πi +� +γr +dw +θx−y +wx−y+1 +1 +ψ(w). +(4.8) +4.2. The biorthogonalization problem. For k ∈ �n�, we define +Ψn +n−k(x) = RQ−1 +(k,n](x, yk) = θx−y +2πi +� +γr +dw +ψ(w) +wx−yk+n−k+1 +n +� +i=k+1 +vi − w +αiai−1(w). +(4.9) +We extend this definition to k > n by setting +Ψn +n−k(x) = RQ(n,k](x, yk) = θx−y +2πi +� +γr +dw +ψ(w) +wx−yk+n−k+1 +k +� +i=n+1 +αiai−1(w) +vi − w +. +(4.10) +We consider a family of functions (Φn +k)k=0,...,n−1 characterized by: +(⋆) The biorthogonality relation � +x∈Z Ψn +ℓ (x)Φn +k(x) = 1k=ℓ for each k, ℓ = 0, . . . , n − 1. +(⋆⋆) span{x ∈ Z �−→ Φn +k(x) : 0 ≤ k < n} = Vn(⃗v, θ), where the set Vn(⃗v, θ) is defined in (3.5). +When all values vi are equal to 1, this biorthogonalization problem simplifies to the one considered in +[MR23, Sec. 5.2]. +Existence and uniqueness of the solution to this biorthogonalization problem is proved in Lem. 4.6 +below, while an exact solution is provided in Thm. 4.9. +It will be convenient in the following computations to employ a different basis of the space (3.5): +Bn(⃗v, θ) = +� +en +k,ℓ(x) : 1 ≤ k ≤ ν(n), 0 ≤ ℓ < βk(n) +� +, +(4.11) +where the basis functions are +en +k,ℓ(x) = (x)ℓ(uk(n)/θ)x, +(4.12) + +EXACT SOLUTION OF TASEP AND VARIANTS WITH INHOMOGENEOUS SPEEDS AND MEMORY LENGTHS +13 +(x)ℓ = x(x − 1) · · · (x − ℓ + 1) is the falling factorial and, we recall, ν(n) and βk(n) where defined in +the paragraph preceding (4.11). Then the space (3.5) can be expressed as follows: +Vn(⃗v, θ) = span +� +x ∈ Z �−→ f(x) : f ∈ Bn(⃗v, θ) +� +. +(4.13) +In the following two lemmas we demonstrate how convolutions with the kernels (Q∗ +n)−1, R∗ and +(R−1)∗ act on the functions (4.12). +Lemma 4.4. Fix n ≥ 1. For each 1 ≤ k ≤ ν(n) and 0 ≤ ℓ < βk(n), there exist real values cn +k(ℓ, m), +0 ≤ m < βk(n), such that +(Q∗ +n)−1en +k,ℓ(x) = +ℓ +� +m=0 +cn +k(ℓ, m)en +k,m(x). +(4.14) +Moreover, cn +k(ℓ, ℓ) ̸= 0 if vn ̸= uk(n). In the case vn = uk(n) we have cn +k(ℓ, ℓ) = 0 and cn +k(ℓ, ℓ − 1) ̸= +0, where the latter holds if ℓ ≥ 1. In particular, the operator (Q∗ +n)−1 maps Vn(⃗v, θ) to Vn−1(⃗v, θ), with +the convention V0(⃗v, θ) = {0}. +Proof. We need to prove only the expansion (4.14) and the stated properties of the coefficients cn +k(ℓ, m), +since the last statement in the lemma follows from those. +From (4.6) and (4.11) we have +(Q∗ +n)−1en +k,ℓ(z) = +� +x∈Z +en +k,ℓ(x)Q−1 +n (x, z) = +� +x∈Z +(x)ℓ +�uk(n) +θ +�x α−1 +n +2πi +� +γr +dw +θx−z +wx−z+1 +vn − w +wan−1(w). +Changing the summation variable x �−→ x + z and using the binomial identity for falling factorials +(x + z)ℓ = �ℓ +m=0 +� ℓ +m +� +(x)m(z)ℓ−m, we write the preceding expression as +ℓ +� +m=0 +� ℓ +m +� +(z)ℓ−m +�uk(n) +θ +�z � +x∈Z +(x)m +α−1 +n +2πi +� +γr +dw uk(n)x +wx+1 +vn − w +wan−1(w). +(4.15) +Now for any m ∈ Z≥0 and any complex ξ satisfying |ξ| < 1 we have +� +x≥0(x)mξx = ξm dm +dξm +� +x≥0 ξx = ξm dm +dξm +1 +1−ξ = ξm +m! +(1−ξ)m+1 , +and, similarly, in the case |ξ| > 1 we have +� +x<0(x)mξx = ξm dm +dξm +� +x<0 ξx = −ξm dm +dξm +1 +1−ξ = −ξm +m! +(1−ξ)m+1 . +Hence for the sum over x ≥ 0 in (4.15) we can deform the integration contour to γ¯r (thanks to +Assum. 4.2) so that |w| > uk(n) to get +� +x≥0 +(x)m +α−1 +n +2πi +� +γ¯r +dw uk(n)x +wx+1 +vn − w +wan−1(w) = m!α−1 +n +2πi +� +γ¯r +dw +1 +(w − uk(n))m+1 +vn − w +wan−1(w), +while for the sum over x < 0 the contour satisfies |w| < uk(n), so +� +x<0 +(x)m +α−1 +n +2πi +� +γr +dw uk(n)x +wx+1 +vn − w +wan−1(w) = −m!α−1 +n +2πi +� +γr +dw +1 +(w − uk(n))m+1 +vn − w +wan−1(w). +In these computations we used Fubini’s theorem to swap summation and integration. Since r < +uk(n) < ¯r, adding the two expressions we conclude that the sum over x in (4.15) equals +m!α−1 +n +2πi +� +Γuk(n) +dw +1 +(w − uk(n))m+1 +vn − w +wan−1(w) +where the contour Γuk(n) includes only the pole at uk(n). Using this in (4.15) together with Cauchy’s +integral formula we get +(Q∗ +n)−1en +k,ℓ(z) = α−1 +n +ℓ +� +m=0 +� ℓ +m +� +(z)ℓ−m +�uk(n) +θ +�z dm +dwm +� vn − w +wan−1(w) +� ���� +w=uk(n) +. + +EXACT SOLUTION OF TASEP AND VARIANTS WITH INHOMOGENEOUS SPEEDS AND MEMORY LENGTHS +14 +The right-hand side is in the span of the functions en +k,ℓ−m(z) for 0 ≤ m ≤ ℓ, and it can be written as +(4.14) with the constants +cn +k(ℓ, m) = α−1 +n +� +ℓ +ℓ − m +� dℓ−m +dwℓ−m +� vn − w +wan−1(w) +� ���� +w=uk(n) +. +If vn ̸= uk(n), then the preceding formula yields cn +k(ℓ, ℓ) = α−1 +n +vn−uk(n) +uk(n)an−1(uk(n)) ̸= 0. On the other +hand, if vn = uk(n), then we have cn +k(ℓ, ℓ) = 0 and additionally if ℓ ≥ 1, +cn +k(ℓ, ℓ − 1) = α−1 +n ℓ d +dw +� vn − w +wan−1(w) +� ���� +w=vn += − +α−1 +n ℓ +vnan−1(vn) ̸= 0. +□ +Lemma 4.5. The operators R∗ and (R−1)∗ map Vn(⃗v, θ) onto itself. +Proof. It is enough to prove the statement for R∗. The argument is in fact essentially the same as the +one in the proof of Lem. 4.4. Using the definition (4.8) and repeating that argument, +R∗en +k,ℓ(z) = +� +x∈Z +�uk(n) +θ +�x (x)ℓ +2πi +� +γr +dw +θx−z +wx−z+1 ψ(w) = +ℓ +� +m=0 +� ℓ +m +� dm +dwm ψ(w) +��� +w=uk(n)en +k,ℓ−m(z). +As a consequence, R∗ maps Vn(⃗v, θ) into itself, and the matrix of this map with respect to the basis +Bn(⃗v, θ) is block diagonal, with blocks indexed by the index k in (4.11), and these blocks are triangular. +Moreover, the diagonal entries in the k-th block are given by the coefficients with m = 0 in the above +sum for each ℓ; this coefficient equals ψ(uk(n)), and Assum. 4.2 guarantees that ψ(uk(n)) ̸= 0 (since +by the assumption the function ψ is analytic and non-zero in an annulus containing all speeds vi, +and hence all values uk(n)). This implies that the matrix of R∗ with respect to the basis Bn(⃗v, θ) is +non-singular, and hence that the map is onto. +□ +Lemma 4.6. There is a unique family of functions (Φn +k)k=0,...,n−1 satisfying the properties (⋆)–(⋆⋆). +Proof. Lem. 4.5 suggests that solving the problem (⋆)–(⋆⋆) is equivalent to solving the following one: +find functions +�¯Φn +k +� +k=0,...,n−1 such that +(¯⋆) (Q∗ +(n−ℓ,n])−1 ¯Φn +k(yn−k) = 1k=ℓ for each k, ℓ = 0, . . . , n − 1. +(¯⋆¯⋆) span{x ∈ Z �−→ ¯Φn +k(x) : 0 ≤ k < n} = Vn(⃗v, θ). +Indeed, we have Q−1 +(k,n](x, yk) = R−1Ψn +n−k(x) (see (4.9)), so (Q∗ +(n−ℓ,n])−1 ¯Φn +k(yn−k) is equal to +� +x∈Z R−1Ψn +n−k(x)¯Φn +k(x) and thus the solutions to these two problems are related by the one-to-one +correspondence ¯Φn +k = R∗Φn +k. Then the lemma will follow if we prove that this new problem has a +unique solution. +Property (¯⋆¯⋆) means that the solution (¯Φn +k)k=0,...,n−1 which we are looking for has to be given as +¯Φn +k(x) = � +f∈Bn ¯W(k, f)f(x). +for some square matrix ¯W = ( ¯W(k, f) : 0 ≤ k < n, f ∈ Bn), where we write Bn for Bn(⃗v, θ). With +this, showing that +�¯Φn +k +� +k=0,...,n−1 satisfies (¯⋆)–(¯⋆¯⋆) reduces to proving that the matrix ¯W can be chosen +so that property (¯⋆) is satisfied and, moreover, that the matrix is uniquely characterized by that property. +From the above formula for ¯Φn +k we have for each k, ℓ = 0, . . . , n − 1 that +(Q∗ +(n−ℓ,n])−1 ¯Φn +k(yn−k) = � +f∈Bn ¯W(k, f)(Q∗ +(n−ℓ,n])−1f(yn−k) +For a fixed k, consider the square matrix +� +Ff,ℓ +� +f∈Bn,0≤ℓ k we write (Q−1 +(n−ℓ,n])∗ = (Q−1 +(n−ℓ,n−k])∗(Q−1 +(n−k,n])∗ and (4.17a) yields +(Q−1 +(n−ℓ,n])∗hn +k(0, yn−ℓ) = (Q−1 +(n−ℓ,n−k])∗hn +k(k, yn−ℓ) = (Q−1 +(n−ℓ,n−k))∗(Q−1 +n−k)∗hn +k(k, yn−ℓ). Using +(4.17b) we have (Q−1 +n−k)∗hn +k(k, yn−ℓ) = � +z∈Z(Q−1 +(n−ℓ,n−k])∗(yn−ℓ, z)(θ/vn−k)yn−k−z, and this van- +ishes using (4.6) after a simple computation, so (Q−1 +(n−ℓ,n])∗hn +k(0, yn−ℓ) = 0. +Now we turn to (⋆⋆). From (4.17d) we have span{x ∈ Z �−→ hn +k(0, x) : ℓ ≤ k < n} = Vn(⃗v, θ) +and Lem. 4.5 implies that the same holds if we convolve the functions with (R∗)−1. +□ +4.4. Main result: representation in terms of random walk hitting times. + +EXACT SOLUTION OF TASEP AND VARIANTS WITH INHOMOGENEOUS SPEEDS AND MEMORY LENGTHS +17 +4.4.1. Preliminaries. Let, for ℓ ≥ 1, +Qℓ(x, y) = +1 +2πi +� +γr +dw θx−y−1 +wx−y +vℓ − θ +vℓ − w = vℓ − θ +θ +(θ/vℓ)x−y1x>y, +(4.23) +and for 0 < ℓ ≤ n +Q(ℓ,n](x, y) = Qℓ+1 · · · Qn(x, y) = +1 +2πi +� +γr +dw +θx−y−n+ℓ +wx−y−n+ℓ+1 +n +� +i=ℓ+1 +vℓ − θ +vi − w, +(4.24) +which coincide with the functions (4.7) if we set ai ≡ 1 and αi = 1 for all i. Then the kernels (4.3) and +(4.7) can be written as +Qm = QmAm−1, +Q(k,m] = Q(k,m]Ak · · · Am−1 +(4.25) +for m ∈ �N� and 0 ≤ k < m, with +Aℓ(x, y) = +1 +2πi +� +γr +dw +θx−y +wx−y+1 +aℓ(w) +aℓ(θ) , +(4.26) +ℓ = 0, . . . , N − 1. Note that, in view of (4.2), +A0 = I +(and thus Q1 = Q1). We also have +A−1 +ℓ (x, y) = +1 +2πi +� +γr +dw +θx−y +wx−y+1 +aℓ(θ) +aℓ(w). +(4.27) +Note that, since the contour γr does not include any of the vi’s, Q(ℓ,n](x, y) vanishes whenever +y > x + ℓ − n. Now let +¯Q(ℓ,n](x, y) = − 1 +2πi +� +Γ⃗v +dw +θx−y−n+ℓ +wx−y−n+ℓ+1 +n +� +i=ℓ+1 +vi − θ +vi − w, +(4.28) +where Γ⃗v is a simple, positively oriented contour enclosing all the vi’s but not the origin. We denote for +1 ≤ m ≤ n +(⃗v)n +m = (vm, . . . , vn), +and we claim that, for fixed x ∈ Z, +¯Q(ℓ,n](x, ·) ∈ Vn−ℓ((⃗v)n +ℓ+1, θ) +and +Q(ℓ,n](x, y) = ¯Q(ℓ,n](x, y) +∀ y < x; +in this sense, we think of ¯Q(x, ·) as an extension of Q(x, ·) to Vn−ℓ((⃗v)n +ℓ+1, θ). To see the identity +simply note that if x > y then the residue at infinity of the integrand in (4.24) vanishes and thus +by Cauchy’s formula, (4.24) equals (4.28). That ¯Q(ℓ,n](x, ·) ∈ Vn−ℓ((⃗v)n +ℓ+1, θ) also follows from +Cauchy’s formula, since the integral in (4.28) is a sum of residues computed at the different values +among vℓ+1, . . . , vn. Moreover, one can readily compute +Q−1 +(k,n]¯Q(ℓ,n] = ¯Q(ℓ,n]Q−1 +(k,n] = ¯Q(ℓ,k] for ℓ < k, +Q−1 +(k,n]¯Q(ℓ,n] = ¯Q(ℓ,n]Q−1 +(k,n] = 0 for ℓ ≥ k. +(4.29) +Next we introduce a new kernel +Q+ +ℓ (x, y) = QℓAℓ(x, y) = A−1 +ℓ−1QℓAℓ(x, y) = α+ +ℓ +2πi +� +γr +dw θx−y +wx−y +aℓ(w) +vℓ − w, +(4.30) +ℓ ∈ �N − 1�, where +α+ +ℓ = vℓ − θ +aℓ(θ)θ = +1 +� +i∈Z(θ/vℓ)iqℓ(i), + +EXACT SOLUTION OF TASEP AND VARIANTS WITH INHOMOGENEOUS SPEEDS AND MEMORY LENGTHS +18 +which is well-defined thanks to Assump. 4.1(ii). As above we also write Q+ +(ℓ,n] = Q+ +ℓ+1 · · · Q+ +n = +Q(ℓ,n]Aℓ+1 · · · An, for ℓ ≤ n in �N − 1�, and we have +Q+ +(ℓ,n](x, y) = +1 +2πi +� +γr +dw +θx−y +wx−y−n+ℓ+1 +n +� +i=ℓ+1 +α+ +i ai(w) +vi − w , +(Q+ +(ℓ,n])−1(x, y) = +1 +2πi +� +γr +dw +θx−y +wx−y+n−ℓ+1 +n +� +i=ℓ+1 +vi − w +α+ +i ai(w). +We define an extension of Q(ℓ,n] to Vn−ℓ((⃗v)n +ℓ+1, θ) as follows: +¯Q+ +(ℓ,n](x, y) = ¯Q(ℓ,n]Aℓ+1 · · · An(x, y) = − 1 +2πi +� +Γ⃗v +dw +θx−y +wx−y−n+ℓ+1 +n +� +i=ℓ+1 +α+ +i ai(w) +vi − w . +(4.31) +Here we choose Γ⃗v as above, with the additional restriction that the contour is contained in {w ∈ +C : |w| > r}. Each ai(w) is analytic in this region, so exactly as for ¯Q(ℓ,n], we have ¯Q+ +(ℓ,n](x, ·) ∈ +Vn−ℓ((⃗v)n +ℓ+1). Moreover, the definition (4.1) of aℓ implies that the coefficient of wk in the Laurent +series for �n +i=ℓ+1 ai(w) vanishes for all k ≥ �n +i=ℓ+1 κi, and hence arguing again as for ¯Q(ℓ,n] we have +¯Q+ +m(x, y) = Q+ +m(x, y) ∀x−y > κm, +¯Q+ +(ℓ,n](x, y) = Q+ +(ℓ,n](x, y) ∀x−y > �n +i=ℓ+1 κi. (4.32) +We also use the notation Q+ +[ℓ,n] = Q+ +(ℓ−1,n], and respectively for the other kernels. +The kernel Q+ +m is Markov; we let B+ +m be the time-inhomogeneous random walk which has transitions +from time m − 1 to time m, m ∈ �N − 1�, with step distribution Q+ +m. We also define the stopping time +τ + = min{m = 0, . . . , N − 1 : B+ +m > ym+1}. +(4.33) +Next for n ≥ 1 and 0 ≤ m < n define the kernels +S−n(z1, z2) = an(θ)(R(Q+ +[1,n])−1An)∗(z1, z2) = an(θ)(RQ−1 +[1,n])∗(z1, z2) +(4.34) += +1 +2πi +� +γr +dw +θz2−z1 +wz2−z1+n+1 ψ(w) +�n +i=1(vi − w) +�n +i=1 α+ +i +�n−1 +i=1 ai(w) +, +¯S(m,n](z1, z2) = an(θ)−1 ¯Q+ +(m,n]A−1 +n R−1(z1, z2), +(4.35) += − 1 +2πi +� +Γ⃗v +dw +θx−y +wx−y−n+m+1 ψ(w)−1 +�n +i=m+1 α+ +i +�n−1 +i=m+1 ai(w) +�n +i=m+1(vi − w) +, +and +¯Sepi(⃗y) +n +(z1, z2) = EB+ +0 =z1 +� ¯S(τ +,n](B+ +τ +, z2)1τ +yn−k +PB+ +n−k−1=η(B+ +m ≤ ym+1 for n − k < m < n − ℓ, B+ +n−ℓ = z), +(4.39) +which can also be thought of as a hitting time distribution for the walk B+ +m moving backwards in time: +more precisely, it corresponds to starting with the walk at z at time n − ℓ, and moving backwards in +time hitting the strict epigraph of (ym+1)≥0 exactly at time m = n − k − 1. In the next result we will +find a Vk−ℓ+1((⃗v)n−ℓ +n−k, θ) extension of this function to all z ∈ Z, which we denote by ¯pn +k(ℓ, z): +Lemma 4.12. Assume yj − yj+1 ≥ κj for each j ∈ �N − 1� and let +¯pn +k(ℓ, z) = +� +η>yn−k +¯Q+ +[n−k,n−ℓ](η, z) +− 1ℓyn−k +� +η′∈Z +Q+ +n−k(η, η′)EB+ +n−k=η′ +� ¯Q+ +(τ +,n−ℓ](B+ +τ +, z)1τ +yn−k PBn−k−1=η(B+ +n−k = z) = � +η>yn−k Q+ +n−k(η, z) = � +η>yn−k ¯Q+ +n−k(η, z), +where the last equality follows from (4.32), since inside the sum we have η − z > κn−k, showing that +¯pn +k(k, z) = pn +k(k, z) for such z. On the other hand, we know already that ¯Q+ +n−k(η, ·) ∈ V1(vn−k, θ), +i.e. that ¯Q+ +n−k(η, z) = c(v1/θ)z−η for some c ∈ R, from which it is straightforward to deduce that +¯pn +k(k, z) = � +η>yn−k ¯Q+ +n−k(η, z) is in V1(vn−k, θ). +Next we turn to the case ℓ < k. For z ≤ yn−ℓ − κn−ℓ, pn +k(ℓ, z) equals +� +η>yn−k +� +η′≤yn−k+1 Q+ +n−k(η, η′)PB+ +n−k=η′(B+ +m ≤ ym+1 for n − k < m < n − ℓ, B+ +n−ℓ = z). + +EXACT SOLUTION OF TASEP AND VARIANTS WITH INHOMOGENEOUS SPEEDS AND MEMORY LENGTHS +20 +The last probability can be written as (using the stopping time τ + defined in (4.33)) +Q+ +(n−k,n−ℓ](η′, z) − PB+ +n−k=η′(hit on (n − k, n − ℓ), B+ +n−ℓ = z) += Q+ +(n−k,n−ℓ](η′, z) − �n−ℓ−1 +m=n−k+1 PB+ +n−k=η′(τ + = m, B+ +n−ℓ = z), += Q+ +(n−k,n−ℓ](η′, z) − �n−ℓ−1 +m=n−k+1 +� +η′′>ym+1 PB+ +n−k=ηk−1(τ + = m, B+ +m = η′′)Q+ +(m,n−ℓ](η′′, z) += Q+ +(n−k,n−ℓ](η′, z) − EB+ +n−k=η′ +� +Q+ +(τ +,n−ℓ](B+ +τ +, z)1τ +yn−k Q+ +n−k ¯χyn−k+1Q+ +(n−k,n−ℓ](η, z) +− � +η>yn−k +� +η′≤yn−k+1 Q+ +n−k(η, η′)EB+ +n−k=η′ +� +Q+ +(τ +,n−ℓ](B+ +τ +, z)1τ + yn−k+1 we have τ + = n − k in the last expectation, which then equals +Q+ +(n−k,n−ℓ](η′, z). Thus +pn +k(ℓ, z) = � +η>yn−k Q+ +n−k ¯χyn−k+1Q+ +(n−k,n−ℓ](η, z) + � +η>yn−k Q+ +n−kχyn−k+1Q+ +(n−k,n−ℓ](η, z) +− � +η>yn−k +� +η′∈Z Q+ +n−k(η, η′)EB+ +n−k=η′ +� +Q+ +(τ +,n−ℓ](B+ +τ +, z)1τ +yn−k Q+ +[n−k,n−ℓ](η, z) +− � +η>yn−k +� +η′∈Z Q+ +n−k(η, η′)EB+ +n−k=η′ +� +Q+ +(τ +,n−ℓ](B+ +τ +, z)1τ +yn−k ¯Q+ +[n−k,n−ℓ](η, z) +− � +η>yn−k +� +η′∈Z Q+ +n−k(η, η′)EB+ +n−k=η′ +� ¯Q+ +(τ +,n−ℓ](B+ +τ +, z)1τ + yn−k − yn−ℓ + κn−ℓ ≥ +�n−ℓ +i=n−k κi for the first sum while inside the expectation in the second one we have B+ +τ + > yn−τ ++1 +so B+ +τ + − z ≥ yn−τ ++1 − yn−ℓ + κn−ℓ ≥ �n−ℓ +i=n−τ ++1 κi. This shows that ¯pn +k(ℓ, z) = pn +k(ℓ, z) +for z ≤ yn−ℓ − κn−ℓ. To see that ¯pn +k(ℓ, ·) ∈ Vk−ℓ+1((⃗v)n−k +n−ℓ , θ) we proceed as in the case ℓ = k, +using for the first sum on the right hand side of (4.40) that ¯Q+ +[n−k,n−ℓ](η, ·) ∈ Vk−ℓ+1((⃗v)n−ℓ +n−k, θ) +while, for the second term, using that ¯Q+ +(τ +,n−ℓ](η, ·) ∈ Vn−ℓ−τ +((⃗v)n−ℓ +τ + , θ), which is a subspace of +Vk−ℓ+1((⃗v)n−ℓ +n−k, θ) for n − k ≤ τ + < n − ℓ. +□ +Now we can show that the functions ¯pn +k(ℓ, z) yield a solution to the system (4.17). +Lemma 4.13. yj − yj+1 ≥ κj for each j ∈ �N − 1� and let, for 0 ≤ ℓ ≤ k ≤ n ≤ N and z ∈ Z, +hn +k(ℓ, z) = (A−1 +n−ℓ)∗¯pn +k(ℓ, z) += +� +η>yn−k +¯Q+ +[n−k,n−ℓ]A−1 +n−ℓ(η, z) +(4.43) +− 1ℓyn−k +� +η′∈Z +Q+ +n−k(η, η′)EB+ +n−k=η′ +� ¯Q+ +(τ,n−ℓ]A−1 +n−ℓ(B+ +τ +, z)1τ +yn−k +¯Q+ +[n−k,n]A−1 +n R−1(η, x) +− 1k>0 +� +η>yn−k +� +η′∈Z +Q+ +n−k(η, η′)EB+ +n−k=η′ +� ¯Q+ +(τ +,n]A−1 +n R−1(B+ +τ +, x)1τ +yn−k ¯Q+ +[n−k,n−ℓ]A−1 +n−ℓQ−1 +n−ℓ(η, z) +− � +η>yn−k +� +η′∈Z Q+ +n−k(η, η′)EB+ +n−k=η′ +� ¯Q+ +(τ,n−ℓ]A−1 +n−ℓQ−1 +n−ℓ(B+ +τ +, z)1τ +yn−k +� ¯Q+ +[n−k,n−ℓ−1]A−1 +n−ℓ−1 +� +(η, z) +− 1ℓyn−k +� +η′∈Z Q+ +n−k(η, η′)EB+ +n−k=η′ +� ¯Q+ +(τ,n−ℓ−1]A−1 +n−ℓ−1(B+ +τ +, z)1τ +yn−k ¯Q+ +n−kA−1 +n−k(η, z) = − 1 +2πi +� +Γvn−k dw θyn−k−z +wyn−k−z +vn−k−θ +(vn−k−w)(w−θ). +The contour encloses only the simple pole at vn−k, and evaluating the residue we get hn +k(k, z) = +(θ/vn−k)yn−k−z, which is what we want. +Next we check (4.17c). From (4.1) we have that that the coefficient of wk in the Laurent series for +�n−ℓ−1 +i=m +ai(w) vanishes for all k ≥ �n−ℓ−1 +i=m +κi, and as a consequence that the integrand in (4.44) has a +vanishing residue at infinity for x − y > �n−ℓ−1 +i=m +κi. Cauchy’s formula, (4.30), (4.7), and (4.44) then +imply that for such x, y, +¯Q+ +[m,n−ℓ]A−1 +n−ℓ(x, y) = +1 +2πi +� +γr +θx−y +wx−y−n+ℓ+m +�n−ℓ +i=m +α+ +i ai(w) +vi−w +an−ℓ(θ) +an−ℓ(w) = Q+ +[m,n−ℓ]A−1 +n−ℓ(x, y) += Q+ +[m,n−ℓ)Qn−ℓ(x, y). +Then for z ≤ yn−ℓ the first term in (4.43) equals � +η>yn−k Q+ +[n−k,n−ℓ)Qn−ℓ(η, z), because η − z > +yn−k − yn−ℓ ≥ �n−ℓ−1 +i=n−k κi by our assumption on the yi’s. Arguing similarly, since B+ +τ + > yτ ++1, for +z ≤ yn−ℓ the expectation in (4.43) equals EB+ +n−k=η′ +� +Q+ +(τ,n−ℓ)Qn−ℓ(B+ +τ +, z)1τ +yn−k Q+ +[n−k,n−ℓ)Qn−ℓ(η, z) +− � +η>yn−k +� +η′∈Z Q+ +n−k(η, η′)EB+ +n−k=η′ +� +Q+ +(τ,n−ℓ)Qn−ℓ(B+ +τ +, z)1τ +yn−k P(B+)(n−ℓ) +n−k−1=η((B+)(n−ℓ) +m +≤ ym+1 for n − k < m < n − ℓ, (B+)(n−ℓ) +n−ℓ += z). +The last probability can be written as +� +η′≤yn−ℓ P(B+)(n−ℓ) +n−k−1=η((B+)(n−ℓ) +m +≤ ym+1 for n − k < m < n − ℓ, (B+)(n−ℓ) +n−ℓ−1 = η′)Qn−ℓ(η′, z), +and setting z = yn−ℓ we get the required identity hn +k(ℓ, yn−ℓ) = 0 because, from (4.23), Qn−ℓ(η′, yn−ℓ) = +0 for η′ − yn−ℓ ≤ 0. + +EXACT SOLUTION OF TASEP AND VARIANTS WITH INHOMOGENEOUS SPEEDS AND MEMORY LENGTHS +22 +It remains to prove (4.17d), i.e., that for each 0 ≤ ℓ < n the functions hn +k(ℓ, ·), ℓ ≤ k < n, span +Vn−ℓ(⃗v, θ). For this, we use the definitions (4.26), (4.28) and (4.31) to compute +� +η>ym ¯Q+ +[m,n−ℓ]A−1 +n−ℓ(η, z) = � +η>ym ¯Q[m,n−ℓ]Am · · · An−ℓ−1(η, z) +(4.45) += − 1 +2πi +� +Γ⃗v dw +θym−z(vn−ℓ−θ) +wym−z−n+ℓ+m(w−θ) +�n−ℓ−1 +i=m +α+ +i ai(w) +�n−ℓ +i=m(vi−w) , +where the contour Γ⃗v is such that |w| > θ (it can be taken such due to Assump. 4.1). Cauchy’s formula +implies that this is an element of Vn−ℓ−m+1((⃗v)n−ℓ +m , θ) as a function of z. The functions (4.43) can +be written as linear combinations of (4.45), for n − k ≤ m ≤ n − ℓ, and hence each function hn +k(ℓ, ·) +belongs to Vk−ℓ+1((⃗v)n−ℓ +n−k, θ). Since Vk−ℓ+1((⃗v)n−ℓ +n−k, θ) ⊂ Vn−ℓ(⃗v, θ), we get the required inclusion +(4.17d). +□ +4.4.4. Proof of Thm. 4.10. Consider the one-point kernel K(n)(z1, z2) +K(n)(z1, z2) = K(n, z1; n, z2) = +n +� +k=1 +Ψn +n−k(z1)Φn +n−k(z2). +(4.46) +From the definition (4.9)/(4.10) of the functions Ψn +n−k we readily get +K(ni, xi; nj, xj) = −Q(ni,nj](xi, xj)1ni n. In view of this and the definition (4.34) of S−n, (4.36) will +follow if we show that for any n ∈ �N�, +K(n) = (S−n)∗ ¯Sepi(⃗y) +n +. +(4.47) +Using (4.9) and (4.22), we rewrite the right hand side of (4.46) as +K(n)(z1, z2) = �n +k=1 Ψn +n−k(z1)Φn +n−k(z2) = �n +k=1 RQ−1 +[1,n]G(k) +0,nR−1(z1, z2) +(4.48) +with +G(k) +0,n(z1, z2) = Q[1,k](z1, yk)hn +n−k(0, z2). +Let also +ˆG(k) +0,n(z1, z2) = A−1 +k−1G(k) +0,nAn(z1, z2) = A−1 +k−1Q[1,k](z1, yk)¯pn +n−k(0, z2), +(4.49) +where we used (4.43). Using Lem. 4.12 and (4.41) together with (4.25) we get, for z2 ≤ yn − κn, +ˆG(k) +0,n(z1, z2) = Q[1,k)Qk(z1, yk) � +η>yk +� +η′≤yk+1 Q+ +k (η, η′)Q+ +(k,n](η′, z2) +− Q[1,k)Qk(z1, yk)(z1, yk) � +η>yk +� +η′≤yk+1 Q+ +k (η, η′)EB+ +k =η′ +� +Q+ +(τ +,n](B+ +τ +, z2)1τ +z2. On the other hand, as in (4.5) we have Q+ +k (z1, z2) = +α+ +k (θ/vk)z1−z2 for z1 − z2 > κk. Thus, since yk − yk+1 ≥ κk we have, for η′ ≤ yk+1, +Qk(z, yk) � +η>yk Q+ +k (η, η′) = vk−θ +θ +(θ/vk)z−yk +θ +vk−θα+ +k (θ/vk)yk−η′1z>yk = Q+ +k (z, η′)1z>yk. +Using this identity in our last expresion for ˆG(k) +0,n we get, still for z2 ≤ yn − κn +ˆG(k) +0,n(z1, z2) = Q[1,k)χykQ+ +k ¯χyk+1Q+ +(k,n](z1, z2) +− � +η′≤yk+1 Q[1,k)χykQ+ +k (z1, η′)EB+ +k =η′ +� +Q+ +(τ +,n](B+ +τ +, z2)1τ + 0 and the remaining ones have jump rate 1. The focus +of that paper was the case of 2-periodic initial state X0(i) = 2(M − i), i ≥ 1 (which is chosen so that +rate 1 particles start on the negative even integers and the additional rate α particles are placed to the +right of those), for which the associated biorthogonal functions Ψn +k and Φn +k were computed explicitly, +leading to a Fredholm determinant formula for the multipoint distribution of the process. This formula +was further used in that paper to compute the long time scaling limit of the process, which leads to +an explicit “process diagram” for the model describing how its asymptotic fluctuations depend on the +value of α and the characteristic direction used in the scaling. +In this section we will consider the two-speed setting for discrete and continuous time TASEP. More +precisely, we consider the kernel K studied in Sec. 4 with aℓ(w) = (1 + w)κ, κ ∈ {0, 1} and speeds +given by +vi = +� +α, +1 ≤ i ≤ M, +β, +i > M, +(5.1) +for some M ≥ 1 and two real parameters α, β > 0. The function ψ will initially remain general +(subject to the assumptions of Sec. 4). In the case ψ(w) = etw and κ = 0, the kernel corresponds to the +two-speed version of continuous time TASEP, with the values α and β corresponding to the jump rates +of the particles in the respective blocks. In the case ψ(w) = (1 + w)t the kernel describes two-speed +right Bernoulli TASEP in discrete time, with either sequential (κ = 0) or parallel (κ = 1) update; in +this last case, and recalling that the values vi are equal to pi/qi, where pi is the probability of the i-th +particle making a jump, the choice (5.1) can be written as +pi = +� +α +1+α, +1 ≤ i ≤ M, +β +1+β, +i > M. +By employing other choices of ψ and aℓ one recovers, in the same way, two-speed versions of the other +TASEP variants, but the formulas which we will derive in what follows depend on the specific choice +of aℓ, so for simplicity we restrict to this setting. +Our main goal here will be to show how versions of the formulas from [BFS09] for the 2-periodic +initial state yi = 2(M − i), i ≥ 1, can be derived in the current setting using our results. With those +formulas in hand, a similar analysis can be performed to recover their process diagrams for general +two-speed TASEP variants. More generally, one could attempt to use these formulas to study the +process diagram in the case of general (right-finite) initial data. We leave this for future work. + +EXACT SOLUTION OF TASEP AND VARIANTS WITH INHOMOGENEOUS SPEEDS AND MEMORY LENGTHS +25 +To simplify notation in this section, we are going to derive a formula only for the one-point kernel +(4.37); the formula for the multi-point kernel then follows from (4.38). The kernel can be written as +K(n)(x, x′) = (S−n)∗χy1 ¯S[1,n](x, x′) + (S−n)∗ ¯χy1 ¯Sepi(⃗y) +n +(x, x′), +(5.2) +where the functions (4.34) and (4.35) are given for n > M by +(S−n)∗(z1, z2) = +��n−1 +i=1 α+ +i +�−1 θz1−z2 +2πi +� +Γ0 du +[ u(α−u) +1+κu ]M[ u(β−u) +1+κu ]n−M +uz1−z2+2n+1 +(1 + κu)ψ(u), +(5.3) +¯S[1,n](z1, z2) = − +��n−1 +i=1 α+ +i +� θz1−z2 +2πi +� +Γα,β dw +wz2−z1+2n−1 +[ w(α−w) +1+κw ]M[ w(β−w) +1+κw ]n−M (1 + κw)−1ψ(w)−1,(5.4) +where we used the trivial identity (1 + u)κ = 1 + κu for κ = 0 or 1. Here and throughout the section +we use the subscripts in the integration contours to indicate which poles they include. The first part of +the kernel (5.2) can be written as +(S−n)∗χy1 ¯S[1,n](x, x′) = θx−x′ +(2πi)2 +� +Γ0 du +� +Γα,β dw +[ u(α−u) +1+κu ]M[ u(β−u) +1+κu ]n−M +[ w(α−w) +1+κw ]M[ w(β−w) +1+κw ]n−M +wx′−y1+2n +ux−y1+2n+1 +1 +u−w +(1+κu)ψ(u) +(1+κw)ψ(w). +(5.5) +For the other part it is convenient to decompose the product according to the two blocks: +(S−n)∗ ¯χy1 ¯Sepi(⃗y) +n +(x, x′) = � +z≤y1(S−n)∗(x, z)EB+ +0 =z +� ¯S(τ +,n](B+ +τ +, x′)1τ + − log(vi/θ) we define the functions +ri(λ) = log EB+ +i−1=0[eλB+ +i ] = log +� +(vi−θ)(eλ+κθ) +(1+θ)κ(vieλ−θ)eλ +� +and rk,ℓ(λ) = �ℓ +i=k ri(λ). For B+ +0 = z ≤ y1 the process (eλB+ +m−r1,m(λ))m≥0 is a martingale, where +we postulate r1,0(λ) = 0. Applying the optional stopping theorem, we get EB+ +0 =z[eλB+ +τ+−r1,τ+(λ)] = +eλz for λ > maxi{− log(vi/θ)}, where the stopping time τ + is defined in (4.33). The definition of +the initial state ⃗y yields B+ +τ + = yτ ++1 + 1 = 2(M − τ +) − 1, and hence EB+ +0 =z[e−2λτ +−r1,τ+(λ)] = +eλ(z−2M+1) for λ > maxi{− log(vi/θ)}. +Introducing a new variable u = e−λ, the preceding +identity may be written as EB+ +0 =z[uτ + �τ + +i=1 +(1+θ)κ(vi−θu) +(vi−θ)(1+κθu)] = u2M−z−1. Defining the functions +p(u) = (1+θ)κu(α−θu) +(α−θ)(1+κθu) and q(u) = (1+θ)κu(β−θu) +(β−θ)(1+κθu) , in the two-speed case (5.1) the preceding identity is +equivalent to +EB+ +0 =z +� +p(u)τ +1τ + 0 and h ̸≡ −∞, endowed with the local Hausdorff topology (see [MQR21, Sec. 3] + +EXACT SOLUTION OF TASEP AND VARIANTS WITH INHOMOGENEOUS SPEEDS AND MEMORY LENGTHS +29 +for more details) and consider TASEP initial data (Xε +0(i))i≥1 such that for some h0 ∈ UC satisfying +h0(x) = −∞ for x > 0, +−ε1/2� +Xε +0(ε−1x) + 2ε−1x +� +−−−→ +ε→0 h0(−x) +(5.14) +in UC2. Then one expects that there be explicit constants α, β, γ, σ > 0 (which are not universal, in +particular they differ between the sequential and parallel cases, see the end of this subsection for their +explicit values) so that +−γ−1σ−1ε1/2� +Xε−3/2t(αε−3/2t − σ2ε−1x) − βε−3/2t − 2σ2ε−1x +� +−→ h(t, x; h0) +(5.15) +as a process in t > 0 and x ∈ R, in distribution in UC; the limiting process h(t, x) is the KPZ fixed +point, which is a UC-valued Markov process with initial data h0 (which in indicated in the notation +h(t, x; h0)). The KPZ fixed point has explicit transition probabilities, which we introduce below. A full +proof of this convergence involves some heavy asymptotic analysis, and has actually not appeared in +detail in the literature for these models (restricting to convergence of finite dimensional distributions, in +the sequential case, [BFP07] proved it for fixed time marginals and periodic initial data, and [Ara20] +gave a proof at the level of critical point computations for the general result; while in the parallel case a +proof for fixed time marginals and periodic initial data appears in [BFS08]), but there is no doubt that it +can be achieved by starting with the Fredholm determinant formulas derived in [MR23] and suitably +adapting the arguments of [MQR21] for continuous time TASEP. +By KPZ universality one expects that (5.15) should also hold for the mixed sequential/parallel +version of TASEP (for different choices of α, β, γ, σ). Our goal here will be sketch the proof of this, for +fixed t > 0 and at the level of finite dimensional distributions, and in particular to work out the right +scaling. We will proceed only at the level of a critical point analysis, and do not attempt a rigorous +derivation. It is worth stressing that this analysis could also be performed, in analogous way, for mixed +sequential/parallel versions of other TASEP variants, such as those described in Sec. 2 of [MR23], as +well as for more general mixtures of caterpillars with different lengths. +Before getting started with the derivation, let us introduce the explicit formula for the KPZ fixed +point transition probabilities. We restrict the discussion to one-sided initial data h0, which are such +that h0(x) = −∞ for all x > 0 (which is the class arising naturally from the class of TASEP initial +conditions being considered in this paper where there is a rightmost particle). Introduce the kernels +St,x(u, v) = t−1/3e +2x3 +3t2 − (u−v)x +t +Ai(t−1/3(v − u) + t−4/3x2) +for t ̸= 0, where Ai is the Airy function. Then for any t > 0 and any x1, . . . , xm one has [MQR21] +P +� +h(t, xi) ≤ ai, i ∈ �m� +� += det +� +I − χaKhypo(h0) +t,ext +χa +� +L2({x1,...,xm}×R) +(5.16) +where (here ey∂2(u, v), y > 0, is the heat kernel, corresponding to the transition density of a Brownian +motion with diffusivity 2a) +Khypo(h0) +t,ext +(xi, ·; xj, ·) = −e(xj−xi)∂21xi 0 and x, a ∈ Rm +and study the quantity P(Xε−3/2t(αε−3/2t−σ2ε−1xi) > βε−3/2t+2σ2ε−1xi−γσε−1/2ai, i ∈ �m�), +which is equal to det(I − ¯χrKt ¯χr)ℓ2({n1,...,nm}×Z), for the kernel Kt given in (2.1b) with the choices +specified above and with +t = ε−3/2t, +ni = αε−3/2t − σ2ε−1xi, +ri = βε−3/2t + 2σ2ε−1xi − γσε−1/2ai. +We introduce the following change of variables in the kernel Kt(ni, xi; nj, xj): +xi = βε−3/2t + 2σ2ε−1xi + γσε−1/2ui +(5.17) +(more properly, xi should be taken to be the integer part of the right hand side, but we ignore this +here and below). After the change of variables in the Fredholm determinant we are on a lattice of size +ε1/2 and the limiting Fredholm determinant will be computed on L2({x1, . . . , xm} × R), while the +projection ¯χr becomes ¯χ−a and the kernel Kt gets multiplied by γσε−1/2. Our goal then is to compute, +under this scaling +¯K(xi, ui; xj, uj) := lim +ε→0 γσε−1/2Kt(ni, xi; nj, xj), +(5.18) +which will identify the limit of the scaled multipoint distributions of the left hand side of (5.15) as +det(I − ¯χ−a ¯K¯χ−a)L2({x1,...,xm}×R) (and which could be upgraded to a rigorous proof of the limit if +the above kernel convergence were upgraded to a rigorous proof of convergence in trace norm, or of +pointwise convergence with suitable uniform tail control). The parameters appearing in the scaling +have to be chosen as follows (recall q = 1 − p): +α = +(p − qθ)2 +pq(1 + θ)2 + ρ(p − qθ)2 , +β = +p(q(1 + θ)2 − 1) +pq(1 + θ)2 + ρ(p − qθ)2 +γ = +� +pqθ +2(p − qθ)2 + +θρ +2(1 + θ)2 +�1/2 +, +σ = +� +2pqθ(1 + θ)(p − qθ) +γ(pq(1 + θ)2 + ρ(p − qθ)2) +�1/3 +, +(5.19) +with +ρ = +b +a + b, +θ = +� +p2(1 − ρ)2 + 4q + p(1 − ρ) − 2q +2q(2 − ρ) +. +(5.20) +This last choice also sets the value of the free parameter θ in the definition of Kt (different choices of θ +in that definition would require an additional conjugation on the right hand side of (5.18), which anyway +would not change the value of the associated Fredholm determinants, see also [MR23, Rem. 5.16(b)]), +while ρ is simply the macroscopic proportion of parallel particles. We will see below where the choices +of θ and γ come from; α, β and σ could in principle be derived from KPZ scaling theory by studying +the invariant measure of the process [Spo14], but we will not attempt that here and instead choose these +parameters based directly on the asymptotics of our formulas. +5.2.3. Asymptotics. We begin by studying the kernel Q(ni,nj](xi, xj) appearing in (4.36) for ni < nj. +This kernel corresponds to the transition matrix of the random walk Bm so, using the scaled variables, +Q(ni,nj](xi, xj) is the probability that Bm has moved by 2σ2ε−1(xj −xi)+γσε−1/2(uj −ui) by time +m = σ2ε−1(xi − xj) (note ni < nj implies xi > xj). We want to use the central limit theorem to +show that this to converges to a Gaussian density. For this we need the mean of the random walk to be +−2 (this choice in our scaling comes from the choice of average density 1/2 in (5.14)). Fix θ and let +ˆθ = qθ/p. Then the mean of the jump distribution Qℓ corresponding to sequential particles is − +1 +1−ˆθ +while the one corresponding to parallel particles is − 1−p(1−ˆθ)2 +(q+pˆθ))(1−ˆθ), as can be computed directly from +their definition (4.3) (or (4.4)) with the current choices. Recalling that ρ denotes the density of parallel +particles, the average mean of the jump distribution of the (inhomogeneous) random walk for the mixed +case is +−(1 − ρ) +1 +1−ˆθ − ρ 1−p(1−ˆθ)2 +(q+pˆθ)(1−ˆθ) = − +1 +1−ˆθ − +p ˆθρ +q+pˆθ. +Hence we need to choose ˆθ to be the solution of +1 +1−ˆθ + p ˆθρ +q+pˆθ = 2, which is explicitly given by ˆθ = qθ/p +with θ as chosen in (5.20). With this choice of the parameter θ and the above scaling, the central limit + +EXACT SOLUTION OF TASEP AND VARIANTS WITH INHOMOGENEOUS SPEEDS AND MEMORY LENGTHS +31 +theorem implies that +γσε−1/2Q(ni,nj](xi, xj) −−−→ +ε→0 eυ(xi−xj)/(2γ2)∂2(ui, uj), +where υ is the average variance of the random walk jump distribution, which can be computed similarly, +and equals (1 − ρ) +ˆθ +(1−ˆθ)2 + ρ +qˆθ+pˆθ3 +(q+pˆθ)(1−ˆθ)2 = +pqθ +(p−qθ)2 + +θρ +(1+θ)2 . The above choice of γ implies that +υ/γ2 = 2, which means that the right hand side above equals e(xi−xj)∂2(ui, uj) as desired. +Next we need to compute the limit of the scaled kernel γσε−1/2(S−ni)∗ ¯Sepi(Xε +0) +nj +(xi, xj). This +composition equals γσε−1/2 � +y∈Z(S−ni)∗(xi, y) ¯Sepi(Xε +0) +nj +(y, xj). We replace the sum by an integral +and change variables y �−→ γσε−1/2u, which yields an extra factor of γσε−1/2, so now both factors +have such a multiplier in front. We focus on the limit of (γσε−1/2S−ni)∗(xi, y). Using (4.34) (and +recalling also that vℓ = p/q for all ℓ in this case while κℓ equals 0 for sequential particles and 1 parallel +ones) we get +S−ni(y, xi) = +1 +2πi +� +γr dw +θxi−y +wxi−y+ni+2 (1 + w)t �ni +ℓ=1 +p/q−w +(1+w)κℓ +(1+θ)κℓθ +p/q−θ += +� +θ +p/q−θ +�ni p/q +2πi +� +γqr/p dw θxi−y(p/q−pw/q)ni +(pw/q)xi−y+ni+1 (1 + pw/q)t �n1 +ℓ=1 +� +1+θ +1+pw/q +�κℓ += +� +ˆθ +1−ˆθ +�ni q/p +2πi +� +γˆr dw +ˆθxi−y(1−w)ni +wxi−y+ni+1 (1 + pw/q)t � +q+pˆθ +q+pw +��ni +ℓ=1 κℓ , +(5.21) +with ˆθ = qθ/p and ˆr = qr/p. Note that �ni +ℓ=1 κℓ is approximately ρni; the difference is bounded +by a + b and will not make any difference in the limit, so we will simply replace the sum by its +approximation. With this we get +θγσε−1/2(1 + θ)−tS−ni(y, xi) = (γσ/ˆθ)ε−1/2 1 +2πi +� +γr dw eFε(w) = +1 +2πi +� +Γε dv eFε(ˆθ(1+ε1/2v/(γσ))), +where Γε is a circle of radius ε−1/2(γσ/ˆθ)r centered at −ε−1/2γσ and +Fε(w) = ni log( +ˆθ +1−ˆθ) + (xi − y + 1) log ˆθ + ni log(1 − w) +− (xi − y + ni + 2) log w + (t − ρni) log( q+pw +q+pˆθ ), +Fε(ˆθ(1 + ε1/2v/(γσ))) = −(xi − y + ni + 1) log(1 + ε1/2v/(γσ)) + ni log(1 − ε1/2Av/(γσ)) ++ (t − ρni) log(1 + ε1/2Bv/(γσ)), +with A = ˆθ/(1 − ˆθ) = qθ/(p − qθ) and B = pˆθ/(q + pˆθ) = θ/(1 + θ). Using the expansion (valid +for fixed c ∈ R) +log(1 + cε1/2v) = cε1/2v − ε +2(cv)2 + ε3/2 +3 (cv)3 + O(ε2v4) +and the scaling (5.17), we get that Fε(ˆθ(1 + ε1/2v/(γσ))) equals +− +� +(α + β)ε−3/2t + σ2ε−1xi + γσε−1/2(ui − u) +�� +ε1/2v/(γσ) − ε(v/(γσ))2 +2 ++ ε3/2(v/(γσ))3 +3 +� +− +� +αε−3/2t − σ2ε−1xi +�� +ε1/2Av/(γσ) + ε(Av/(γσ))2 +2 ++ ε3/2(Av/(γσ))3 +3 +� ++ +� +(1 − ρα)ε−3/2t + ρσ2ε−1xi +�� +ε1/2Bv/(γσ) − ε(Bv/(γσ))2 +2 ++ ε3/2(Bv/(γσ))3 +3 +� ++ O(ε1/2v). +(5.22) +Then the coefficients of ε−1tv/(γσ), 1 +2ε−1/2t(v/(γσ))2 and ε−1/2xiv/(γσ) are, respectively, +−α(1 + A + ρB) − β + B +α(1 − A2 + ρB2) + β − B2 +and +− 1 + A + ρB, +and they all vanish thanks to our choices of α and β and the fact that our choice of ˆθ satisfies +1 +1−ˆθ + +p ˆθρ +q+pˆθ = 2, which is the same as A + ρB = 1. Similarly, the coefficients of tv3, xiv2 and +(ui − u)v respectively equal +− α+β+αA3−(1−ραB3) +3γ3σ3 += − 1 +3, +1+A2−ρB2 +2γ2 += 1 +and +− 1, + +EXACT SOLUTION OF TASEP AND VARIANTS WITH INHOMOGENEOUS SPEEDS AND MEMORY LENGTHS +32 +where the identities follow again from our parameter choices. Hence +Fε(ˆθ(1 + ε1/2v/(γσ))) = − t +3v3 + xiv2 − (ui − u)v + O(ε1/2v). +and thus, as ε → 0, θγσε−1/2(1 + θ)−tS−ni(y, xi) can be approximated by +1 +2πi +� +Γε dv e− t +3 v3+xiv2−(ui−u)v = +1 +2πi +� +−Γε dv e +t +3 w3+xiw2+(ui−u)w+O(ε1/2w). +Let ⟨ denote a contour formed by rays going off the origin at angles ±π/3 (going up in the imaginary +direction). We can deform the contour −Γε to ⟨ε∪Cε where ⟨ε is the part of ⟨ lying inside −Γε and Cε +is the part of −Γε lying to the right of ⟨ε. A standard argument using the decay of the integrand on Cε +shows that that part can be discarded as ε → 0, and we are left with +θγσε−1/2(1 + θ)−tS−ni(y, xi) −−−→ +ε→0 +1 +2πi +� +⟨ dv e +t +3 w3+xiw2−(u−ui)w = St,xi(u, ui) = S−t,xi(ui, u), +where the first equality comes from the above definition of St,x and a simple change of variables in the +contour integral formula for the Airy function Ai(z) = +1 +2πi +� +⟨ dwew3/3−zw and the second one follows +directly from the same definition. +An analogous argument shows that +θ−1γσε−1/2(1 + θ)t ¯S(0,nj](y, xj) −−−→ +ε→0 S−t,−xj(u, uj). +In fact, the kernel can be written as +¯S(0,nj](y, xj) = +� 1−ˆθ +ˆθ +�nj−1 p/q +2πi +� +γδ dw (1−w)xj−y+nj +ˆθxj−ywnj +((1 − pw)/q)−t � +1−pw +q+pθ +��nj +ℓ=1 κℓ +where δ > 0 is small enough so that only the pole at 0 of the integrand is inside the contour; to get this +formula from (4.35) we have changed variables w �−→ pw/q as above and afterwards w �−→ 1 − w. +Proceeding as before we may write +θ−1γσε−1/2(1 + θ)t ¯S(0,nj](y, xj) = +1 +2πi +� +¯Γε dv e ¯Fε(1−ˆθ(1+ε1/2v/(γσ))), +where now +¯Fε(1 − ˆθ(1 + ε1/2v/(γσ))) = (xj − y + nj) log(1 + ε1/2v/(γσ)) − nj log(1 − ε1/2Av/(γσ)) +− (t − ρnj) log(1 + ε1/2pv/(γσ)), +and the same asymptotic analysis goes through. On the other hand, under this scaling the random walk +B+ inside the expectation defining Sepi(Xε +0) +−t,nj +becomes +γ−1σ−1ε1/2(B+ +σ2ε−1x + 2σ2ε−1x), +which converges to a Brownian motion B(x) with diffusivity 2; this is obtained by studying the +associated transition probabilities Q+ +(ni,nj](xi, xj) using the same argument we used for the term +Q(ni,nj](xi, xj) (the only difference between Q and Q+ in this context is that the arrangement of +sequential and parallel particles is shifted by 1, but this makes no difference in the argument). The +hitting time τ + of the walk B+ to the epigraph of Xε +0 similarly becomes the hitting time of B to +the epigraph of the curve −h− +0 since, by (5.14), the initial data Xε +0 rescales to this function. The +conclusion of all this is that θ−1γσε−1/2(1 + θ)t ¯Sepi(Xε +0) +nj +(y, xj) −−−→ +ε→0 Sepi(−h− +0 ) +−t,xj +(u, uj), with Sepi(g) +t,x +defined analogously to Shypo(h) +t,x +except that τ is now the hitting time of the epigraph of g. +5.2.4. Conclusion. Putting the above computations together we deduce that the limiting kernel ¯K +defined in (5.18) is given by +¯K(xi, ui; xj, uj) = −e(xi−xj)∂21xi>xj + (S−t,xi)∗Sepi(−h− +0 ) +−t,−xj . +The right hand side is an “upside down” version of Khypo(h0) +t,ext +: one has Khypo(h0) +t,ext +(xi, ui; xj, uj) = +¯K∗(xi, −ui; xj, −uj), which also implies +det(I − ¯χ−a ¯K¯χ−a) = det(I − χaKhypo(h0) +t,ext +χa), +(5.23) + +EXACT SOLUTION OF TASEP AND VARIANTS WITH INHOMOGENEOUS SPEEDS AND MEMORY LENGTHS +33 +see [MQR21, Sec. 3.3] for the details. +Putting all of this together, and in view of (5.16), we conclude that: +For Xt the mixed sequential/parallel version of right Bernoulli TASEP, with blocks of +a sequential particles followed by blocks of b parallel particles, one has +−γ−1σ−1ε1/2� +Xε−3/2t(αε−3/2t − σ2ε−1x) − βε−3/2t − 2σ2ε−1x +� +−→ h(t, x) +for fixed t > 0 and in the sense of finite dimensional distributions, with the parameter +choices specified in (5.19) and (5.20). +Setting ρ = 0 and ρ = 1 we recover the sequential and parallel cases, for which the scaling parameters +simplify somewhat: in the sequential case one has θ = p/2q, α = pq/(1+q)2, β = p2/(1+q)2, γ = 1 +and σ = 22/3(pq)1/3/(1 + q), while the parallel case one has θ = (1 − √q)/√q, α = (1 − √q)/2, +β = 0, γ = q1/4 and σ = 2−1/3p1/3q−1/12. +5.3. One long caterpillar. Finally we consider a situation where the first particle corresponds to a +long caterpillar of length +M = bε−1/2 + c +while all other particles update sequentially (i.e. with L = κ + 1 = 0). More precisely, as in the +previous section we consider discrete time TASEP with right Bernoulli jumps with parameter p ∈ (0, 1) +(so that ψ(w) = 1 + w, vi = p/q for all i ≥ 1), but now take the caterpillar lengths to be +L1 = M +and +Li = 1 +for i ≥ 2. +(so that a1(w) = (1 + w)M while ai ≡ 1 for i ≥ 2). +We want to study the system under a scaling similar to (5.15), and to this end we want to consider +initial data X0 satisfying (5.14). However, in the current situation we face the problem that in general X0 +also has to satisfy X0(i) − X0(i + 1) ≥ κi, which in our case imposes the condition X0(1) − X0(2) ≥ +M − 1. So, in order to keep the frame of reference implicit in (5.14), we choose our initial data as +follows: we consider initial data ( ˆXε +0(i))i≥1 such that for some h0 ∈ UC satisfying h0(x) = −∞ for +x > 0, ˆXε +0 satisfies (5.14), in UC, and then fix the initial data for our system to be +Xε +0(1) = ˆXε +0(1) + M′ +and +Xε +0(i) = ˆXε +0(i − 1) +for i ≥ 2, +for some M′ ∈ N satisfying M′ ≥ M − 1. In words, we are taking TASEP initial data ˆXε +0 satisfying +(5.14) and placing an additional caterpillar of length M at the beginning of the system, at distance +M′ ≥ M − 1. +It is not too hard to check that any choice of M′, possibly depending on ε, Xε +0 defined in this way +still satisfies (5.14), and so the question is whether the system feels the long caterpillar placed to its +right. We will see that, for the type of initial data which our results allow us to probe, which imposes +the condition M′ ≥ M − 1, there is no such effect asymptotically but, based in part on our analysis, we +will conjecture that there will be a such an effect when the caterpillar is placed more closely to the rest +of the system. +5.3.1. Scaling. We will assume that the distance M′ at which the caterpillar is placed satisfies +ε1/2M′ −−−→ +ε→0 b′ ∈ [b, ∞) ∪ {∞}. +(5.24) +The restriction b′ ≥ b comes from the assumption M′ > M = bε−1/2 + c. We also allow for b′ to +take the value ∞ to allow for choices of M′ of order larger than ε−1/2. For simplicity we will also +assume in the derivation that h0(0) = 0, which means that ε1/2 ˆXε +0(1) −−−→ +ε→0 0; the general case can be +recovered by translation and shift invariance of the limit. +For some fixed t > 0 and xi, ai ∈ R, i ∈ �m�, we use the scaling +t = ε−3/2t, +ni = αε−3/2t − σ2ε−1xi, +ri = βε−3/2t + 2σ2ε−1xi − σε−1/2ai +(5.25) + +EXACT SOLUTION OF TASEP AND VARIANTS WITH INHOMOGENEOUS SPEEDS AND MEMORY LENGTHS +34 +for the parameters and +xi = βε−3/2t + 2σ2ε−1xi + σε−1/2ui +(5.26) +for the kernel variables. All particles but the first update sequentially and thus, as we will see, the +correct parameter choices in this case are those used in the previous example, i.e. (5.19), with ρ = 0: +θ = p/(2q) (and thus ˆθ = 1/2) and +α = +pq +(1 + q)2 , +β = +p2 +(1 + q)2 , +σ = 22/3(pq)1/3 +1 + q +. +5.3.2. Asymptotics. Consider first the term Q(ni,nj](xi, xj) appearing in (4.36) for ni < nj. In view +of our scaling we only need to consider ni, nj ≫ 1, so Q(ni,nj](xi, xj) does not see the different +dynamics of the first particle, and thus it simply corresponds to the same kernel as for the sequential +case. Since we have chosen the scaling by setting ρ = 0 in the previous example, the conclusion is that +σε−1/2Q(ni,nj](xi, xj) converges as ε → 0 to e(xi−xj)∂2(ui, uj) by the central limit theorem. +Next we need to study the scaled kernel σε−1/2(S−ni)∗ ¯Sepi(Xε +0) +(0,nj] (x1, x2). Proceeding as before, we +focus first on the limit of σε−1/2(S−ni)∗(xi, y) with y = σε−1/2u, for which we get as in (5.21) +S−ni(y, xi) = q/p +2πi +� +γˆr dw +(1−w)ni +2xi−ywxi−y+ni+1 (1 + pw/q)t � +q+p/2 +q+pw +�M−1 +with ˆr = qr/p. Continuing the argument in the same way leads to +θσε−1/2(1 + θ)−tS−ni(y, x1) = +1 +2πi +� +Γε dv eFε( 1 +2 (1+ε1/2v/σ)), +where Γε is a circle of radius 2ε−1/2σr centered at −ε−1/2σ and +Fε( 1 +2(1 + ε1/2v/σ)) = −(xi − y + ni + 2) log(1 + ε1/2v/(γσ)) + ni log(1 − ε1/2Av/σ) ++ (t − M + 1) log(1 + ε1/2Bv/σ), +where in this case we have A = 1 and B = p/(1 + q). Note that this expression coincides with the +one we got for the mixed sequential/parallel case after replacing ρni by M − 1 and setting γ = 1 +there. Hence using the same expansions as in (5.22) and the scaling (5.26) together with our choice +M = bε−1/2 + c, we get that Fε( 1 +2(1 + ε1/2v/σ)) equals +− +� +(α + β)ε−3/2t + σ2ε−1xi + σε−1/2(ui − u) +�� +ε1/2v/σ − ε(v/σ)2 +2 ++ ε3/2(v/σ)3 +3 +� +− +� +αε−3/2t − σ2ε−1xi +�� +ε1/2Av/σ + ε(Av/σ)2 +2 ++ ε3/2(Av/σ)3 +3 +� ++ +� +ε−3/2t − bε−1/2 − c + 1 +�� +ε1/2Bv/σ − ε(Bv/σ)2 +2 ++ ε3/2(Bv/σ)3 +3 +� ++ O(ε1/2v). +Exactly the same argument as in the previous example shows that all terms of order ε−1/2 or higher +cancel, while the coefficients of tv3, xiv2 and (ui − u)v are −1/3, 1 and −1. But in this case there +is an additional term of order 1, coming from the last line above, which equals −bBv/σ. Writing +b = bB/σ = 2−2/3p2/3q−1/3b and continuing the argument as in the previous section we get +θσε−1/2(1 + θ)−tS−ni(y, xi) −−−→ +ε→0 +1 +2πi +� +Γε dv e− t +3 v3+xiv2−(ui+b−u)v = S−t,xi(ui + b, u). +Now we turn to ¯Sepi(Xε +0) +nj +(y, xj) = EB+ +0 =y[ ¯S(τ +,nj](B+ +τ +, xj)1τ + ˆXε +0(1) + M′ then in the expectation we have τ + = 0. Using this and generalizing the +definition of ¯Sepi(⃗y) +n +to +¯Sepi(⃗y) +(m,n](y, x) = EB+ +m=y[ ¯S(τ +,n](B+ +τ +, x)1τ +xj + SbS∗ +−t,xi +� +χb′/σS−t,−xj + ¯χb′/σSepi(−h− +0 +b) +−t,−xj +� +S∗ +b. +(5.28) +Now b = Bb/σ < b/σ ≤ b′/σ because B = p/(1 + q) < 1, while we assumed h0(0) = 0, so +S−t,−xj(v1, v2) = Sepi(−h− +0 (0)+b) +−t,−xj +(v1, v2) if v1 > b′/σ, and thus the above can be rewritten as +¯K(xi, ·; xj, ·) = −e(xi−xj)∂21xi>xj + SbS∗ +−t,xiSepi(−h− +0 +b) +−t,−xj +S∗ +b. +(5.29) +But SbS∗ +−t,xiSepi(−h− +0 +b) +−t,−xj +S∗ +b = S∗ +−t,xiSepi(−h− +0 ) +−t,−xj , so what this tells us is that, with this choice of scaling +and initial data, P(Xt(ni) > ri, i ∈ �m�) converges to +det(I − ¯χ−a ¯K¯χ−a) = det(I − χaKhypo(h0) +t,ext +¯χa) = Ph0(h(t, xi) ≤ ri, i ∈ �m�), +where the first equality is as in (5.23). We conclude that: +For Xt the version of right Bernoulli TASEP with sequential update and an additional +caterpillar of length M = bε−1/2 + c placed at distance M′ ≥ M − 1 of the rightmost +particle, +−σ−1ε1/2� +Xε−3/2t(αε−3/2t − σ2ε−1x) − βε−3/2t − 2σ2ε−1x +� +−→ h(t, x; h0) +(5.30) +for fixed t > 0, in the sense of finite dimensional distributions, with the parameter +choices specified in (5.25). +In words, a caterpillar of length ε−1/2 placed at distance bigger than or equal to its length has no +effect on the one point distributions of the system. +Now consider again the limiting kernel (5.28). We are restricted to work under the assumption that +b′ ≥ b (because our Fredholm determinant formulas have such a restriction on the distance between + +EXACT SOLUTION OF TASEP AND VARIANTS WITH INHOMOGENEOUS SPEEDS AND MEMORY LENGTHS +36 +caterpillars), which is what simplified the kernel to (5.29). On the other hand, if we had been in a +situation with b′/σ < b then this simplification would not have occurred, and the scaling limit would +have been different. Based on this and some exploratory Mathematica computations, we formulate the +following: +Conjecture 5.1. In the setting of this section, with a caterpillar of length bε−1/2 is placed at distance +b′ε−1/2, there is a b′ +0 > 0 such that the scaling limit (5.30) holds if and only if b′ ≥ b′ +0. +From the (formal) scaling limit derived in this section we see that the phase transition necessarily has +to occur at b′ +0 ≤ b. On the other hand, the discussion in the previous paragraph may suggest that, since +b = Bb/σ and B = p/(1 + q), the critical value is b′ +0 = p/(1 + q)b, but we have no strong evidence +for this stronger version of the conjecture. +Remark 5.2. A similar analysis can be used to study the case when the caterpillar placed at the right +of the system has length of order ε−1. 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Phys. 108.5-6 (2002), pp. 1071–1106. +[QS23] +J. Quastel and S. Sarkar. Convergence of exclusion processes and the KPZ equation to the +KPZ fixed point. J. Amer. Math. Soc. 36.1 (2023), pp. 251–289. +[Sas05] +T. Sasamoto. Spatial correlations of the 1D KPZ surface on a flat substrate. Journal of +Physics A: Mathematical and General 38.33 (2005), p. L549. +[Sch97] +G. M. Schütz. Exact solution of the master equation for the asymmetric exclusion process. J. +Statist. Phys. 88.1-2 (1997), pp. 427–445. +[Spo14] +H. Spohn. KPZ scaling theory and the semidiscrete directed polymer model. In: Random +matrix theory, interacting particle systems, and integrable systems. Vol. 65. Math. Sci. Res. +Inst. Publ. Cambridge Univ. Press, New York, 2014, pp. 483–493. +[TW94] +C. A. Tracy and H. Widom. Level-spacing distributions and the Airy kernel. Comm. Math. +Phys. 159.1 (1994), pp. 151–174. +[TW96] +C. A. Tracy and H. Widom. On orthogonal and symplectic matrix ensembles. Comm. Math. +Phys. 177.3 (1996), pp. 727–754. +[Vir20] +B. Virág. The heat and the landscape I. 2020. arXiv: 2008.07241. +(K. Matetski) DEPARTMENT OF MATHEMATICS, MICHIGAN STATE UNIVERSITY, 619 RED CEDAR ROAD, EAST +LANSING, MI 48824, USA +Email address: matetski@msu.edu +(D. Remenik) DEPARTAMENTO DE INGENIERÍA MATEMÁTICA AND CENTRO DE MODELAMIENTO MATEMÁTICO +(IRL-CNRS 2807), UNIVERSIDAD DE CHILE, AV. BEAUCHEF 851, TORRE NORTE, PISO 5, SANTIAGO, CHILE +Email address: dremenik@dim.uchile.cl + diff --git a/nNFST4oBgHgl3EQfLDje/content/tmp_files/load_file.txt b/nNFST4oBgHgl3EQfLDje/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..cb7be0591fc6f17db42d1a5dbff929d67df7bd67 --- /dev/null +++ b/nNFST4oBgHgl3EQfLDje/content/tmp_files/load_file.txt @@ -0,0 +1,1756 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf,len=1755 +page_content='EXACT SOLUTION OF TASEP AND VARIANTS WITH INHOMOGENEOUS SPEEDS AND MEMORY LENGTHS KONSTANTIN MATETSKI AND DANIEL REMENIK ABSTRACT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' In [MQR21;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' MR23] an explicit biorthogonalization method was developed that applies to a class of determinantal measures which describe the evolution of several variants of classical interacting particle systems in the KPZ universality class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' The method leads to explicit Fredholm determinant formulas for the multipoint distributions of these systems which are suitable for asymptotic analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' In this paper we extend the method to a broader class of determinantal measures which is applicable to systems where particles have different jump speeds and different memory lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' As an application of our results we study three particular examples: some variants of TASEP with two blocks of particles having different speeds, a version of discrete time TASEP which mixes particles with sequential and parallel update, and a version of sequential TASEP where a single particle with long memory length (equivalently, a long “caterpillar”) is added to the right of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' In the last two cases we also include a formal asymptotic analysis which shows convergence to the KPZ fixed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' CONTENTS 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' Introduction 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' Motivating examples: interacting particle systems 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' Continuous time TASEP 6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' Discrete time TASEPs with right Bernoulli jumps 6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' Caterpillars with right Bernoulli jumps 6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' Other types of caterpillars 7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' PushASEP 7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' A biorthogonal ensemble formula for determinantal measures 8 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' An explicit biorthogonalization scheme 11 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' Setting 11 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' The biorthogonalization problem 12 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' The boundary value problem 15 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' Main result: representation in terms of random walk hitting times 16 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' Application to particle systems 24 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' Two-speed variants of TASEP 24 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' Mixed sequential and parallel TASEP and KPZ fixed point limit 28 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' One long caterpillar 33 References 36 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' INTRODUCTION In the (continuous time, one-dimensional) totally asymmetric simple exclusion process (TASEP), particles perform totally asymmetric nearest neighbour random walks on the integer lattice Z subject to the exclusion rule: each particle independently attempts jumps to the neighbouring site to the right at rate 1, the jumps being allowed only when the destination site is empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' Despite its simplicity, TASEP 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content='13739v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content='PR] 31 Jan 2023 EXACT SOLUTION OF TASEP AND VARIANTS WITH INHOMOGENEOUS SPEEDS AND MEMORY LENGTHS 2 presents a very rich asymptotic behavior, and due to its tractability it has become a paradigmatic model in out-of-equilibrium statistical physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' Much of the interest in TASEP arises from the central role it plays as a member of the KPZ universality class, a broad collection of physical and probabilistic models including particle systems, one-dimensional random growth models, directed polymers, stochastic reaction-diffusion equations, and random stirred fluids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' Models in the KPZ class share a common asymptotic fluctuation behavior, identified by their (in general, conjectural) convergence, under the characteristic 1:2:3 scaling, to a universal, scale-invariant Markov process known as the KPZ fixed point, which was first constructed in [MQR21] as the scaling limit of TASEP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' This 1:2:3 scaling refers to the ratios between the exponents used to rescale the fluctuations, space and time: for KPZ models, as t → ∞ one has fluctuations growing like t1/3 with non-trivial spatial correlations arising at a scale of t2/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' What makes TASEP special in this context is that its distribution can be expressed as a marginal of an (in general, signed) determinantal point process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' For general initial data, this was first discovered in [Sas05;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' Bor+07] (building on exact determinantal formulas for the transition probabilities of the system derived in [Sch97] using the coordinate Bethe ansatz), where it was used to study the special case of periodic initial data, with particles initially occupying sites at 2Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' There the associated spatial fluctuations in the long time 1:2:3 scaling limit were derived;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' they lead to the Airy1 process, whose marginals are given by the Tracy-Widom GOE distribution from Random Matrix Theory [TW96].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' For another choice of special initial data known as step, where particles initially occupy sites at Z<0, there is an even richer algebraic structure, and the analogous scaling limit had been known since the early 2000’s [Joh00;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' PS02;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' Joh03], leading to the Airy2 process and Tracy-Widom GUE marginals [TW94].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' The method employed in [Sas05;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' Bor+07] leads to an expression for the multi-point distribution of TASEP as the Fredholm determinant of a kernel defined implicitly as the solution of a certain biorthogonalization problem which depends on the initial data of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' For step initial data, the biorthogonalization turns out to be (in a certain, concrete sense) trivial, while for periodic initial data the authors were able to solve it explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' The solution of the biorthogonalization problem for general initial data was discovered in [MQR21], and leads to a kernel which can be expressed in terms of the hitting time of a certain random walk to a curve defined by the initial data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' In the 1:2:3 scaling limit, this kernel naturally rescales to an analogous kernel defined in terms of Brownian hitting times, whose Fredholm determinants yield the finite dimensional distributions of the KPZ fixed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' TASEP is part of a family of exactly solvable models in the KPZ class for which a description in terms of biorthogonal ensembles is available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' Besides continuous time TASEP, this family includes discrete time TASEP with both sequential and parallel update, with pushing and blocking dynamics, and with Bernoulli and geometric jumps, as well several generalizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' In [MR23] we extended the explicit biorthogonalization method of [MQR21] to a general class of determinantal measures which includes these models and several others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' The purpose of this paper is to develop a further generalization of the method to cover extensions of these models to the case where particles have different speeds and different memory lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' By the speed of a particle we mean, in the context of continuous time TASEP, simply its jump rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' The memory length of a particle, on the other hand, is easier to interpret in the case of discrete time TASEP: it refers to the amount of time a site remains blocked after a particle occupying it leaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' Memory lengths 0 and 1 translate respectively into the standard discrete time TASEPs with sequential and parallel updates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' For more general memory lengths, the system is no longer Markovian, but it can be reinterpreted as a Markovian system of interacting caterpillars, which occupy a variable number of sites in the lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' The case of systems of caterpillars with equal lengths associated to TASEP and its variants was studied in [MR23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' The biorthogonal ensemble representation for TASEP-like systems in the case of inhomogeneous speeds is well known [BF08;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' BFS08].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' Those papers focus on two particle systems, PushASEP (a combination of TASEP with blocking and pushing dynamics) and TASEP with parallel update, for which they compute scaling limits in the case of periodic initial data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' They actually obtain more general multipoint distributions along “space-like paths” (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' the distribution of collections of particles at different times, but subject to a certain ordering in space-time).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' As we will explain in the next section, the case of TASEP with general memory lengths, or caterpillars, can be recovered by considering an EXACT SOLUTION OF TASEP AND VARIANTS WITH INHOMOGENEOUS SPEEDS AND MEMORY LENGTHS 3 extension of this setting to one where particles are also allowed to start evolving at different times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' The biorthogonal ensemble representation in this setting was obtained in some generality in [MR23], but the explicit solution of the biorthogonalization problem in that paper was restricted to the case corresponding to equal caterpillar lengths and equal speeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' Our goal in this paper is thus to complete this program in the general setting of inhomogeneous speeds and memory lengths, by providing an explicit formula for the biorthogonal kernel appearing in these formulas which is amenable to asymptotic analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' As in [MR23], we will actually work in a more general, abstract setting, which will cover all the examples we have mentioned so far, and several more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' To illustrate the use of such formulas we include three applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' In the first one we will consider the two-speed setting studied in [BFS09].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' In that paper, the authors considered continuous time TASEP with a leading block of particles with a different speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' They obtained explicit contour integral formulas in the case of periodic initial data, for which they were able to perform the asymptotic analysis necessary to describe its limiting behavior depending on the parameters of the model (the length and speed of the leading block).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' Here we will obtain similar formulas for systems in a slightly more general setting which includes continuous and discrete time TASEP with both sequential and parallel update.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' Our result is of course applicable to more general initial data, and the resulting formulas can be used to perform asymptotic analysis of these formulas in the general case, but we leave this for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' In the next two applications we consider discrete time TASEP with equal speeds but different lengths: in the first one we study a system which mixes particles updating sequentially and in parallel, while in the second one we consider the case where a single long caterpillar is placed at the right of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' In both cases we derive, formally, their limits under the proper KPZ 1:2:3 rescaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' We finish this introduction by mentioning that in the particular case of discrete time TASEP with right Bernoulli jumps, the explicit kernel for inhomogeneous speeds was obtained recently, and independently, by Bisi, Liao, Saenz, and Zygouras [Bis+22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' In that paper they also provide a new derivation of the biorthogonal ensemble representation of the system which uses combinatorial properties of the Robinson-Schensted-Knuth correspondence together with intertwining relations to express the transition kernel of the system in terms of an ensemble of non-intersecting lattice paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' Outline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' 2 we describe several interacting particle systems (and some of their generalizations to systems of caterpillars) in the KPZ universality class, whose distributions are particular cases of the determinantal measure considered in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' Under quite general assumptions, we prove in Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content='3 that a marginal of this measure can be written as a Fredholm determinant of a kernel described implicitly through the solution of a biorthogonalization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' 4 is devoted to the explicit solution of this problem in an abstract setting, leading to an explicit formula for the kernel in Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' Finally, in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' 5 we study this kernel and its KPZ scaling limit for the particular examples mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' Notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' We will use the same notation and conventions employed in [MR23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' We use the standard notation N for the set of natural numbers {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' }, and we use N0 = N ∪ {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' For n ∈ N we define the set �n� = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' For N ≥ 2 the Weyl chamber is ΩN = {⃗x ∈ ZN : xN < xN−1 < · · · < x1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' Throughout the paper we consider various kernels K : Z × Z −→ R, which we identify with integral operators acting on suitable families of functions f : Z → C as Kf(x) = � y∈Z K(x, y)f(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content='1) We prefer not to specify precisely the domains of such operators and always interpret them in terms of absolutely convergent sums (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' The composition of two such operators K and L is defined as KL(x, y) = � z∈Z K(x, z)L(z, y), provided that the sum is absolutely convergent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' Then we say that K−1 is an inverse of an operator K if KK−1(x, y) = K−1K(x, y) = I(x, y), where I is the identity operator I(x, y) = 1x=y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' We use the standard notation K∗(x1, x2) = K(x2, x1) for the adjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' EXACT SOLUTION OF TASEP AND VARIANTS WITH INHOMOGENEOUS SPEEDS AND MEMORY LENGTHS 4 Our kernels will often be defined in terms of functions written as contour integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' The contours in these integrals will be usually γr, a circle in the complex plane with radius r and centered at the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' Whenever the contour is different, it will be specified explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' For a closed subset U of C we say that a complex function f is analytic on U if it is analytic on some open domain which contains U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' A particular case of interest will be when U is the closed annulus on the complex plane centered at the origin and with radii 0 < r < R, which we denote by Ar,R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' Finally, for a fixed vector a ∈ Rm and indices n1 < · · · < nm we let χa(nj, x) = 1x>aj and ¯χa(nj, x) = 1x≤aj, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content='2) which we also regard as multiplication operators acting on the space ℓ2({n1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' , nm} × Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' MOTIVATING EXAMPLES: INTERACTING PARTICLE SYSTEMS The main result of this paper will be stated in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' 4 in an abstract setting which, in general, does not necessarily originate from determinantal measures connected to particle systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' In order to motivate that setting and to provide some physical intuition, we begin by presenting in this section some particular cases of that result, stated in the context of the variants of TASEP and systems of interacting caterpillars with inhomogeneous jump speeds and lengths (equivalently, discrete time TASEPs with inhomogeneous speeds and memory lengths) discussed in the introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' We will begin by presenting the general formula which we will obtain for the multipoint distribution of this type of systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' At this stage we will not be precise about the details of and assumptions on the systems to which this result will apply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' Later on we will introduce particular cases corresponding to several particle systems and systems of caterpillars to which the result will apply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' The precise setting for our general result will be provided in Secs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' 3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' A (forward) caterpillar of length L ≥ 1 is an element X of the set KL = � (X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' , XL) ∈ ZL : Xi − Xi+1 ∈ {0, 1}, i ∈ �L − 1� � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' A caterpillar thus has L ordered sections X1 ≥ X2 ≥ · · · ≥ XL;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' we will call X1 the head of the caterpillar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' A system of N ≥ 2 interacting caterpillars of lengths ⃗L = (L1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' , LN) ≥ 1 takes values in the set ΩN,⃗L = � X = (X(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' , X(N)): X(i) ∈ KLi : X1(i + 1) < XLi(i), i ∈ �N − 1� � , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=', the caterpillar X(i) has length Li and no two caterpillars overlap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' For X ∈ ΩN,⃗L we define Xhead = (X1(i) : i ∈ �N�) ∈ ΩN to be the vector of heads of the caterpillars, which can be thought of as N particles located at the sites X1(i) for i ∈ �N�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' Now for fixed speeds vi > 0, i ∈ �N�, we will consider certain specific dynamics for caterpillars Xt ∈ ΩN,⃗L in time t, which is either in R+ or in N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' The simplest example is the case of continuous time TASEP, where all caterpillars have length 1 and the i-th one jumps to the right at rate vi except that jumps onto already occupied sites are forbidden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' We provide below other examples of dynamics of caterpillars to which our results are applicable;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' those with lengths 2 or more all evolve in discrete time (we remark that there is also a generalized version of continuous time TASEP which has the flavor of a length-2 system, but its definition does not quite fit the setting of this section, although it can be analyzed in the framework of Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' 4, see [MR23, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content='3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' We say that the system of caterpillars Xt has initial condition ⃗y ∈ ΩN if X0 ∈ ΩN,⃗L is given by X1 0(k) = · · · = XLk 0 (k) = yk for each k ∈ �N�;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' in words, the k-th caterpillar starts with all its sections at yk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' With a little ambiguity, we will write in this case X0 = ⃗y ∈ ΩN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' Throughout the paper we will be restricted to work in the case when the initial condition ⃗y is in the set ΩN(⃗L) = {⃗x ∈ ZN : xi − xi+1 ≥ (Li − 1) ∨ 1 for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' , N − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' The following holds for all of the systems of caterpillars considered in this paper: for fixed initial condition ⃗y ∈ ΩN(⃗L), for any t ≥ 0 and 1 ≤ n1 < · · · < nm ≤ N, and for any real a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' , am, the EXACT SOLUTION OF TASEP AND VARIANTS WITH INHOMOGENEOUS SPEEDS AND MEMORY LENGTHS 5 distribution function of the heads of the system can be written in the form P � Xhead t (i) > ai, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' , m � = det � I − ¯χaKt ¯χa � ℓ2({n1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=',nm}×Z), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content='1a) where the kernel Kt is given implicitly in terms of the solution of a certain biorthogonalization problem which involves the initial data ⃗y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' The precise form of the biorthogonal kernel Kt is presented in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' We will see shortly one way to interpret the restriction to X0 = ⃗y with ⃗y ∈ ΩN(⃗L) in our setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' The restriction actually arises as a requirement in the proof of this representation (which for systems of caterpillars can be found in [MR23]), but it appears in general to be necessary for this representation to hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' The same restriction will be crucial in the proof of our main abstract result, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content='10, from which the results presented in this section will be corollaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' Our main result will provide an explicit formula for the kernel Kt appearing in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content='1a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' The formula follows from solving explicitly the biorthogonalization problem defining the kernel for general initial condition and representing the result in terms of a hitting problem for a certain random walk to a curve defined by the initial data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' This representation is such that the appropriate scaling limits can be obtained naturally, by computing the limits of the kernels involved in the formula and recognizing that the random walk hitting problem converges to a similar problem for a Brownian motion;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' we present examples of this in Secs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content='2 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content='3 (see also [MQR21] where the scheme was implemented in detail for continuous time TASEP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' In order to state our formula we first need to make several definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' Consider a meromorphic function ϕ : U −→ C, where the domain U ⊆ C contains 0 and all values vi, which is analytic and non-zero in an annulus Ar,¯r with radii 0 < r < min vi and ¯r > max vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' Fix also a real parameter θ ∈ (r, min vi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' We introduce the kernels Q(ℓ,n](x, y) = 1 2πi � γr dw θx−y wx−y−n+ℓ+1 n � i=ℓ+1 αiϕ(w)Li−1−1 vi − w with αi = vi−θ θ ϕ(θ)1−Li−1, integer 0 ≤ ℓ < n and L0 = 1, and Q+ (ℓ,n](x, y) = 1 2πi � γr dw θx−y wx−y−n+ℓ+1 n � i=ℓ+1 α+ i ϕ(w)Li−1 vi − w , with α+ i = vi−θ θ ϕ(θ)1−Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' These two kernels are Markov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' We let B+ m be the time-inhomogeneous random walk which has transitions from time m−1 to time m, m ≥ 1, with step distribution Q+ (m−1,m].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' For a fixed initial condition ⃗y ∈ ΩN(⃗L) we define the stopping time τ + = min{m = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' , N − 1 : B+ m > ym+1}, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=', τ + is the hitting time of the strict epigraph of the “curve” (ym+1)m=0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=',n−1 by the random walk (B+ m)m≥0 (we set τ + = ∞ if the walk does not go above the curve by time N − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNFST4oBgHgl3EQfLDje/content/2301.13739v1.pdf'} +page_content=' Next for integer n ≥ 1 and 0 ≤ m < n and for a real t ≥ 0 we define the kernels S−n(x, y) = 1 2πi � γr dw θy−x wy−x+n+1 ϕ(w)t �n i=1(vi − w) �n−1 i=1 α+ i ϕ(w)Li−1 , ¯S(m,n](x, y) = − 1 2πi � Γ⃗v dw θx−y wx−y−n+m+1 ϕ(w)−t �n−1 i=m+1 α+ i ϕ(w)Li−1 �n i=m+1(vi − w) , and ¯Sepi(⃗y) n (x, y) = EB+ 0 =x � ¯S(τ +,n](B+ τ +, y)1τ + Ic. Here η represents efficiency of a superconducting diode which changes its sign when the polarity of +magnetic field (B) is reversed. (d) Strength of superconducting diode effect. (Left) The superconducting rectification becomes +maximal when +Ic (or −Ic) remains finite but −Ic (or +Ic) becomes zero. As defined by the diode efficiency, equation (7), +the maximum difference of critical depairing currents +Ic and −Ic can be about a factor of 2. (Right) In the junction-free +noncentrosymmetric bulk material, rectification can be induced by applying magnetic field perpendicular to the directions of +both the polar axis and the current. +I. +MECHANISMS OF SUPERCONDUCTING +DIODE EFFECT +The origin of SDE manifests in a number of physi- +cal phenomena, imposed by transport mechanisms, sym- +metry constraints, and underlying quantum functionali- +ties of superconducting materials. In this section, it is +explicitly demonstrated how nonreciprocity of supercur- +rent is intertwined with underlying symmetries of non- +centrosymmetric systems, e.g., nonreciprocity driven by +MCA in time-reversal asymmetric systems and that in- +duced by shift current or Coulomb interactions in time- +reversal symmetric systems. It is also highlighted how +nonreciprocity of supercurrent is associated with nonre- +ciprocal behaviour of physical quantities characterizing +current-voltage (I-V) and current-phase relation (CPR), +e.g., resistance and inductance respectively. +Quantum +functionalities of SCs, such as SOI, Berry phase, band- +topology and their effects on the SDE efficiency are also +discussed. +Finally, intertwining between nonreciproc- +ity and helical superconductivity with finite-momentum +spin-singlet or spin-triplet Cooping pairing is presented. + +a) +VA +c) +VA +VA +B=0 +Noncentrosymmetric +M +B=0 +B=0 +n=0 +Semiconductor +B>0 +B>0 +n>0 +Ic+ +- +B<0 +B<0 +n> N +Ys +b) +d) 0.0 +Ict +Aeis +VA +0.0 +B +Noncentrosymmetric +B±0 +Superconductor +1.5 +1.0 +'c- I'r. +0.5 +-Ic- +Ir+ +Ic+ +0.0 +-0.5 +-1.0 +Bar +fUpwardSweep +-1.5 +Sc +Lrrier +SC +Downward Sweep +Aeid. +Aeigr4 +A. +Nonreciprocity and magnetochirality +In condensed matters, nonreciprocity refers to the +spatial-dependence of physical quantities. A prototypical +example of nonreciprocal transport is a diode effect which +refers to a highly direction-selective electron transport +in systems with a lack of spatial inversion center. Un- +til recently, nonracirocity was thought to be a transport +phenomenon associated with dissipative materials. For +instance, in conventional semiconductors, where resis- +tance is the nonreciprocal quantity, nonreciprocity refers +to charge transport that is sensitive to the polarity of cur- +rent or bias potential. Such nonreciprocal charge trans- +port leads to a diode effect in a spatially asymmetric pn +junction [4, 5], in which, spatial asymmetry of the junc- +tion is associated with electron-hole asymmetry across +the contact of n- and p-type semiconductors. +In modern quantum condensed matter physics, in addi- +tion to electron-hole asymmetric junctions, nonreciprocal +charge transport can be induced in spatially symmetric +devices, in which resistance is direction-selective when in- +version and/or time-reversal symmetry are broken. This +can be realized, for instance, by externally applying an +electric and/or a magnetic field orthogonal to each other +and to the direction along which current is traversing. It +implies that nonreciprocal transport can be treated as a +bulk property of noncentrosymmetric quantum materials +[57, 58]. In noncentrosymmetric systems, i.e., in which +inversion symmetry is broken, nonreciprocal responses +can be classified into four categories [57]: (i) linear- and +(ii) nonlinear-response in time-reversal symmetric sys- +tems, and (iii) linear- and (iv) nonlinear-response in time- +reversal asymmetric systems. +When both inversion and time-reversal symmetry are +simultaneously broken, a closely related phenomenon +leading to nonlinear nonreciprocal response is MCA +[16, 27, 28, 59–70]. In the linear response regime of non- +centrosymmetric systems, broken time-reversal symme- +try produces finite magnetochiral effect, as recognized +by the Onsager’s reciprocal theorem [57, 66, 71, 72], and +the longitudinal transport coefficients become dependent +on the polarity of the current. The Onsager’s reciprocal +theorem, and thus the magnetochiral effect and direction- +selective transport, can be generalized to the nonlinear +regime of both (semi)conductors [59, 62] and SCs [27, 28]. +1. +Nonreciprocity of supercurrent +In 1996, before even prediction/observation of nonre- +ciprocity in (semi)conductors by Rikken et al. [59, 62], +V. M. Edelstein [28] proposed nonreciprocity in the crit- +ical supercurrent. Followed by his earlier work charac- +terizing Cooper pairing in noncentrosymmetric SCs [26] +and describing magnetoelectric effects in polar SCs [27], +V. M. Edelstein [28] proposed that if the mixed product +(c × B) · ˆjc is non-vanishing in polar SCs, then the mag- +nitude of the critical current jc(B) depends on the sign +of this mixed product, i.e., the critical current appears +to be different for two opposite directions. By employ- +ing GL theory for a thin film of polar superconductor, +expression for the nonreciprocity in the critical current +reads [28] +jc(B) = jc(0)[1 + γj(c × B) · ˆj] +(1) +Here c is the unit vector along the polar axis, ˆj is the +unit vector along the supercurrent, and B is an in-plane +magnetic field. The exact expression for the observable +can be found in the reference [28]. +MCA +and +nonreciprocity +has +been +observed +in +(semi)conductors [60–64, 69] that show resistive current +as well as in SCs [16, 67, 68, 70] that display dissipation- +less supercurrent. So a question arises naturally: how +nonreciprocity can uniquely be defined in these two sys- +tems with completely contrasting behaviour? +As first +pointed by Rikken et al. [59], when both inversion and +time-reversal symmetries are broken, the finite MCA co- +efficient γ gives rise to different resistance for electric cur- +rents traversing in different (opposite) directions. That +is, MCA can be defined as the inequivalence of R(+I) +and R(−I). In (semi)conductors, resistances along op- +posite directions differ, i.e. R(+I) ̸= R(−I), but both +R(+I) and R(−I) normally take finite values. On the +other hand, in SCs, such a situation becomes more dras- +tic: either one of R(±I) remains finite while the other +vanishes completely. +With this consideration in SCs, it becomes more appro- +priate to define nonreciprocity in terms of (super)current. +That is, as shown in figure [1], nonreciprocity in SCs +means supercurrent flows along one direction while a +normal current along the other(opposite). Observation +of such a situation is more probable near critical tem- +perature Tc, i,e., in the fluctuation regime of metal- +superconductor resistive transition, where the critical +current is different along opposite directions, i.e. Ic+ ̸= +Ic−. Thus, if the current is tuned between Ic+ and Ic−, +the system displays zero resistance for supercurrent but +nonzero for the normal current. +It can be understood how conductance varies while go- +ing from normal to a superconducting phase. The lin- +ear resistance R0 is normally scaled by the Fermi en- +ergy EF , i.e., the kinetic energy of the electrons, while +the MCA coefficient γ depends upon the strength of SOI +and the magnetic field. Correspondingly, nonlinear resis- +tance induced by MCA may be treated as a perturbation +to R0. In the normal conducting phase, because the SOI +energy (Esoi) and the Zeeman energy (µBB) is usually +much smaller (by many orders of magnitude) than EF , +MCA coefficient γ → γN is typically very tiny, usually of +the order of ∼ 10−3 to 10−2 T−1 A−1 in typical metals +[59, 61, 67]. However, as the superconducting phase de- +velops, superconducting transition temperature Tc or the +superconducting gap ∆sc appear as a new energy scale. +That is, the energy scale in SCs, to which the strength of +SOI has to be compared with, is superconducting gap and +not the Fermi energy. Since the energy scale (∼meV) in + +5 +the SCs is much smaller than the Fermi energy (∼eV) in +metals, the effects of SOI and Zeeman energy greatly en- +hance in the superconducting phase [66, 67]. As a result, +near the superconducting transition temperature T ≳ Tc, +MCA coefficient becomes reasonably large [57] and, thus, +the paraconductivity [73] above Tc becomes nonrecipro- +cal. In the superconducting fluctuation region, i.e. when +T → Tc and the superconducting order parameter ∆sc +develops, a sizable enhancement in MCA coefficient γS +is found (ref.[16, 57, 66, 67]) and a robust non-reciprocal +charge transport is demonstrated in noncentrosymmetric +SCs [67, 70]. For instance, by employing GL theory for +an Ising type SC MoS2, R. Wakatsuki et al. [67] showed +that the ratio of MCA coefficients in the superconducting +resistive region (γS) and the normal resistive region (γN) +is quite large +γS +γN +∼ +� EF +kBTc +�3 +(2) +Such anomalous enhancement of the MCA coefficient, as +it is associated with the energy scale difference between +the superconducting gap and the Fermi energy, can be +considered an intrinsic feature of both Rashba and Ising +type noncentrosymmetric SCs [66]. However, mainly due +to a gradual decrease in the linear resistance R0 during +the metal-superconducting transition, R0 remains larger +(by orders of magnitude) than the nonlinear resistance in +low-dimensional superconducting materials such as MoS2 +(ref. [67]), WS2 (ref. [68]) and Bi2Te3/FeTe (ref. [16]). +As a result, low rectification ratio in these superconduct- +ing materials does not suffice for device implementation. +In this regard, it is highly desired to search for novel +mechanisms/principles to enlarge the rectification effect +and guide the design of efficient SDE. +2. +From resistance to supercurrent +Rikken et al. +[59, 62] generalized the Onsager’s re- +ciprocal theorem to the nonlinear regime and gave a +heuristic argument for nonreciprocity and MCA in two- +dimensional diffusive conductors. In their seminal pro- +posal of MCA in (semi)conductors, Rikken et al. +[59] +suggested that nonreciprocal nonlinear resistive response, +characterized by the directional IV-characteristics, can +be described by a current-dependent resistance R(I) as +R(I) = R0[1 + βB2 + γ(B × r) · I] +(3) +Here R, B, and I are the resistance, magnetic field, and +the electric current, respectively. The unit vector r rep- +resents the direction along which mirror symmetry is bro- +ken. On the right-hand side, first term is the resistance +at zero magnetic field, second term denotes the normal +magnetoresistance, and the third term corresponds to the +MCA. Dependence of MCA coefficient γ on electric cur- +rent, magnetic field, as well as their mutual orientation, +relative to the direction along which mirror symmetry +is broken, allows us to access various functionalities and +aspects of noncentrosymmetric materials. +First, dependence of MCA on electric current leads to +a current-dependent resistance which generally causes a +nonlinear nonreciprocal transport, i.e. nonlinear voltage- +drop. Such nonlinear nonreciprocal transport can be de- +tected by measuring the second harmonic signal through +lock-in techniques, see further details in section (V A). +Second, dependence of MCA on magnetic field implies +that its coefficient γ remains non-zero only when time- +reversal symmetry is broken. In addition, the orientation +of magnetic field must be orthogonal to both current and +the direction along which mirror symmetry is broken. It +implies that, not only finite magnetic field is required, +but its orientation is also important depending upon the +nature of SOI associated broken mirror symmetry. Here +we discuss the key mechanisms associated with nonre- +ciprocity in superconducting systems. +The conventional semiconducting diode is not fa- +vorable for energy-efficient technologies with ultralow +power consumption. At high temperatures relevant for +thermionic transport, owning to their finite resistance, +energy loss is inevitable in semiconductors. At low (sub- +Kelvin) temperatures, on the other hand, relevant for +cryogenic electronics [7] and ultrasensitive (sub-THz fre- +quencies) optoelectronics and detection [74], semicon- +ductors cease to work due to their large energy gap. +Therefore, owning to their dissipationless supercurrents, +intrinsically low impedance and thereby very high rec- +tification of supercurrents, and low energy scales as- +sociated with superconducting gap (∼meV) as com- +pared to semiconductor energy gap (∼eV), a supercon- +ducting diode is highly desired for energy-efficient cryo- +genic electronic/optoelectronic devices [6, 7]. However, +as broken electron-hole symmetry is required, physi- +cal realization of a junction-free superconducting diode +turns out to be difficult with electron-hole symmet- +ric Bardeen–Cooper–Schrieffer (BCS) superconducting +state. +In light of this, SCs with broken spatial-inversion and +time-reversal symmetry can offer bright perspectives for +supercurrent diode effect via MCA. However, for the +implementation of a simplest possible device displaying +SDE intrinsically, it is worthy to pin down intertwining +between superconductivity and MCA. First, unlike recti- +fication due to self-field effects in asymmetric supercon- +ducting quantum interference devices (SQUIDs) [75, 76], +MCA is expected to induce an intrinsic SDE in sym- +metric devices with spatially homogeneous supercurrent +density. Second, intertwining between superconductivity +and MCA can lead to a spin-filtering diode effect in a +spin-selective Al/EuS/Cu superconducting tunnel junc- +tion [77] and thus superconducting spintronic technolo- +gies [8]. +However, even such promising ferromagnetic +superconducting structure, in which electron-hole sym- +metry can possibly be broken when both spin-filtering +and spin-splitting are present to induce opposite shift in +BCS density of state (DOS), are not desired for the in- + +6 +trinsic SDE with nonreciprocal supercurrent transport. +Finally, we could see the light at the end of the tunnel: +Intrinsic SDE with nonreciprocal supercurrent transport +can be realized in a helical superconductor with finite- +momentum Cooper pairing which can be induced by anti- +symmetric Rashba/Ising SOI and Zeeman exchange spin- +splitting. Further details on this key mechanism are dis- +cussed in section I D. +3. +From inductance to supercurrent +Nonreciprocity in the fluctuation regime of metal- +superconductor resistive transition confines SDE to a +narrow temperature window near Tc. Baumgartner et al. +[13] pointed that the temperature window in which MCA +coefficient becomes sizeable must be widened for a sus- +tainable fabrication of devices showing SDE. To achieve +this milestone, the authors demonstrated supercurrent +rectification in the superconducting phase, i.e., far be- +low the transition temperature Tc. Since d.c. measure- +ment of resistance–current (R–I) curve is not viable at +low temperatures, as the resistance vanishes, supercur- +rent response to an alternating-current (a.c.) excitation +is studied, which is described by its superfluid stiffness, +and thus, can be detected through kinetic inductance +measurements. +If mirror symmetry is broken along out-of-plane direc- +tion (ˆez), whereas the current I and magnetic field B are +directed in-plane, MCA or nonreciprocity for the super- +fluid can be described by an equation similar to that for +the resistance (3), i.e., +L(I) = L0[1 + γLˆez(B × I)] +(4) +Here resistance (R) is substituted for the kinetic induc- +tance (L). +The nonraciproocity in supercurrent could +then be characterized by a new observable, i.e., MCA +coefficient γL. +B. +Nonreciprocity without magnetochirality +In noncentrosymmetric but time-reversal symmetric +systems, nonreciprocal nonlinear response can be real- +ized via shift current (photovoltaic effect) [78–80], via +Coulomb interactions [81], and asymmetric Hall effect +of vortices and antivortices [82]. Shift current is a non- +trivial contribution by the Berry phase of the electronic +states [83]. That is, unlike conventional charge transport +which comes form intraband transition [78–80] and de- +pends only on the energy dispersion, interband shift cur- +rent depends not only on the energy dispersion but also +on the Bloch wavefunction and plays an essential role +in modern quantum transport phenomena [83, 84]. Fol- +lowed by theoretical proposals [78, 79], shift current has +been studied for semiconductor (GaAs) [85], ferroelec- +tric semiconductor (SbSI) [86], and Dirac surface states +of a 3D topological insulator (Bi2X3(X=Te, Se)) with a +hexagonal warping [87]. It shows that shift current is an +ubiquitous phenomenon in noncentrosymmetric quantum +materials, and the nonreciprocal nonlinear response can +also be realized without breaking of time-reversal sym- +metry. +T. Morimoto and N. Nagaosa [81] theoretically showed +that nonreciprocal nonlinear I–V characteristics can be +induced by electron correlations in noncentrosymmet- +ric multiband systems without time-reversal symmetry +breaking. According to general symmetry considerations, +nonreciprocal nonlinear response in such time-reversal +symmetric systems is generally constrained by the pres- +ence of two ingredients: (i) dissipation, and (ii) inter- +actions (e.g., electron-electron and electron-phonon in- +teractions). First, generalization of Onsager’s reciprocal +theorem to nonlinear current responses shows that dis- +sipation is crucial for nonreciprocity. Second, gauge in- +variant formulation of Keldysh Green’s function shows +that nonreciprocity disappears without interactions. A +general formula of the nonreciprocity ratio (γc), and de- +rived by employing nonequilibrium Green’s functions for +two-band systems with onsite Coulomb interaction, reads +[81] +γc = δJ +J ≃ +U +Eg,kF +eEa +W +(5) +where U is Coulomb interaction energy (γc → 0 for +U → 0), Eg,kF is the band gap, kF is the Fermi mo- +mentum, e is charge of electron, E is the applied electric +field, a is the lattice constant, and W is the bandwidth. +Here J is the linear current response (the part of current +response proportional to E) while δJ is the nonlinear cur- +rent response (the part of current response proportional +to E2). When U ≈ Eg,kF , nonreciprocal response can be +estimated by quantifying the ratio eEa/W between the +electric potential (eEa) in the unit cell and the band- +width (W). +First the nonreciprocity induced by electron correla- +tion [81] is relatively smaller than that induced by MCA, +in both typical metals [59, 61] as well as resistive semi- +conductors [62]. Second, the requirement of dissipation +means nonreciprocal response induced by Coulomb in- +teractions is only measurable in the resistive fluctua- +tion regime of metal-superconductor transition, and not +in the superconducting phase below transition temper- +ature. On the other hand, nonreciprocity of supercur- +rent by asymmetric Hall effect of vortices and antivor- +tices in time-reversal symmetric trigonal superconductors +(PbTaSe2) [82] promise another nonlinear transport phe- +nomena to study SDE. However, thus far, experimental +observation of nonreciprocity of supercurrent has only +been reported in noncentrosymmetric systems with bro- +ken time-reversal symmetry, while the observation of su- +percurrent nonreciprocity in time-reversal symmetric SCs +is scarce. + +7 +C. +Role of spin-orbit coupling +Apart from the strength of SOI, since broken inversion +symmetry is assumed/required (γ = 0 for centrosymmet- +ric systems), MCA coefficient γ also depends on the na- +ture of associated SOI. That is, based on the lattice sym- +metry, finite γ may be realized in noncentrosymmetric +condensed matter systems [58] such as polar or Rashba +SCs and trigonal or Ising SCs. In polar systems, where +Rashba SOI generated from broken Mz and electron’s +spin is locked to in-plane orientations, nonreciprocal su- +percurrent is controlled by an in-plane magnetic field. On +the other hand, in trigonal systems with D3h symmetry, +where Ising or valley-Zeeman SOI is originated from bro- +ken Mx/y and electron’s spin is locked to out-of-plane +orientations, nonreciprocal supercurrent is controlled by +an out-of-plane magnetic field. +In addition, it would be interesting to study effects on +SDE due to a crossover between various SOI types as- +sociated with broken inversion symmetry. For instance, +Baumgartner et al. [42] studied effects of Rashba and +Dresselhaus SOI on supercurrent rectification and MCA +by fabricating Al/InAs-2DEG/Al ballistic JJs. Similarly, +Pekerten et al. [44] studied an interplay between Rashba +and Dresselhaus SOI and investigated effects of magnetic +and crystalline anisotropies on the topological supercon- +ductivity in JJs. If only Rashba-type SOI is present in +the JJs, the topological phase diagram strongly depends +on the magnetic field orientation but remains insensi- +tive to the supercurrent polarity. +On the other hand, +when both Rashba- and Dresselhaus-type SOIs coexist, +the phase diagram exhibits a strong dependence on the +magnetic field as well as junction crystallographic orien- +tations. These studies illustrate the role of SOI, both for +the material search leading to SDE with the best perfor- +mance and probing phase diagram of topological/helical +SCs. +Furthermore, H. Yi recently showed a crossover from +Ising- to Rashba-type superconductivity in epitaxial +topological insulator and monolayer Ising superconduc- +tor heterostructure [88] (Bi2Se3/NbSe2). By altering the +thickness of Bi2Se3 film, emergence of topological super- +conductivity coincides with a considerable suppression of +the upper critical in-plane magnetic field. While the for- +mer transition is marked by the emergence of spin-non- +degenerate surface states and Rashba-type quantum-well +bands in the bulk, the later signatures a crossover from +Ising- to Rashba-type superconductivity. +This system +represents a classic example and sheds light on the role +of SOI while searching new systems to engineer SDE. +Based on the above discussion, one can conclude that +Ising/trigonal topological SCs, such as NbSe2 which dis- +play exceptional upper critical-fields exceeding the Pauli +limit [89–91], can be identified as suitable materials for +the realization of SDE via magnetic field driven MCA. +On the other hand, owning to the nontrivial Berry phase +intertwined with band topology, time-reversal symmetric +polar/Rashba SCs can be identified as promising mate- +rials for the realization of SDE via shift current. This +qualitative analogy needs further quantitative investiga- +tion, as the performance of SDE also depends upon the +strength of SOI, interband transition, and photoresponse +etc. +D. +Helical superconductivity +To observe SDE via MCA in noncentrosymmetric su- +perconductor, breaking of time-reversal reversal symme- +try (T ) is necessary but not sufficient. First, SDE is not +necessarily present in all the magnetic SCs but rather +the orientation of magnetic field or magnetization should +be such that it breaks all possible inversion symmetries +Pi (i = x, y, z). Second, time-reversal reversal symmetry +should be broken such that a finite-momentum Cooper +pairing or a helical superconductivity emerges. +Third, +magnetic field (or magnetization) should have a com- +ponent perpendicular to the polarity of applied current +such that finite pairing momentum emerges parallel/anti- +parallel to the current direction. In this section, after a +brief overview of helical superconductivity, desired ori- +entation of magnetic field or magnetization, and its in- +tertwining with the nature of SOI, polarity of applied +current, direction along which structural mirror symme- +try is broken, and the momentum space orientation of +Cooper pairing momentum is discussed. +1. +Fulde–Ferrell–Larkin–Ovchinnikov state +In the field of conventional superconductivity, follow- +ing from the fact that Cooper pairing is formed between +Kramers partners and most known conventional SCs are +characterized by the Bardeen–Cooper–Schrieffer theory +[93], presence of time-reversal symmetry is a key ingre- +dient and the preserved Kramers degeneracy is the fun- +damental reason/criterion that stabilize superconducting +phase in so many systems at sufficiently low temperatures +[94–96]. Thus, such a conventional superconducting state +with a spin-singlet pairing is suppressed or destroyed by +time-reversal symmetry breaking perturbations — as a +consequence of applied magnetic field, doped magnetic +impurities, or intrinsic magnetic instability leading to +spontaneous magnetization — due to electron pair break- +ing. +On +the +other +hand, +beyond +conventional +BCS +paradigm, unconventional superconductivity allows co- +existence of more exotic superconducting order param- +eters with magnetic order. +For instance, as predicted +independently by Peter Fulde and Richard Ferrell (FF) +[97] and Anatoly Larkin and Yuri Ovchinnikov (LO) +[98], magnetic fields can give rise to a superconduct- +ing state with FF-type order parameter ∆(x) = ∆eiqx +and/or spatially inhomogeneous LO-type pair potential +∆(x) = ∆ cos qx. +The underlying physical mechanism +of the Fulde–Ferrell–Larkin–Ovchinnikov (FFLO) state + +8 +FIG. 2. abc (A) Schematics of the Ising- and Rashba-type superconducting pairing symmetry. a Ising-type pairing symmetry +originates from spin-singlet Cooper pairs formed between the electrons near the K and K′ valleys with opposite spins pinned +to the out-of-plane direction. (b) Rashba-type pairing symmetry originates from spin-singlet Cooper pairs formed between +the electrons near the Γ point with opposite momentum and opposite spins pinned to the in-plane direction. Figure (A) is +reproduced with permission from ref. +[88]. +(B) (a) Schematic sketch showing magnetic field driven spin-splitting of free- +electron parabola, inducing Pauli paramagnetism, and leading to different Fermi momenta for spin-up (k↑ +F ) and spin-down +(k↓ +F ) electrons. (b) Schematic representation of the conventional spin-singlet BCS pairing state (left) with zero center-of-mass +momentum and the spin-singlet FFLO pairing state (right) with a finite center-of-mass momentum (q). The red (blue) circle +represents the Fermi surface for electrons with spin-up (spin-down). Figure (B) is reproduced with permission from ref. [92]. +(C) Band splitting and Fermi contours under Rashba SOI and exchange field in a JJ Nb/Pt/Nb with a Pt barrier proximity- +magnetized by a ferrimagnetic insulating Y3Fe5O12 (YIG) film. a Rashab SOI splits the conduction bands laterally (along +momentum (k) axis) by ∆kR while the Zeeman exchange field splits them vertically (along energy (E) axis) by ∆Eex such that +the Kramers degeneracy is removed. Here EF represents the Fermi level while kx/y stands for in-plane momentum components. +bI-V curve representing SDE at T = 2 K (< Tc) for different orientations of the Pt magnetization (MP t), parallel (yellow) and +antiparallel (cyan) with respect to the x-axis. Here Pt magnetization orientations, and, thus the the direction of the exchange +field, reverses when the magnetization orientation of the proximity-coupled YIG is inverted. The diode symbols in the yellow +(cyan) shaded regime indicates that the Josephson supercurrent flows only in the positive (negative) y-direction. Figure (C) is +reproduced with permission from ref. [19], Springer Nature Ltd. (D) Supercurrent diode effect under external current source J +and in-plane magnetic field B in a noncentrosymmetric Rashba superconductor. (a and c) Schematics of device plots showing +Rashba- and Zeeman-split normal state Fermi surfaces (denoted by circles) and the directions along which J and B are applied. +(b and d) Schematic phase diagrams in the B-J plane corresponding to device configurations shown in (A and B), respectively. +Figure (D) is taken from ref. [30]. +[97, 98], owning to the opposite energy-shift in the elec- +tronic spin bands as shown in Fig. 2(B), induces non-zero +centre-of-mass momentum of Cooper pairs and leads to +a spatially-modulated order parameter. +The FF state +ubiquitously exist in noncentrosymmetric SCs and is par- +ticularly known as the helical superconductivity [99–111]. +The FFLO states, and/or the implications of the he- +lical superconductivity, have been obtained in heavy- +fermion SCs CeCoIn5 [112–114], organic SCs [92], pure +single crystals of FeSe [115, 116], thin films of Pb [110] +and doped SrTiO3 [117], a heavy-fermion Kondo super- +lattice [118, 119], and a three-dimensional topological + +K" +Ising type +(b) +Rashba type +(A) +(a) +(B) +(a) +E +ki +2ugB +Kk +(b) +BCS +FFLO +K" +k; =-k +E +k;=-kt+q +(a) +(b) +(C) +(a) +(q) +(D) +Rashba SOC + exchange spin-splitting +0.3 +E +T = 2.0 K +0.2 +0.1 +I (mA) +0 +0.1 +(c) +(d) +0.2 +B +Mpt +0.3 +0.A +V (mV)9 +insulator Bi2Se3 [120]. +While the existence of FFLO- +like states is well established in proximity-coupled SCs +and ferromagnets [121], the experimental observation of +FFLO states has been reported in nonmagnetic SCs by +applying external magnetic fields [112, 113] as well as +intrinsic ferromagnetic SCs [122–128]. +2. +Pairing in Rashba/Ising SCs +To understand SDE via finite-momentum Cooper pair- +ing in noncentrosymmetric superconducting materials, it +is instructive to quickly review pairing phenomenon in +Ising- and Rashba-type superconductivity. In this regard, +(5QL)Bi2Se3/NbSe2(ML) heterostructure is a promising +example where a crossover from Ising- to Rashba-type +superconductivity is reported recently [88]. NbSe2 bulk +crystal with 2H phase is a well-studied superconduc- +tor with Fermi surface sheet-dependent s-wave super- +conductivity [129]. 2H-NbSe2 bulk crystals covered by +molecular-beam epitaxy (MBE)-grown films of Bi2Se3 +or Bi2Te3 topological insulators are the most success- +ful topological superconductor interfaces [130–135]. It is +also well-known that monolayer NbSe2 with the Se-Nb- +Se trilayer structure, with preserved out-of-plane mir- +ror symmetry but broken in-plane inversion symmetry, +is a prototypical Ising-type superconductor [89, 90], and +are preferred over 2H-NbSe2 bulk crystals for for device +fabrication and technological applications. On the other +hand, Bi2Se3 with the Se-Bi-Se-Bi-Se Quintuple-layered +structure is a prototypical 3D strong TI hosting a sin- +gle surface Dirac cone intertwined with nontrivial bulk +Rashba bands at the Γ-point [136, 137]. +In monolayer NbSe2, broken in-plane inversion sym- +metry generates an out-of-plane spin polarization and +originates the Ising-type SOI which induces a valley- +dependent Zeeman-type spin-splitting, as shown in fig- +ure 2(A-a). +Such opposite spin-splitting in the bulk +valence bands around valleys K and K′ leads to Ising- +type superconducting pairing symmetry in monolayer +NbSe2, i.e., which refer to the intervalley spin-momentum +locked spin-singlet Cooper pairing between two electrons +with opposite momenta and opposite out-of-plane spins. +On the other hand, as shown in figure 2(A-b), own- +ing to the emergence of Rashba-split low-energy con- +duction bands at Γ-point and corresponding Dirac sur- +face states, (5QL)Bi2Se3/NbSe2(ML) heterostructures +become proximity-coupled topological SCs with Rashba- +type superconducting pairing symmetry, i.e., which refer +to the Cooper pairing between two Rashba-split electrons +with opposite momenta and opposite spins pinned to the +in-plane direction. +3. +Nonreciprocity in FFLO states +Such momentum-dependent spin-splitting of the low- +energy electronic bands, caused by broken inversion sym- +metry in the noncentrosymmetric bulk crystals, surfaces, +and interfaces, is crucial for the emergence of nonrecip- +rocal transport. +However, in order to avoid the can- +cellation of this effect due to superposition of degen- +erate Kramers pairs, one also needs energy-dependent +spin-splitting such that electrons with opposite momenta +and opposite spin become Kramers non-degenerate, or +simply non-equivalent. +Typically, this can achieve by +breaking time-reversal symmetry. +In the presence of +external magnetic field or intrinsic magnetization, par- +allel to the electron’s spin-orientation, BSC-type zero- +momentum Cooper pairing (symmetric around Γ-point) +become asymmetric around Γ-point due to opposite +energy-shift and FFLO-type finite-momentum Cooper +pairing originates in both Ising- and Rashba-type super- +conducting phase. +Recent theoretical studies [29–32] revealed how a non- +trivial interplay of antisymmetric Rashba SOI, magnetic +field, and helical supercurrent leads to an intrinsic SDE in +noncentrosymmetric bulk SCs. It implies that intrinsic +SDE is closely related to the FFLO state [97, 98] with +a periodically modulating phase of the superconducting +order parameter ∆sc(r) = ∆sceiq·r: Rashba SOI splits +Fermi surfaces while the finite pairing momentum q0 is +induced by the magnetic field and varies continuously +with its strength and orientation. +In terms of charge +transport, when an in-plane magnetic field is applied, +Cooper pairs in noncentrosymmetric Rashba SCs acquire +a finite-momentum q0, and, as a result, critical currents +traversing along the direction parallel and antiparallel to +q0 become unequal. +Recently, N. Yuan and L. Fu [30] explicitly demon- +strated the effect of in-plane magnetic field on Rashba +spin-split bands and, thus, the emergence of finite- +momentum Cooper pairing. As shown in figure 2 (D-a) +and (D-c), a finite magnetic field (B) displaces the cen- +ters of Rashba-split inner(+) and outer(-) Fermi pockets +from k = 0 to opposite momenta, ±k0 = ±ˆz × B/vF , re- +spectively, and leads to a finite intrapocket Cooper pair +momentum q0 = ±2k0. Owning to the larger DOS in +the outer pocket, usually, energetically favored state is +the one with the Cooper pair momentum q0 = −2k0. +Figure 2 (BR-b) and (BR-d) show a magnetic field de- +pendence of the depairing critical current in the fluctu- +ation regime of a metal-superconductor resistive transi- +tion. +When B ∥ J, as shown in figure 2 (BR-b), the +phase diagram in the B-J plane remains symmetric with +respect to both B and J axes and thus, no nonreciproc- +ity in the critical current Jc and critical magnetic field +Bc. However, when B ⊥ J, as shown in figure 2 (BR-d), +the phase diagram becomes asymmetric/skewed, indicat- +ing nonreciprocity in the critical current J+ +c ̸= J− +c +and +polarity-dependence of critical field B+ +c +̸= B− +c . +That +is, the maximum critical current flowing in the direction +parallel and antiparallel to q0 are different, which leads +to SDE. On the same footing, in the presence of a su- +percurrent, the polarity-dependence of in-plane critical +fields is also a direct consequence of the finite-momentum + +10 +Cooper pairing. Similar mechanism has been realized in +Rashba SCs with intrinsic magnetization. Figure 2(BL) +demonstrates Rashba spin-splitting and SDE by control- +ling magnetization orientation in a JJ Nb/Pt/Nb with a +proximity-magnetized Pt barrier (Pt/Y3Fe5O12 (YIG)) +[19]. +4. +From spin-singlet to spin-triplet pairing +It is also crucial to consider competition between spin- +singlet and spin-triplet pairing. +Note that, in the ab- +sence of magnetic field, pairing momentum remains zero +for spin-singlet symmetry even in the presence of SOI, +whereas SOI induces finite momentum for spin-triplet +symmetry, q± = q± +0 . However, owning to the symmet- +ric shift of pairing momentum (q+ +0 = q− +0 ) in the absence +of magnetic field, even finite-momentum of spin-triplet +pairing does not induce nonreciprocity of supercurrent. +It can be explained by noticing that the q-linear term in +the kinetic energy of Cooper pairs could only shift the +momentum space positions of the optimal critical cur- +rents (maximum I+ and minimum I−) while keeping I± +values unchanged, and thus, could not induced nonre- +ciprocity. To induce nonreciprocity, one needs magnetic +field dependent (higher order) q-terms in the GL free en- +ergy of a SC [31]. +When magnetic field is applied, energy-dependent +spin-splitting lifts the Kramers degeneracy, such that +electrons +with +opposite +spin +are +not +momentum- +symmetric around Γ = 0, and nonrecirpocity emerges +in both cases. +As a result, contribution from Cooper +pairs with opposite center of mass momentum becomes +non-equivalent and the cancellation of their effect is +avoided. In the spin-singlet pairing symmetry, magnetic +field changes both the magnitude and the momentum- +space position of center of mass momentum: enlarging +and moving q+ +0 along the current direction while reduc- +ing and moving q− +0 opposite to the current direction. On +the other hand, with the spin-triplet pairing symmetry, +SOI shifts momentum of Cooper pairs from q0 = 0 (sym- +metric around Γ-point) to q = q± +0 (asymmetric around +Γ-point) due to opposite momentum-shift. Unlike spin- +singlet case, magnetic field cannot change the momentum +space position of q = q± +0 but rather modifies their mag- +nitude: q+ +0 enlarges whereas q− +0 reduces with increasing +magnetic field. +There is a threshold limit, certainly, for magnetic field. +For the spin-triplet pairing, when bottom of one CB +passes above FL, q− +0 → 0 while q+ +0 become maximal. On +the other hand, for the spin-singlet pairing, magnitude +of q− +0 become constant when in the inner Fermi circle +shifted completely on one-side of Γ-point. Furthermore, +one needs to keep an eye on the curvature of parabolic +bands as it is key to understand change in momentum +when magnetic field is increased, means the Fermi veloc- +ity plays central role. +II. +THEORY OF SUPERCONDUCTING DIODE +EFFECTS +Before jumping onto the recently reported theoreti- +cal analysis of SDE, it is important to have a quick +review of theoretical studies in which nonreciprocity of +supercurrent is reported and the interesting functionali- +ties of SCs intertwined with broken inversion symmetry +and SOI are highlighted. Interestingly, V. M. Edelstein +has discussed the characteristics of the Cooper pairing +in two-dimensional noncentrosymmetric electron systems +[26], magnetoelectric effect in polar SCs [27], and nonre- +ciprocity in the supercurrent by studying the Ginzburg- +Landau equation for SCs of polar symmetry [28]. In other +words, SDE has been there since 1990s, and only recently +demonstrated experimentally. +In 1996, followed by his earlier work characterizing +Cooper pairing in noncentrosymmetric SCs [26] and de- +scribing magnetoelectric effects polar SCs [27], V. M. +Edelstein [28] explicitly proposed nonreciprocity in the +supercurrent, that is, when applied magnetic field (B), +electric current (j), and polar axis (ˆr) are orthogonal to +each other, the magnitude of the critical current jc(B) +depends on the sign of the mixed product (ˆr × ˆB) · ˆjc, +i.e., the critical current should be different for two oppo- +site directions. +Recently, intriguing experimental demonstrations of +SDE, especially in the Rashba-type bulk superconduct- +ing [V/Nb/Ta]n superlattice [12] or Al/InAs-2DEG/Al +JJs [13] and in the Ising-type superconducting JJs such +as NbSe2 constriction [22] or Nb/NiTe2/Nb junction [23], +has stimulated theoretical research on nonreciprocal su- +percurrent transport in a number of exotic quantum ma- +terials. In addition, it also sparked the discussion on fun- +damental mechanisms that cause nonreciprocal charge +transport in SCs. For instance, how nonracirocal charge +transport in a semiconductor with finite resistance could +be generalized to a superconductor allowing supercurrent +with zero-resistance? More specifically, which physical +quantity display nonreciprocal behaviour in the +By employing mean-field (MF), Bogoliubov–de Gennes +(BdG), and time-dependent Ginzburg-Landau (GL) the- +ories, Daido et al. [29], J. He et al. [31], N. Yuan and L. +Fu [30], and S. Ili´c and F. S. Bergeret [32] theorized SDE +in junction-free Rashba/polar SCs. A. Daido et al. [29] +studied Rashba-Zeeman-Hubbard model for the helical +superconductivity and proposed that nonreciprocity in +the depairing critical current is the intrinsic mechanism +of SDE in the fluctuation regime of metal-superconductor +resistive transition. A. Daido et al [29] also showed that +such mechanism of intrinsic SDE can be employed as a +microscopic probe to study and explore the phase dia- +gram of helical superconductivity. Similar proposal has +been made by N. Yuan and L. Fu [30], who studied effec- +tive Rashba-Zeeman-Hubbard model and reported that +nonreciprocal depairing critical current and the polarity- +dependent critical magnetic field are the consequences of +finite-momentum Cooper pairing. On the same footing, + +11 +mainly using the GL theory and phenomenological theory +of SDE, J. He et al. [31] presented a detailed discussion +on symmetry breaking phenomenon and an intertwining +between polar axis, magnetic field orientation, and cur- +rent direction that is desired for the realization of SDE. +The theory of SDE has been generalized for Rashba SCs +with arbitrary disorder by S. Ili´c and F. S. Bergeret [32]. +Thus far, theoretical discussion on nonreciprocal su- +percurrent and prediction of intrinsic SDE has also been +extended for other junction-free polar superconducting +systems. For instance, H. D. Scammell et al. presented +theory of zero-field SDE in twisted trilayer graphene [33]. +Zhai et al. [34] predicted reversible SDE in ferroelectric +SCs. The experimental demonstration of nonreciprocal +transport in chiral SCs, e.g., Ru-Sr2RuO4 eutectic system +[138, 139] and WS2 nanotubes [68], is recently followed by +B. Zinkl et al. [35] who discussed the detailed symmetry +conditions for the SDE in various chiral superconduct- +ing models/systems. The theory of nonreciprocal charge +transport and intertwining between SDE and band topol- +ogy has also been presented for topological SCs [36–38]. +For instance, N. Yuan and L. Fu [36] uncovered an inter- +twining between finite-momentum superconductivity and +topological band theory, i.e., Cooper pairing with finite +momentum depends closely on the nontrivial topologi- +cal spin texture of nondegenerate Fermi surfaces, driven +by combined effect of SOI and Zeeman fields. Recently, +H. F. Legg et al. +[37] theorized SDE due to MCA in +topological insulators and Rashba nanowires, while K. +Takasan et al. [38] discussed supercurrent-induced topo- +logical phase transitions. +In addition, the basic mechanisms of SDE (first en- +visioned by J. Hu et al. +[39]) has also been theo- +rized for JJs [40], e.g., conventional superconducting +NbSe2/Nb3Br8/NbSe2 JJ [41] and Al/InAs-2DEG/Al JJ +[42], graphene-based JJ [43], and topological supercon- +ducting JJ [44, 45]. Furthermore, the effect of Rashba +and Dresselhaus SOI on supercurrent rectification and +MCA have also been studied for JJs based on conven- +tional SCs [42] topological SCs [44]. In the recent the- +oretical studies on the topological JJ dS/FI/dS (dS: d- +wave superconductor, FI: ferromagnetic insulator) on a +3D topological insulator surface, Y. Tanaka and N. Na- +gaosa [45] also demonstrated the relevance of the Ma- +jorana bound states (MBS), i.e., spin-momentum locked +energy-zero Andreev bound states (ABS) at the interface +[140, 141]. +III. +MATERIALS FOR SUPERCONDUCTING +DIODE EFFECTS +In the last two years, SDE has been experimentally ob- +served in a number of superconducting structures, rang- +ing from junction-free SCs [12, 14–18], JJs [13, 19–24], +and other engineered structures such as superconduct- +ing tunnelling junctions [77] and superconducting de- +vices with pinning centres of asymmetric pattern [25]. +JJs, mainly due to the presence of a junction, can be +though as symmetric and superconducting analogue of +asymmetric semiconducting pn junction. On the other +hand, junction-free SCs can be though as symmetric and +superconducting analogue of symmetric semiconductors. +The observation of SDE originated from nonrecipro- +cal charge transport driven by MCA in symmetric SCs +, whether junction-free or JJs, relies on simultaneously +broken spatial-inversion and time-reversal symmetries, +similar to that in symmetric semiconductors. Further- +more, similar that in topologically nontrivial semicon- +ductor/semimetals, SDE can be realized in time-reversal +symmetric systems where nonreciprocal charge transport +is associated with nontrivial Berry phase. +Since both +MCA and nontrivial Berry phase are strongly associated +the strength and nature of SOI originated due to bro- +ken inversion symmetry, noncentrosymmetric SCs can be +classified as Rashba SCs [12–17, 19, 20] or Ising SCs [21– +23]. +If spatial-inversion symmetry is broken, SDE can be +realized in three-dimensional bulk materials, quasi-two- +dimensional thin films and van der Waals heterostruc- +tures, and atomically-thin superconducting materials. +Thus far, SDE has been reported in several materials, +ranging from conventional SCs such as [Nb/V/Ta]n su- +perlattice [12, 14], Al/InAs-2DEG/Al junction [13], Nb +SCs [20], Cu/EuS/Al tunnel junction [77], and super- +conducting thin films with conformal-mapped nanoholes +[25], ferromagnetic SCs [15], twisted-angle bilayer [24] +and trilayer [18] graphene with unconventional super- +conductivity and, TMDCs with Ising superconductiv- +ity [21–23], and topological superconducting materials +[16, 17, 23] where superconductivity coexists with non- +trivial band topology. +For device fabrication of superconducting electronics, +and especially for the search/utilization of novel su- +perconducting materials with high workable tempera- +ture and large magnetic field, it is important to cat- +egorize materials hosting SDE. Superconducting ma- +terials/structures displaying SDE can be classified as +junction-free or JJs based on device structure, Rashba or +Ising SCs based on the nature of SOI, and trivial or non- +trivial based on band topology. Furthermore, SDE can +be classified as magnetic-field-driven or field-free SDE +depending on the magnetic character of superconduct- +ing materials. Furthermore, depending upon the origin +of nanoraciprocity of charge transport, whether MCA or +nontrivial Berry phase, SDE materials can be classified +as time-reversal-symmetric or time-reversal-asymmetric. +IV. +EFFICIENCY OF SUPERCONDUCTING +DIODE +Let’s consider a superconducting sheet with pairing po- +tential ∆(q), where q = qˆx is the center-of-mass momen- +tum. The metal-superconducting transition, and thus a +distinction between supercurrent, depairing current, and + +12 +a normal current, can be conveniently described by in- +troducing condensation energy F(q) ≡ Fn(q) − Fs(q) for +each q, i.e., the difference between free energy per unit +area in the normal (n) and superconducting (s) states. +The sheet current density, as an expectation value of the +current operator, can be obtained by j(q) = 2∂qF(q). If +a current source supplies an electric current jex, a super- +conducting state with pairing momentum q should be +realized when jex = j(q). However, when jex < j− +c +≡ +minqj(q) or jex > j+ +c ≡ maxqj(q), the superconducting +state can not sustain jex and turns into a normal state. +Thus, the depairing critical current along a direction par- +allel (+ˆx) and antiparallel (−ˆx) to the pairing momen- +tum q is given by the maximum (j+ +c ) and minimum (j− +c ) +value of j(q). The SDE in such helical superconductor is +identified and characterized with a finite ∆jc given by +∆jc ≡ j+ +c + j− +c = j+ +c − |j− +c | +(6) +Although a huge current density is generally required +to achieve the depairing limit in a typical superconduc- +tor, depairing critical current density (jc) has recently +been reported in the superconducting microbridge de- +vices [142–144]. +For an optimal performance of SDE, +it is instructive to analyse the behavior of depairing jc +and ∆jc(T) through various perspectives. For instance, +dependence of critical current density jc on tempera- +ture and the orientation of magnetic field reported for +Fe-based Ba0.5K0.5Fe2As2 microbridge with nanoscale +thickness, see, e.g. +Fig. +3 and Fig. +4 in ref. +[143], +critical current density as a function of bridge width +and length reported for Cu-based YBa2Cu3O7−δ micro- +bridge with nanoscale thickness, see, e.g. Fig. 3 in ref. +[142], a comparison of critical current density obtained +from Ginzburg-Landau (GL) theory [∝ (Tc − T)3/2] to +that from Kupriyanov-Lukichev (KL) theory for Fe-based +Fe1+yTe1−xSex microbridge with microscale thickness, +see figure 3 in ref. [144], and the sign reversal of ∆jc by +increasing the magnetic field at low temperatures, see, +e.g. Fig. 4 and Fig. 5 in ref. [29]. As a figure of merit, +the strength of the nonreciprocal response or the super- +conducting diode efficiency can be expressed as a ratio +between ∆jc and the averaged critical current javg +c +[29– +32] +η ≡ j+ +c − |j− +c | +j+ +c + |j− +c | = ∆jc +2javg +c +(7) +Recent theoretical studies [29–32] show that the strength +of η depends on a range of relevant system parameters: +applied magnetic field, working temperature, induced +Cooper pairing momentum, intrinsic SOI, and an inter- +twining between them [29–32]. In addition, strength of η +also depends on two other related but distinct parame- +ters, chemical potential [31] and next-nearest neighbour +hopping [29] that break the particle-hole symmetry. Fur- +thermore, though the SDE persists even in the presence +of disorder, strength of η is also affected by disorder as +it may cause changes in the nature of the two helical +bands by introducing mixing between them [32]. Thus, +for energy-efficient and high performance superconduct- +ing device application, it is crucial to find certain opti- +mal system parameter regimes where the strength of η +is maximal. Recently, Ilice et al. [32] theoretically pre- +dicted that SDE efficiency may exceed η = 40% (in the +ballistic limit) at optimal magnetic field, temperature, +and SOI in Rashba SCs. Interestingly, SDE with opti- +mal efficiency can be engineered by steering the exotic +characteristics and the design of a JJ [13, 45]. +At some fixed temperature, SDE efficiency shows non- +monotonic magnetic field dependence [29, 30, 32]: η in- +creases (almost linearly) for (weak) moderate fields and +then suppresses beyond a certain breakdown/threshold +field Bmax,η, see, e.g. Fig. 3(D) in ref. [30] and Fig. 4 in +ref. [32]. For Rashba SCs, threshold field is theoretically +[30, 32] predicted to be of the order of the Pauli param- +agnetic limit, i.e., much larger than the breakdown limit +observed in recent experiments [12, 13, 22]. Along with +this nonmonotonic behavior, SDE efficiency changes its +sign with increase in magnetic field [29, 30, 32]. Such +change in sign of SDE efficiency appears approximately +at the Pauli limit B ≈ BP , see, e.g. Fig. 3(F) in ref. +[30], Fig. 3 in ref. [32], and Fig. 4 in ref [29]. Such +magnetic field driven sign reversal of the SDE, accom- +panied by the crossover between weak and strong helical +phase, is a general feature of helical SCs irrespective of +their details [145]. +Unlike magnetic field dependence, recent theoretical +studies predict quite diverse behaviour for the tem- +perature dependence of SDE efficiency. +For instance, +at some fixed magnetic field, Rashba-Zeeman-Hubbard +model [29–31] predict that SDE efficiency shows a mono- +tonic square-root-like temperature dependence near the +transition temperature which saturates at low tempera- +tures, see, e.g. Fig. 2 in ref. [29], and Fig. 3 in ref [31]. +On the other hand, quasiclassical Eilenberger equation +for a 2D disordered Rashba superconductor [111] shows +that the temperature dependence of SDE efficiency is +critically affected by the strength of fixed magnetic field +and may display nonmonotonic temperature-dependence +[32]. +For instance, SDE efficiency shows a monotonic +temperature-dependence for B ⪆ BP but it becomes +nonmonotonic when B ⪅ BP , see, e.g. +Fig. +4 in ref. +[32]. That is, in the later case, first SDE efficiency in- +creases with decrease in temperature but it is gradually +suppressed when temperature is further lowers after cer- +tain breakdown limit. Recent observation of SDE shows +that the monotonic [13, 22] and the nonmonotonic [22] +temperature-dependence of SDE efficiency may also de- +pend on the sample fabrication [22]. Similar transition +from monotonic to nonmonotonic temperature depen- +dence may also be realized by varying strength of dis- +order, see, e.g. Fig. 6 in ref. [32] +. +Next, we turn to the dependence of SDE efficiency on +the momentum of Cooper pairs or the nature of helical +phase. For spin-orbit coupled Rashba SCs in magnetic + +13 +FIG. 3. Dependence of superconducting diode effect on the system parameters and its optimization (L) Rashba- +Zeeman-Hubbard model for a Rashba superconductor. (a) The temperature dependence of ∆jc(T) (red closed circles) and +∆(T) (open blue circles) with arb. units. The red and blue dashed lines represent the fitting curves of ∆jc(T) (with (Tc − T)2) +and ∆(T) (with √Tc − T) respectively. Inset: Enlarged view near Tc. Here Zeeman exchange parameter is set as h = 0.03 and +the transition temperature reads Tc ≈ 0.036 respectively. (b) The h-T phase diagram depicting temperature and the magnetic +field dependence of ∆jc(h, T) with t2 = 0 (left) and with t2 = 0.2 (right). Here t2 denotes next-nearest-neighbour hopping +while the red (blue) color indicates positive (negative) values of ∆jc. (c) Pairing momentum q0 (left) and SDE efficiency +(right), represented by r here, for various values of h and T. Figure (L) is reproduced with permission from ref. [29]. (R) +Quasiclassical Eilenberger equation for a 2D Rashba superconductor (a) Helical modulation vector q0 as a function of magnetic +field, where q0v ≈ 2(α/v)h corresponding to the “weak helical phase” at low fields, whereas q0v ≈ 2h corresponding to the +“strong helical phase” at high fields. Here q0 is calculated in the vicinity of the upper critical field (hc2) at different strengths +of spin-orbit interaction (α/v). (b) Supercurrent j (red) and the superconducting gap ∆ (black) plotted as a function of the +phase gradient, for different magnetic fields at fixed values of temperature T = 0.01Tc and spin-orbit interaction α/v = 0.25. +Both quantities are calculated self-consistently and the curves are normalized with j0 and ∆0, respectively, which represent the +critical current and the superconducting gap at T = h = 0. (c) Temperature-dependence of superconducting diode efficiency +η, calculated for different values of the magnetic field and fixed spin-orbit interaction α/v = 0.25. (d) Superconducting diode +efficiency η, calculated for different strengths of spin-orbit interaction in the ballistic limit, corresponding to every point in +the h-T phase diagram. Here black curve corresponds to the upper critical field hc2 while the purple (orange) color indicates +positive (negative) values of η. Figure is reproduced with permission from ref. [32]. +field, the nature of helical phase can be characterized +by quantifying the contribution of two helical bands to +the helical superconductivity [111]. Owning to the op- +posite energy shift induced by magnetic field, the two +helical bands denoted with the index λ = ± and char- +acterized by the same Fermi velocity v = +� +2µ/m + α2 +but different densities of states νλ = ν(1 − λα/v), pre- +fer opposite modulation vectors: qλ +0 v = −2λh. +Here, +m is the effective electron mass, µ is the chemical po- +tential, ν = m/(2π), and α = ∆so/√2mµ characterizes +the SOI strength. Figure 3(R-a) illustrates the crossover +from a “weak” to “strong” helical phase for different ra- +tios of Fermi velocity (v) and the velocity associated with +Rashba SOI (α). In the “weak” or long-wavelength heli- +cal phase subject to low magnetic fields, contribution of +both bands to helical superconductivity yields a modu- +lation vector q0v ≈ 2(α/v)h. In the “strong” or short- +wavelength helical phase at large magnetic fields, owning +to the dominance (suppression) of contribution from the +band with higher (lower) density of states, only one of +the bands contributes to helical superconductivity which +leads to the modulation vector q0v ≈ 2h. +S. Ili´c and F. S. Bergeret [32], based on the quasiclassi- +cal Eilenberger equation for a 2D Rashba superconductor +[111], predicted that the maximum of η emerges when +both the bands contribute to the helical superconductiv- +ity and the magnetic field is close to the critical value h∗ +at which a crossover between “weak” and “strong” he- + +(L) +(R) +Ai(T) +△(T) +(a) +(c) +×10-3 +(a) +0.5 +20 +1.5 +0.25 +2 +6 +15 +"L/aob +0.1 +1.75 +0.05 +h +1.5 +5 +1 +2 +0 +0.5 +0.5 +0 +1 +2 +3 +4 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +h/Te +T/T。 +0 +0 +0.01 +0.02 +0.03 +0.04 +T +(b) +(d) +(b) +-1.2x10-30 +2.5×10-3 +-6.0×1030 +1.0×10-2 +1.0f +3.0 += 0.05 + 0.1 +0.2 +2.5 +n[%] +0.1 +4jc +4jc +2.0 +L/y +二 0.05 +1.5 +40 +0.1 +0.5 +1.0 +0.5 +30 +0 +g = 0% +0 +0.02 +0.04 +0 +0.05 +0.1 +1.0 +T +0 +1.0 +3.0F +-8.8×10-2 +-0.46 +0.31 + 0.25 +(c) +0 +0 +2.5 +10 +00.5 +2.0 +0.1 +qo +0.1 +r +1.5 +0 +- +0.05 +0.5 +1.0 + 1.75 +0.5 +10 +↑ = 22% +7 = 6% +0 +0 +-1.06 +0.0L +0.02 +0.04 +0 +0.02 +0.04 +-6-420 +6-420 +2 +4 +6 +0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 +T +T +qu/T +qu/T +T/T +T/Te14 +lical superconducting phase occurs. It can be explained +from the self-consistent calculation of ∆(q), j(q) and η +vs Cooper pairing momentum under various magnetic +field strength as shown in Fig. 3(R-b) or from the h-T +phase diagram under various strengths of Rashba SOI as +shown in Fig. 3(R-d). In the absence of magnetic field +(h = 0), as shown in the upper left panel of Fig. 3(R-b), +there is no helical phase (q0 = 0) and thus no nonre- +ciprocity of the critical current. In the presence of finite +magnetic field (h ̸= 0), finite Cooper pairing momentum +(q0 ̸= 0) leads to nonreciprocity of the critical current in +both the “weak” helical state induced by sufficiently low +h, as shown in the upper right panel of Fig. 3(R-b), and +the “strong” helical state induced by large h, as as shown +in the two lower panels of Fig. +3(R-b). +The momen- +tum dependence of ∆(q) and j(q) is markedly different +in these three superconducting states, and thus, depict a +completely different supercurrent transport: no SDE in +the BCS state (j+ +c = |j− +c |, η = 0), whereas negative SDE +in the “weak” helical state (j+ +c < |j− +c |, η < 0) while posi- +tive SDE in the “strong” helical states (j+ +c > |j− +c |, η > 0). +In addition, different strength and opposite sign of SDE +under different magnetic field values hint that there must +be some optimal field at which η should be maximum. +It can be depicted by plotting η for every point in the +h-T phase diagram, and in addition, effect of other pa- +rameters can be visualised. +For instance, as shown in +Fig. 3(R-d), S. Ili´c and F. S. Bergeret [32] plotted the h- +T phase diagram and calculated η for different strengths +of SOI. Here black curve corresponds to the upper crit- +ical field hc2 while the orange and purple colors clearly +illustrate the two distinct regimes in which SDE is driven +by the “weak” and “strong” helical phases, respectively. +First, it showcases that the maximum efficiency appears +at the crossover between “weak” and “strong” helical +phases. Second, maximum efficiency exceeding 40% at +the crossover corresponds to the optimal SOI. Third, the +maximum efficiency also corresponds to optimal temper- +ature in the superconducting phase. +Such momentum dependence, yielding maximum η +with optimal magnetic field and SOI driving system +at the crossover between “weak” and “strong” helical +phases, implies that the competition and the contribu- +tion of both helical bands is central for the SDE. This can +be explained by noticing that the MCA is proportional to +magnetic field and SOI, and thus become strongest when +both of these parameters are maximal. +The maximal +magnetic field and SOI borne by the system, along with +the constraint of contribution from both helical bands, is +ensured at the crossover between “weak” and “strong” +helical phases. +This can further be explained by the +analysing the h-T phase diagram regimes, as illustrated +in Fig. 3(R-d), where too large magnetic field and too +large SOI both suppress the SDE efficiency. For instance, +SDE efficiency vanishes when magnetic field is increased, +beyond the crossover to the “strong” phase, where only +one of the helical bands dominates. Similarly, when SOI +is increased — such that α/v → 1, only one helical band +with a large DOS (ν− ≈ 2ν) exists while the helical band +with vanishingly small DOS (ν+ → 0) other is fully sup- +pressed, and the SDE disappears. +This phase diagram also helps to understand the in- +tertwining of optimal temperature with magnetic field +and SOI. At weak SOI, such as depicted in the upper +left panel of Fig. 3(R-d), SDE becomes strongest at the +tricritical point (T ∗, h∗) where the “weak” helical phase +meets the “strong” helical phase and the normal phase. +It is in good qualitative agreement with the results pre- +dicted by N. Yuan and L. Fu [30] where (T ∗, h∗) denotes +tricritical point at which the FF phase meets the nor- +mal phase and the BCS phase. However, with increasing +strength of SOI, h-T phase diagram regime hosting max- +imum SDE moves towards zero-temperature, i.e., where +T ≪ T ∗. +Similarly, as shown in Fig. 3(L-b), Daido et al. [29] +plotted the h-T phase diagram and calculated η for dif- +ferent strengths of next-nearest neighbour hopping t2 in +the Rashba-Zeeman-Hubbard model. It depicts the sign +change of η with increasing magnetic field. In addition, +at some magnetic field, the sign of SDE efficiency found +at t2 = 0 (left panel) also switches when a finite t2 ̸= 0 +is considered (right panel). Furthermore, magnetic field +dependence of pairing momentum as shown in the left +panel of Fig. +3(L-c) and the SDE efficiency as shown +in the right panel of Fig. +3(L-c) showcases that the +maximum η appears at the crossover between “weak” +and “strong” helical phase. It implies that the results +obtained from the numerical study of Rashba-Zeeman- +Hubbard model [29] and that from quasiclassical Eilen- +berger equation [32] are in good qualitative agreement. +However, as mentioned above, there is considerable dif- +ferences between these two studies when it comes to the +temperature dependence of SDE efficiency, i.e., numerical +study of Rashba-Zeeman-Hubbard model shows mono- +tonic behaviour while quasiclassical Eilenberger equation +shows temperature dependence could be either mono- +tonic or nonmonotonic depending on the strength of mag- +netic field. Based on the above analysis, one can conclude +that the nonmonotonic behaviour, for both magnetic field +and temperature dependence, and the change of sign of +η with increasing magnetic field is related to the mag- +netic field-driven evolution of the helical phase. That is, +η becomes maximum at a particular field h∗ and optimal +temperature, and then lowers for other values. +Similar to the dependence on next-nearest neighbour +hopping [29], and consistent with the analogy discussed +for magnetic field and SOI intertwined with the varia- +tion in the DOS of two helical bands [32], He et. al [31] +theoretically predicted that SDE efficiency show strong +dependence on the chemical potential. For a Rashba su- +perconductor with Zeeman field, where the free energy +includes all terms up to the linear order in h√ϵ, GL the- + +15 +ory results in the SDE efficiency [31]: +η = 2.7λR +|λR| +h√ϵ +Tc +× +� +(1 + ˜µ)−1/2 +if ˜µ > 0 +8 +7 + 16 +21 ˜µ + (1 + ˜µ)−1/2 +if − 1 < ˜µ < 0 +(8) +Here λR is the Rashba SOI strength, ϵ = 1 − T/Tc, and +˜µ = µ/ER where ER = 1 +2mλ2 +R is the energy difference be- +tween band crossing point (µ = 0) of Rashba-split bands +and the conduction band edge (µ = −ER) and m denotes +the effective electron mass. At some fixed magnetic field, +temperature, and SOI, SDE efficiency shows maximum +strength at µ = 0, whereas it decrease when the Fermi +level moves away from the band crossing point, either +towards the large µ limit (µ ≫ ER) or towards the con- +duction band edge (µ = −ER). It is important to note +that there are several constraints, and thus limitations, +on these GL theory calculations. For instance, the ex- +pression (8) is derived by assuming |h| ≪ Tc ≪ ER +and treating the problem in the band basis where only +the intra-band pairing ∆t is considered while the inter- +band pairing ∆s is neglected. As a consequence of tak- +ing the limit Tc/ER → 0 and neglecting the inter-band +pairing, there exists a discontinuity in η at µ = 0. In +addition, owning to the consideration of intra-band pair- +ing only, such a discontinuity appears also due to the +flip of spin-momentum locking helicity. +However, the +features of SDE efficiency obtained numerically from a +self-consistent Bogoliubov–de Gennes mean-field Hamil- +tonian [31] are in good qualitative agreement with those +displayed by SDE efficiency obtained from the analytic +generalized GL theory calculations. In addition, the dis- +continuity of η at µ = 0 is smoothed out when Tc/ER is +not so small and it shows square root dependence on µ, +η ∼ µ1/2, when µ is large. +Finally, it is important to emphasise that SDE effi- +ciency also depends upon the characteristics and the de- +sign of a JJ [13, 45]. +In general, with a macroscopic +phase difference φ between two SCs, the standard CPR +of the Josephson supercurrent I(φ) between two SCs is +I(φ) ∼ sin φ. That is, when either space-inversion sym- +metry or time-reversal symmetry is preserved, purely si- +nusoidal terms leads to an antisymmetric CPR, I(φ) = +−I(−φ), and the Josephson current vanishes for φ = 0. +On the other hand, when both time-reversal symmetry +and space-inversion symmetry are simultaneously bro- +ken, an anomalous CPR [46, 49, 141, 146–160] (displaying +finite anomalous Josephson current even at zero phase +difference) contains cosine terms as well. However, even +the presence of such cosine term does not suffice to obtain +SDE because it simply introduces an anomalous phase +shift in the purely sinusoidal CPR and thus the Joseph- +son inductance remains reciprocal (symmetric across the +zero-current). Thus, in order to realize SDE, it is manda- +tory that an asymmetry is induced in the CPR by higher +order phase (especially sine) terms such that the cosine +terms are not absorbed in a mere phase shift [13, 45]. +By fabricating Al/InAs-2DEG/Al ballistic JJs, Baum- +gartner et al. +[13] observed supercurrent rectification. +When an in-plane magnetic field is applied perpendicular +to the current, Rashba superconducting system shows an +anomalous Josephson supercurrent due to even (cosine) +terms in the CPR [156]. Such anomalous CPR contains +higher harmonic sine terms if the junction transparency +is high [159, 161], and thus leads to SDE. By theoreti- +cal studying a JJ dS/FI/dS made with d-wave SCs (dS) +and a ferromagnetic insulator (FI) on the surface of a +3D topological insulator, Y. Tanaka and N. Nagaosa [45] +showed that asymmetric CPR containing a wide variety +of phase terms leads to high quality SDE [45]. +Apart +from the conventional sin φ phase term in the Joseph- +son current, energy-zero Andreev bound state (ABS) at +the dS/FI/dS interface enhances the sin 2φ component of +I(φ) [162, 163]. When the junction dS/FI/dS is placed +on the surface of topological insulator [164], simultane- +ous space-inversion and time-reversal symmetry break- +ing allows a cos φ phase term [141, 148] leading to an +exotic current-phase relation with I(φ) ̸= −I(−φ) [155] +while the energy-zero ABS become MBS due to the spin- +momentum locking [140, 141]. +The simultaneous exis- +tence, with almost the same order, of sin φ, cos φ, and +sin 2φ phase terms promises a maximum value of SDE +efficiency (η = ±2) for the d-wave SCs junction on the +surface of topological insulator [45]. In light of this, opti- +mal supercurrent rectification effect of a JJ can be real- +ized by exploiting exotic characteristics of unconventional +SCs as well as optimizing junction transparency. +V. +OBSERVATION OF SUPERCURRENT +DIODE EFFECT +SDE +is +associated +with +the +literal +metal- +superconductor transition and defined as nonreciprocity +of depairing critical current, +i.e., +depairing critical +current in the direction parallel (j+ +c ) and antiparallel +(j− +c ) to the pairing momentum differ (j+ +c +̸= j− +c ). +An +ideal SDE would be either j+ +c +or j− +c +is zero so that +one has maximum ∆Jc. +Such a resistive transition +between a supercurrent and a normal current can be +realized either by extrinsic stimuli or via mechanisms +that are intrinsic to the superconducting materials. +For instance, the resistive transition can be caused by +the vortex motion, usually realized under out-of-plane +magnetic fields. Owning to the dependence of dynamics +and the statistical mechanics of the vortex system on +the device setup such as impurity concentrations and +the thermal/quantum fluctuations [165], such extrinsic +mechanism promise tunability of the resistive transition +by the nanostructure engineering [56, 166]. Apart from +the resistance caused by extrinsic mechanisms, +the +metal-superconductor resistive transition can literally be +caused by the dissociation of the Cooper pairs resulting +in a transition from supercurrent to a normal current +[6, 167]. This occurs at the maximum critical current, +which is known as depairing current. +In other words, +the depairing critical current is directly associated with + +16 +FIG. 4. +Magnetochiral anisotropy of the resistance. (T) Nonreciprocal transport measurements of critical current in +the resistive fluctuation regime of [Nb/V/Ta]n superlattice. a Magnetic field dependence of first-harmonic (Rω) and second- +harmonic (R2ω) sheet resistances. +Rω vanishes in the superconducting region (white shadings) while become finite in the +normal conducting region (blue shadings). R2ω enhances when the magnetic field orientation is orthogonal to the current +direction and becomes maximal in the fluctuation region. b Temperature dependence of second-harmonic sheet resistance. c +Temperature dependence of the coefficient of magnetochiral anisotropy (γ) calculated from R2ω/Rω. The plot roughly shows +that γ increases with temperature and become maximal in the vicinity of Tc, except a a dip appearing at 4.2 K and 4.3 K +reflecting small R2ω values at these temperatures. Figure is reproduced with permission from ref.[12]. (B) Nonreciprocal +transport measurements of critical current in the resistive fluctuation regime of Rashba-type Al/InAs-2DEG/Al JJ array. +(a) Temperature dependence of first-harmonics Rω(T, θ) showing resistive transition for different angles (θ) of the in-plane +magnetic field (Bip). (b) Temperature dependence of second-harmonics R2ω(T, θ) = V2ω(T, θ)/Iac of the I-V characteristics +for different θ values with the a.c. current bias of Iac= 20 nA. c The coefficient of magnetochiral anisotropy 2Rmax +2ω /Rω versus +orientation/angle θ of the in-plane magnetic field. Here Rmax +2ω +are the maxima of second-harmonics displayed in (b) and Rω +is the corresponding linear resistance displayed in (a). The red data point shown at θ = 90◦ is obtained by switching the +orientation of Bip at θ = 90◦ (the data point in blue), which is equivalent to setting θ = 270◦. The maximal coefficient of +magnetochiral anisotropy, extracted from a sine fit of the data, is γS ≃ 4.1 × 106 T−A−. Figure is reprinted with permission +from ref.[13] +the closing of the superconducting gap, which reduces +and eventually closes with increasing supercurrent. As +the depairing limit or the upper limit of the critical +current is unique to each superconducting material, +depairing current is an intrinsic material parameter for +characterizing SCs [165]. Thus, the intrinsic mechanism +responsible for SDE ties around the nonreciprocity in +the depairing critical current in the fluctuation regime +of metal-superconductor resistive transition. +In this +picture, like many exotic characteristics of quantum ma- +terials, intrinsic SDE is a nontrivial quantum mechanical +effect. +Based on the working temperature, +or a work- +ing +regime +of +phase +diagram +representing +metal- +superconductor resistive transition, observation of SDE +can be classified into two main categories: (i) SDE based +on the nonreciprocity of depairing current near the su- +perconducting transition temperature (T ≈ Tc), i.e., in +the fluctuation regime of metal-superconductor resistive +transition, and (ii) SDE based on the nonreciprocity of +supercurrent at sub-Kelvin temperatures (T ≪ Tc), i.e., +deep in the superconducting phase regime. +A. +Magnetochiral anisotropy of the resistance +In the fluctuation regime of resistive transition close +to Tc, SDE can be described by MCA of the resistance +(γS, as defined in equation (3)), similar to that in semi- +conductors, and may be characterized by I-V curves. In +this regime, MCA coefficient γS can be found by measur- +ing the second harmonic signal in lock-in measurements. + +(1) +a +4.2K +b +Amplitude2.0mA +600 +Amplitude2.0mA ++y +-y +C +0.10 +0.2 +3.5 K +500 +4.0 K +4.1K +3 +0.05 +0.1 +4.2 K +400 +4.3 K +4.35K +2 +0.00 +0.0 +R2m +200 +1 +0.05 ++y +0.1 +R. +-V +100 +0 +0.10 +0.2日 +0.4 +0.2 +0.0 +0.2 +0.4 +0.60.40.20.0 +0.20.40.6 +oF +Magnetic field (M) +Magnetic field () +0.7 +0.8 +0.9 +1.0 +T/T。 +(B) +a +b +120 +200 +0 +6 +21Rma*/R,BI (T-" μA-") +B,lIII +00 +B._III +80 +22.5° +100 +22.5° +45° +45° +3 +(sy) "y +(s) +67.5° +0 +2 +006 +40 +67.5° +270°Bipll +-- 7s sin(0) +100 +90° +B.y = +90 mT +270°BiplI +Bp = +90 mT +-200 +Bpy = 90 mT +0 +0 +1.25 +1.35 +1.45 +1.55 +1.25 +1.35 +1.45 +1.55 +00 +22.5° +45.0° +67.5° +90.0° +Temperature (K) +Temperature (K) +Angle 17 +That is, for an ac current (Iin = I sin ωt) with an ampli- +tude of I and a frequency of ω applied as input, the non- +linear voltage-drop and current-dependent resistance can +be derived from the nonlinear resistance term in equation +(3) as: +V2ω(t) = γBRωI2 sin2 ωt += 1 +2γBRωI2 � +1 + sin +� +2ωt − π +2 +�� +R2ω = 1 +2γBRωI +(9) +Here Rω corresponds to the current-independent linear +resistance R0, while R2ω represents the second-order non- +linear resistance, which is dependent on both the cur- +rent and the magnetic field. +Thus by measuring the +first- (Rω) and second-harmonic R2ω sheet/junction re- +sistances through 2ω voltage response, γS can be esti- +mated as γS = +2R2ω +BIRω. +However, such resistive measurements cannot realis- +tically simulate the intrinsic SDE at temperatures well +below Tc due to no measurable resistance in this regime +(R0 = 0). +Thus, the efficiency of SDE is expected to +be finite only at T ≈ Tc while negligibly small both at +temperatures well below Tc and above Tc (γN ≪ γS). +For instance, Ando et al. [12] measured MCA of the re- +sistance by performing an a.c. harmonic measurements +for Rashba-type bulk superconducting [V/Nb/Ta]n su- +perlattice [12]. MCA coefficient γS show sharp increase +in the fluctuation regime and reaches to its maximal value +γS ≃ 550 T−A− at Tc. +However, γS remains negligi- +bly small at temperatures well below Tc. +Though the +observation seems to be at variance with the theoreti- +cal predictions for intrinsic SDE [29–32] and the tem- +perature dependence of experimentally measured MCA +in JJs [13, 22], but it is an expected outcome of resis- +tive measurements. On the other hand, by fabricating +symmetric Rashba-type Al/InAs-2DEG/Al JJs, Baum- +gartner et al. [13] measured MCA for both the induc- +tance (γL) and the resistance (γS). Finite MCA coeffi- +cient γS ≃ 4.1 × 106 T−A− observed through resistive +measurements near Tc ∼ 1.45 K is of the same order +(namely, in the range of 106 T−A−) of the corresponding +MCA coefficient observed for the inductance (measured +at T = 100 mK), γL ≃ 0.77 × 106 T−A−. +B. +Magnetochiral anisotropy of the inductance +Unlike fluctuation regime, where nonreciprocity of de- +pairing critical current is tied to the nonlinear resistance, +nonreciprocity of sub-Kelvin supercurrent promise fully +superconducting/dissipationless nonreciprocal circuit el- +ement. Deep in the sub-Kelvin superconducting regime +of the phase diagram, i.e., far below the transition tem- +perature where resistance is zero (so DC measurements +are not feasible), supercurrent MCA and a corresponding +SDE (supercurrent rectification/nonreciprocity) is char- +acterized rather by measuring kinetic (or Josephson) in- +ductance (clearly with AC measurements). By measuring +Josephson inductance, nonreciprocal supercurrent can be +linked to an asymmetry in the current–phase relation, +induced by simultaneous breaking of inversion and time- +reversal symmetry such that B is not parallel to I, and +the MCA coefficient (γL) for the supercurrent can be di- +rectly derived from the equation (4). +This mechanism can be understood from a semiquanti- +tative model [13, 42, 161] in which Josephson inductance +can be derived from the CPR relation I = Ic0f(ϕ) (where +f is a 2ϕ-periodic function) and second Josephson equa- +tion ˙ϕ = 2πV/Φ0 (where Φ0 = h/(2e) is the magnetic +flux quantum) as +L(I) = V +dI +dt += +V +dI +dϕ ˙ϕ = +Φ0 +2πIc0 +df(ϕ) +dϕ += Φ0 +2π +�dI(ϕ) +dϕ +�−1 +(10) +It shows that Josephson inductance is a convenient probe +to study CPR symmetry by investigating the effects of +space-inversion/time-reversal symmetry breaking on the +current–phase relation (CPR). Let’s assume a JJ con- +figuration in which electric current is flowing along x- +direction, while inversion and time-reversal symmetry is +broken by applying out-of-plane electric field E = Ezˆz +and in-plane magnetic field Bip = Bxˆx + Byˆy, respec- +tively. +Equation (10) shows that L(I) is inversely propor- +tional to the derivative of the CPR, therefore, the min- +imum of Josephson inductance occurs at the inflection- +point of the CPR. In the absence of in-plane magnetic +field component along y-direction (By = 0), CPR re- +mains symmetric around inflection-point appearing at +zero-phase, that is (i, ϕ) = (0, 0). As a result, the min- +imum inductance occurs at zero-current, around which +L(I) appears to be symmetric. On the other hand, in +the presence of in-plane magnetic field component along +y-direction (By ̸= 0), CPR become asymmetric around +inflection-point (i∗, ϕ∗), mainly associated with the bro- +ken Kramers degeneracy between the oppositely polar- +ized spin components of Andreev bound states (ABS) +leading to a finite-momentum pairing. As a result, cur- +rent dependence of the Josephson inductance L(I) also +become asymmetric and the minimum of L(I) appears +at some finite current i∗, corresponding to the shifted +inflection point (i∗, ϕ∗) in the CPR. +Such a pronounced asymmetry in the skewed CPR and, +thus, in the Josephson inductance L(I), signals the super- +current MCA (as defined in equation (4)) and hence su- +percurrent SDE. First, with a given orientation of electric +field and polarity of applied current, the shift in inflec- +tion point switches along with the sign of By: (i∗, ϕ∗) +for +By and (−i∗, −ϕ∗) for −By, as shown in figure +5(Top(d,e)). Second, for a given orientation of By, the +CPR gets more skewed with increasing By implying in- +crease in the value of i∗ with increasing strength of By, +as shown in figure 5(Bottom-a). As a result, as shown +in figure 5(Top-d), the extremal values of i∗ (which are + +18 +FIG. 5. +Current-phase relation and nonreciprocity of inductance in a JJ array. (T) Device fabrication, current +phase relation, and measurement of inductance. (a) A JJ array is made of 2,250 Al islands (grey), of width w=3.15 µm, +length a=1.0 µm and separated by d=0.1 µm, on top of a Rashba-type InAs quantum well (yellow) sandwiched between +InGaAs barriers. Red and blue arrows represent the spontaneous supercurrents, with zero phase difference, via spin-split pairs +of Andreev bound states, denoted by black and white particle representing oppositely spin-polarized electron and hole. The +strength and direction of these spontaneous supercurrents depend on that of an in-plane magnetic field Bip. Counterpropagating +circles of black arrows represent the Rashba spin-texture in the InAs quantum well. (b) Fabricated device showing growth +sequence of the heterostructure. The Al layer induces a superconducting gap ∆∗, via proximity effect, in the InAs quantum +well. (c) Scanning electron micrograph of the array with a scale bar of 1 µm. (d) Illustrative current-phase relation for a short- +ballistic JJ, with high transparency (τ=0.94) and strong SOI, in the absence (black) and presence (red/blue) of an in-plane +magnetic field By ∥ ˆy (red, By > 0; blue, By < 0). The finite magnetic field (±By) reduces the critical current by a factor +0.8, Ic = 0.8Ic0, and adds a cosine term ±0.2Ic cos(φ) to the current-phase relation’s Fourier series. The red dots represent +the inflection points (i∗, φ∗) of the current-phase relation. (e) Josephson inductance (Φ0/2πIc0) as function of current (Ic0), +corresponding to the current-phase relation in (d). (f) Resonance curves for the RLC circuit, measured at 500 mK, for different +values of the bias current. (g) Current dependence of measured Josephson inductance (at B = 0). Coloured dots correspond +to the spectra in (f). (B) Measurements of inductance and supercurrent anisotropy. (a) Kinetic inductance versus current, +for different orientations of in-plane magnetic field of 100 mT. (b,c) Constant (b) and linear (c) coefficients of the polynomial +expansion of kinetic inductance L(I) as a function of the angle (θ) between in-plane magnetic field Bip and the supercurrent +density directed along ˆx. (d) Measured supercurrent magnetochiral anisotropy (coloured lines and symbols) −2L′ +0/(L0Bip) +versus in-plane magnetic field orientation (θ). The maximum magnetochiral anisotropy, coefficient γL ≃ 0.77 × 106 T−A−, is +extracted from a sinusoidal fit of the data. Fitted supercurrent magnetochiral anisotropy (Grey scale lines) is computed within +semiquantitative model (eq. (10)) for different values of the confinement potential Vconf. The three fitted curves are perfect +sinusoidal functions. All measurements are performed at T = 100 mK. Figure is reproduced with permission from ref. [13]. +the critical currents I+ +c and I− +c ) differ for positive (ϕ+ +c ) +and negative (ϕ− +c ) phase difference, signaling the exis- +tence of a certain bias-current range in which SDE can +be observed for a supercurrent which become different +for opposite phase difference polarities. That is, junction +allows supercurrent (I < I+ +c +(red curve) or |I| < |I− +c | +(blue curve)) along one current direction while it enters +in a resistive state (|I| > |I− +c | (red curve) or I > I+ +c (blue + +(I) +a +p +1.0 +3 +B,=0 +bc (μA) +Current (lo) +2 +0 +B,<0 +1.0 +1.2 +B(90°) +1.3 +1.4 +1.5 +1.0 +B(270°) +1.0 +0.5 +0 +0.5 +1.0 +2.75 +3.00 +3.25 +Phase (s) +Frequency (MHz) +e +2.25 +g +410 +b +Inductance (Φ 2x/lo) +Inductance (nH) +T= 500 mK +Etched2DEG +4110 +2.00 +370 +AI +Al +1.75 +330 +InGaAs +Al +InAs +4 +InGaAs +1.5 +290 +1.0 +0.5 +0 +0.5 +1.0 +1.0 +0 +1.0 +Etched2DEG +Current (lco) +Current bias (μA) +(B) +a +370 +b +:06 +c +90° +p +16 +1.0 +390 +45° +IL"l (nH μA-") +12 +45° +Experiment +350 +180° +Inductance (nH) +(Hu) +0.5 +330 +360 +350 ++ +0 +4 +0.4 +1.6 +315° +-0.8 +340 +310 +0 +0° +0 +45° +Theory +350 +320 +Veoer (meV) 7, = (T- μA") +270° +Vgate (M) +0.5 +0 +0.09 +330 +90° +0 +100 +0.39 +315° +0.4 +315* +200 +0.72 +310 +0.8 +1.2 +0.8 +0.4 +0.4 +0.8 +1.2 +270° +1.6 +270° +1.0 +0 +270° +315° +0° +45° +90° +Current bias (μA) +Constant part +Linear part +Angle 19 +curve)) along the other current-direction. +The MCA of the inductance, can be quantified by mea- +suring the constant (L0) and the and the linear (L′ +0) +junction inductance, which appear as the leading terms +in the polynomial expansion of L(I) around zero cur- +rent: L(I) ≈ L0 + L′I + L′′I2/2 with L′ ≡ ∂IL|I=0 and +L′′ ≡ ∂2 +IL|I=0. As shown in figure 5(Bottom(b,c)), L0 +and L′ +0 are plotted as functions of the angle θ between +the direction of supercurrent ˆx and the orientation of ap- +plied in-plane magnetic field Bip. In the Hall-bar geom- +etry of Al/InAs-2DEG/Al junctions with a Ti-Au global +top gate, the constant term L0 strongly depends on the +gate voltage, reaches its maximum when By = 0, and +shows relatively small anisotropy. In contrast, the linear +term L′ +0 shows relatively weak dependence on the gate- +voltage, completely vanishes when By = 0 and reaches +its maximum when Bx = 0, and thus shows strongly +anisotropic behaviour. As shown in figure 5(Bottom-d), +MCA coefficient for the inductance γL = 2L′ +0/(L0Bip) +shows sinusoidal θ-dependence, that is, proportional to +(B × I) · ˆz = BI sin θ and agrees with the numerical re- +sults obtained from semiquantitative model. In addition, +γL remains nearly independent of the gate-voltage and +its maximum extracted from the amplitude of the sine +reads γL ≃ 0.77 × 106 T−A−. This value of γL, obtained +from measurements performed at T = 100 mK, far below +the transition temperature (Tc), is of the same order as +that of γS calculated for resistive measurements at Tc. +VI. +OUTLOOK +SDE is a captivating phenomenon and could be a +promising building block of the superconducting dissi- +pationless technologies. Thus far, by characterizing the +type/nature of SOI and optimizing/matching the SOI +energy with the characteristic energy scale (supercon- +ducting gap) of charge carriers [57], SDE has been ob- +served in both Rashba SCs and Ising SCs. Recent theo- +retical studies show that SDE is the strongest (i) when +the Cooper pairing momentum lies at the crossover be- +tween weak and the strong helical superconducting phase +in the vicinity of high critical field, which may be real- +ized via optimizing magnetic field (or intrinsic magneti- +zation), temperature, and SOI [32] and/or (ii) when the +Fermi level lies at the band crossing point of two heli- +cal bands, which may be tuned by gating [31]. +From +here on, one of the prime goals is to expand the existing +platforms and mechanisms for the observation of SDE. +For instance, considering the discussion on the optimiza- +tion of SDE originated from MCS, one of the remaining +challenge is to identify suitable superconducting mate- +rial which may provide the best performance. Thus far, +in addition to conventional superconducting structures, +the SDE has also been predicted and/or observed in un- +conventional superconducting structures such as twisted +few-layer graphene, ferroelectric materials, topological +semimetals, and topological insulators. +Recent obser- +vation of extremely long-range and high-temperature +Josephson coupling across a half-metallic ferromagnet +[168] and the prediction of SDE in a JJ with half-metals +[169] opens another rout for the search and utilization of +promising quantum material class, known as spin-gapless +materials [170–173]. +In passing, it is interesting to note that the realization +of SDE via Rashba SOI and Zeeman exchange interaction +in ferromagnetic SCs has a close connection to the real- +ization of QAHE via Rashba SOI and Zeeman exchange +interaction in ferromagnetic topological insulators. +In +the later case, a combined effect of Rashba SOI and Zee- +man exchange leads to a spin-splitting in the low-energy +bands such that only one of spin sectors display nontriv- +ial band topology with inverted band structure while the +other spin sector becomes/remains trivial with normal +band structure. As a result, when Fermi level is tuned +inside the energy band gap, spin-momentum locked chi- +ral edge state leads to a quantized conductance. In the +former case, however, low energy bands in both of the +spin sectors play role, mainly due to formation of intra- +(Fermi)surface and inter-(conduction)bands spin-singlet +Cooper pairing. As a result, when Fermi level is tuned +inside the superconducting gap, locking between mag- +netization orientation and finite-momentum of Cooper +pairing leads to finite MCA and nonraciprocity in the +supercurrent. Such a fundamental connection between +the realization of QAHE and SDE may allow searching +suitable topological superconducting materials based on +heterostructure of s-wave SCs and QAH insulators [174– +177]. In addition, intrinsic iron-based SCs where Rashba +SOI-driven band topology and superconductivity coexist +[178] may also provide promising platform for the real- +ization of SDE in topological superconducting materials +[36–38, 54–56]. +However, regarding orientation and the strength of ex- +change interaction, it is important to remember two dif- +ferences between the realization of SDE and QAHE. (i) +Magnetization orientation needs to be in-plane (at an +angel to the polar axis) for SDE while out-of-plane for +QAHE. (ii) Nontrivial QAH gap saturates after a criti- +cal strength of exchange interaction. On the other hand, +strength of SDE decreases after the critical value of ex- +change interaction h∗, yielding crossover between weak +and strong helical phase, and vanishes for too high val- +ues. +On the other hand, considering the reliance of SDE on +intrinsic system parameters, search of novel mechanisms +may open new rout towards the observation of ideal SDE. +In a broader sense, SDE is a manifestation of the in- +terplay between superconductivity and spatial inversion +asymmetry. Apart from its realization via MCA induced +by time-reversal symmetry breaking, it could also be real- +ized via shift currents induced by nontrivial Berry phase +in a time-reversal symmetric systems. Furthermore, for +JJs, M. Davydova et al. [40] recently proposed that finite- +momentum Cooper pairing, which elucidates the origin +of SDE, can also be achieved without relying on SOI. + +20 +Similar to the gate-controllability of Fermi level and +thus the tunability of SDE strength [31], it would +be intriguing to understand electric field-effects on +the intrinsic properties of a superconducting structure, +switching of SDE, and its utilization for dissipationless +logic/memory applications. For instance, from the ma- +terial aspect, SOI, critical current, and pair-breaking are +the most important intrinsic properties directly impact- +ing the SDE. Antisymmetric SOI, Rashba and Zeeman +SOI, and thus the corresponding spin-splitting can be +tuned via electric field. Superconducting pair-breaking +shows strong dependence on the strength and the fre- +quency/wavelength of electric field [179]. +Similarly, it +is shown that a gate tunable critical current in a NbN +micro- and nano superconducting bridges [180] can be +enhanced up to 30%. +Electric field tunability of su- +perconducting properties has recently been discussed for +various ionic-gated superconducting materials, includ- +ing cuprates, iron-based SCs, and honeycomb structures +such as transition-metal dichalcogenides and bilayer SCs +[181, 182]. For the device prospects, it would be intrigu- +ing to replicate magnetic field (or intrinsic magnetiza- +tion) driven switching of SDE with electric field driven +switching via electrical control of magnetization orienta- +tion. Electric field driven switching of SDE may also be +realized by devising reversible SDE via electric switch of +ferroelectricity [34]. +Furthermore, gate-controlled bar- +rier transparency in Rashba semiconductor based JJ +(Al/InAs/Al) [159] and the gate-controlled asymmetry of +highly skewed CPR in topological insulator (BiSbTeSe2) +based JJ [183] demonstrate potential rout of controlling +SDE in gate-controlled JJs. +The plausible electric field controllability of SDE +and +the +intertwining +between +band +topology +and +superconductivity may allow searching new mecha- +nisms/functionalities [184–187] of topological quantum +materials for steering the engineering of low-power and +low-dimensional topological superconducting technolo- +gies. 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Medhekar, Y. Yin, and J. Cole, Proposal +for a negative capacitance topological quantum field- +effect transistor, in 2021 IEEE International Electron +Devices Meeting (IEDM) (2021) pp. 38.2.1–38.2.4. + diff --git a/otFRT4oBgHgl3EQfcjcx/content/tmp_files/load_file.txt b/otFRT4oBgHgl3EQfcjcx/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..dfcafbfb3f54b129688a4ba97e99ca026e257b20 --- /dev/null +++ b/otFRT4oBgHgl3EQfcjcx/content/tmp_files/load_file.txt @@ -0,0 +1,2613 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf,len=2612 +page_content='Superconducting Diode Effect “Fundamental Concepts, Material Aspects, and Device Prospects” Muhammad Nadeem∗ Institute for Superconducting and Electronic Materials (ISEM), Australian Institute for Innovative Materials (AIIM), University of Wollongong, Wollongong, New South Wales 2525, Australia and ARC Centre of Excellence in Future Low-Energy Electronics Technologies (FLEET), University of Wollongong, Wollongong, New South Wales 2525, Australia Michael S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Fuhrer School of Physics and Astronomy, Monash University, Clayton, Victoria 3800, Australia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' and ARC Centre of Excellence in Future Low-Energy Electronics Technologies (FLEET), Monash University, Clayton, Victoria 3800, Australia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Xiaolin Wang† Institute for Superconducting and Electronic Materials (ISEM), Australian Institute for Innovative Materials (AIIM), University of Wollongong, Wollongong, New South Wales 2525, Australia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' and ARC Centre of Excellence in Future Low-Energy Electronics Technologies (FLEET), University of Wollongong, Wollongong, New South Wales 2525, Australia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Superconducting diode effect, in analogy to the nonreciprocal resistive charge transport in semi- conducting diode, is a nonreciprocity of dissipationless supercurrent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Such an exotic phenomenon originates from intertwining between symmetry-constrained supercurrent transport and intrinsic quantum functionalities of helical/chiral superconductors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' In this article, research progress of su- perconducting diode effect including fundamental concepts, material aspects, device prospects, and theoretical/experimental development is reviewed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' First, fundamental mechanisms to cause super- conducting diode effect including simultaneous space-inversion and time-reversal symmetry breaking, magnetochiral anisotropy, interplay between spin-orbit interaction energy and the characteristic en- ergy scale of supercurrent carriers, and finite-momentum Cooper pairing are discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Second, the progress of superconducting diode effect from theoretical predictions to experimental observations are reviewed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Third, interplay between various system parameters leading to superconducting diode effect with optimal performance is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Then, it is explicitly highlighted that nonreciprocity of supercurrent can be characterized either by current-voltage relation obtained from resistive direct- current measurements in the metal-superconductor fluctuation region (T ≈ Tc) or by current-phase relation and nonreciprocity of superfluid inductance obtained from alternating-current measure- ments in the superconducting phase (T < Tc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Finally, insight into future directions in this active research field is provided with a perspective analysis on intertwining between band-topology and helical superconductivity, which could be useful to steer the engineering of emergent topological superconducting technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Keywords: Superconducting diode effect, Josephson diode effect, Nonreciprocal transport, Mag- netochiral anisotropy, Spin-orbit coupling, Helical superconductivity, Chiral superconductors Superconducting diode effect (SDE), a recently ob- served quantum phenomenon in noncentrosymmetric su- perconductors (SCs) with finite-momentum Cooper pair- ing, refers to the nonreciprocity of supercurrent [1–3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' As depicted by the word ‘nonreciprocity’ and ‘diode ef- fect’, the system allows supercurrent to flow only in one direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Similar to the role of semiconducting diode [4, 5], which is one of the central building blocks for (opto-)electronic technologies, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=', current rectifiers, voltage-controlled oscillators, alternating–direct current converters, LEDs, photodetectors, and solar cells etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=', SDE envisions novel device applications in superconduct- ∗ mnadeem@uow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='au † xiaolin@uow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='au ing electronics [6, 7], superconducting spintronics [8, 9], and quantum information and communication technol- ogy (QICT) [10, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' After recent observation of SDE for the critical cur- rent (fluctuation regime) in symmetric superconductor [12] and for the supercurrent (far below the fluctu- ation regime) in the Josephson junction (JJ) version [13], nonreciprocity has emerged as an active research topic in the field of superconductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' For instance, after seminal observation of SDE in artificially fabri- cated junction-free superconducting [Nb/V/Ta]n super- lattice, reported first time by F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Ando et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' [12] in 2020, SDE has been experimentally observed in a num- ber of junction-free SCs [14–18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Similarly, JJ version of SDE in symmetric Al/InAs-2DEG/Al junction, first reported by Baumgartner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' [13] in 2022, is fol- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='13564v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='supr-con] 31 Jan 2023 2 lowed by SDE experiments on various JJs utilizing dif- ferent materials acting as a normal barrier or weak link sandwiched between conventional SCs [19–24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' In addi- tion, observation of SDE has also been demonstrated in engineered superconducting systems, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=', superconduct- ing thin films with conformal-mapped nanoholes [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' The interest in the nonreciprocal supercurrent transport has been further advanced by the recent demonstration of SDE in unconventional/topological superconducting materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' For instance, apart from conventional SCs, SDE has also been observed in unconventional SCs such as magic-angle-twisted bilayer-graphene (MATBLG) [24] and small-twist-angle trilayer graphene (STATLG) [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Furthermore, SDE has also been demonstrated in topo- logical SCs [16, 17, 23] where superconductivity coex- ists with nontrivial band-topology, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=', topological JJ [23] where type-II Dirac semimetal NiTe2 is sandwiched between conventional s-wave spin-singlet superconduc- tor Nb, and a topological insulator-superconductor in- terface such as Bi2Te3/FeTe heterostructure [16] and Bi2Te3/PdTe2 heterostructure [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Following these intriguing experimental observations, and inspired by theoretical work by V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Edelstein [26– 28], SDE has been theorized by a number of research groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' For instance, by employing mean-field (MF), Bo- goliubov–de Gennes (BdG) and Ginzburg-Landau (GL) theories, theoretical insights have recently been presented for SDE in junction-free bulk SCs [29–38] as well as for its JJ version [39–45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Though, the intrinsic mechanism to cause SDE in junction-free SCs is recently clarified, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=', nonreciprocity of depairing critical current, theoret- ical modelling for potential spin-orbit coupled bulk SCs is still enjoying its infancy [29–38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' In comparison, the underlying mechanism of nonreciprocal supercurrent and SDE is better understood in engineered systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' For in- stance, diode effect can be engineered in a JJ by control- ling Andreev bound states in the normal metal barrier or a weak-link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Davydova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' [40] showed that such effects in a short JJ can arise from both the Doppler energy shift in the Andreev bound states due to finite- momentum Cooper pairing and the asymmetric current from the continuum of states due to phase-independent contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' It has also been shown that SDE in JJ [13] and conformal-mapped nanoholes [25] is well simu- lated by BdG [13] and time-dependent GL theories [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Furthermore, before even experimental demonstration of SDE in artificial devices [13, 25], similar nonreciprocal effects have been recognized in several engineered sys- tems [46–56], e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=', conventional JJs [46–51], domain-wall superconducting state [52], ferromagnetic JJ with a spin- flipper weak-link acting as a quantized Josephson phase battery [53], and topological JJs [54–56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' In nonreciprocal quantum materials (NRQM) lack- ing space-inversion symmetry, direction-selective charge transport can generally be realized whether time-reversal symmetry is broken or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' However, thus far, exper- imental observation of nonreciprocal supercurrent has only been reported in SCs with simultaneous space- inversion and time-reversal symmetry breaking leading to magnetochiral effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Space-inversion symmetry is either intrinsically broken or it can be broken by apply- ing an electric field externally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Similarly, time-reversal symmetry can be broken either by applying an exter- nal magnetic-field or through intrinsic magnetization, leading to an observation of field-free SDE [15, 18– 21, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' In time-reversal asymmetric SCs, nonreciprocity of supercurrent is guaranteed if the following symmetry- imposed constraint, inducing finite-momentum Cooper pairing, is satisfied: both the orientation along which inversion/mirror symmetry is broken and the direction along which (super)current is flowing must be perpen- dicular to the magnetic-field orientation or magnetiza- tion polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Thus, owning to the magnetochi- ral effects, nonreciprocal supercurrent can be switched by reversing the orientation/polarization of magnetic- field/magnetization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' In this article, recent theoretical and experimental progress on SDE including fundamental concepts, mate- rial aspects, and device prospects, is reviewed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' In section I, fundamental concepts and various mechanisms to cause SDE, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=', nonreciprocal charge transport, magnetochi- ral anisotropy (MCA), breaking of space-inversion/time- reversal symmetry, type of associated spin-orbit interac- tion (SOI), origin and orientation of magnetization, up- per critical field, and finite-momentum Cooper pairing or helical/chiral superconductivity are discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' We high- lighted how all these mechanisms are closely related with each other and, especially, their intertwining with SOI, which is a fundamental relativistic quantum functional- ity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' In section II, theoretical progress is reviewed for bulk and engineered SCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Section III covers the material as- pects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Various superconducting materials, in which ob- servation of SDE has been reported, are classified based on the geometric structure of diode device, nature of SOI, origin and orientation of magnetization, and their topological character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Section IV demonstrates how the strength/efficiency of SDE depend on a range of parame- ters such as magnetic field, temperature, Cooper pairing momentum, SOI, chemical potential [31], next-nearest neighbour hopping [29], disorder [32], and design or char- acteristics of a JJ [13, 45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Section V covers two main techniques employed for the observation of SDE: non- reciprocity of critical current via resistive direct-current (dc) measurements and nonreciprocity of supercurrent via inductive alternating-current (ac) measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' In section VI, the article is concluded with a perspective on future directions and device prospects of SDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Since SDE is a novel quantum mechanical phenomenon and the system hosting this effect may prove to be a key compo- nent of emergent quantum technologies, we hope this re- view article may shed light on profound understanding of fundamental mechanism/concepts of SDE and may allow search of novel superconducting systems for emergent su- perconducting technologies such as electronics, spintron- ics, optoelectronics, and fault-tolerant quantum comput- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' 3 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Diode effect in semiconductors and SCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Here straight black lines represent supercurrent flowing due to coherent Cooper pairs while the wiggly black lines represent normal current flowing due to depaired electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' (a) Diode effects in noncentrosymmetric bulk semiconductors and pn junctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' (Left) Diode effects, such as rectification, can be realized in a junction-free and noncentrosymmetric bulk electrical conductor (top) and at a semiconducting pn junction (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' (Right) I-V curve for noncentrosymmetric bulk semiconductors (dashed) and pn junctions (solid).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' (b) Superconducting diode effect in junction-free noncentrosymmetric bulk crystals and JJs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' (Left) SDE in a bulk crystal with an order parameter ∆eiφ (top) and SDE in a JJ between two SCs with phases φL and φR, which are separated by a normal barrier (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' (Right) I-V curves for SDE in a bulk crystal (solid red lines) and a JJ (solid red lines and dashed red curves) show that a SDE occurs when Ic+ ̸= Ic− while the superconductor becomes a normal metal when I is larger than the critical current Ic+ (along positive direction) or Ic− (along negative direction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' For a JJ version, Ir+ and Ir− represent two critical return currents in the downward sweep measurements and leads to another non-reciprocal effect when Ir+ ̸= Ir−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' (c) Schematic illustration of nonreciprocal current/supercurrent in noncentrosymmetric bulk crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' (Left) In the normal state of a noncentrosymmetric crystal, whose MCA coefficient (γN) is usually tiny, non-linear I-V curves (red and blue) show a small deviation form the linear I-V curve (gray), indicating a small nonreciprocal current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' (Right) In the fluctuation regime (resistive superconducting state) of a noncentrosymmetric crystal, whose MCA coefficient (γS) becomes much larger than that of the normal state, I-V curve shows large nonreciprocal current below the critical current (Ic), whereas it resembles to that for the normal state and remains unchanged at I > Ic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Here η represents efficiency of a superconducting diode which changes its sign when the polarity of magnetic field (B) is reversed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' (d) Strength of superconducting diode effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' (Left) The superconducting rectification becomes maximal when +Ic (or −Ic) remains finite but −Ic (or +Ic) becomes zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' As defined by the diode efficiency, equation (7), the maximum difference of critical depairing currents +Ic and −Ic can be about a factor of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' (Right) In the junction-free noncentrosymmetric bulk material, rectification can be induced by applying magnetic field perpendicular to the directions of both the polar axis and the current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' MECHANISMS OF SUPERCONDUCTING DIODE EFFECT The origin of SDE manifests in a number of physi- cal phenomena, imposed by transport mechanisms, sym- metry constraints, and underlying quantum functionali- ties of superconducting materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' In this section, it is explicitly demonstrated how nonreciprocity of supercur- rent is intertwined with underlying symmetries of non- centrosymmetric systems, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=', nonreciprocity driven by MCA in time-reversal asymmetric systems and that in- duced by shift current or Coulomb interactions in time- reversal symmetric systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' It is also highlighted how nonreciprocity of supercurrent is associated with nonre- ciprocal behaviour of physical quantities characterizing current-voltage (I-V) and current-phase relation (CPR), e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=', resistance and inductance respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Quantum functionalities of SCs, such as SOI, Berry phase, band- topology and their effects on the SDE efficiency are also discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Finally, intertwining between nonreciproc- ity and helical superconductivity with finite-momentum spin-singlet or spin-triplet Cooping pairing is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' a) VA c) VA VA B=0 Noncentrosymmetric M B=0 B=0 n=0 Semiconductor B>0 B>0 n>0 Ic+ B<0 B<0 n> N Ys b) d) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='0 Ict Aeis VA 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='0 B Noncentrosymmetric B±0 Superconductor 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content="0 'c- I'r." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='5 Ic- Ir+ Ic+ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='0 Bar fUpwardSweep 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='5 Sc Lrrier SC Downward Sweep Aeid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Aeigr4 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Nonreciprocity and magnetochirality In condensed matters, nonreciprocity refers to the spatial-dependence of physical quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' A prototypical example of nonreciprocal transport is a diode effect which refers to a highly direction-selective electron transport in systems with a lack of spatial inversion center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Un- til recently, nonracirocity was thought to be a transport phenomenon associated with dissipative materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' For instance, in conventional semiconductors, where resis- tance is the nonreciprocal quantity, nonreciprocity refers to charge transport that is sensitive to the polarity of cur- rent or bias potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Such nonreciprocal charge trans- port leads to a diode effect in a spatially asymmetric pn junction [4, 5], in which, spatial asymmetry of the junc- tion is associated with electron-hole asymmetry across the contact of n- and p-type semiconductors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' In modern quantum condensed matter physics, in addi- tion to electron-hole asymmetric junctions, nonreciprocal charge transport can be induced in spatially symmetric devices, in which resistance is direction-selective when in- version and/or time-reversal symmetry are broken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' This can be realized, for instance, by externally applying an electric and/or a magnetic field orthogonal to each other and to the direction along which current is traversing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' It implies that nonreciprocal transport can be treated as a bulk property of noncentrosymmetric quantum materials [57, 58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' In noncentrosymmetric systems, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=', in which inversion symmetry is broken, nonreciprocal responses can be classified into four categories [57]: (i) linear- and (ii) nonlinear-response in time-reversal symmetric sys- tems, and (iii) linear- and (iv) nonlinear-response in time- reversal asymmetric systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' When both inversion and time-reversal symmetry are simultaneously broken, a closely related phenomenon leading to nonlinear nonreciprocal response is MCA [16, 27, 28, 59–70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' In the linear response regime of non- centrosymmetric systems, broken time-reversal symme- try produces finite magnetochiral effect, as recognized by the Onsager’s reciprocal theorem [57, 66, 71, 72], and the longitudinal transport coefficients become dependent on the polarity of the current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' The Onsager’s reciprocal theorem, and thus the magnetochiral effect and direction- selective transport, can be generalized to the nonlinear regime of both (semi)conductors [59, 62] and SCs [27, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Nonreciprocity of supercurrent In 1996, before even prediction/observation of nonre- ciprocity in (semi)conductors by Rikken et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' [59, 62], V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Edelstein [28] proposed nonreciprocity in the crit- ical supercurrent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Followed by his earlier work charac- terizing Cooper pairing in noncentrosymmetric SCs [26] and describing magnetoelectric effects in polar SCs [27], V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Edelstein [28] proposed that if the mixed product (c × B) · ˆjc is non-vanishing in polar SCs, then the mag- nitude of the critical current jc(B) depends on the sign of this mixed product, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=', the critical current appears to be different for two opposite directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' By employ- ing GL theory for a thin film of polar superconductor, expression for the nonreciprocity in the critical current reads [28] jc(B) = jc(0)[1 + γj(c × B) · ˆj] (1) Here c is the unit vector along the polar axis, ˆj is the unit vector along the supercurrent, and B is an in-plane magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' The exact expression for the observable can be found in the reference [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' MCA and nonreciprocity has been observed in (semi)conductors [60–64, 69] that show resistive current as well as in SCs [16, 67, 68, 70] that display dissipation- less supercurrent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' So a question arises naturally: how nonreciprocity can uniquely be defined in these two sys- tems with completely contrasting behaviour?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' As first pointed by Rikken et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' [59], when both inversion and time-reversal symmetries are broken, the finite MCA co- efficient γ gives rise to different resistance for electric cur- rents traversing in different (opposite) directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' That is, MCA can be defined as the inequivalence of R(+I) and R(−I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' In (semi)conductors, resistances along op- posite directions differ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' R(+I) ̸= R(−I), but both R(+I) and R(−I) normally take finite values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' On the other hand, in SCs, such a situation becomes more dras- tic: either one of R(±I) remains finite while the other vanishes completely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' With this consideration in SCs, it becomes more appro- priate to define nonreciprocity in terms of (super)current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' That is, as shown in figure [1], nonreciprocity in SCs means supercurrent flows along one direction while a normal current along the other(opposite).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Observation of such a situation is more probable near critical tem- perature Tc, i,e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=', in the fluctuation regime of metal- superconductor resistive transition, where the critical current is different along opposite directions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Ic+ ̸= Ic−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Thus, if the current is tuned between Ic+ and Ic−, the system displays zero resistance for supercurrent but nonzero for the normal current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' It can be understood how conductance varies while go- ing from normal to a superconducting phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' The lin- ear resistance R0 is normally scaled by the Fermi en- ergy EF , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=', the kinetic energy of the electrons, while the MCA coefficient γ depends upon the strength of SOI and the magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Correspondingly, nonlinear resis- tance induced by MCA may be treated as a perturbation to R0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' In the normal conducting phase, because the SOI energy (Esoi) and the Zeeman energy (µBB) is usually much smaller (by many orders of magnitude) than EF , MCA coefficient γ → γN is typically very tiny, usually of the order of ∼ 10−3 to 10−2 T−1 A−1 in typical metals [59, 61, 67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' However, as the superconducting phase de- velops, superconducting transition temperature Tc or the superconducting gap ∆sc appear as a new energy scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' That is, the energy scale in SCs, to which the strength of SOI has to be compared with, is superconducting gap and not the Fermi energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Since the energy scale (∼meV) in 5 the SCs is much smaller than the Fermi energy (∼eV) in metals, the effects of SOI and Zeeman energy greatly en- hance in the superconducting phase [66, 67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' As a result, near the superconducting transition temperature T ≳ Tc, MCA coefficient becomes reasonably large [57] and, thus, the paraconductivity [73] above Tc becomes nonrecipro- cal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' In the superconducting fluctuation region, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' when T → Tc and the superconducting order parameter ∆sc develops, a sizable enhancement in MCA coefficient γS is found (ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' [16, 57, 66, 67]) and a robust non-reciprocal charge transport is demonstrated in noncentrosymmetric SCs [67, 70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' For instance, by employing GL theory for an Ising type SC MoS2, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Wakatsuki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' [67] showed that the ratio of MCA coefficients in the superconducting resistive region (γS) and the normal resistive region (γN) is quite large γS γN ∼ � EF kBTc �3 (2) Such anomalous enhancement of the MCA coefficient, as it is associated with the energy scale difference between the superconducting gap and the Fermi energy, can be considered an intrinsic feature of both Rashba and Ising type noncentrosymmetric SCs [66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' However, mainly due to a gradual decrease in the linear resistance R0 during the metal-superconducting transition, R0 remains larger (by orders of magnitude) than the nonlinear resistance in low-dimensional superconducting materials such as MoS2 (ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' [67]), WS2 (ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' [68]) and Bi2Te3/FeTe (ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' [16]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' As a result, low rectification ratio in these superconduct- ing materials does not suffice for device implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' In this regard, it is highly desired to search for novel mechanisms/principles to enlarge the rectification effect and guide the design of efficient SDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' From resistance to supercurrent Rikken et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' [59, 62] generalized the Onsager’s re- ciprocal theorem to the nonlinear regime and gave a heuristic argument for nonreciprocity and MCA in two- dimensional diffusive conductors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' In their seminal pro- posal of MCA in (semi)conductors, Rikken et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' [59] suggested that nonreciprocal nonlinear resistive response, characterized by the directional IV-characteristics, can be described by a current-dependent resistance R(I) as R(I) = R0[1 + βB2 + γ(B × r) · I] (3) Here R, B, and I are the resistance, magnetic field, and the electric current, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' The unit vector r rep- resents the direction along which mirror symmetry is bro- ken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' On the right-hand side, first term is the resistance at zero magnetic field, second term denotes the normal magnetoresistance, and the third term corresponds to the MCA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Dependence of MCA coefficient γ on electric cur- rent, magnetic field, as well as their mutual orientation, relative to the direction along which mirror symmetry is broken, allows us to access various functionalities and aspects of noncentrosymmetric materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' First, dependence of MCA on electric current leads to a current-dependent resistance which generally causes a nonlinear nonreciprocal transport, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' nonlinear voltage- drop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Such nonlinear nonreciprocal transport can be de- tected by measuring the second harmonic signal through lock-in techniques, see further details in section (V A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Second, dependence of MCA on magnetic field implies that its coefficient γ remains non-zero only when time- reversal symmetry is broken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' In addition, the orientation of magnetic field must be orthogonal to both current and the direction along which mirror symmetry is broken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' It implies that, not only finite magnetic field is required, but its orientation is also important depending upon the nature of SOI associated broken mirror symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Here we discuss the key mechanisms associated with nonre- ciprocity in superconducting systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' The conventional semiconducting diode is not fa- vorable for energy-efficient technologies with ultralow power consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' At high temperatures relevant for thermionic transport, owning to their finite resistance, energy loss is inevitable in semiconductors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' At low (sub- Kelvin) temperatures, on the other hand, relevant for cryogenic electronics [7] and ultrasensitive (sub-THz fre- quencies) optoelectronics and detection [74], semicon- ductors cease to work due to their large energy gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Therefore, owning to their dissipationless supercurrents, intrinsically low impedance and thereby very high rec- tification of supercurrents, and low energy scales as- sociated with superconducting gap (∼meV) as com- pared to semiconductor energy gap (∼eV), a supercon- ducting diode is highly desired for energy-efficient cryo- genic electronic/optoelectronic devices [6, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' However, as broken electron-hole symmetry is required, physi- cal realization of a junction-free superconducting diode turns out to be difficult with electron-hole symmet- ric Bardeen–Cooper–Schrieffer (BCS) superconducting state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' In light of this, SCs with broken spatial-inversion and time-reversal symmetry can offer bright perspectives for supercurrent diode effect via MCA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' However, for the implementation of a simplest possible device displaying SDE intrinsically, it is worthy to pin down intertwining between superconductivity and MCA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' First, unlike recti- fication due to self-field effects in asymmetric supercon- ducting quantum interference devices (SQUIDs) [75, 76], MCA is expected to induce an intrinsic SDE in sym- metric devices with spatially homogeneous supercurrent density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Second, intertwining between superconductivity and MCA can lead to a spin-filtering diode effect in a spin-selective Al/EuS/Cu superconducting tunnel junc- tion [77] and thus superconducting spintronic technolo- gies [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' However, even such promising ferromagnetic superconducting structure, in which electron-hole sym- metry can possibly be broken when both spin-filtering and spin-splitting are present to induce opposite shift in BCS density of state (DOS), are not desired for the in- 6 trinsic SDE with nonreciprocal supercurrent transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Finally, we could see the light at the end of the tunnel: Intrinsic SDE with nonreciprocal supercurrent transport can be realized in a helical superconductor with finite- momentum Cooper pairing which can be induced by anti- symmetric Rashba/Ising SOI and Zeeman exchange spin- splitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Further details on this key mechanism are dis- cussed in section I D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' From inductance to supercurrent Nonreciprocity in the fluctuation regime of metal- superconductor resistive transition confines SDE to a narrow temperature window near Tc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Baumgartner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' [13] pointed that the temperature window in which MCA coefficient becomes sizeable must be widened for a sus- tainable fabrication of devices showing SDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' To achieve this milestone, the authors demonstrated supercurrent rectification in the superconducting phase, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=', far be- low the transition temperature Tc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Since d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' measure- ment of resistance–current (R–I) curve is not viable at low temperatures, as the resistance vanishes, supercur- rent response to an alternating-current (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=') excitation is studied, which is described by its superfluid stiffness, and thus, can be detected through kinetic inductance measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' If mirror symmetry is broken along out-of-plane direc- tion (ˆez), whereas the current I and magnetic field B are directed in-plane, MCA or nonreciprocity for the super- fluid can be described by an equation similar to that for the resistance (3), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=', L(I) = L0[1 + γLˆez(B × I)] (4) Here resistance (R) is substituted for the kinetic induc- tance (L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' The nonraciproocity in supercurrent could then be characterized by a new observable, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=', MCA coefficient γL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Nonreciprocity without magnetochirality In noncentrosymmetric but time-reversal symmetric systems, nonreciprocal nonlinear response can be real- ized via shift current (photovoltaic effect) [78–80], via Coulomb interactions [81], and asymmetric Hall effect of vortices and antivortices [82].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Shift current is a non- trivial contribution by the Berry phase of the electronic states [83].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' That is, unlike conventional charge transport which comes form intraband transition [78–80] and de- pends only on the energy dispersion, interband shift cur- rent depends not only on the energy dispersion but also on the Bloch wavefunction and plays an essential role in modern quantum transport phenomena [83, 84].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Fol- lowed by theoretical proposals [78, 79], shift current has been studied for semiconductor (GaAs) [85], ferroelec- tric semiconductor (SbSI) [86], and Dirac surface states of a 3D topological insulator (Bi2X3(X=Te, Se)) with a hexagonal warping [87].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' It shows that shift current is an ubiquitous phenomenon in noncentrosymmetric quantum materials, and the nonreciprocal nonlinear response can also be realized without breaking of time-reversal sym- metry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Morimoto and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Nagaosa [81] theoretically showed that nonreciprocal nonlinear I–V characteristics can be induced by electron correlations in noncentrosymmet- ric multiband systems without time-reversal symmetry breaking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' According to general symmetry considerations, nonreciprocal nonlinear response in such time-reversal symmetric systems is generally constrained by the pres- ence of two ingredients: (i) dissipation, and (ii) inter- actions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=', electron-electron and electron-phonon in- teractions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' First, generalization of Onsager’s reciprocal theorem to nonlinear current responses shows that dis- sipation is crucial for nonreciprocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Second, gauge in- variant formulation of Keldysh Green’s function shows that nonreciprocity disappears without interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' A general formula of the nonreciprocity ratio (γc), and de- rived by employing nonequilibrium Green’s functions for two-band systems with onsite Coulomb interaction, reads [81] γc = δJ J ≃ U Eg,kF eEa W (5) where U is Coulomb interaction energy (γc → 0 for U → 0), Eg,kF is the band gap, kF is the Fermi mo- mentum, e is charge of electron, E is the applied electric field, a is the lattice constant, and W is the bandwidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Here J is the linear current response (the part of current response proportional to E) while δJ is the nonlinear cur- rent response (the part of current response proportional to E2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' When U ≈ Eg,kF , nonreciprocal response can be estimated by quantifying the ratio eEa/W between the electric potential (eEa) in the unit cell and the band- width (W).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' First the nonreciprocity induced by electron correla- tion [81] is relatively smaller than that induced by MCA, in both typical metals [59, 61] as well as resistive semi- conductors [62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Second, the requirement of dissipation means nonreciprocal response induced by Coulomb in- teractions is only measurable in the resistive fluctua- tion regime of metal-superconductor transition, and not in the superconducting phase below transition temper- ature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' On the other hand, nonreciprocity of supercur- rent by asymmetric Hall effect of vortices and antivor- tices in time-reversal symmetric trigonal superconductors (PbTaSe2) [82] promise another nonlinear transport phe- nomena to study SDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' However, thus far, experimental observation of nonreciprocity of supercurrent has only been reported in noncentrosymmetric systems with bro- ken time-reversal symmetry, while the observation of su- percurrent nonreciprocity in time-reversal symmetric SCs is scarce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' 7 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Role of spin-orbit coupling Apart from the strength of SOI, since broken inversion symmetry is assumed/required (γ = 0 for centrosymmet- ric systems), MCA coefficient γ also depends on the na- ture of associated SOI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' That is, based on the lattice sym- metry, finite γ may be realized in noncentrosymmetric condensed matter systems [58] such as polar or Rashba SCs and trigonal or Ising SCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' In polar systems, where Rashba SOI generated from broken Mz and electron’s spin is locked to in-plane orientations, nonreciprocal su- percurrent is controlled by an in-plane magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' On the other hand, in trigonal systems with D3h symmetry, where Ising or valley-Zeeman SOI is originated from bro- ken Mx/y and electron’s spin is locked to out-of-plane orientations, nonreciprocal supercurrent is controlled by an out-of-plane magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' In addition, it would be interesting to study effects on SDE due to a crossover between various SOI types as- sociated with broken inversion symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' For instance, Baumgartner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' [42] studied effects of Rashba and Dresselhaus SOI on supercurrent rectification and MCA by fabricating Al/InAs-2DEG/Al ballistic JJs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Similarly, Pekerten et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' [44] studied an interplay between Rashba and Dresselhaus SOI and investigated effects of magnetic and crystalline anisotropies on the topological supercon- ductivity in JJs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' If only Rashba-type SOI is present in the JJs, the topological phase diagram strongly depends on the magnetic field orientation but remains insensi- tive to the supercurrent polarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' On the other hand, when both Rashba- and Dresselhaus-type SOIs coexist, the phase diagram exhibits a strong dependence on the magnetic field as well as junction crystallographic orien- tations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' These studies illustrate the role of SOI, both for the material search leading to SDE with the best perfor- mance and probing phase diagram of topological/helical SCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Furthermore, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Yi recently showed a crossover from Ising- to Rashba-type superconductivity in epitaxial topological insulator and monolayer Ising superconduc- tor heterostructure [88] (Bi2Se3/NbSe2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' By altering the thickness of Bi2Se3 film, emergence of topological super- conductivity coincides with a considerable suppression of the upper critical in-plane magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' While the for- mer transition is marked by the emergence of spin-non- degenerate surface states and Rashba-type quantum-well bands in the bulk, the later signatures a crossover from Ising- to Rashba-type superconductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' This system represents a classic example and sheds light on the role of SOI while searching new systems to engineer SDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Based on the above discussion, one can conclude that Ising/trigonal topological SCs, such as NbSe2 which dis- play exceptional upper critical-fields exceeding the Pauli limit [89–91], can be identified as suitable materials for the realization of SDE via magnetic field driven MCA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' On the other hand, owning to the nontrivial Berry phase intertwined with band topology, time-reversal symmetric polar/Rashba SCs can be identified as promising mate- rials for the realization of SDE via shift current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' This qualitative analogy needs further quantitative investiga- tion, as the performance of SDE also depends upon the strength of SOI, interband transition, and photoresponse etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Helical superconductivity To observe SDE via MCA in noncentrosymmetric su- perconductor, breaking of time-reversal reversal symme- try (T ) is necessary but not sufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' First, SDE is not necessarily present in all the magnetic SCs but rather the orientation of magnetic field or magnetization should be such that it breaks all possible inversion symmetries Pi (i = x, y, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Second, time-reversal reversal symmetry should be broken such that a finite-momentum Cooper pairing or a helical superconductivity emerges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Third, magnetic field (or magnetization) should have a com- ponent perpendicular to the polarity of applied current such that finite pairing momentum emerges parallel/anti- parallel to the current direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' In this section, after a brief overview of helical superconductivity, desired ori- entation of magnetic field or magnetization, and its in- tertwining with the nature of SOI, polarity of applied current, direction along which structural mirror symme- try is broken, and the momentum space orientation of Cooper pairing momentum is discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Fulde–Ferrell–Larkin–Ovchinnikov state In the field of conventional superconductivity, follow- ing from the fact that Cooper pairing is formed between Kramers partners and most known conventional SCs are characterized by the Bardeen–Cooper–Schrieffer theory [93], presence of time-reversal symmetry is a key ingre- dient and the preserved Kramers degeneracy is the fun- damental reason/criterion that stabilize superconducting phase in so many systems at sufficiently low temperatures [94–96].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Thus, such a conventional superconducting state with a spin-singlet pairing is suppressed or destroyed by time-reversal symmetry breaking perturbations — as a consequence of applied magnetic field, doped magnetic impurities, or intrinsic magnetic instability leading to spontaneous magnetization — due to electron pair break- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' On the other hand, beyond conventional BCS paradigm, unconventional superconductivity allows co- existence of more exotic superconducting order param- eters with magnetic order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' For instance, as predicted independently by Peter Fulde and Richard Ferrell (FF) [97] and Anatoly Larkin and Yuri Ovchinnikov (LO) [98], magnetic fields can give rise to a superconduct- ing state with FF-type order parameter ∆(x) = ∆eiqx and/or spatially inhomogeneous LO-type pair potential ∆(x) = ∆ cos qx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' The underlying physical mechanism of the Fulde–Ferrell–Larkin–Ovchinnikov (FFLO) state 8 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' abc (A) Schematics of the Ising- and Rashba-type superconducting pairing symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' a Ising-type pairing symmetry originates from spin-singlet Cooper pairs formed between the electrons near the K and K′ valleys with opposite spins pinned to the out-of-plane direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' (b) Rashba-type pairing symmetry originates from spin-singlet Cooper pairs formed between the electrons near the Γ point with opposite momentum and opposite spins pinned to the in-plane direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Figure (A) is reproduced with permission from ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' [88].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' (B) (a) Schematic sketch showing magnetic field driven spin-splitting of free- electron parabola, inducing Pauli paramagnetism, and leading to different Fermi momenta for spin-up (k↑ F ) and spin-down (k↓ F ) electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' (b) Schematic representation of the conventional spin-singlet BCS pairing state (left) with zero center-of-mass momentum and the spin-singlet FFLO pairing state (right) with a finite center-of-mass momentum (q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' The red (blue) circle represents the Fermi surface for electrons with spin-up (spin-down).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Figure (B) is reproduced with permission from ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' [92].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' (C) Band splitting and Fermi contours under Rashba SOI and exchange field in a JJ Nb/Pt/Nb with a Pt barrier proximity- magnetized by a ferrimagnetic insulating Y3Fe5O12 (YIG) film.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' a Rashab SOI splits the conduction bands laterally (along momentum (k) axis) by ∆kR while the Zeeman exchange field splits them vertically (along energy (E) axis) by ∆Eex such that the Kramers degeneracy is removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Here EF represents the Fermi level while kx/y stands for in-plane momentum components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' bI-V curve representing SDE at T = 2 K (< Tc) for different orientations of the Pt magnetization (MP t), parallel (yellow) and antiparallel (cyan) with respect to the x-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Here Pt magnetization orientations, and, thus the the direction of the exchange field, reverses when the magnetization orientation of the proximity-coupled YIG is inverted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' The diode symbols in the yellow (cyan) shaded regime indicates that the Josephson supercurrent flows only in the positive (negative) y-direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Figure (C) is reproduced with permission from ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' [19], Springer Nature Ltd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' (D) Supercurrent diode effect under external current source J and in-plane magnetic field B in a noncentrosymmetric Rashba superconductor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' (a and c) Schematics of device plots showing Rashba- and Zeeman-split normal state Fermi surfaces (denoted by circles) and the directions along which J and B are applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' (b and d) Schematic phase diagrams in the B-J plane corresponding to device configurations shown in (A and B), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Figure (D) is taken from ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' [97, 98], owning to the opposite energy-shift in the elec- tronic spin bands as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' 2(B), induces non-zero centre-of-mass momentum of Cooper pairs and leads to a spatially-modulated order parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' The FF state ubiquitously exist in noncentrosymmetric SCs and is par- ticularly known as the helical superconductivity [99–111].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' The FFLO states, and/or the implications of the he- lical superconductivity, have been obtained in heavy- fermion SCs CeCoIn5 [112–114], organic SCs [92], pure single crystals of FeSe [115, 116], thin films of Pb [110] and doped SrTiO3 [117], a heavy-fermion Kondo super- lattice [118, 119], and a three-dimensional topological K" Ising type (b) Rashba type (A) (a) (B) (a) E ki 2ugB Kk (b) BCS FFLO K" k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' =-k E k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='=-kt+q (a) (b) (C) (a) (q) (D) Rashba SOC + exchange spin-splitting 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='3 E T = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='0 K 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='1 I (mA) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='1 (c) (d) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='2 B Mpt 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='A V (mV)9 insulator Bi2Se3 [120].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' While the existence of FFLO- like states is well established in proximity-coupled SCs and ferromagnets [121], the experimental observation of FFLO states has been reported in nonmagnetic SCs by applying external magnetic fields [112, 113] as well as intrinsic ferromagnetic SCs [122–128].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Pairing in Rashba/Ising SCs To understand SDE via finite-momentum Cooper pair- ing in noncentrosymmetric superconducting materials, it is instructive to quickly review pairing phenomenon in Ising- and Rashba-type superconductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' In this regard, (5QL)Bi2Se3/NbSe2(ML) heterostructure is a promising example where a crossover from Ising- to Rashba-type superconductivity is reported recently [88].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' NbSe2 bulk crystal with 2H phase is a well-studied superconduc- tor with Fermi surface sheet-dependent s-wave super- conductivity [129].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' 2H-NbSe2 bulk crystals covered by molecular-beam epitaxy (MBE)-grown films of Bi2Se3 or Bi2Te3 topological insulators are the most success- ful topological superconductor interfaces [130–135].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' It is also well-known that monolayer NbSe2 with the Se-Nb- Se trilayer structure, with preserved out-of-plane mir- ror symmetry but broken in-plane inversion symmetry, is a prototypical Ising-type superconductor [89, 90], and are preferred over 2H-NbSe2 bulk crystals for for device fabrication and technological applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' On the other hand, Bi2Se3 with the Se-Bi-Se-Bi-Se Quintuple-layered structure is a prototypical 3D strong TI hosting a sin- gle surface Dirac cone intertwined with nontrivial bulk Rashba bands at the Γ-point [136, 137].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' In monolayer NbSe2, broken in-plane inversion sym- metry generates an out-of-plane spin polarization and originates the Ising-type SOI which induces a valley- dependent Zeeman-type spin-splitting, as shown in fig- ure 2(A-a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Such opposite spin-splitting in the bulk valence bands around valleys K and K′ leads to Ising- type superconducting pairing symmetry in monolayer NbSe2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=', which refer to the intervalley spin-momentum locked spin-singlet Cooper pairing between two electrons with opposite momenta and opposite out-of-plane spins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' On the other hand, as shown in figure 2(A-b), own- ing to the emergence of Rashba-split low-energy con- duction bands at Γ-point and corresponding Dirac sur- face states, (5QL)Bi2Se3/NbSe2(ML) heterostructures become proximity-coupled topological SCs with Rashba- type superconducting pairing symmetry, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=', which refer to the Cooper pairing between two Rashba-split electrons with opposite momenta and opposite spins pinned to the in-plane direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Nonreciprocity in FFLO states Such momentum-dependent spin-splitting of the low- energy electronic bands, caused by broken inversion sym- metry in the noncentrosymmetric bulk crystals, surfaces, and interfaces, is crucial for the emergence of nonrecip- rocal transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' However, in order to avoid the can- cellation of this effect due to superposition of degen- erate Kramers pairs, one also needs energy-dependent spin-splitting such that electrons with opposite momenta and opposite spin become Kramers non-degenerate, or simply non-equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Typically, this can achieve by breaking time-reversal symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' In the presence of external magnetic field or intrinsic magnetization, par- allel to the electron’s spin-orientation, BSC-type zero- momentum Cooper pairing (symmetric around Γ-point) become asymmetric around Γ-point due to opposite energy-shift and FFLO-type finite-momentum Cooper pairing originates in both Ising- and Rashba-type super- conducting phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Recent theoretical studies [29–32] revealed how a non- trivial interplay of antisymmetric Rashba SOI, magnetic field, and helical supercurrent leads to an intrinsic SDE in noncentrosymmetric bulk SCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' It implies that intrinsic SDE is closely related to the FFLO state [97, 98] with a periodically modulating phase of the superconducting order parameter ∆sc(r) = ∆sceiq·r: Rashba SOI splits Fermi surfaces while the finite pairing momentum q0 is induced by the magnetic field and varies continuously with its strength and orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' In terms of charge transport, when an in-plane magnetic field is applied, Cooper pairs in noncentrosymmetric Rashba SCs acquire a finite-momentum q0, and, as a result, critical currents traversing along the direction parallel and antiparallel to q0 become unequal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Recently, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Yuan and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Fu [30] explicitly demon- strated the effect of in-plane magnetic field on Rashba spin-split bands and, thus, the emergence of finite- momentum Cooper pairing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' As shown in figure 2 (D-a) and (D-c), a finite magnetic field (B) displaces the cen- ters of Rashba-split inner(+) and outer(-) Fermi pockets from k = 0 to opposite momenta, ±k0 = ±ˆz × B/vF , re- spectively, and leads to a finite intrapocket Cooper pair momentum q0 = ±2k0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Owning to the larger DOS in the outer pocket, usually, energetically favored state is the one with the Cooper pair momentum q0 = −2k0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Figure 2 (BR-b) and (BR-d) show a magnetic field de- pendence of the depairing critical current in the fluctu- ation regime of a metal-superconductor resistive transi- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' When B ∥ J, as shown in figure 2 (BR-b), the phase diagram in the B-J plane remains symmetric with respect to both B and J axes and thus, no nonreciproc- ity in the critical current Jc and critical magnetic field Bc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' However, when B ⊥ J, as shown in figure 2 (BR-d), the phase diagram becomes asymmetric/skewed, indicat- ing nonreciprocity in the critical current J+ c ̸= J− c and polarity-dependence of critical field B+ c ̸= B− c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' That is, the maximum critical current flowing in the direction parallel and antiparallel to q0 are different, which leads to SDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' On the same footing, in the presence of a su- percurrent, the polarity-dependence of in-plane critical fields is also a direct consequence of the finite-momentum 10 Cooper pairing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Similar mechanism has been realized in Rashba SCs with intrinsic magnetization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Figure 2(BL) demonstrates Rashba spin-splitting and SDE by control- ling magnetization orientation in a JJ Nb/Pt/Nb with a proximity-magnetized Pt barrier (Pt/Y3Fe5O12 (YIG)) [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' From spin-singlet to spin-triplet pairing It is also crucial to consider competition between spin- singlet and spin-triplet pairing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Note that, in the ab- sence of magnetic field, pairing momentum remains zero for spin-singlet symmetry even in the presence of SOI, whereas SOI induces finite momentum for spin-triplet symmetry, q± = q± 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' However, owning to the symmet- ric shift of pairing momentum (q+ 0 = q− 0 ) in the absence of magnetic field, even finite-momentum of spin-triplet pairing does not induce nonreciprocity of supercurrent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' It can be explained by noticing that the q-linear term in the kinetic energy of Cooper pairs could only shift the momentum space positions of the optimal critical cur- rents (maximum I+ and minimum I−) while keeping I± values unchanged, and thus, could not induced nonre- ciprocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' To induce nonreciprocity, one needs magnetic field dependent (higher order) q-terms in the GL free en- ergy of a SC [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' When magnetic field is applied, energy-dependent spin-splitting lifts the Kramers degeneracy, such that electrons with opposite spin are not momentum- symmetric around Γ = 0, and nonrecirpocity emerges in both cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' As a result, contribution from Cooper pairs with opposite center of mass momentum becomes non-equivalent and the cancellation of their effect is avoided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' In the spin-singlet pairing symmetry, magnetic field changes both the magnitude and the momentum- space position of center of mass momentum: enlarging and moving q+ 0 along the current direction while reduc- ing and moving q− 0 opposite to the current direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' On the other hand, with the spin-triplet pairing symmetry, SOI shifts momentum of Cooper pairs from q0 = 0 (sym- metric around Γ-point) to q = q± 0 (asymmetric around Γ-point) due to opposite momentum-shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Unlike spin- singlet case, magnetic field cannot change the momentum space position of q = q± 0 but rather modifies their mag- nitude: q+ 0 enlarges whereas q− 0 reduces with increasing magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' There is a threshold limit, certainly, for magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' For the spin-triplet pairing, when bottom of one CB passes above FL, q− 0 → 0 while q+ 0 become maximal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' On the other hand, for the spin-singlet pairing, magnitude of q− 0 become constant when in the inner Fermi circle shifted completely on one-side of Γ-point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Furthermore, one needs to keep an eye on the curvature of parabolic bands as it is key to understand change in momentum when magnetic field is increased, means the Fermi veloc- ity plays central role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' THEORY OF SUPERCONDUCTING DIODE EFFECTS Before jumping onto the recently reported theoreti- cal analysis of SDE, it is important to have a quick review of theoretical studies in which nonreciprocity of supercurrent is reported and the interesting functionali- ties of SCs intertwined with broken inversion symmetry and SOI are highlighted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Interestingly, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Edelstein has discussed the characteristics of the Cooper pairing in two-dimensional noncentrosymmetric electron systems [26], magnetoelectric effect in polar SCs [27], and nonre- ciprocity in the supercurrent by studying the Ginzburg- Landau equation for SCs of polar symmetry [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' In other words, SDE has been there since 1990s, and only recently demonstrated experimentally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' In 1996, followed by his earlier work characterizing Cooper pairing in noncentrosymmetric SCs [26] and de- scribing magnetoelectric effects polar SCs [27], V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Edelstein [28] explicitly proposed nonreciprocity in the supercurrent, that is, when applied magnetic field (B), electric current (j), and polar axis (ˆr) are orthogonal to each other, the magnitude of the critical current jc(B) depends on the sign of the mixed product (ˆr × ˆB) · ˆjc, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=', the critical current should be different for two oppo- site directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Recently, intriguing experimental demonstrations of SDE, especially in the Rashba-type bulk superconduct- ing [V/Nb/Ta]n superlattice [12] or Al/InAs-2DEG/Al JJs [13] and in the Ising-type superconducting JJs such as NbSe2 constriction [22] or Nb/NiTe2/Nb junction [23], has stimulated theoretical research on nonreciprocal su- percurrent transport in a number of exotic quantum ma- terials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' In addition, it also sparked the discussion on fun- damental mechanisms that cause nonreciprocal charge transport in SCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' For instance, how nonracirocal charge transport in a semiconductor with finite resistance could be generalized to a superconductor allowing supercurrent with zero-resistance?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' More specifically, which physical quantity display nonreciprocal behaviour in the By employing mean-field (MF), Bogoliubov–de Gennes (BdG), and time-dependent Ginzburg-Landau (GL) the- ories, Daido et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' [29], J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' [31], N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Yuan and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Fu [30], and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Ili´c and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Bergeret [32] theorized SDE in junction-free Rashba/polar SCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Daido et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' [29] studied Rashba-Zeeman-Hubbard model for the helical superconductivity and proposed that nonreciprocity in the depairing critical current is the intrinsic mechanism of SDE in the fluctuation regime of metal-superconductor resistive transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Daido et al [29] also showed that such mechanism of intrinsic SDE can be employed as a microscopic probe to study and explore the phase dia- gram of helical superconductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Similar proposal has been made by N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Yuan and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Fu [30], who studied effec- tive Rashba-Zeeman-Hubbard model and reported that nonreciprocal depairing critical current and the polarity- dependent critical magnetic field are the consequences of finite-momentum Cooper pairing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' On the same footing, 11 mainly using the GL theory and phenomenological theory of SDE, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' [31] presented a detailed discussion on symmetry breaking phenomenon and an intertwining between polar axis, magnetic field orientation, and cur- rent direction that is desired for the realization of SDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' The theory of SDE has been generalized for Rashba SCs with arbitrary disorder by S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Ili´c and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Bergeret [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Thus far, theoretical discussion on nonreciprocal su- percurrent and prediction of intrinsic SDE has also been extended for other junction-free polar superconducting systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' For instance, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Scammell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' presented theory of zero-field SDE in twisted trilayer graphene [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Zhai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' [34] predicted reversible SDE in ferroelectric SCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' The experimental demonstration of nonreciprocal transport in chiral SCs, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=', Ru-Sr2RuO4 eutectic system [138, 139] and WS2 nanotubes [68], is recently followed by B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Zinkl et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' [35] who discussed the detailed symmetry conditions for the SDE in various chiral superconduct- ing models/systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' The theory of nonreciprocal charge transport and intertwining between SDE and band topol- ogy has also been presented for topological SCs [36–38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' For instance, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Yuan and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Fu [36] uncovered an inter- twining between finite-momentum superconductivity and topological band theory, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=', Cooper pairing with finite momentum depends closely on the nontrivial topologi- cal spin texture of nondegenerate Fermi surfaces, driven by combined effect of SOI and Zeeman fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Recently, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Legg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' [37] theorized SDE due to MCA in topological insulators and Rashba nanowires, while K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Takasan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' [38] discussed supercurrent-induced topo- logical phase transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' In addition, the basic mechanisms of SDE (first en- visioned by J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' [39]) has also been theo- rized for JJs [40], e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=', conventional superconducting NbSe2/Nb3Br8/NbSe2 JJ [41] and Al/InAs-2DEG/Al JJ [42], graphene-based JJ [43], and topological supercon- ducting JJ [44, 45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Furthermore, the effect of Rashba and Dresselhaus SOI on supercurrent rectification and MCA have also been studied for JJs based on conven- tional SCs [42] topological SCs [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' In the recent the- oretical studies on the topological JJ dS/FI/dS (dS: d- wave superconductor, FI: ferromagnetic insulator) on a 3D topological insulator surface, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Tanaka and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Na- gaosa [45] also demonstrated the relevance of the Ma- jorana bound states (MBS), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=', spin-momentum locked energy-zero Andreev bound states (ABS) at the interface [140, 141].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' MATERIALS FOR SUPERCONDUCTING DIODE EFFECTS In the last two years, SDE has been experimentally ob- served in a number of superconducting structures, rang- ing from junction-free SCs [12, 14–18], JJs [13, 19–24], and other engineered structures such as superconduct- ing tunnelling junctions [77] and superconducting de- vices with pinning centres of asymmetric pattern [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' JJs, mainly due to the presence of a junction, can be though as symmetric and superconducting analogue of asymmetric semiconducting pn junction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' On the other hand, junction-free SCs can be though as symmetric and superconducting analogue of symmetric semiconductors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' The observation of SDE originated from nonrecipro- cal charge transport driven by MCA in symmetric SCs , whether junction-free or JJs, relies on simultaneously broken spatial-inversion and time-reversal symmetries, similar to that in symmetric semiconductors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Further- more, similar that in topologically nontrivial semicon- ductor/semimetals, SDE can be realized in time-reversal symmetric systems where nonreciprocal charge transport is associated with nontrivial Berry phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Since both MCA and nontrivial Berry phase are strongly associated the strength and nature of SOI originated due to bro- ken inversion symmetry, noncentrosymmetric SCs can be classified as Rashba SCs [12–17, 19, 20] or Ising SCs [21– 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' If spatial-inversion symmetry is broken, SDE can be realized in three-dimensional bulk materials, quasi-two- dimensional thin films and van der Waals heterostruc- tures, and atomically-thin superconducting materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Thus far,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' SDE has been reported in several materials,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' ranging from conventional SCs such as [Nb/V/Ta]n su- perlattice [12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' 14],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Al/InAs-2DEG/Al junction [13],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Nb SCs [20],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Cu/EuS/Al tunnel junction [77],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' and super- conducting thin films with conformal-mapped nanoholes [25],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' ferromagnetic SCs [15],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' twisted-angle bilayer [24] and trilayer [18] graphene with unconventional super- conductivity and,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' TMDCs with Ising superconductiv- ity [21–23],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' and topological superconducting materials [16,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' 17,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' 23] where superconductivity coexists with non- trivial band topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' For device fabrication of superconducting electronics, and especially for the search/utilization of novel su- perconducting materials with high workable tempera- ture and large magnetic field, it is important to cat- egorize materials hosting SDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Superconducting ma- terials/structures displaying SDE can be classified as junction-free or JJs based on device structure, Rashba or Ising SCs based on the nature of SOI, and trivial or non- trivial based on band topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Furthermore, SDE can be classified as magnetic-field-driven or field-free SDE depending on the magnetic character of superconduct- ing materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Furthermore, depending upon the origin of nanoraciprocity of charge transport, whether MCA or nontrivial Berry phase, SDE materials can be classified as time-reversal-symmetric or time-reversal-asymmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' EFFICIENCY OF SUPERCONDUCTING DIODE Let’s consider a superconducting sheet with pairing po- tential ∆(q), where q = qˆx is the center-of-mass momen- tum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' The metal-superconducting transition, and thus a distinction between supercurrent, depairing current, and 12 a normal current, can be conveniently described by in- troducing condensation energy F(q) ≡ Fn(q) − Fs(q) for each q, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=', the difference between free energy per unit area in the normal (n) and superconducting (s) states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' The sheet current density, as an expectation value of the current operator, can be obtained by j(q) = 2∂qF(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' If a current source supplies an electric current jex, a super- conducting state with pairing momentum q should be realized when jex = j(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' However, when jex < j− c ≡ minqj(q) or jex > j+ c ≡ maxqj(q), the superconducting state can not sustain jex and turns into a normal state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Thus, the depairing critical current along a direction par- allel (+ˆx) and antiparallel (−ˆx) to the pairing momen- tum q is given by the maximum (j+ c ) and minimum (j− c ) value of j(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' The SDE in such helical superconductor is identified and characterized with a finite ∆jc given by ∆jc ≡ j+ c + j− c = j+ c − |j− c | (6) Although a huge current density is generally required to achieve the depairing limit in a typical superconduc- tor, depairing critical current density (jc) has recently been reported in the superconducting microbridge de- vices [142–144].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' For an optimal performance of SDE, it is instructive to analyse the behavior of depairing jc and ∆jc(T) through various perspectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' For instance, dependence of critical current density jc on tempera- ture and the orientation of magnetic field reported for Fe-based Ba0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='5K0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='5Fe2As2 microbridge with nanoscale thickness, see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' 3 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' 4 in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' [143], critical current density as a function of bridge width and length reported for Cu-based YBa2Cu3O7−δ micro- bridge with nanoscale thickness, see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' 3 in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' [142], a comparison of critical current density obtained from Ginzburg-Landau (GL) theory [∝ (Tc − T)3/2] to that from Kupriyanov-Lukichev (KL) theory for Fe-based Fe1+yTe1−xSex microbridge with microscale thickness, see figure 3 in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' [144], and the sign reversal of ∆jc by increasing the magnetic field at low temperatures, see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' 4 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' 5 in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' As a figure of merit,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' the strength of the nonreciprocal response or the super- conducting diode efficiency can be expressed as a ratio between ∆jc and the averaged critical current javg c [29– 32] η ≡ j+ c − |j− c | j+ c + |j− c | = ∆jc 2javg c (7) Recent theoretical studies [29–32] show that the strength of η depends on a range of relevant system parameters: applied magnetic field,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' working temperature,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' induced Cooper pairing momentum,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' intrinsic SOI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' and an inter- twining between them [29–32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' In addition, strength of η also depends on two other related but distinct parame- ters, chemical potential [31] and next-nearest neighbour hopping [29] that break the particle-hole symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Fur- thermore, though the SDE persists even in the presence of disorder, strength of η is also affected by disorder as it may cause changes in the nature of the two helical bands by introducing mixing between them [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Thus, for energy-efficient and high performance superconduct- ing device application, it is crucial to find certain opti- mal system parameter regimes where the strength of η is maximal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Recently, Ilice et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' [32] theoretically pre- dicted that SDE efficiency may exceed η = 40% (in the ballistic limit) at optimal magnetic field, temperature, and SOI in Rashba SCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Interestingly, SDE with opti- mal efficiency can be engineered by steering the exotic characteristics and the design of a JJ [13, 45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' At some fixed temperature, SDE efficiency shows non- monotonic magnetic field dependence [29, 30, 32]: η in- creases (almost linearly) for (weak) moderate fields and then suppresses beyond a certain breakdown/threshold field Bmax,η, see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' 3(D) in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' [30] and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' 4 in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' For Rashba SCs, threshold field is theoretically [30, 32] predicted to be of the order of the Pauli param- agnetic limit, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=', much larger than the breakdown limit observed in recent experiments [12, 13, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Along with this nonmonotonic behavior, SDE efficiency changes its sign with increase in magnetic field [29, 30, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Such change in sign of SDE efficiency appears approximately at the Pauli limit B ≈ BP , see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' 3(F) in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' [30], Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' 3 in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' [32], and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' 4 in ref [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Such magnetic field driven sign reversal of the SDE, accom- panied by the crossover between weak and strong helical phase, is a general feature of helical SCs irrespective of their details [145].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Unlike magnetic field dependence, recent theoretical studies predict quite diverse behaviour for the tem- perature dependence of SDE efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' For instance, at some fixed magnetic field, Rashba-Zeeman-Hubbard model [29–31] predict that SDE efficiency shows a mono- tonic square-root-like temperature dependence near the transition temperature which saturates at low tempera- tures, see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' 2 in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' [29], and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' 3 in ref [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' On the other hand, quasiclassical Eilenberger equation for a 2D disordered Rashba superconductor [111] shows that the temperature dependence of SDE efficiency is critically affected by the strength of fixed magnetic field and may display nonmonotonic temperature-dependence [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' For instance, SDE efficiency shows a monotonic temperature-dependence for B ⪆ BP but it becomes nonmonotonic when B ⪅ BP , see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' 4 in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' That is, in the later case, first SDE efficiency in- creases with decrease in temperature but it is gradually suppressed when temperature is further lowers after cer- tain breakdown limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Recent observation of SDE shows that the monotonic [13, 22] and the nonmonotonic [22] temperature-dependence of SDE efficiency may also de- pend on the sample fabrication [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Similar transition from monotonic to nonmonotonic temperature depen- dence may also be realized by varying strength of dis- order, see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' 6 in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' [32] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Next, we turn to the dependence of SDE efficiency on the momentum of Cooper pairs or the nature of helical phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' For spin-orbit coupled Rashba SCs in magnetic 13 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Dependence of superconducting diode effect on the system parameters and its optimization (L) Rashba- Zeeman-Hubbard model for a Rashba superconductor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' (a) The temperature dependence of ∆jc(T) (red closed circles) and ∆(T) (open blue circles) with arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' The red and blue dashed lines represent the fitting curves of ∆jc(T) (with (Tc − T)2) and ∆(T) (with √Tc − T) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Inset: Enlarged view near Tc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Here Zeeman exchange parameter is set as h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='03 and the transition temperature reads Tc ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='036 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' (b) The h-T phase diagram depicting temperature and the magnetic field dependence of ∆jc(h, T) with t2 = 0 (left) and with t2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='2 (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Here t2 denotes next-nearest-neighbour hopping while the red (blue) color indicates positive (negative) values of ∆jc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' (c) Pairing momentum q0 (left) and SDE efficiency (right), represented by r here, for various values of h and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Figure (L) is reproduced with permission from ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' (R) Quasiclassical Eilenberger equation for a 2D Rashba superconductor (a) Helical modulation vector q0 as a function of magnetic field, where q0v ≈ 2(α/v)h corresponding to the “weak helical phase” at low fields, whereas q0v ≈ 2h corresponding to the “strong helical phase” at high fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Here q0 is calculated in the vicinity of the upper critical field (hc2) at different strengths of spin-orbit interaction (α/v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' (b) Supercurrent j (red) and the superconducting gap ∆ (black) plotted as a function of the phase gradient, for different magnetic fields at fixed values of temperature T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='01Tc and spin-orbit interaction α/v = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Both quantities are calculated self-consistently and the curves are normalized with j0 and ∆0, respectively, which represent the critical current and the superconducting gap at T = h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' (c) Temperature-dependence of superconducting diode efficiency η, calculated for different values of the magnetic field and fixed spin-orbit interaction α/v = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' (d) Superconducting diode efficiency η, calculated for different strengths of spin-orbit interaction in the ballistic limit, corresponding to every point in the h-T phase diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Here black curve corresponds to the upper critical field hc2 while the purple (orange) color indicates positive (negative) values of η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Figure is reproduced with permission from ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' field, the nature of helical phase can be characterized by quantifying the contribution of two helical bands to the helical superconductivity [111].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Owning to the op- posite energy shift induced by magnetic field, the two helical bands denoted with the index λ = ± and char- acterized by the same Fermi velocity v = � 2µ/m + α2 but different densities of states νλ = ν(1 − λα/v), pre- fer opposite modulation vectors: qλ 0 v = −2λh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Here, m is the effective electron mass, µ is the chemical po- tential, ν = m/(2π), and α = ∆so/√2mµ characterizes the SOI strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Figure 3(R-a) illustrates the crossover from a “weak” to “strong” helical phase for different ra- tios of Fermi velocity (v) and the velocity associated with Rashba SOI (α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' In the “weak” or long-wavelength heli- cal phase subject to low magnetic fields, contribution of both bands to helical superconductivity yields a modu- lation vector q0v ≈ 2(α/v)h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' In the “strong” or short- wavelength helical phase at large magnetic fields, owning to the dominance (suppression) of contribution from the band with higher (lower) density of states, only one of the bands contributes to helical superconductivity which leads to the modulation vector q0v ≈ 2h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Ili´c and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Bergeret [32], based on the quasiclassi- cal Eilenberger equation for a 2D Rashba superconductor [111], predicted that the maximum of η emerges when both the bands contribute to the helical superconductiv- ity and the magnetic field is close to the critical value h∗ at which a crossover between “weak” and “strong” he- (L) (R) Ai(T) △(T) (a) (c) ×10-3 (a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='5 20 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='25 2 6 15 "L/aob 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='05 h 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='5 5 1 2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='5 0 1 2 3 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='0 h/Te T/T。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='01 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='0f 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='5 n[%] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='1 4jc 4jc 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='0 L/y 二 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='5 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='5 30 0 g = 0% 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='04 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='0 T 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='0F 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='8×10-2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='46 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='31 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='25 (c) 0 0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='5 10 00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='1 qo 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='1 r 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='5 10 ↑ = 22% 7 = 6% 0 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='0L 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='04 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='04 6-420 6-420 2 4 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='0 T T qu/T qu/T T/T T/Te14 lical superconducting phase occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' It can be explained from the self-consistent calculation of ∆(q), j(q) and η vs Cooper pairing momentum under various magnetic field strength as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' 3(R-b) or from the h-T phase diagram under various strengths of Rashba SOI as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' 3(R-d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' In the absence of magnetic field (h = 0), as shown in the upper left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' 3(R-b), there is no helical phase (q0 = 0) and thus no nonre- ciprocity of the critical current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' In the presence of finite magnetic field (h ̸= 0), finite Cooper pairing momentum (q0 ̸= 0) leads to nonreciprocity of the critical current in both the “weak” helical state induced by sufficiently low h, as shown in the upper right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' 3(R-b), and the “strong” helical state induced by large h, as as shown in the two lower panels of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' 3(R-b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' The momen- tum dependence of ∆(q) and j(q) is markedly different in these three superconducting states, and thus, depict a completely different supercurrent transport: no SDE in the BCS state (j+ c = |j− c |, η = 0), whereas negative SDE in the “weak” helical state (j+ c < |j− c |, η < 0) while posi- tive SDE in the “strong” helical states (j+ c > |j− c |, η > 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' In addition, different strength and opposite sign of SDE under different magnetic field values hint that there must be some optimal field at which η should be maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' It can be depicted by plotting η for every point in the h-T phase diagram, and in addition, effect of other pa- rameters can be visualised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' For instance, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' 3(R-d), S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Ili´c and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Bergeret [32] plotted the h- T phase diagram and calculated η for different strengths of SOI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Here black curve corresponds to the upper crit- ical field hc2 while the orange and purple colors clearly illustrate the two distinct regimes in which SDE is driven by the “weak” and “strong” helical phases, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' First, it showcases that the maximum efficiency appears at the crossover between “weak” and “strong” helical phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Second, maximum efficiency exceeding 40% at the crossover corresponds to the optimal SOI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Third, the maximum efficiency also corresponds to optimal temper- ature in the superconducting phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Such momentum dependence, yielding maximum η with optimal magnetic field and SOI driving system at the crossover between “weak” and “strong” helical phases, implies that the competition and the contribu- tion of both helical bands is central for the SDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' This can be explained by noticing that the MCA is proportional to magnetic field and SOI, and thus become strongest when both of these parameters are maximal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' The maximal magnetic field and SOI borne by the system, along with the constraint of contribution from both helical bands, is ensured at the crossover between “weak” and “strong” helical phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' This can further be explained by the analysing the h-T phase diagram regimes, as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' 3(R-d), where too large magnetic field and too large SOI both suppress the SDE efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' For instance, SDE efficiency vanishes when magnetic field is increased, beyond the crossover to the “strong” phase, where only one of the helical bands dominates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Similarly, when SOI is increased — such that α/v → 1, only one helical band with a large DOS (ν− ≈ 2ν) exists while the helical band with vanishingly small DOS (ν+ → 0) other is fully sup- pressed, and the SDE disappears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' This phase diagram also helps to understand the in- tertwining of optimal temperature with magnetic field and SOI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' At weak SOI, such as depicted in the upper left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' 3(R-d), SDE becomes strongest at the tricritical point (T ∗, h∗) where the “weak” helical phase meets the “strong” helical phase and the normal phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' It is in good qualitative agreement with the results pre- dicted by N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Yuan and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Fu [30] where (T ∗, h∗) denotes tricritical point at which the FF phase meets the nor- mal phase and the BCS phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' However, with increasing strength of SOI, h-T phase diagram regime hosting max- imum SDE moves towards zero-temperature, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=', where T ≪ T ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Similarly, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' 3(L-b), Daido et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' [29] plotted the h-T phase diagram and calculated η for dif- ferent strengths of next-nearest neighbour hopping t2 in the Rashba-Zeeman-Hubbard model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' It depicts the sign change of η with increasing magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' In addition, at some magnetic field, the sign of SDE efficiency found at t2 = 0 (left panel) also switches when a finite t2 ̸= 0 is considered (right panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Furthermore, magnetic field dependence of pairing momentum as shown in the left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' 3(L-c) and the SDE efficiency as shown in the right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' 3(L-c) showcases that the maximum η appears at the crossover between “weak” and “strong” helical phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' It implies that the results obtained from the numerical study of Rashba-Zeeman- Hubbard model [29] and that from quasiclassical Eilen- berger equation [32] are in good qualitative agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' However, as mentioned above, there is considerable dif- ferences between these two studies when it comes to the temperature dependence of SDE efficiency, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=', numerical study of Rashba-Zeeman-Hubbard model shows mono- tonic behaviour while quasiclassical Eilenberger equation shows temperature dependence could be either mono- tonic or nonmonotonic depending on the strength of mag- netic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Based on the above analysis, one can conclude that the nonmonotonic behaviour, for both magnetic field and temperature dependence, and the change of sign of η with increasing magnetic field is related to the mag- netic field-driven evolution of the helical phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' That is, η becomes maximum at a particular field h∗ and optimal temperature, and then lowers for other values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Similar to the dependence on next-nearest neighbour hopping [29], and consistent with the analogy discussed for magnetic field and SOI intertwined with the varia- tion in the DOS of two helical bands [32], He et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' al [31] theoretically predicted that SDE efficiency show strong dependence on the chemical potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' For a Rashba su- perconductor with Zeeman field, where the free energy includes all terms up to the linear order in h√ϵ, GL the- 15 ory results in the SDE efficiency [31]: η = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='7λR |λR| h√ϵ Tc × � (1 + ˜µ)−1/2 if ˜µ > 0 8 7 + 16 21 ˜µ + (1 + ˜µ)−1/2 if − 1 < ˜µ < 0 (8) Here λR is the Rashba SOI strength, ϵ = 1 − T/Tc, and ˜µ = µ/ER where ER = 1 2mλ2 R is the energy difference be- tween band crossing point (µ = 0) of Rashba-split bands and the conduction band edge (µ = −ER) and m denotes the effective electron mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' At some fixed magnetic field, temperature, and SOI, SDE efficiency shows maximum strength at µ = 0, whereas it decrease when the Fermi level moves away from the band crossing point, either towards the large µ limit (µ ≫ ER) or towards the con- duction band edge (µ = −ER).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' It is important to note that there are several constraints, and thus limitations, on these GL theory calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' For instance, the ex- pression (8) is derived by assuming |h| ≪ Tc ≪ ER and treating the problem in the band basis where only the intra-band pairing ∆t is considered while the inter- band pairing ∆s is neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' As a consequence of tak- ing the limit Tc/ER → 0 and neglecting the inter-band pairing, there exists a discontinuity in η at µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' In addition, owning to the consideration of intra-band pair- ing only, such a discontinuity appears also due to the flip of spin-momentum locking helicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' However, the features of SDE efficiency obtained numerically from a self-consistent Bogoliubov–de Gennes mean-field Hamil- tonian [31] are in good qualitative agreement with those displayed by SDE efficiency obtained from the analytic generalized GL theory calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' In addition, the dis- continuity of η at µ = 0 is smoothed out when Tc/ER is not so small and it shows square root dependence on µ, η ∼ µ1/2, when µ is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Finally, it is important to emphasise that SDE effi- ciency also depends upon the characteristics and the de- sign of a JJ [13, 45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' In general, with a macroscopic phase difference φ between two SCs, the standard CPR of the Josephson supercurrent I(φ) between two SCs is I(φ) ∼ sin φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' That is, when either space-inversion sym- metry or time-reversal symmetry is preserved, purely si- nusoidal terms leads to an antisymmetric CPR, I(φ) = −I(−φ), and the Josephson current vanishes for φ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' On the other hand, when both time-reversal symmetry and space-inversion symmetry are simultaneously bro- ken, an anomalous CPR [46, 49, 141, 146–160] (displaying finite anomalous Josephson current even at zero phase difference) contains cosine terms as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' However, even the presence of such cosine term does not suffice to obtain SDE because it simply introduces an anomalous phase shift in the purely sinusoidal CPR and thus the Joseph- son inductance remains reciprocal (symmetric across the zero-current).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Thus, in order to realize SDE, it is manda- tory that an asymmetry is induced in the CPR by higher order phase (especially sine) terms such that the cosine terms are not absorbed in a mere phase shift [13, 45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' By fabricating Al/InAs-2DEG/Al ballistic JJs, Baum- gartner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' [13] observed supercurrent rectification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' When an in-plane magnetic field is applied perpendicular to the current, Rashba superconducting system shows an anomalous Josephson supercurrent due to even (cosine) terms in the CPR [156].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Such anomalous CPR contains higher harmonic sine terms if the junction transparency is high [159, 161], and thus leads to SDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' By theoreti- cal studying a JJ dS/FI/dS made with d-wave SCs (dS) and a ferromagnetic insulator (FI) on the surface of a 3D topological insulator, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Tanaka and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Nagaosa [45] showed that asymmetric CPR containing a wide variety of phase terms leads to high quality SDE [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Apart from the conventional sin φ phase term in the Joseph- son current, energy-zero Andreev bound state (ABS) at the dS/FI/dS interface enhances the sin 2φ component of I(φ) [162, 163].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' When the junction dS/FI/dS is placed on the surface of topological insulator [164], simultane- ous space-inversion and time-reversal symmetry break- ing allows a cos φ phase term [141, 148] leading to an exotic current-phase relation with I(φ) ̸= −I(−φ) [155] while the energy-zero ABS become MBS due to the spin- momentum locking [140, 141].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' The simultaneous exis- tence, with almost the same order, of sin φ, cos φ, and sin 2φ phase terms promises a maximum value of SDE efficiency (η = ±2) for the d-wave SCs junction on the surface of topological insulator [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' In light of this, opti- mal supercurrent rectification effect of a JJ can be real- ized by exploiting exotic characteristics of unconventional SCs as well as optimizing junction transparency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' OBSERVATION OF SUPERCURRENT DIODE EFFECT SDE is associated with the literal metal- superconductor transition and defined as nonreciprocity of depairing critical current, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=', depairing critical current in the direction parallel (j+ c ) and antiparallel (j− c ) to the pairing momentum differ (j+ c ̸= j− c ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' An ideal SDE would be either j+ c or j− c is zero so that one has maximum ∆Jc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Such a resistive transition between a supercurrent and a normal current can be realized either by extrinsic stimuli or via mechanisms that are intrinsic to the superconducting materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' For instance, the resistive transition can be caused by the vortex motion, usually realized under out-of-plane magnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Owning to the dependence of dynamics and the statistical mechanics of the vortex system on the device setup such as impurity concentrations and the thermal/quantum fluctuations [165], such extrinsic mechanism promise tunability of the resistive transition by the nanostructure engineering [56, 166].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Apart from the resistance caused by extrinsic mechanisms, the metal-superconductor resistive transition can literally be caused by the dissociation of the Cooper pairs resulting in a transition from supercurrent to a normal current [6, 167].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' This occurs at the maximum critical current, which is known as depairing current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' In other words, the depairing critical current is directly associated with 16 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Magnetochiral anisotropy of the resistance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' (T) Nonreciprocal transport measurements of critical current in the resistive fluctuation regime of [Nb/V/Ta]n superlattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' a Magnetic field dependence of first-harmonic (Rω) and second- harmonic (R2ω) sheet resistances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Rω vanishes in the superconducting region (white shadings) while become finite in the normal conducting region (blue shadings).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' R2ω enhances when the magnetic field orientation is orthogonal to the current direction and becomes maximal in the fluctuation region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' b Temperature dependence of second-harmonic sheet resistance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' c Temperature dependence of the coefficient of magnetochiral anisotropy (γ) calculated from R2ω/Rω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' The plot roughly shows that γ increases with temperature and become maximal in the vicinity of Tc, except a a dip appearing at 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='2 K and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='3 K reflecting small R2ω values at these temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Figure is reproduced with permission from ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='[12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' (B) Nonreciprocal transport measurements of critical current in the resistive fluctuation regime of Rashba-type Al/InAs-2DEG/Al JJ array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' (a) Temperature dependence of first-harmonics Rω(T, θ) showing resistive transition for different angles (θ) of the in-plane magnetic field (Bip).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' (b) Temperature dependence of second-harmonics R2ω(T, θ) = V2ω(T, θ)/Iac of the I-V characteristics for different θ values with the a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' current bias of Iac= 20 nA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' c The coefficient of magnetochiral anisotropy 2Rmax 2ω /Rω versus orientation/angle θ of the in-plane magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Here Rmax 2ω are the maxima of second-harmonics displayed in (b) and Rω is the corresponding linear resistance displayed in (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' The red data point shown at θ = 90◦ is obtained by switching the orientation of Bip at θ = 90◦ (the data point in blue), which is equivalent to setting θ = 270◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' The maximal coefficient of magnetochiral anisotropy, extracted from a sine fit of the data, is γS ≃ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='1 × 106 T−A−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Figure is reprinted with permission from ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' [13] the closing of the superconducting gap, which reduces and eventually closes with increasing supercurrent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' As the depairing limit or the upper limit of the critical current is unique to each superconducting material, depairing current is an intrinsic material parameter for characterizing SCs [165].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Thus, the intrinsic mechanism responsible for SDE ties around the nonreciprocity in the depairing critical current in the fluctuation regime of metal-superconductor resistive transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' In this picture, like many exotic characteristics of quantum ma- terials, intrinsic SDE is a nontrivial quantum mechanical effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Based on the working temperature, or a work- ing regime of phase diagram representing metal- superconductor resistive transition, observation of SDE can be classified into two main categories: (i) SDE based on the nonreciprocity of depairing current near the su- perconducting transition temperature (T ≈ Tc), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=', in the fluctuation regime of metal-superconductor resistive transition, and (ii) SDE based on the nonreciprocity of supercurrent at sub-Kelvin temperatures (T ≪ Tc), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=', deep in the superconducting phase regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Magnetochiral anisotropy of the resistance In the fluctuation regime of resistive transition close to Tc, SDE can be described by MCA of the resistance (γS, as defined in equation (3)), similar to that in semi- conductors, and may be characterized by I-V curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' In this regime, MCA coefficient γS can be found by measur- ing the second harmonic signal in lock-in measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' (1) a 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='2K b Amplitude2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='0mA 600 Amplitude2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='0mA +y y C 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='5 K 500 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='0 K 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='1K 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='05 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='1 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' V 100 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='2日 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='6 oF Magnetic field (M) Magnetic field () 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='0 T/T。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' (B) a b 120 200 0 6 21Rma*/R,BI (T-" μA-") B,lIII 00 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='_III 80 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='5° 100 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='5° 45° 45° 3 (sy) "y (s) 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='5° 0 2 006 40 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='5° 270°Bipll -- 7s sin(0) 100 90° B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='y = +90 mT 270°BiplI Bp = +90 mT 200 Bpy = 90 mT 0 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='35 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='45 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='55 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='35 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='45 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='55 00 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='5° 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='0° 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='5° 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='0° Temperature (K) Temperature (K) Angle 17 That is,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' for an ac current (Iin = I sin ωt) with an ampli- tude of I and a frequency of ω applied as input,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' the non- linear voltage-drop and current-dependent resistance can be derived from the nonlinear resistance term in equation (3) as: V2ω(t) = γBRωI2 sin2 ωt = 1 2γBRωI2 � 1 + sin � 2ωt − π 2 �� R2ω = 1 2γBRωI (9) Here Rω corresponds to the current-independent linear resistance R0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' while R2ω represents the second-order non- linear resistance,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' which is dependent on both the cur- rent and the magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Thus by measuring the first- (Rω) and second-harmonic R2ω sheet/junction re- sistances through 2ω voltage response, γS can be esti- mated as γS = 2R2ω BIRω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' However, such resistive measurements cannot realis- tically simulate the intrinsic SDE at temperatures well below Tc due to no measurable resistance in this regime (R0 = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Thus, the efficiency of SDE is expected to be finite only at T ≈ Tc while negligibly small both at temperatures well below Tc and above Tc (γN ≪ γS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' For instance, Ando et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' [12] measured MCA of the re- sistance by performing an a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' harmonic measurements for Rashba-type bulk superconducting [V/Nb/Ta]n su- perlattice [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' MCA coefficient γS show sharp increase in the fluctuation regime and reaches to its maximal value γS ≃ 550 T−A− at Tc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' However, γS remains negligi- bly small at temperatures well below Tc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Though the observation seems to be at variance with the theoreti- cal predictions for intrinsic SDE [29–32] and the tem- perature dependence of experimentally measured MCA in JJs [13, 22], but it is an expected outcome of resis- tive measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' On the other hand, by fabricating symmetric Rashba-type Al/InAs-2DEG/Al JJs, Baum- gartner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' [13] measured MCA for both the induc- tance (γL) and the resistance (γS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Finite MCA coeffi- cient γS ≃ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='1 × 106 T−A− observed through resistive measurements near Tc ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='45 K is of the same order (namely, in the range of 106 T−A−) of the corresponding MCA coefficient observed for the inductance (measured at T = 100 mK), γL ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='77 × 106 T−A−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Magnetochiral anisotropy of the inductance Unlike fluctuation regime, where nonreciprocity of de- pairing critical current is tied to the nonlinear resistance, nonreciprocity of sub-Kelvin supercurrent promise fully superconducting/dissipationless nonreciprocal circuit el- ement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Deep in the sub-Kelvin superconducting regime of the phase diagram, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=', far below the transition tem- perature where resistance is zero (so DC measurements are not feasible), supercurrent MCA and a corresponding SDE (supercurrent rectification/nonreciprocity) is char- acterized rather by measuring kinetic (or Josephson) in- ductance (clearly with AC measurements).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' By measuring Josephson inductance, nonreciprocal supercurrent can be linked to an asymmetry in the current–phase relation, induced by simultaneous breaking of inversion and time- reversal symmetry such that B is not parallel to I, and the MCA coefficient (γL) for the supercurrent can be di- rectly derived from the equation (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' This mechanism can be understood from a semiquanti- tative model [13,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' 42,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' 161] in which Josephson inductance ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='can be derived from the CPR relation I = Ic0f(ϕ) (where ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='f is a 2ϕ-periodic function) and second Josephson equa- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='tion ˙ϕ = 2πV/Φ0 (where Φ0 = h/(2e) is the magnetic ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='flux quantum) as ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='L(I) = V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='dI ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='dt ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='dI ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='dϕ ˙ϕ = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='Φ0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='2πIc0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='df(ϕ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='dϕ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='= Φ0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='2π ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='�dI(ϕ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='dϕ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='�−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='(10) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='It shows that Josephson inductance is a convenient probe ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='to study CPR symmetry by investigating the effects of ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='space-inversion/time-reversal symmetry breaking on the ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='current–phase relation (CPR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Let’s assume a JJ con- figuration in which electric current is flowing along x- direction, while inversion and time-reversal symmetry is broken by applying out-of-plane electric field E = Ezˆz and in-plane magnetic field Bip = Bxˆx + Byˆy, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Equation (10) shows that L(I) is inversely propor- tional to the derivative of the CPR, therefore, the min- imum of Josephson inductance occurs at the inflection- point of the CPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' In the absence of in-plane magnetic field component along y-direction (By = 0), CPR re- mains symmetric around inflection-point appearing at zero-phase, that is (i, ϕ) = (0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' As a result, the min- imum inductance occurs at zero-current, around which L(I) appears to be symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' On the other hand, in the presence of in-plane magnetic field component along y-direction (By ̸= 0), CPR become asymmetric around inflection-point (i∗, ϕ∗), mainly associated with the bro- ken Kramers degeneracy between the oppositely polar- ized spin components of Andreev bound states (ABS) leading to a finite-momentum pairing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' As a result, cur- rent dependence of the Josephson inductance L(I) also become asymmetric and the minimum of L(I) appears at some finite current i∗, corresponding to the shifted inflection point (i∗, ϕ∗) in the CPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Such a pronounced asymmetry in the skewed CPR and, thus, in the Josephson inductance L(I), signals the super- current MCA (as defined in equation (4)) and hence su- percurrent SDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' First, with a given orientation of electric field and polarity of applied current, the shift in inflec- tion point switches along with the sign of By: (i∗, ϕ∗) for +By and (−i∗, −ϕ∗) for −By, as shown in figure 5(Top(d,e)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Second, for a given orientation of By, the CPR gets more skewed with increasing By implying in- crease in the value of i∗ with increasing strength of By, as shown in figure 5(Bottom-a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' As a result, as shown in figure 5(Top-d), the extremal values of i∗ (which are 18 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Current-phase relation and nonreciprocity of inductance in a JJ array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' (T) Device fabrication, current phase relation, and measurement of inductance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' (a) A JJ array is made of 2,250 Al islands (grey), of width w=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='15 µm, length a=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='0 µm and separated by d=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='1 µm, on top of a Rashba-type InAs quantum well (yellow) sandwiched between InGaAs barriers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Red and blue arrows represent the spontaneous supercurrents, with zero phase difference, via spin-split pairs of Andreev bound states, denoted by black and white particle representing oppositely spin-polarized electron and hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' The strength and direction of these spontaneous supercurrents depend on that of an in-plane magnetic field Bip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Counterpropagating circles of black arrows represent the Rashba spin-texture in the InAs quantum well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' (b) Fabricated device showing growth sequence of the heterostructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' The Al layer induces a superconducting gap ∆∗, via proximity effect, in the InAs quantum well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' (c) Scanning electron micrograph of the array with a scale bar of 1 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' (d) Illustrative current-phase relation for a short- ballistic JJ, with high transparency (τ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='94) and strong SOI, in the absence (black) and presence (red/blue) of an in-plane magnetic field By ∥ ˆy (red, By > 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' blue, By < 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' The finite magnetic field (±By) reduces the critical current by a factor 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='8, Ic = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='8Ic0, and adds a cosine term ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='2Ic cos(φ) to the current-phase relation’s Fourier series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' The red dots represent the inflection points (i∗, φ∗) of the current-phase relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' (e) Josephson inductance (Φ0/2πIc0) as function of current (Ic0), corresponding to the current-phase relation in (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' (f) Resonance curves for the RLC circuit, measured at 500 mK, for different values of the bias current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' (g) Current dependence of measured Josephson inductance (at B = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Coloured dots correspond to the spectra in (f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' (B) Measurements of inductance and supercurrent anisotropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' (a) Kinetic inductance versus current, for different orientations of in-plane magnetic field of 100 mT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' (b,c) Constant (b) and linear (c) coefficients of the polynomial expansion of kinetic inductance L(I) as a function of the angle (θ) between in-plane magnetic field Bip and the supercurrent density directed along ˆx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' (d) Measured supercurrent magnetochiral anisotropy (coloured lines and symbols) −2L′ 0/(L0Bip) versus in-plane magnetic field orientation (θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' The maximum magnetochiral anisotropy, coefficient γL ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='77 × 106 T−A−, is extracted from a sinusoidal fit of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Fitted supercurrent magnetochiral anisotropy (Grey scale lines) is computed within semiquantitative model (eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' (10)) for different values of the confinement potential Vconf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' The three fitted curves are perfect sinusoidal functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' All measurements are performed at T = 100 mK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Figure is reproduced with permission from ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' the critical currents I+ c and I− c ) differ for positive (ϕ+ c ) and negative (ϕ− c ) phase difference, signaling the exis- tence of a certain bias-current range in which SDE can be observed for a supercurrent which become different for opposite phase difference polarities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' That is, junction allows supercurrent (I < I+ c (red curve) or |I| < |I− c | (blue curve)) along one current direction while it enters in a resistive state (|I| > |I− c | (red curve) or I > I+ c (blue (I) a p 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='0 3 B,=0 bc (μA) Current (lo) 2 0 B,<0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='2 B(90°) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='0 B(270°) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='75 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='00 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='25 Phase (s) Frequency (MHz) e 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='25 g 410 b Inductance (Φ 2x/lo) Inductance (nH) T= 500 mK Etched2DEG 4110 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='00 370 AI Al 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='75 330 InGaAs Al InAs 4 InGaAs 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='5 290 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='0 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='0 Etched2DEG Current (lco) Current bias (μA) (B) a 370 b :06 c 90° p 16 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='0 390 45° IL"l (nH μA-") 12 45° Experiment 350 180° Inductance (nH) (Hu) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='5 330 360 350 + 0 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='6 315° 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='8 340 310 0 0° 0 45° Theory 350 320 Veoer (meV) 7, = (T- μA") 270° Vgate (M) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='09 330 90° 0 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='39 315° 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='4 315* 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='72 310 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='2 270° 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='6 270° 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='0 0 270° 315° 0° 45° 90° Current bias (μA) Constant part Linear part Angle 19 curve)) along the other current-direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' The MCA of the inductance, can be quantified by mea- suring the constant (L0) and the and the linear (L′ 0) junction inductance, which appear as the leading terms in the polynomial expansion of L(I) around zero cur- rent: L(I) ≈ L0 + L′I + L′′I2/2 with L′ ≡ ∂IL|I=0 and L′′ ≡ ∂2 IL|I=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' As shown in figure 5(Bottom(b,c)), L0 and L′ 0 are plotted as functions of the angle θ between the direction of supercurrent ˆx and the orientation of ap- plied in-plane magnetic field Bip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' In the Hall-bar geom- etry of Al/InAs-2DEG/Al junctions with a Ti-Au global top gate, the constant term L0 strongly depends on the gate voltage, reaches its maximum when By = 0, and shows relatively small anisotropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' In contrast, the linear term L′ 0 shows relatively weak dependence on the gate- voltage, completely vanishes when By = 0 and reaches its maximum when Bx = 0, and thus shows strongly anisotropic behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' As shown in figure 5(Bottom-d), MCA coefficient for the inductance γL = 2L′ 0/(L0Bip) shows sinusoidal θ-dependence, that is, proportional to (B × I) · ˆz = BI sin θ and agrees with the numerical re- sults obtained from semiquantitative model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' In addition, γL remains nearly independent of the gate-voltage and its maximum extracted from the amplitude of the sine reads γL ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='77 × 106 T−A−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' This value of γL, obtained from measurements performed at T = 100 mK, far below the transition temperature (Tc), is of the same order as that of γS calculated for resistive measurements at Tc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' OUTLOOK SDE is a captivating phenomenon and could be a promising building block of the superconducting dissi- pationless technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Thus far, by characterizing the type/nature of SOI and optimizing/matching the SOI energy with the characteristic energy scale (supercon- ducting gap) of charge carriers [57], SDE has been ob- served in both Rashba SCs and Ising SCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Recent theo- retical studies show that SDE is the strongest (i) when the Cooper pairing momentum lies at the crossover be- tween weak and the strong helical superconducting phase in the vicinity of high critical field, which may be real- ized via optimizing magnetic field (or intrinsic magneti- zation), temperature, and SOI [32] and/or (ii) when the Fermi level lies at the band crossing point of two heli- cal bands, which may be tuned by gating [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' From here on, one of the prime goals is to expand the existing platforms and mechanisms for the observation of SDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' For instance, considering the discussion on the optimiza- tion of SDE originated from MCS, one of the remaining challenge is to identify suitable superconducting mate- rial which may provide the best performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Thus far, in addition to conventional superconducting structures, the SDE has also been predicted and/or observed in un- conventional superconducting structures such as twisted few-layer graphene, ferroelectric materials, topological semimetals, and topological insulators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Recent obser- vation of extremely long-range and high-temperature Josephson coupling across a half-metallic ferromagnet [168] and the prediction of SDE in a JJ with half-metals [169] opens another rout for the search and utilization of promising quantum material class, known as spin-gapless materials [170–173].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' In passing, it is interesting to note that the realization of SDE via Rashba SOI and Zeeman exchange interaction in ferromagnetic SCs has a close connection to the real- ization of QAHE via Rashba SOI and Zeeman exchange interaction in ferromagnetic topological insulators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' In the later case, a combined effect of Rashba SOI and Zee- man exchange leads to a spin-splitting in the low-energy bands such that only one of spin sectors display nontriv- ial band topology with inverted band structure while the other spin sector becomes/remains trivial with normal band structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' As a result, when Fermi level is tuned inside the energy band gap, spin-momentum locked chi- ral edge state leads to a quantized conductance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' In the former case, however, low energy bands in both of the spin sectors play role, mainly due to formation of intra- (Fermi)surface and inter-(conduction)bands spin-singlet Cooper pairing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' As a result, when Fermi level is tuned inside the superconducting gap, locking between mag- netization orientation and finite-momentum of Cooper pairing leads to finite MCA and nonraciprocity in the supercurrent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Such a fundamental connection between the realization of QAHE and SDE may allow searching suitable topological superconducting materials based on heterostructure of s-wave SCs and QAH insulators [174– 177].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' In addition, intrinsic iron-based SCs where Rashba SOI-driven band topology and superconductivity coexist [178] may also provide promising platform for the real- ization of SDE in topological superconducting materials [36–38, 54–56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' However, regarding orientation and the strength of ex- change interaction, it is important to remember two dif- ferences between the realization of SDE and QAHE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' (i) Magnetization orientation needs to be in-plane (at an angel to the polar axis) for SDE while out-of-plane for QAHE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' (ii) Nontrivial QAH gap saturates after a criti- cal strength of exchange interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' On the other hand, strength of SDE decreases after the critical value of ex- change interaction h∗, yielding crossover between weak and strong helical phase, and vanishes for too high val- ues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' On the other hand, considering the reliance of SDE on intrinsic system parameters, search of novel mechanisms may open new rout towards the observation of ideal SDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' In a broader sense, SDE is a manifestation of the in- terplay between superconductivity and spatial inversion asymmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Apart from its realization via MCA induced by time-reversal symmetry breaking, it could also be real- ized via shift currents induced by nontrivial Berry phase in a time-reversal symmetric systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Furthermore, for JJs, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Davydova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' [40] recently proposed that finite- momentum Cooper pairing, which elucidates the origin of SDE, can also be achieved without relying on SOI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' 20 Similar to the gate-controllability of Fermi level and thus the tunability of SDE strength [31], it would be intriguing to understand electric field-effects on the intrinsic properties of a superconducting structure, switching of SDE, and its utilization for dissipationless logic/memory applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' For instance, from the ma- terial aspect, SOI, critical current, and pair-breaking are the most important intrinsic properties directly impact- ing the SDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Antisymmetric SOI, Rashba and Zeeman SOI, and thus the corresponding spin-splitting can be tuned via electric field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Superconducting pair-breaking shows strong dependence on the strength and the fre- quency/wavelength of electric field [179].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Similarly, it is shown that a gate tunable critical current in a NbN micro- and nano superconducting bridges [180] can be enhanced up to 30%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Electric field tunability of su- perconducting properties has recently been discussed for various ionic-gated superconducting materials, includ- ing cuprates, iron-based SCs, and honeycomb structures such as transition-metal dichalcogenides and bilayer SCs [181, 182].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' For the device prospects, it would be intrigu- ing to replicate magnetic field (or intrinsic magnetiza- tion) driven switching of SDE with electric field driven switching via electrical control of magnetization orienta- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Electric field driven switching of SDE may also be realized by devising reversible SDE via electric switch of ferroelectricity [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Furthermore, gate-controlled bar- rier transparency in Rashba semiconductor based JJ (Al/InAs/Al) [159] and the gate-controlled asymmetry of highly skewed CPR in topological insulator (BiSbTeSe2) based JJ [183] demonstrate potential rout of controlling SDE in gate-controlled JJs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' The plausible electric field controllability of SDE and the intertwining between band topology and superconductivity may allow searching new mecha- nisms/functionalities [184–187] of topological quantum materials for steering the engineering of low-power and low-dimensional topological superconducting technolo- gies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' We hope this article may provide a route to un- derstand/achieve the optimal performance of SDE and its utilization for superconducting logic/memory device applications.' metadata={'source': 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M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Edmonds, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Culcer, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Nadeem, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Wang, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Medhekar, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Yin, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' Cole, Proposal for a negative capacitance topological quantum field- effect transistor, in 2021 IEEE International Electron Devices Meeting (IEDM) (2021) pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content=' 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='1–38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFRT4oBgHgl3EQfcjcx/content/2301.13564v1.pdf'} diff --git a/rtAzT4oBgHgl3EQf6P4r/content/tmp_files/load_file.txt b/rtAzT4oBgHgl3EQf6P4r/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e50371b3fadbe3dc64100ae8ce58e0276e6cb874 --- /dev/null +++ b/rtAzT4oBgHgl3EQf6P4r/content/tmp_files/load_file.txt @@ -0,0 +1,1324 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf,len=1323 +page_content='Hypotheses Tree Building for One-Shot Temporal Sentence Localization Daizong Liu1,2, Xiang Fang3, Pan Zhou1*, Xing Di4, Weining Lu5, Yu Cheng6 1Hubei Key Laboratory of Distributed System Security, Hubei Engineering Research Center on Big Data Security, School of Cyber Science and Engineering, Huazhong University of Science and Technology 2Peking University 3Nanyang Technological University 4ProtagoLabs Inc 5Tsinghua University 6Microsoft Research dzliu@hust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='cn, xfang9508@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='com, panzhou@hust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='cn, xing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='di@protagolabs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='com, luwn@tsinghua.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='cn, yu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='cheng@microsoft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='com Abstract Given an untrimmed video, temporal sentence localization (TSL) aims to localize a specific segment according to a given sentence query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Though respectable works have made decent achievements in this task, they severely rely on dense video frame annotations, which require a tremendous amount of hu- man effort to collect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' In this paper, we target another more practical and challenging setting: one-shot temporal sentence localization (one-shot TSL), which learns to retrieve the query information among the entire video with only one an- notated frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Particularly, we propose an effective and novel tree-structure baseline for one-shot TSL, called Multiple Hy- potheses Segment Tree (MHST), to capture the query-aware discriminative frame-wise information under the insufficient annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Each video frame is taken as the leaf-node, and the adjacent frames sharing the same visual-linguistic seman- tics will be merged into the upper non-leaf node for tree build- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' At last, each root node is an individual segment hypoth- esis containing the consecutive frames of its leaf-nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Dur- ing the tree construction, we also introduce a pruning strat- egy to eliminate the interference of query-irrelevant nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' With our designed self-supervised loss functions, our MHST is able to generate high-quality segment hypotheses for rank- ing and selection with the query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Experiments on two chal- lenging datasets demonstrate that MHST achieves competi- tive performance compared to existing methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Introduction Temporal sentence localization (TSL) (Anne Hendricks et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' 2017) has drawn increasing atten- tion in recent years, which aims to retrieve a temporal video segment that semantically corresponds to a given sentence query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' An illustrative example of TSL is shown in Figure 1 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Clearly, this task involves both computer vision and nat- ural language processing techniques for multi-modal encod- ing and cross-modal reasoning, to correctly locate the seg- ment boundaries from an untrimmed video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Most previous TSL works (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' 2018b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Ge et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Yuan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' 2019a, 2020a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' 2020a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Yuan, Mei, and Zhu 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' 2020b,a, 2021a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Liu, Qu, and Zhou 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' 2022d,b) pro- posed for this task are under fully-supervised setting, where Corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Copyright © 2023, Association for the Advancement of Artificial Intelligence (www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='aaai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='org).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' All rights reserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' (a) An illustration of the temporal sentence localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Sentence Query: We see the man in khakis stumble and fail repeatedly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Ground Truth | | 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='05s 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='87s Sentence Query: We see the man in khakis stumble and fail repeatedly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Ground Truth | | 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='05s 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='87s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' One-shot Label .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Tree Building Process Segment Hypothesis Segment Hypothesis Negative Proposal Positive Proposal (b) An example of our proposed tree building process for segment generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Positive Proposal Segment Hypothesis Ranking and Selection Query-Guided Figure 1: (a) The example of the temporal sentence localiza- tion (TSL) task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' (b) Illustration of the one-shot TSL setting and our proposed tree-structure method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' each frame is manually labeled as the query-relevant or query-irrelevant frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' In spite of their great advances, these methods severely rely on abundant video-query annotations, which is labor-intensive and time-consuming to collect in real-world scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' To alleviate this problem, some recent works explore a weakly-supervised setting (Mithun, Paul, and Roy-Chowdhury 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' 2020c;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' 2022c;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' 2020b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Song et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' 2020) with only the video-query correspondence rather than the dense frame annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' However, their performances are validated to be less satisfied with such weak supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' In this paper, we make the first attempt to explore whether a TSL model can be learned with a limited frame annota- tion budget rather than previous full annotation or no an- notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Considering that humans have the ability to gen- eralize and identify the query-related semantics from more frames once they comprehend a paired single frame and the sentence query, we tackle a more practical and challenging scenario for TSL task, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=', one-shot TSL, which only has the label of one query-relevant frame in each video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' As for one-shot TSL, we find that previous supervised TSL meth- ods can not be directly applicable to this novel setting due to the following limitations: 1) Firstly, most existing works generally pre-define multiple segment proposals and utilize the dense annotations to score these proposals for ranking arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='01871v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='CV] 5 Jan 2023 EVAEand selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' However, one-shot TSL has only one frame la- bel in each video, lacking sufficient knowledge to build and score the segment proposals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' 2) Secondly, there are many unlabeled frames that may be either irrelevant to the query or the labeled frame within each video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' These frames may bring confounding between the frame label and irrelevant visual features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Therefore, how to discriminate their frame- wise representations for precise segment boundary estima- tion is also a challenging issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' 3) Thirdly, with limited su- pervision signals, traditional supervised loss functions are ineffective enough to train the one-shot TSL models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Thus, an appropriate framework and the training strategy should be well-designed for this one-shot setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' To this end, we propose the first and novel baseline model for one-shot TSL task, called Multiple Hypotheses Seg- ment Tree (MHST), which adopts a tree structure to gen- erate learnable segment hypothesis by merging the adjacent frames sharing the same visual-linguistic semantics with the query and the labeled frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' As shown in Figure 1 (b), the tree building process is taken as the segment hypothesis con- struction process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Specifically, we start from treating video frames as the initial leaf-nodes, and then merge the adja- cent nodes to the upper non-leaf nodes based on the visual relevance of themselves and their linguistic relevance with the query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' During the tree building, we further introduce a tree pruning strategy to selectively weaken the impact of the query-irrelevant nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' The final hypotheses tree will have several root nodes, which represent corresponding segment hypotheses constructed by the frames of its contained leaf- nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' At last, we take the segments containing the labeled frame as positive samples and the others as the negative ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' We devise several self-supervised training losses to jointly learn discriminative frame-wise representations for accurately segment hypotheses scoring and selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Dur- ing the inference stage, we directly choose the segment hy- pothesis with the highest confidence score as the final pre- diction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' To sum up, our main contributions are as follows: To the best of our knowledge, this is the first work to propose and address temporal sentence localization in a novel one-shot setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Specifically, we devise a new tree- structure framework for one-shot TSL, called MHST, to construct query-related segment hypothesis based on the solely one labeled frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' As for the hypotheses tree building, we propose to merge the adjacent nodes into their upper node based on both visual and linguistic relevances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' We also introduce a tree pruning strategy to filter out the query-irrelevant nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' During the training, we apply the self-supervised losses to learn the model under limited annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' We conduct comprehensive experiments on two chal- lenging datasets (ActivityNet Captions and Charades- STA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' The results demonstrate the effectiveness of our proposed method, where MHST achieves decent results and outperforms most fully-supervised methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Related Work Temporal sentence localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Most of the existing TSL methods refer to fully-supervised setting where all video- query pairs are annotated in details, including correspond- ing segment boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Therefore, the main challenge in such setting is how to align multi-modal features well to pre- dict precise boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Some works (Liu, Qu, and Hu 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Liu and Hu 2022b,a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' 2018b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Ge et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Guo, Liu, and Zhou 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' 2019a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' 2021b, 2022a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Fang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' 2021, 2022b,a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' 2022e) integrate sentence information with each fine-grained video clip unit, and predict the scores of candi- date segments by gradually merging the fusion feature se- quence over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Although these methods achieve good performances, they severely rely on the quality of the seg- ment proposals and are time-consuming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Without using pro- posals, some latest methods (Nan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' 2020a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' 2020a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Yuan, Mei, and Zhu 2019) are pro- posed to leverage the interaction between video and sentence to directly predict the starting and ending frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' However, the above methods heavily rely on the datasets that require numerous manually labelled annotations for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' To ease the human labelling efforts, several recent works (Mithun, Paul, and Roy-Chowdhury 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' 2020b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Song et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' 2020c) consider a weakly-supervised setting which only access the information of matched video-query pairs without accurate segment boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' However, their performance is less sat- isfied with such weak supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Considering that we are more likely to have a limited annotation budget rather than full annotation or no annotation in practice, in this paper, we introduce a new practical setting for TSL task, called one- shot TSL, with solely one labeled query-relevant frame in each video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Multiple hypotheses construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Multiple hypotheses strategies are firstly used in the field of object tracking (Blackman and Popoli 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Blackman 2004) in videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Their widely used hypotheses tracking (Cox and Hingo- rani 1996) algorithm originally evaluates its usefulness in the context of visual tracking and motion correspondence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' To further improve the tracking quality, multiple hypotheses tracking in (Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' 2015) propose a tracking tree with a scoring function to prune the hypothesis space efficiently and accurately which is suited to current visual tracking con- text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' In our method, we adapt the multiple hypotheses strat- egy to the video segment construction scenario, where prop- agation between the adjacent frames is newly determined by their visual-linguistic relevance instead of the unreliable appearance similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' We build such multiple hypotheses tree to generate query-related segment proposals within each video by grouping the query-aware semantically closed ad- jacent frames, and measure their scores for selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Differ- ent from (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' 2022), we propose newly designed self- supervised losses for TSL task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' The Proposed Model Overview Problem definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Given an untrimmed video V = {vi}Nv i=1 and the natural language query Q = {qi}Nq i=1, where Nv and Nq are the number of frames and words, the tem- poral sentence localization (TSL) task aims to localize the Sentence Query: We see the man in khakis stumble and fail repeatedly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Video Encoder Labeled Frame Query Encoder Merge or not ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Multi-step Query-Guided .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Pruning Pruning Selection Segment Hypothesis Ranking Loss Inter- Constraint Intra- Constraint | | Preparation | | | Tree Building Pruning & Selection Supervision Segment Hypothesis Scoring Figure 2: An overview of the proposed architecture for the one-shot TSL task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Given the video-query input, we first take each video frame as leaf-node, and then iteratively build the hypotheses segment tree by merging the nodes sharing the same visual-linguistic semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' During the tree building, we also employ a tree pruning strategy to remove and down-weight the query-irrelevant nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' After that, we take the final tree root nodes as the segment hypotheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' At last, we utilize several self-supervised losses to score and rank these segments for learning more discriminative representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' query-described activity segment from the video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Particu- larly, in one-shot TSL setting, there is solely one matched frame annotated with the query in video V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Therefore, it is more challenging than previous TSL works to predict the accurate segment with the limited supervision signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' In this section, considering the few supervision in one-shot TSL task, we propose a novel pipeline named Multiple Hypotheses Segment Tree (MHST) to construct multiple segment hypotheses that are visual-related and linguistic-related to the labeled frame and the query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Specifi- cally, as shown in Figure 2, MHST contains four main steps: Firstly, given a video-query pair, we extract frame-wise fea- tures via a video encoder and extract sentence-level features via a query encoder for preparation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Secondly, we build the hypotheses tree to discriminate frame-wise information for segment construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' At the beginning, each video frame is taken as a leaf-node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Then, the adjacent nodes will be it- eratively merged into an upper node based on their visual- relevance and the query-relevance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Thirdly, to avoid the tree explosion and the negative impact of unnecessary nodes, we further introduce a tree pruning strategy to remove and down-weight the nodes conditioned on the query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' At last, the final generated root nodes are the segment hypotheses constructed by the frames of their contained leaf-nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' We develop a scoring head to rank these segments and design two self-supervised losses to train the whole hypotheses tree for discriminating the frame-wise representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Preparation Before building the hypotheses tree, we first extract both video and query features for preparation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Video encoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Following previous works (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' 2019b, 2020a), given the video V , we first extract its frame- wise features by a pre-trained C3D network (Tran et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' 2015), and then employ a multi-head self-attention (Vaswani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' 2017) module to capture the long-range dependencies among video frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' We denote the extracted video features as V = {vi}Nv i=1 ∈ RNv×d, where d is feature dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Query encoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Given the query Q, we first utilize the Glove (Pennington, Socher, and Manning 2014) model to embed each word into dense vector, and then employ Bi- GRU (Chung et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' 2014) to encode its sequential informa- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' The final sentence-level feature q ∈ Rd can be obtained by concatenating the last hidden unit outputs in Bi-GRU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Hypotheses Tree Building Here, we will illustrate how to construct the hypotheses tree for generating the possible query-related segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Given the frame features {vi}Nv i=1, we treat each frame as the initial leaf-node, and iteratively merge the adjacent frames as the upper non-leaf node if they share the same visual-linguistic semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' We repeat this merging process until there is none matched adjacent nodes and take the final node as the root node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Therefore, the leaf-nodes contained in the root node are the consecutive frames for constructing the possi- ble query-related segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' To generate multiple root nodes in once tree building process, we try to measure and merge each pair of adjacent leaf-nodes at the beginning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Specifically, we take two leaf-nodes vi, vj as an example to illustrate the node-wise merging process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' To determine whether the node pair vi, vj should be merged into the same segment (or upper node), we calculate both the linguistic rel- evance between each node pair with the language query and the visual relevance between the nodes as the judgments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' For the linguistic relevance calculation, we first compute the query-guided information relation between the query feature and each frame feature as follows: rqv vi = sigmoid((W1vi)(W2q)⊤), (1) rqv vj = sigmoid((W1vj)(W2q)⊤), (2) where W1, W2 are the projection matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Then, we mea- sure the query-aware difference between the node pair via rqv = |rqv vi − rqv vj |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' For the visual relevance calculation, we directly compute the appearance relation between two frame features via a cosine similarity function as follows: rvv = viv⊤ j ||vi||2||vj||2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' (3) Therefore, we can utilize both linguistic and visual rele- vances as scores to evaluate the semantic difference between the adjacent node pair vi, vj by: r = λ1rqv + λ2rvv, (4) where λ1, λ2 are used to control the balance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' With the relevant difference value r, we can rank all node pairs in each step and pick out the top α, which is a hyper- parameter to be set as a percentage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' For the top α node pair vi, vj, we merge them into a new ancestor non-leaf node: vnew i,j = W3vi + W3vj + b, (5) where W3, b are the learnable weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' During the hypothe- ses tree construction, we repeat this node-merging process until there is no-relevant node pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Hypotheses Tree Pruning During the building process of the hypotheses tree, not all the non-leaf nodes in the tree branches are closely related to the sentence query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Therefore, we have to take a prun- ing step to remove these non-leaf nodes and their descen- dant non-leaf nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' In other words, we need to determine the mostly likely query-relevant non-leaf nodes for accu- rately constructing most query-relevant video segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' To this end, we first calculate the semantic relevance r for each non-lead node and the query following the Equation (4) and then remove the nodes that are with semantic relevance less than a hyperparameter τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' To reserve enough frame nodes (leaf-nodes) for other hypotheses construction, we do not remove their leaf-nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Instead, we down-weight the con- tribution of these lead-nodes to the final segment as follows: (vi)′ = λτrqv vi vi, (6) where rqv vi is the linguistic relevance in Equation (1) to mea- sure the similarity between frame node and query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' In partic- ular, we apply this down-weighting process to all leaf-nodes, where λτ of the lead-nodes in the removed non-leaf nodes are set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='5, λτ of other leaf-nodes are set to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' We take a L-scan pruning method to prune the disturbing non-leaf nodes gradually instead of pruning the whole tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' In particular, for every L step, we apply the pruning pro- cess to remove the query-irrelevant non-leaf nodes in current step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Then, we track and remove their descendant non-leaf nodes in previous L − 1 step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Finally, we down-weight their leaf-nodes via Equation (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Segment Hypothesis Selection and Training After the hypotheses tree building and pruning, we can get the final segment hypotheses tree in which each root node represents a segment hypothesis containing the consecutive frames of its descendant leaf-nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' We directly select the root nodes that contain the ground-truth annotated leaf-node as the positive hypotheses, and denote their corresponding segment proposals as P = {pi}Np i=1 where Np is the pro- posal number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' To generate confidence scores S = {si}Np i=1 for these segment proposals, we feed their merged node- wise features (obtained by Equation (5)) into a fully con- nected layer with further sigmoid function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Then, we apply the selection algorithm that considers the confidence score to select the top-K proposals P K = {pK i }K i=1 and give their corresponding confidence scores as SK = {sK i }K i=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' In order to learn and correct the above confidence scores, we apply a ranking loss based on a reward policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Specifically, we define the reward RK i for the proposal pK i with a reward function to encourage proposals with higher linguistic relevance rqv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Then the strategy of policy gradient is used to correct the scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Note that the confidence scores are normalized by a softmax layer, which is an extremely important operation to highlight the semantically matching proposals and weaken the mismatched ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' The ranking loss can be formulated by: Lrank = K � i=1 −RK i log( exp(sK i ) �K j=1 exp(sK j ) ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' (7) Beside, to further assist the node-wise representation learning during the hypotheses tree construction, we addi- tionally develop an inter-constrain and an intra-constrain losses to train the tree-structure framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' As for the inter- constrain loss,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' we take the matched video-sentence pairs as positive samples and the unmatched video-sentence pairs as negative samples,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' and use the weighted binary cross-entropy loss to supervise the query-relevance rqv of the top-K hy- pothesis segments as: Linter = J � j=1 K � i=1 (−yjlog(rqv i ) − (1 − yj)log(1 − rqv i ))),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' (8) where J is the number of all video-sentence pairs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' yj the label of the j-th sample that equals 1 for the matched video- sentence pairs and 0 for the unmatched video-sentence pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' As for the intra-constrain loss, within the same video, we take the top-K segment proposals as positive samples and define the random segments unoverlaped with the top-K segment proposals as negative samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' We adopt a hinge loss to supervise their semantics as: Lintra = K � i=1 max(0, β − rqv i + ¯rqv i ), (9) where ¯rqv i denotes the query-relevance of negative samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' During the training, we utilize three fixed weights to balance the values of above three losses to joint learn the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Methods Type ActivityNet Captions R@1 R@1 R@5 R@5 IoU=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='3 IoU=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='5 IoU=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='3 IoU=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='5 Random FS 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='64 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='73 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='78 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='49 VSA-RNN FS 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='28 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='43 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='84 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='52 VSA-STV FS 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='71 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='01 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='05 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='62 CTRL FS 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='01 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='17 TGN FS 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='81 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='93 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='56 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='20 2D-TAN FS 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='45 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='51 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='53 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='13 IVG-DCL FS 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='22 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='84 DRN FS 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='45 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='97 CTF WS 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='30 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='60 ICVC WS 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='62 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='52 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='92 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='61 MARN WS 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='01 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='95 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='02 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='49 SCN WS 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='23 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='22 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='45 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='69 LCNet WS 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='49 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='33 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='51 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='66 CCL WS 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='12 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='07 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='36 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='29 VCA WS 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='45 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='00 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='79 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='83 WSTAN WS 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='45 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='01 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='38 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='42 CRM WS 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='26 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='19 MHST OS 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='34 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='68 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='92 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='75 Table 1: Performance comparisons for temporal sentence lo- calization on ActivityNet Captions dataset, where FS: fully- supervised setting, WS: weakly-supervised setting, and OS: one-shot setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' During the testing, we directly choose the root node from the constructed tree with the maximum confidence score, and then generate corresponding segment as the final prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Experiments Dataset ActivityNet Captions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' This dataset is built from Activi- tyNet v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='3 dataset (Caba Heilbron et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' 2015) for dense video captioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' It contains 20000 YouTube videos with 100000 queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' On average, videos are about 120 seconds and queries are about 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='5 words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' We follow the public split of the dataset that contains a training set and two validation sets val 1 and val 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Following common settings, we use val 1 as our validation set and use val 2 as our testing sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Charades-STA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' This dataset is built from the Charades (Sigurdsson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' 2016) dataset and transformed into tem- poral sentence localization task by (Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' It con- tains 16128 video-sentence pairs with 12408 pairs used for training and 3720 for testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' The videos are about 30 sec- onds on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' The annotations are generated by sentence decomposition and keyword matching with manually check.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Experimental Settings Evaluation metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' We adopt “R@n, IoU=m” as the eval- uation metrics, following previous works (Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' 2018a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' 2020a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' The “R@n, IoU=m” denotes the percentage of language queries having at least one result whose Intersection over Union (IoU) with ground truth is larger than m in top-n retrieved segment hypotheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Implementation details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' In order to make a fair compar- ison with previous works, we utilize the pre-trained C3D Methods Type Charades-STA R@1 R@1 R@5 R@5 IoU=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='5 IoU=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='7 IoU=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='5 IoU=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='7 Random FS 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='51 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='03 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='12 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='06 VSA-RNN FS 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='50 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='32 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='43 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='21 VSA-STV FS 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='91 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='81 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='89 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='58 CTRL FS 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='62 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='89 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='92 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='52 2D-TAN FS 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='81 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='25 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='33 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='15 DRN FS 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='40 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='40 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='01 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='38 IVG-DCL FS 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='24 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='88 SCN WS 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='58 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='97 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='80 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='87 CTF WS 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='30 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='90 WSTAN WS 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='35 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='28 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='13 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='53 ICVC WS 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='02 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='53 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='53 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='91 MARN WS 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='94 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='18 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='00 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='40 CCL WS 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='21 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='68 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='50 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='87 CRM WS 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='76 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='37 VCA WS 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='13 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='57 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='75 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='75 LCNet WS 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='19 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='17 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='56 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='24 MHST OS 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='62 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='48 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='29 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='50 Table 2: Performance comparisons for temporal sentence localization on Charades-STA dataset, where FS: fully- supervised setting, WS: weakly-supervised setting, and OS: one-shot setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' (Tran et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' 2015) model to extract video features and em- ploy the Glove model (Pennington, Socher, and Manning 2014) to obtain word embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' As some videos are too long, we set the length of video feature sequences to 128 for Charades-STA and 256 for ActivityNet Captions, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' We fix the query length to 10 in Charades-STA and 20 in ActivityNet Captions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' The feature dimension d is set to 512.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' For the hyper-parameters, we set the percentage α to 60%, and set the pruning threshold τ as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' The bal- anced weights λ1, λ2 are set to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='0,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' The step L in L-scan pruning is set to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' During training, the learning rate is by default 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='00005, and decays by a factor of 10 for every 35 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' The batch size is 1 and the maximum training epoch is 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' For one-shot setting, we randomly selection one la- beled frame from the ground-truth as the annotation for each video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' All the experiments are implemented by PyTorch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Comparison with State-of-the-Art Compared methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' We compare the proposed MHST with state-of-the-art TSL methods on two datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' These meth- ods are grouped into two categories: 1) Fully-supervised setting: Random Selection (Gidaris, Singh, and Komodakis 2018), VSA-RNN and VSA-STV (Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' 2017), CTRL (Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' 2017), TGN (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' 2018), 2D-TAN (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' 2020b), IVG-DCL (Nan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' 2021), DRN (Zeng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' 2) Weakly-supervised setting: CTF (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' 2020b), ICVC (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' 2022), MARN (Song et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' 2020), SCN (Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' 2020), LCNet (Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' 2021), CCL (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' 2020c), VCA (Wang, Chen, and Jiang 2021), WSTAN (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' 2021), CRM (Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Comparison and analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' As shown in Table 1 and 2, we compare our method with existing works on both Ac- tivityNet Captions and Charades-STA datasets, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' From the tables, we have the following findings: Segment Tree Self- R@1 R@1 R@5 R@5 Tree Pruning supervision IoU=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='3 IoU=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='5 IoU=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='3 IoU=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='5 × × × 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='97 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='95 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='82 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='47 × × ✓ 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='64 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='71 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='38 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='99 ✓ × × 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='84 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='76 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='17 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='93 ✓ × ✓ 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='27 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='05 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='53 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='28 ✓ ✓ × 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='16 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='22 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='39 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='41 ✓ ✓ ✓ 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='34 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='68 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='92 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='75 Table 3: Main ablation study on ActivityNet Captions dataset, where we remove each key individual component to investigate its effectiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Firstly, compared to the weakly-supervised methods, our method outperforms them by a large margin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Specif- ically, on ActivityNet Captions dataset, compared to the SOTA method CRM, we bring the improvement of 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='08 and 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='49 in terms of R@1, IoU=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='3 and R@1, IoU=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' On Charades-STA dataset, we also outperform the SOTA method LCNet by 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='43, 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='31, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='73 and 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='26 on all metrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Note that, our one-shot setting costs similar human labors as the weakly-supervised setting since the latter also requests the annotators to determine the matched video-query pair with at least one query- relevant frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' However, our performance is much better than the weakly-supervised one, demonstrating the effec- tiveness of the proposed tree-structure framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Secondly, compared to the fully-supervised methods, our one-shot setting has much less annotations (one labeled frame vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' dense video annotations) for model training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' However, as shown in the tables, our method achieves very competitive performances, which are even better than the previous SOTA methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Specifically, on Ac- tivityNet Captions dataset, compared to DRN, we out- perform it by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='23 in R@1, IoU=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' On Charades-STA dataset, compared to IVG-DCL, we bring improvement of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='60 in R@1, IoU=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' This demonstrates that our framework is robust to the weak annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Although we have much less annotations than the fully-supervised works, our well-designed tree-structure framework can group the query- and labeled frame-related adjacent frames into the same segment under the supervision of the proposed effective self-supervised losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Ablation Study In this section, we conduct ablation study to validate the ef- fectiveness of each components in our methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' All experi- ments are conducted on ActivityNet Captions dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Main ablation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' To analyze how each model component con- tributes to the task, we perform main ablation study as shown in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' We start from the baseline model which does not rely on both the tree-structure framework and the self- supervised strategy to address the one-shot TSL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Specifi- cally, this baseline model directly generates multiple coarse segment proposals like previous supervised methods and then utilizes the rank loss in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' (7) for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' It shows that this baseline achieves relatively worse performance com- pared to most weakly- and fully-supervised methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' By applying the self-supervised constraint of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' (8)(9) to the Methods Type R@1 R@1 R@5 R@5 IoU=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='3 IoU=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='5 IoU=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='3 IoU=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='5 2D-TAN FS 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='45 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='51 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='53 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='13 2D-TAN* OS 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='76 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='29 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='88 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='02 WSTAN WS 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='45 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='01 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='38 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='42 WSTAN* OS 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='69 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='17 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='93 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='58 MHST OS 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='34 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='68 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='92 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='75 Table 4: Applying the one-shot setting to both fully- and weakly-supervised methods on ActivityNet Captions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Visual Linguistic R@1 R@1 R@5 R@5 Similarity Similarity IoU=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='3 IoU=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='5 IoU=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='3 IoU=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='5 ✓ × 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='96 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='17 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='58 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='42 × ✓ 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='10 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='34 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='66 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='25 ✓ ✓ 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='34 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='68 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='92 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='75 Table 5: Effect of visual- and linguistic similarities for node merging during the tree building.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Hyperparameters R@1 R@1 R@5 R@5 IoU=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='3 IoU=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='5 IoU=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='3 IoU=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='5 α = 50% 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='28 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='39 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='56 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='31 α = 60% 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='34 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='68 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='92 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='75 α = 70% 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='32 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='70 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='84 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='69 τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='6 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='47 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='82 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='83 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='95 τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='7 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='34 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='68 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='92 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='75 τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='8 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='96 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='33 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='60 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='41 L = 1 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='58 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='01 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='20 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='12 L = 3 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='34 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='68 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='92 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='75 L = 5 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='26 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='75 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='07 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='83 Table 6: Effect of different hyperparameters for node merg- ing during and tree pruning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' baseline, the performance improves a lot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Applying the tree- structure framework brings the largest improvement since our well-designed merging strategy helps to distinguish the ambiguous adjacent frames for constructing more accurate segments under the limited supervision signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Moreover, the self-supervised constraint seems to be more robust to the tree-structure framework due to the high quality of the segment hypotheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' The tree pruning strategy also helps the model to reduce the negative influence of the query- irrelevant frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Overall, the whole tree-structure frame- work with both pruning strategy and self-supervised con- straints achieves the best results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Effect of our one-shot pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' To fairly compare with the existing fully- and weakly-supervised methods, we re- implement some works with our one-shot setting as shown in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Specifically, for 2D-TAN, we replace the loss of 2D map with our rank loss in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' (7) for handling the weak labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' For WSTAN, we also add the rank loss for enriching the model contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' We keep all the other settings being the same as their original works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' From this table, we can find that, although one-shot setting brings additional one frame label to the weakly setting, the improvement of WSTAN* is still limited since it lacks sufficient self-supervision for dis- Query: A bull knocks a man down onto the ground.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Ground Truth | | 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='61s 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='45s MHST | 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='58s | 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='49s Query: The person jumps up on a skateboard and grinds a rail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Ground Truth | | 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='91s 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='70s MHST | 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='92s | 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='65s Query: The person puts down the bag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Ground Truth | | 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='40s 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='20s MHST | 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='40s | 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='20s Query: We see the man in khakis stumble and fail repeatedly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Ground Truth | | 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='05s 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='87s MHST | 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='06s | 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='91s Figure 3: Qualitative results on both ActivityNet Captions and Charades-STA datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' The red rectangle denotes the single labeled frame in each video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Lrank Linter Lintra R@1 R@1 R@5 R@5 IoU=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='3 IoU=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='5 IoU=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='3 IoU=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='5 ✓ × × 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='16 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='22 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='39 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='41 ✓ ✓ × 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='98 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='01 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='27 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='44 ✓ × ✓ 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='46 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='49 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='83 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='02 ✓ ✓ ✓ 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='34 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='68 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='92 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='75 Table 7: Ablation study on the supervision losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' criminating the frame-wise representations and cannot gen- erate accurate segment by measuring both visual-linguistic relevances like us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Moreover, the performance of 2D-TAN* degenerates a lot, demonstrating that previous fully setting is not robust to the weak label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Overall, it indicates that the one-shot setting is worth being investigated, and our pro- posed tree-structure framework is effectiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Effect of the visual-linguistic relevance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' As shown in Ta- ble 5, we investigate the effect of the visual- and linguis- tics relevances for node merging during the tree building process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' It shows that both of them are crucial for segment construction, since the visual relevance helps to determine the visual similar adjacent frames near the labeled frame while the linguistic relevance helps to filter our the query- irrelevant frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' By applying both of them for node merg- ing, our model can achieve the best performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Hyper-parameters of tree building and pruning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' More- over, we investigate the robustness of the proposed model to different hyper-parameters in Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' For the hyperparam- eter α in node merging, we find that the model achieves the best result when α is set to 60%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Smaller α will lead nega- tive node merging while larger α will filter out some positive nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' As for τ in tree pruning, the model achieves the best result when τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Larger τ will filter out positive nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Besides, we also investigate the effect of different number L in the pruning process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' It shows that the model achieves the best result when L = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' However, lager L gets the punish- ment in speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Therefore, we set L = 3 in all experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Effect of the supervision losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' At last, we investigate the effectiveness of the proposed self-supervised losses of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' (8) and (9) in Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Here, the ranking loss Lrank is the baseline for addressing the one-shot TSL setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' It shows that the ranking loss Lrank is well-designed to the one-shot setting, and achieves great performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Moreover, by ap- plying the inter-constraint loss or the intra-constraint loss to the baseline, the model has significant performance by learn- ing more discriminative frame-wise representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Over- all, both two self-constraint losses Linter and Lintra con- tributes a lot to the final performance, and we can achieve the best performance by jointly utilizing them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Qualitative Results As shown in Figure 3, we give the visualization of the local- ization results on both ActivityNet Captions and Charades- STA datasets for examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' In this figure, each video has only one frame (in red rectangle) labeled according to the query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' It shows that our proposed MHST is robust to the one- shot setting and can well predict the accurate query-related segment boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Conclusion In this paper, we introduce a new one-shot setting into the temporal sentence localization task to reduce the labeling cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' To achieve this goal, the model needs to make full use of the solely one labeled frame in each video to retrieve the target segment according to the query semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Consider- ing that the invalid frames unrelated to the query sentence or the labeled frame may bring confounding to the one-shot training process, we design a novel tree-structure framework called Multiple Hypotheses Segment Tree (MHST) to avoid this issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Specifically, the hypotheses tree module merges adjacent frames sharing the similar visual-linguistic seman- tics into a new upper node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Then, by iteratively building the tree and pruning the invalid nodes, we can get the com- plete and query-related root nodes which represent the seg- ment hypotheses constructed by their contained consecutive frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Finally, we develop several self-supervised losses to train the segment tree and predict the confidence scores for each segment hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' During the inference, we directly choose the segment with highest score as the final predic- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Experimental results on two challenging datasets (Ac- tivityNet Captions and Charades-STA) demonstrate that our MHST achieves a competitive performance compared to ex- isting fully- and weakly-supervised methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' This work is supported by National Natural Science Foundation of China (NSFC) under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAzT4oBgHgl3EQf6P4r/content/2301.01871v1.pdf'} +page_content=' 61972448.' 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0000000000000000000000000000000000000000..25b99c36975cc23f2d1461224a12763dedac5e23 --- /dev/null +++ b/vNE5T4oBgHgl3EQfLw6U/content/tmp_files/2301.05476v1.pdf.txt @@ -0,0 +1,2615 @@ +arXiv:2301.05476v1 [math.KT] 13 Jan 2023 +A COMBINATORIAL CHARACTERISATION OF d-KOSZUL AND (D, A)-STACKED +MONOMIAL ALGEBRAS THAT SATISFY (FG) +RUAA JAWAD, NICOLE SNASHALL, AND RACHEL TAILLEFER +ABSTRACT. Condition (Fg) was introduced in [6] to ensure that the theory of support varieties of a +finite dimensional algebra, established by Snashall and Solberg, has some similar properties to that of +a group algebra. In this paper we give some easy to check combinatorial conditions that are equivalent +to (Fg) for monomial d-Koszul algebras. We then extend this to monomial (D, A)-stacked algebras. +We also extend the description of the Yoneda algebra of a d-Koszul algebra in [11] to (D, A)-stacked +monomial algebras. +INTRODUCTION +Let Λ be an indecomposable finite dimensional algebra over a field K. In this Introduction, we +shall assume that K is algebraically closed. +Support varieties for modules over Λ were introduced by Snashall and Solberg in [23], as a +geometric tool to study the representation theory of Λ, using the Hochschild cohomology HH∗(Λ). +It was then proved in [6] that many of the properties of support varieties for group algebras have +analogues in this more general case, providing some finiteness conditions hold. These are now +known as (Fg) and can be expressed in the following way. Let r be the Jacobson radical of Λ and +let E(Λ) = Ext∗ +Λ(Λ/r, Λ/r) be its Yoneda algebra. Then Condition (Fg) states that: +(Fg1) there is a commutative noetherian graded subalgebra H of HH∗(Λ) with H0 = HH0(Λ) +such that +(Fg2) E(Λ) is a finitely generated H-module. +In particular, it was shown in [6] that if Λ satisfies Condition (Fg), then Λ is necessarily Gorenstein, +that the variety of a module is trivial if and only if the module has finite projective dimension, and +periodic modules can be characterised up to projective summands as those whose support variety +is a line. Moreover, the converse of the first result mentioned was proved for monomial algebras +in [4], that is, a Gorenstein monomial algebra satisfies (Fg). +Support varieties for group algebras have been very effective in the study of the representations +of these algebras. Therefore Condition (Fg) has been much studied as it ensures a similarly useful +theory of support varieties for finite dimensional algebras. For instance, Condition (Fg) is invariant +under various constructions, such as derived equivalence or singular equivalence of Morita type, +see [16, 22, 19]. Condition (Fg) has been studied or shown to hold for large families of algebras in +[24, 26, 25, 5] among others, and support varieties have been studied for algebras that satisfy (Fg), +see for instance [8, 20]. +Since Hochschild cohomology is generally very difficult to compute, Condition (Fg) can be diffi- +cult to establish for a given algebra. It is therefore useful to have necessary, sufficient or equivalent +conditions for (Fg) to hold for a given algebra. One such result was proved by Erdmann and +Solberg in [7], where they showed that if (Fg) holds for Λ, then the graded centre Zgr(E(Λ)) of +the Yoneda algebra is a noetherian algebra and E(Λ) is a finitely generated Zgr(E(Λ))-module; +moreover, they proved that this is an equivalence when the algebra Λ is Koszul. For monomial al- +gebras, (Fg) was proved in [4] to be equivalent to the related condition that the A∞-centre Z∞(E(Λ)) +is a noetherian algebra and E(Λ) is a finitely generated Z∞(E(Λ))-module. We note that if Λ has +Date: 16th January 2023. +2020 Mathematics Subject Classification. 16G20, 16E40, 16S37, 16E65, 16E30. +Key words and phrases. d-Koszul, Ext algebra, Hochschild cohomology, finiteness condition, (D, A)-stacked. +Some of these results formed part of the first author’s PhD thesis at the University of Leicester, which was supported +by The Higher Committee For Education Development in Iraq (HCED) Reference D1201116. +1 + +2 +JAWAD, SNASHALL, AND TAILLEFER +finite global dimension then both E(Λ) and HH∗(Λ) are finite dimensional as vector spaces, and +Λ has (Fg). Thus we are particularly interested in algebras of infinite global dimension. +The aim of this paper is to prove that a number of conditions are equivalent to (Fg) for a large +category of algebras, namely finite dimensional d-Koszul, and more generally (D, A)-stacked, +monomial algebras. This is motivated in particular by a result of the first author in her PhD thesis, +who gave a sufficient and not difficult to check condition for d-Koszul monomial algebras to satisfy +(Fg); this result is Theorem 2.7 in this paper. +Berger introduced d-Koszul algebras in [3] as a natural generalisation of Koszul algebras (which +occur as 2-Koszul algebras). They are the algebras such that the n-th projective module in a min- +imal projective resolution of Λ/r as a Λ-module is generated in a specific degree denoted by δ(n) +(with δ(n) = n if d = 2). Moreover, they were characterised in [11] as the algebras Λ that are +d-homogeneous (that is, their ideal of relations can be generated by a set of homogeneous ele- +ments of degree d) and such that E(Λ) is generated in degrees 0, 1 and 2. The (D, A)-stacked +monomial algebras, where D ⩾ 2 and A ⩾ 1 are integers, were introduced by Green and Snashall +in [13], and those of infinite global dimension were characterised by the same authors in [12] as the +monomial algebras such that the n-th projective module in a minimal projective resolution of Λ/r +as a Λ-module is generated in precisely one degree and such that E(Λ) is finitely generated (in +which case E(Λ) is generated in degrees 0, 1, 2 and 3). In particular, when A = 1, a (D, 1)-stacked +monomial algebra is D-Koszul. Thus (D, A)-stacked monomial algebras are natural generalisa- +tions of d-Koszul and indeed Koszul monomial algebras. +In this paper, we consider Condition (Fg) for d-Koszul monomial algebras and more generally +for (D, A)-stacked monomial algebras. We introduce some combinatorial conditions 2.4 and 3.6 +that are easy to check in terms of a minimal set of relations for the algebra Λ, and we prove that +they are equivalent to (Fg). This gives a very practical way of checking whether a monomial d- +Koszul or (D, A)-stacked algebra satisfies (Fg), because it is easy to check that a monomial algebra +is d-Koszul using [11, Theorem 10.2] (recalled in Property 2.1) or (D, A)-stacked using [13, Section +3] (recalled in Property 3.2). +To summarise, if Λ is a finite dimensional monomial K-algebra that is d-Koszul with d ⩾ 2 or +(D, A)-stacked with D ̸= 2A whenever A > 1, then the following conditions are equivalent: +(C1) Λ satisfies (Fg). +(C2) Λ satisfies some combinatorial conditions defined in Conditions 2.4 and 3.6 (this follows +from Theorems 2.8 and 3.10). +(C3) Zgr(E(Λ)) is noetherian and E(Λ) is a finitely generated Zgr(E(Λ))-module (by [7] and +Theorems 2.8 and 3.10). +(C4) E(Λ) is a finitely generated Zgr(E(Λ))-module (again by Theorems 2.8 and 3.10). +(C5) Z∞(E(Λ)) is noetherian and E(Λ) is a finitely generated Z∞(E(Λ))-module (by [4]). +(C6) Λ is Gorenstein (by [4]). +The paper is organised as follows. In Section 1 we give some background on monomial algebras +and the notion of overlaps, as well as on the Yoneda algebra and the Hochschild cohomology of a +monomial algebra. Section 2 is devoted to the proof of the implications (C2)⇒(C1) and (C4)⇒(C2) +for d-Koszul monomial algebras, which completes the equivalence of all the conditions above. +The first implication relies on a presentation of the Hochschild cohomology for (D, A)-stacked +monomial algebras from [13] and the second one uses a description of the Yoneda algebra E(Λ) of +a d-Koszul algebra Λ as a graded subspace of the Koszul dual algebra Λ +! +from [11]. In Section 3, we +extend these results to (D, A)-stacked monomial algebras where D ̸= 2A whenever A > 1. Here +again, we use a description of E(Λ) as a subspace of an analogue Λ +♮ +of the Koszul dual of Λ; this +description is detailed and proved in the Appendix, and is a generalisation of the corresponding +result of [11] to (D, A)-stacked monomial algebras. +General assumptions. Throughout the paper, Λ is an indecomposable finite dimensional algebra +over a field K with char(K) ̸= 2 that is not necessarily algebraically closed. Moreover, we assume +that Λ = KQ/I where Q is a finite quiver (it has a finite number of vertices and arrows) and I is an +admissible ideal in KQ. If Λ = KQ/I is also a monomial algebra then I is generated by a minimal +set ρ of paths (monomials) and Λ is graded by the length of paths; we denote by ℓ(p) the length + +A COMBINATORIAL CHARACTERISATION OF (FG) +3 +of a path p. Note that paths in any algebra given by quiver and relations are written from left to +right. For any j ⩾ 0, we shall denote by Qj the set of paths of length j in Q. +In order to use the results in [13], we shall need to assume that gldim Λ ⩾ 4. However, if Λ is +a monomial algebra with finite global dimension, all the conditions (C1)–(C6) hold for Λ (we note +that Condition (C2) is necessarily empty in this case). Therefore we do not lose any generality in +making this assumption. +1. SOME BACKGROUND ON MONOMIAL ALGEBRAS AND THEIR COHOMOLOGY +1.1. Overlaps +Keeping the above assumptions, let Λ = KQ/I be a monomial algebra so that Λ = ⊕i⩾0Λi is a +graded algebra with the length grading. We denote by r = ⊕i⩾1Λi the radical of Λ. An arrow α +starts at the vertex o(α) and ends at the vertex t(α). If p = α1α2 · · · αn is a path with α1, α2, . . . , αn +in Q1 then o(p) = o(α1) and t(p) = t(αn). +A path p is a prefix of a path q if there is some path p′ such that q = pp′; if an arrow α is a prefix +of q then we say that q begins with α. A path p is a suffix of a path q if there is some path p′ such +that q = p′p; if an arrow α is a suffix of q then we say that q ends with α. +We use the concept of overlaps of [9] and [14] to describe the minimal projective resolution of +Λ0 ∼= Λ/r over Λ, and to describe the minimal projective resolution of Λ over Λe, where Λe is the +enveloping algebra Λop ⊗K Λ of Λ. We recall the relevant definitions here using the notation of +[13]. +Definition 1.1. +(1) A path q overlaps a path p with overlap pu if there are paths u and v such +that pu = vq and 1 ⩽ ℓ(u) < ℓ(q). We illustrate the definition with the following diagram. +✤ +p +✤ +� +v +�✤ +q +✤ +� +u +� +Note that we allow ℓ(v) = 0 here. +(2) A path q properly overlaps a path p with overlap pu if q overlaps p and ℓ(v) ⩾ 1. +(3) A path p has no overlaps with a path q if p does not properly overlap q and q does not +properly overlap p. +We now define sets Rn recursively. Let +R0 += +Q0, the set of vertices of Q +R1 += +Q1, the set of arrows of Q +R2 += +ρ, the minimal generating set for I. +For n ⩾ 3, the construction is as follows. +Definition 1.2. +(1) For n ⩾ 3, we say that R2 ∈ R2 maximally overlaps Rn−1 ∈ Rn−1 with +overlap Rn = Rn−1u if +(a) Rn−1 = Rn−2p for some path p; +(b) R2 overlaps p with overlap pu; +(c) there is no element of R2 which overlaps p with overlap being a proper prefix of pu. +We may also say that Rn is a maximal overlap of R2 ∈ R2 with Rn−1 ∈ Rn−1. +The construction of Rn is illustrated in the following diagram. +✤ +Rn−2 +✤ +✤ +Rn−1 +✤ +� +p +� +✤ +R2 +✤ +� +u +� +(2) For n ⩾ 3, the set Rn is defined to be the set of all overlaps Rn formed in this way. +We also recall from [14] that if Rn +1 p = Rn +2q, for Rn +1, Rn +2 ∈ Rn and paths p, q, then Rn +1 = Rn +2 +and p = q. Any element Rn in Rn may be expressed uniquely as Rn−1 +j +aj and as bkRn−1 +k +for some +Rn−1 +j +, Rn−1 +k +in Rn−1 and paths aj, bk. We say that the elements Rn−1 +j +and Rn−1 +k +occur in Rn. + +4 +JAWAD, SNASHALL, AND TAILLEFER +1.2. The Ext algebra E(Λ) +The Ext algebra E(Λ) is given by E(Λ) = Ext∗ +Λ(Λ0, Λ0) = � +n⩾0 Extn +Λ(Λ0, Λ0) with the Yoneda +product. In the terminology of overlaps, the n-th projective module in a minimal projective Λ- +resolution of Λ0 is � +Rn∈Rn t(Rn)Λ. Then Extn +Λ(Λ0, Λ0) has a basis indexed by Rn and E(Λ) has a +basis indexed by � +n⩾0 Rn (see [14, 9]). We identify Rn +i ∈ Rn with the corresponding element of +Extn +Λ(Λ0, Λ0), that is, with the map � +Rn∈Rn t(Rn)Λ → Λ0 given by +t(Rn)λ �→ +� +t(Rn +i )λ + r +if Rn = Rn +i +0 +otherwise. +1.3. The Hochschild cohomology ring HH∗(Λ) +Let (P∗, ∂∗) be the minimal projective Λe-resolution of Λ from [1]. We write ⊗ for ⊗K through- +out. Then +P n = +� +Rn∈Rn +Λo(Rn) ⊗ t(Rn)Λ. +The maps are given as follows. In odd degrees, if R2n+1 = R2n +j aj = bkR2n +k ∈ R2n+1 then ∂2n+1 : P2n+1 → +P2n is given by +o(R2n+1) ⊗ t(R2n+1) �→ o(R2n +j ) ⊗ aj − bk ⊗ t(R2n +k ) +where the first tensor lies in the summand corresponding to R2n +j +and the second tensor lies in the +summand corresponding to R2n +k . +For even degrees, any element R2n in R2n may be expressed in the form pjR2n−1 +j +qj for some +R2n−1 +j +∈ R2n−1 and paths pj, qj with n ⩾ 1. Let R2n = p1R2n−1 +1 +q1 = · · · = prR2n−1 +r +qr be all +expressions of R2n which contain some element of R2n−1 as a subpath. Then, for R2n ∈ R2n, the +map ∂2n : P2n → P2n−1 is given by +o(R2n) ⊗ t(R2n) �→ +r +∑ +j=1 +pj ⊗ qj +where the tensor pj ⊗ qj lies in the summand of P2n−1 corresponding to R2n−1 +j +. +If not specified, then it will always be clear from the context in which summand of a projective +module our tensors lie. +The Hochschild cohomology ring HH∗(Λ) of Λ is given by +HH∗(Λ) = Ext∗ +Λe(Λ, Λ) = +� +n⩾0 +Extn +Λe(Λ, Λ) +with the Yoneda product. +2. CHARACTERISATIONS OF d-KOSZUL MONOMIAL ALGEBRAS THAT SATISFY (FG) +2.1. Notation and properties of d-Koszul monomial algebras +Let Λ = KQ/I be a monomial algebra, where Q is a finite quiver and I is an admissible ideal +in KQ generated by a minimal set ρ of paths. Recall that the algebra Λ = � +i⩾0 Λi is graded by +the length of paths. We can express Λ as a quotient Λ = TΛ0(Λ1)/I of the tensor algebra, where +Λ/r ∼= Λ0 = KQ0 and Λ1 = KQ1 and I is an ideal generated by a minimal set ρ of monomials. +The algebra Λ0 ∼= K|Q0| is isomorphic to a finite product of copies of the base field K; it is therefore +a semisimple and commutative K-algebra. We denote by ei the idempotent in Λ0 corresponding to +the vertex i. +Let d ⩾ 2 be an integer. We assume that Λ is a d-Koszul algebra, that is, for any minimal +projective right Λ-module resolution of Λ0, the n-th projective module is generated in degree δ(n) +where +δ(n) = +� +n +2 d +if n is even +n−1 +2 d + 1 +if n is odd. +It follows that Λ is d-homogeneous (that is, ρ consists of paths of length d). + +A COMBINATORIAL CHARACTERISATION OF (FG) +5 +The monomial d-Koszul algebras can be characterised as follows. +Property 2.1. [11, Theorem 10.2] A finite dimensional d-homogeneous monomial algebra Λ = KQ/I is +d-Koszul if, and only if, ρ is d-covering, that is, for any paths p, q and r in Q, +(pq ∈ ρ, qr ∈ ρ, ℓ(q) ⩾ 1) ⇒ (all subpaths of pqr of length d are in ρ). +Note that this condition is always satisfied if d = 2; it is indeed well known that all finite +dimensional quadratic monomial algebras are Koszul, see [14] and [18, Corollary 2.4.3]. +Example 1. Let Λ = KQ/I where Q is the quiver +· +γ3 +�❃❃❃❃❃❃❃❃ +· +γ2 +� +· +β +� +γ1 +� +· +α +� +and the ideal I has minimal generating set ρ = {α3, γ1γ2γ3, γ2γ3γ1, γ3γ1γ2}. Then Λ is a 3-Koszul +monomial algebra. +From now on, we assume that Λ = KQ/I is a finite dimensional d-Koszul monomial algebra +with d ⩾ 2. +We have the following consequences of Property 2.1. +Consequence 2.2. [15, Proposition 7.13] Let Rn +i be an element in Rn. Then all subpaths of Rn +i of length +d are in ρ. +Proof. The result is proved by induction. It is clear when n = 2. Moreover, if n = 3, since R3 +i ∈ R3 +is a maximal overlap of two elements in R2, it follows from Property 2.1. +Now let n ⩾ 4 be an integer and take Rn +i ∈ Rn. Then Rn +i is a maximal overlap of R2 +1 ∈ R2 with +Rn−1 +2 +∈ Rn−1 so that Rn +i = Rn−1 +2 +u for some path u, and Rn−1 +2 +is a maximal overlap of R2 +3 ∈ R2 with +Rn−2 +4 +∈ Rn−2 so that Rn−1 +2 += Rn−2 +4 +u′ for some path u′. This can be illustrated as follows: +✤ +Rn +i +✤ +✤ +Rn−2 +4 +✤ +✤ +Rn−1 +2 +✤ +✤ +R2 +3 +✤ +✤ +R2 +1 +✤ +� +u′ +� +� +u � +Moreover, ℓ(u′u) = ℓ(Rn +i ) − ℓ(Rn−2 +4 +) = δ(n) − δ(n − 2) = d so u′u = R2 +1. By induction, every +subpath of Rn−1 +2 +of length d is in ρ. Any other subpath of length d of Rn +i is either u′u = R2 +1 ∈ R2 +or a proper subpath of R2 +3u, therefore it is in ρ by Property 2.1. We have proved the induction +step. +□ +A trail in Q is a path T = α1 · · · αn with n ⩾ 1 such that the arrows αi are all distinct. We say that +the trail is closed when t(αn) = o(α1). A path q is said to lie on the closed trail T if q is a subpath of +Tm for some m ⩾ 1. We say that two trails are distinct if neither lies on the other. +We now have a second consequence of Property 2.1. +Consequence 2.3. [15, Proposition 7.14] Suppose that T = α1 · · · αn is a closed trail in Q and that +d ⩾ n + 1. Then all paths of length d that lie on the closed trail T are in ρ. +Proof. Since Λ is finite dimensional, there is a path R2 ∈ ρ that lies on T. Now, ℓ(R2) = d and +d ⩾ n + 1 so, without loss of generality, we may suppose that R2 = (α1α2 · · · αn)mα1α2 · · · αs for +some 1 ⩽ s ⩽ n with d = nm + s and m ⩾ 1. Let p = (α1α2 · · · αn)m, q = α1α2 · · · αs and r = +(αs+1 · · · αnα1 · · · αs)m. Then pq = R2 = qr and we can apply Property 2.1 so that all subpaths of +pqr of length d are in ρ. Now, any path of length d that lies on the closed trail T is a subpath of pqr +and hence is in ρ. +□ +We now introduce Condition 2.4. Jawad showed in her PhD thesis [15] that this condition is +sufficient for Λ to satisfy (Fg); we give a proof in Theorem 2.7 below. + +6 +JAWAD, SNASHALL, AND TAILLEFER +Condition 2.4. [15, Theorems 7.11 and 7.15] +(1) Let α be a loop in Q1. Then αd ∈ ρ but there is no path in ρ of the form αd−1β or βαd−1 +where β is an arrow that is distinct from α. +(2) Let T = α1 · · · αn be a closed trail in Q with n > 1 and αi ∈ Q1 for all i and such that +ρT := {α1 · · · αd, α2 · · · αdαd+1, . . . , αnα1 · · · αd−1} ⊆ ρ. Then there are no elements in ρ \ ρT +which begin or end with the arrow αi, for all i. +Remark 2.5. If T = α1 · · · αn is a closed trail then the subscript i of αi is taken modulo n within the +range 1 ⩽ i ⩽ n. Thus ρT is the set of all paths of length d that lie on the closed trail T. +Remark 2.6. Suppose that Condition 2.4 is non-empty, that is, there is a loop or a closed trail with +the given properties. Then the description of the projective modules in Section 1.2 using overlaps +shows that Λ0 has infinite projective dimension as a Λ-module, and hence Λ has infinite global +dimension. +2.2. Condition 2.4 is sufficient for Λ to satisfy (Fg) +The proof of Theorem 2.7 uses the description of the Hochschild cohomology ring modulo nil- +potence of a (D, A)-stacked monomial algebra from [13, Theorem 3.4]. We recall the definition of +a (D, A)-stacked monomial algebra in Subsection 3.1. The Hochschild cohomology ring modulo +nilpotence is the quotient HH∗(Λ)/N where N is the ideal of HH∗(Λ) that is generated by the +homogeneous nilpotent elements. It is well-known that HH∗(Λ) is a graded commutative ring, +so, since char(K) ̸= 2, every homogeneous element of odd degree squares to zero. Moreover, N is +the set of all nilpotent elements of HH∗(Λ). Our calculations involving HH∗(Λ) use the minimal +projective Λe-resolution (P∗, ∂∗) of Λ from [1]; see Section 1.3. +Noting that a d-Koszul monomial algebra is a (d, 1)-stacked monomial algebra (see [13]), we +apply [13, Theorem 3.4] in the special case where D = d and A = 1, and this simplifies the +hypotheses. Specifically, if there is a closed path C in Q with CD/A ∈ ρ then Cd ∈ ρ and it is +immediate that C has length 1 and is necessarily a loop. +Theorem 2.7. [15, Theorems 7.11 and 7.15] Let Λ = KQ/I be a finite dimensional d-Koszul monomial +algebra with d ⩾ 2. Assume that Λ satisfies Condition 2.4. Then Λ satisfies (Fg). +Proof. We keep the notation of Condition 2.4. +Let α1, . . . , αu be the loops in the quiver Q, and suppose that αi is a loop at the vertex vi. Since Λ +is a finite dimensional d-Koszul monomial algebra, αd +i is necessarily in the minimal generating set +ρ. By Condition 2.4 (1), for each i = 1, . . . , u, there are no elements in ρ of the form αd−1 +i +β or βαd−1 +i +where β is an arrow that is distinct from αi. +We need to show that there are no overlaps of αd +i with any element of ρ \ {αd +i }. This is immediate +if d = 2, so suppose that d ⩾ 3. If R ∈ ρ \ {αd +i } and R overlaps αd +i , then either R = αs +ib or R = bαs +i +where 1 ⩽ s ⩽ d − 1 and b is a path of length d − s which does not begin (respectively, end) with +the arrow αi. Suppose first that R = αs +ib. Then R overlaps αd +i with overlap of length 2d − s as +follows: +� +αd−s +i +� +✤ +αd +i +✤ +✤ +R +✤ +� +b +� +This is a maximal overlap since αi is not the first arrow of b and thus gives an element R3 +1 ∈ R3. +However, ℓ(R3 +1) = d + 1 since Λ is d-Koszul. Thus 2d − s = d + 1 and so s = d − 1. But then +R = αd−1 +i +b and b is an arrow distinct from αi, which contradicts our hypothesis. The case where +R = bαs +i is similar. So there are no overlaps of αd +i with any element of ρ \ {αd +i }. Moreover, as Λ is a +finite dimensional monomial algebra, it follows that the vertices v1, . . . , vu are distinct. +Let Tu+1, . . . , Tr be the distinct closed trails in Q such that all paths of length d that lie on these +closed trails are contained in ρ. For each i = u + 1, . . . , r, we write Ti = αi,1 · · · αi,mi, where the αi,j +are arrows, and set +ρTi = {αi,1 · · · αi,d, αi,2 · · · αi,d+1, . . . , αi,miαi,1 · · · αi,d−1}. + +A COMBINATORIAL CHARACTERISATION OF (FG) +7 +Then ρTi is contained in ρ. By Condition 2.4 (2), for each closed trail Ti (i = u + 1, . . . , r), there are +no elements in ρ \ ρTi which begin or end with the arrow αi,j, for all j = 1, . . . , mi. So no arrow αi,j +has overlaps with any element in ρ \ ρTi. +For i = u + 1, . . . , r, let Ti,1, . . . , Ti,mi be defined by +Ti,1 = Ti = αi,1αi,2 · · · αi,mi +Ti,2 = αi,2αi,3 · · · αi,miαi,1 +... +Ti,mi = αi,miαi,1 · · · αi,mi−1. +Then the paths Ti,1, . . . , Ti,mi are all of length mi and lie on the closed path Ti. +We now show that Λ satisfies (Fg1). As noted above, Λ is a (d, 1)-stacked monomial algebra. +Moreover, Condition (Fg) is always satisfied if the global dimension of Λ is finite, therefore we +may assume that gldim Λ ⩾ 4. Hence we can apply [13, Theorem 3.4], which gives HH∗(Λ)/N ∼= +K[x1, . . . , xr]/⟨xaxb for a ̸= b⟩, where +• for i = 1, . . . , u, the vertices v1, . . . , vu are distinct and the element xi corresponding to the +loop αi is in degree 2 and is represented by the map P2 −→ Λ where for R2 ∈ R2, +o(R2) ⊗ t(R2) �→ +� +vi +if R2 = αd +i +0 +otherwise +• and for i = u + 1, . . . , r, the element xi corresponding to the closed trail Ti = αi,1 · · · αi,mi +is in degree 2µi such that µi = mi/ gcd(d, mi) and is represented by the map P2µi −→ Λ, +where for R2µi ∈ R2µi, +o(R2µi) ⊗ t(R2µi) �→ +� +o(Ti,k) +if R2µi = Td/ gcd(d,mi) +i,k +for all k = 1, . . . , mi +0 +otherwise. +Let H be the subring of HH∗(Λ) generated by Z(Λ) and {x1, . . . , xr}. Since Z(Λ) = HH0(Λ) +and HH∗(Λ) is graded commutative, it follows that +H = Z(Λ)[x1, . . . , xr]/⟨xaxb for a ̸= b⟩ +and so H is a commutative ring. Moreover, Z(Λ) is finite dimensional so is a commutative Noeth- +erian ring. Thus H is a Noetherian ring (see [21, Corollary 8.11]). Therefore Λ satisfies (Fg1). +The rest of this proof shows that Λ satisfies (Fg2). Following the discussion in Section 1.2, we +identify � +n⩾0 Rn with a basis of E(Λ). The action of a homogeneous element x ∈ HHn(Λ) on +E(Λ) is then given by left multiplication by ∑j Rn +j where the sum is over all j such that x(o(Rn +j ) ⊗ +t(Rn +j )) ̸= 0. Thus if xi ∈ HH2(Λ) corresponds to the loop αi, then the action of xi on E(Λ) is given +by left multiplication by αd +i . And if xi in degree 2µi corresponds to the closed trail Ti, then the +action of xi on E(Λ) is given by left multiplication by ∑mi +k=1 Td/ gcd(d,mi) +i,k +. +Set N = max{3, |x1|, . . . , |xr|, |Q1|}. We show that �N +n=0 Rn is a generating set for E(Λ) as a left +H-module and thus E(Λ) is finitely generated as a left H-module. +Let R ∈ Rn with n > N. Then ℓ(R) = δ(n) ⩾ 2d and we can write R = a1a2 · · · aδ(n) where the +ai are in Q1. From Consequence 2.2, all subpaths of R of length d are in ρ, so we may illustrate R +with the following diagram: +✤ +✤ +a1 +✤ +✤ +a2 · · · ad ✤ ad+1 · · · ✤ +✤ +aδ(n)−d+1 · · · aδ(n) +Now, n > N ⩾ |Q1| so there is some repeated arrow. Choose j, k with k minimal and k ⩾ 1 such +that aj is a repeated arrow, aj, . . . , aj+k−1 are all distinct arrows and aj+k = aj. Write +R = (a1 · · · aj−1)(aj · · · aj+k−1)(ajaj+k+1 · · · aδ(n)). +There are two cases to consider. +Case (1): k = 1. Then aj = aj+1 and so aj is a loop. It follows that +R = (a1 · · · aj−1)(ajaj)(aj+2 · · · aδ(n)). + +8 +JAWAD, SNASHALL, AND TAILLEFER +Suppose first that j ⩽ d − 1. Then j + d − 1 ⩽ δ(n) so from Consequence 2.2, a2 +j aj+2 · · · aj+d−1 +is in ρ. But ad +j ∈ ρ and we have already shown that there are no overlaps of ad +j with any element +of ρ \ {ad +j }. Thus aj = aj+2 = · · · = aj+d−1. Inductively we see that R = (a1 · · · aj−1)aδ(n)−j+1 +j +. +Similarly, a1 · · · aj−1ad−j+1 +j +is in ρ and d − j + 1 ⩾ 2. Again, there are no overlaps of ad +j with any +element of ρ \ {ad +j } so aj = a1 = · · · = aj−1. Thus R = aδ(n) +j +. +Now suppose that j ⩾ d. Then j − d + 1 ⩾ 1, so by Consequence 2.2, aj−d+1 · · · aj−1aj is in ρ. As +there are no overlaps of ad +j with any element of ρ \ {ad +j }, it follows that aj−d+1 = · · · = aj−1 = aj, +and inductively R = aj+1 +j +(aj+2 · · · aδ(n)). Using Consequence 2.2 again, ad−1 +j +aj+2 is in ρ so aj = aj+2. +Inductively, we have R = aδ(n) +j +. +Hence, for all j, +R = aδ(n) +j += +� +(ad +j )(n/2) +if n even +(ad +j )((n−1)/2)aj +if n odd. +Let xi be the generator in H corresponding to the loop aj, so 1 ⩽ i ⩽ u and |xi| = 2. Then xi acts +on E(Λ) as left multiplication by ad +j . Hence +R = +� +(xi)(n/2)o(aj) +if n even +(xi)((n−1)/2)aj +if n odd +with xi ∈ H, o(aj) ∈ R0 and aj ∈ R1, so that o(aj) and aj are in �N +n=0 Rn. +Case (2): k > 1. We note by our choice of j, k that aj · · · aj+k−1 is a closed trail of length k, which +we denote by T. Let ρT be the set of all paths of length d which lie on T. +The first step is to show that ρT is contained in ρ. If d ⩾ k + 1, then this follows from Con- +sequence 2.3. So, suppose that d ⩽ k. Recall that +R = (a1 · · · aj−1)(aj · · · aj+k−1)(ajaj+k+1 · · · aδ(n)). +Then: +ajaj+1 · · · aj+d−1, +aj+1aj+2 · · · aj+d, +... +aj+k−daj+k−d+1 · · · aj+k−1, +aj+k−d+1aj+k−d+2 · · · aj+k−1aj +are all paths of length d which are subpaths of R, and so, by Consequence 2.2, are in ρ. +Now ajaj+1 · · · aj+d−1 overlaps aj+k−d+1aj+k−d+2 · · · aj+k−1aj. So there is an element R2 +1 ∈ ρ such +that R2 +1 maximally overlaps aj+k−d+1aj+k−d+2 · · · aj+k−1aj with maximal overlap of length d + 1. +Then we have that +R2 +1 = aj+k−d+2aj+k−d+3 · · · aj+k−1ajaj+1 +and this maximal overlap is +� +aj+k−d+1aj+k−d+2 · · · aj+k−1aj +� +aj+1 = aj+k−d+1R2 +1. Continuing in this +way, aj+1aj+2 · · · aj+d overlaps R2 +1. So there is an element R2 +2 ∈ ρ such that R2 +2 maximally overlaps +R2 +1 with maximal overlap of length d + 1. So +R2 +2 = aj+k−d+3aj+k−d+4 · · · aj+k−1ajaj+1aj+2 +and this maximal overlap is R2 +1aj+2 = aj+k−d+2R2 +2. Inductively, we see that every path of length d +on the closed trail T is in ρ. Hence ρT is contained in ρ. +It follows from Condition 2.4 (2), that there are no paths in ρ \ ρT which begin or end with any +of the arrows aj, aj+1, . . . , aj+k−1. +Next we show that R can be written in the form R = p1Tqp2, where p1 is a suffix of T and p2 is a +prefix of T. If d = 2 then ajaj+k+1 is a subpath of R of length 2 and hence is in ρ. By Condition 2.4 (2) +ajaj+k+1 must be in ρT and so aj+k+1 = aj+1. Then aj+k+1aj+k+2 = aj+1aj+k+2 and is a subpath of R + +A COMBINATORIAL CHARACTERISATION OF (FG) +9 +of length 2, so we must have that aj+k+2 = aj+2. Inductively, we see that R lies on the closed trail +T. So R = p1Tqp2, where p1 is a suffix of T and p2 is a prefix of T. +So let d ⩾ 3, and suppose first that d ⩽ k. Then aj+k−d+2 · · · aj+k−1ajaj+k+1 is a subpath of R +of length d which begins with the arrow aj+k−d+2 ∈ {aj, aj+1, . . . , aj+k−1}. So, by Consequence 2.2 +and Condition 2.4 (2), this path is in ρT and hence aj+k+1 = aj+1. Inductively, we have aj+k+2 = +aj+2, aj+k+3 = aj+3, . . . . Similarly, aj−1aj · · · aj+d−2 is a subpath of R of length d which ends with +the arrow aj+d−2 ∈ {aj, aj+1, . . . , aj+k−1}. So this path is in ρT and hence aj−1 = aj+k−1. Inductively, +we have aj−2 = aj+k−2, aj−3 = aj+k−3, . . . . So we may write R = p1Tqp2, where p1 is a suffix of T +and p2 is a prefix of T. +Now suppose that d ⩾ k + 1 (with d ⩾ 3). We consider j ⩽ d − 1 and j ⩾ d separately. Let +j ⩽ d − 1. Then j + d < δ(n), so aj+1aj+2 · · · aj+k−1ajaj+k+1 · · · aj+d is a subpath of R of length +d and starts with the arrow aj+1 ∈ {aj, aj+1, . . . , aj+k−1}. +So by Consequence 2.2 and Condi- +tion 2.4 (2), this path is in ρT and hence aj+k+1 = aj+1, aj+k+2 = aj+2, . . . . So, inductively, we +may write R = (a1 · · · aj−1)Tqp2, where p2 is a prefix of T. Now a1a2 · · · aj−1 · · · ad is a subpath +of R of length d and ends with the arrow ad ∈ {aj, aj+1, . . . , aj+k−1}. So by Condition 2.4 (2), this +path is in ρT and hence aj−1 = aj+k−1, aj−2 = aj+k−2, . . . . Thus R = p1Tqp2, where p1 is a suffix +of T and p2 is a prefix of T. Finally, suppose j ⩾ d. Then, we know that aj+k−d · · · aj−1aj · · · aj+k−1 +is a subpath of R of length d and ends with the arrow aj+k−1 ∈ {aj, aj+1, . . . , aj+k−1}. So by Con- +sequence 2.2 and Condition 2.4 (2), this path is in ρT and hence aj−1 = aj+k−1, aj−2 = aj+k−2, . . . . +Also aj+k−d+2 · · · aj−1aj · · · aj+k+1 is a subpath of R of length d and starts with the arrow aj+k−d+2. +But we have just shown that aj+k−d+2 ∈ {aj, aj+1, . . . , aj+k−1}. So again, this path is in ρT and hence +aj+k+1 = aj+1. Inductively, aj+k+2 = aj+2, . . . . Thus R = p1Tqp2, where p1 is a suffix of T and p2 is +a prefix of T. +So, in all cases, R = p1Tqp2, where T = aj · · · aj+k−1, p1 is a suffix of T and p2 is a prefix of T. +Without loss of generality, relabel the trail T and write R = Tqp, where T = a1 · · · ak, p is a prefix +of T, δ(n) = kq + ℓ(p), and we choose ℓ(p) in the range 1 ⩽ ℓ(p) ⩽ k. Note that R has a repeated +arrow so q ⩾ 1, and if q = 1 then ℓ(p) ⩾ 1; moreover if ℓ(p) = k then p = T and R = Tq+1. +Let xi be the generator in H corresponding to this closed trail T, so u + 1 ⩽ i ⩽ r. Let +Ti,1 = T = a1a2 · · · ak; +Ti,2 = a2a3 · · · aka1; +... +Ti,k = aka1 · · · ak−1. +The action of xi on E(Λ) is left multiplication by +Td/ gcd(d,k) +i,1 ++ Td/ gcd(d,k) +i,2 ++ · · · + Td/ gcd(d,k) +i,k +and |xi| = 2k/ gcd(d, k). Consequently, N ⩾ 2k/ gcd(d, k). Now R = Tqp with 1 ⩽ ℓ(p) ⩽ k. Write +q = +d +gcd(d,k)c + w with 0 ⩽ w ⩽ +d +gcd(d,k) − 1. Then +R = +� +Td/ gcd(d,k)�c +(Twp). +Moreover, from the construction of R as a maximal overlap, we see that Twp is also constructed as +a maximal overlap and so corresponds to a basis element of E(Λ). We have ℓ(Twp) = kw + ℓ(p) ⩽ +k +� +d +gcd(d,k) − 1 +� ++ k = kd/ gcd(d, k) = δ(2k/ gcd(d, k)). So Twp corresponds to a basis element of +E(Λ) of degree at most 2k/ gcd(d, k), that is, Twp is in Rm for some m ⩽ N. +Let 2 ⩽ l ⩽ k; we show that Td/ gcd(d,k) +i,l +(Twp) = 0 in E(Λ). We have Ti,l = alal+1 · · · aka1 · · · al−1, +T = a1a2 · · · ak and p = a1a2 · · · aℓ(p) with 1 ⩽ ℓ(p) ⩽ k. If Td/ gcd(d,k) +i,l +(Twp) represents a non-zero +element in E(Λ), then t(al−1) = o(a1) so that a1 · · · al−1 is a closed trail. But l − 1 < k, so this +contradicts the minimality of k. Hence Td/ gcd(d,k) +i,l +(Twp) = 0 in E(Λ) for 2 ⩽ l ⩽ k. + +10 +JAWAD, SNASHALL, AND TAILLEFER +A similar argument also shows that +� +k +∑ +l=1 +Td/ gcd(d,k) +i,l +�c += +k +∑ +l=1 +� +Td/ gcd(d,k) +i,l +�c +. +Thus +R = +� +Td/ gcd(d,k)�c +(Twp) = +k +∑ +l=1 +� +Td/ gcd(d,k) +i,l +�c +(Twp) = +� +k +∑ +l=1 +Td/ gcd(d,k) +i,l +�c +(Twp). +Hence R = xc +i (Twp) with xi in H, and Twp ∈ Rm for some m ⩽ N. +Hence each R ∈ Rn with n > N can be written in the form hr for some h ∈ H and r ∈ �N +n=0 Rn. +It follows that �N +n=0 Rn is a generating set for E(Λ) as a left H-module. Thus Λ satisfies (Fg2) and +we conclude that Λ has (Fg). +□ +Example 2. We return to Example 1. Condition 2.4 is satisfied: the only closed trails that are not +loops are the cycles of length 3 (whose arrows are the γi); for all of these closed trails T, we have +ρT = ρ \ +� +α3� +. Hence by Theorem 2.7 the algebra Λ = KQ/I satisfies (Fg). +2.3. Conditions equivalent to (Fg) for a d-Koszul monomial algebra +Our aim is now to prove the converse, and more precisely, the following theorem. +Theorem 2.8. Let Λ be an indecomposable finite dimensional d-Koszul monomial K-algebra with d ⩾ 2. +Consider the following statements: +(C1) Λ satisfies (Fg); +(C2) Condition 2.4 holds for Λ; +(C3) Zgr(E(Λ)) is noetherian and E(Λ) is a finitely generated Zgr(E(Λ))-module; +(C4) E(Λ) is finitely generated as a module over Zgr(E(Λ)). +Then (C4) implies (C2) which in turn implies (C1). +Moreover, if the field K is algebraically closed, then the four statements are equivalent. +We shall need the description of the Ext algebra E(Λ) from [11], which we recall here. +Let Qop be the opposite quiver of Q, so that Qop +0 = Q0 and Qop +1 = {α: j → i | there is α: i → j in Q1}. +Now consider Λ +! += KQop/J with J = ( ρ +⊥ ), where the orthogonal is taken with respect to the +natural bilinear form KQop +d × KQd → K, that is, ⟨βd · · · β1, α1 · · · αd⟩ is equal to 1 if α1 · · · αd = +β1 · · · βd and is equal to 0 otherwise. (Recall that, for any n ⩾ 0, Qn denotes the set of paths of +length n in Q.) +If d = 2, set B = Λ +! +and if d ⩾ 3 let B = � +n⩾0 Bn be the algebra defined as follows: +• Bn = Λ +! +δ(n) +• for x ∈ Bn and y ∈ Bm, define x · y ∈ Bn+m by +x · y = +� +0 +if n and m are odd +xy +(in Λ +! +) if n or m is even. +Note that if n or m is even, δ(n) + δ(m) = δ(n + m), so that this defines a graded algebra B. +Then by [10, 2, 11], the algebras E(Λ) and B are isomorphic (for d ⩾ 2). +Since Λ is monomial it is easy to see that the algebra Λ +! +is d-homogeneous monomial and that +the set σ of paths αd · · · α1 ∈ Qop +d +such that α1 · · · αd ∈ Qd is not in ρ is a minimal generating set +for J consisting of paths of length d. There is a basis B Λ +! +of Λ +! +consisting of all paths p in Qop such +that no subpath of p is in σ. It follows from Consequence 2.2 that no subpath of length d of R +n +i is in +σ. Therefore R +n +i ∈ B Λ +! . +As we mentioned in Subsection 1.2, there is a basis of E(Λ) indexed by � +n⩾0 Rn that corres- +ponds, via the isomorphism with the algebra B, to the set of paths R +n +i for n ⩾ 0 and Rn +i ∈ Rn. We +then have an embedding of the basis � +n⩾0 R +n of B into B Λ +! +where R +n = +� +R +n +i | Rn +i ∈ Rn� +; denote +by BB its image, which is a basis of B. + +A COMBINATORIAL CHARACTERISATION OF (FG) +11 +We now define several gradings on Λ, Λ +! +and B. +There are natural gradings on Λ and on Λ +! +given by the lengths of paths; denote the length +by ℓ for both algebras. The degree of a homogeneous element x in B will be denoted by |x|, so +x ∈ Λ +! +δ(|x|) or, in other terms, |x| = k if, and only if, ℓ(x) = δ(k). +The algebra Λ +! +is also multi-graded by NQ1: for each path p in Qop, we define an element +d(p) = (dα(p))α∈Q1 ∈ NQ1 as follows: +• if ℓ(p) = 0, then d(p) = (0)α∈Q1 +• if ℓ(p) > 0, then dα(p) is the number of occurrences of α in p (it is 0 if α does not occur in +p). +Since Λ +! +is monomial, the ideal J is homogeneous with respect to this multi-degree and therefore Λ +! +is multi-graded. In B, if x and y are homogeneous and |x| or |y| is even, then dα(xy) = dα(x) + dα(y) +but if both degrees are odd then dα(xy) = 0. +Let Z := Zgr(B) be the graded centre of B. It is generated as a subring of B by the homogeneous +elements z ∈ B such that for all homogeneous y ∈ B, zy = (−1)|y||z|yz. Note that Z ⊂ � +e∈Q0 eBe, +therefore Z is generated by elements that are linear combinations of (non-zero) cycles in Qop. +Moreover, it can be checked easily that the graded centre Z of B is generated by elements z that +are homogeneous with respect to the grading |·| and the multi-degree d and such that, for any +element y ∈ B that is homogeneous with respect to the grading |·|, we have zy = (−1)|y||z|yz. +Remark 2.9. If d = 2 then B ∼= E(Λ) is generated in degrees 0 and 1, so in order to check that +an element of B is in Z, it is sufficient to check that it is a linear combination of cycles and that it +commutes or anti-commutes with all arrows in Qop. +If d ⩾ 3, then B ∼= E(Λ) is generated in degrees 0, 1 and 2. Therefore when checking that an +element is in Z, we need to check that it is a linear combination of cycles and that it commutes or +anti-commutes with paths of degrees 1 and 2, that is, arrows and (non-zero) paths of length d in +Qop. +The proof of Theorem 2.8 relies on some preliminary results. These are Lemma 2.10, Propos- +ition 2.11 and Lemma 2.13. For the first of these, we start with the following observation about +loops. Suppose α is a loop in Q1. Since Λ is finite dimensional, there is some integer N such that +αN ∈ I; therefore αN has a subpath of length d that is in ρ and so αd ∈ ρ. Therefore αd ̸∈ σ and it +follows that αj ̸= 0 in Λ +! +for all j ⩾ 0 and that αδ(j) ̸= 0 in B for all j ⩾ 0. +Lemma 2.10. Let α be a loop in Q1 and let n ⩾ 2. Then +• if d = 2, αn ∈ Z if, and only if, n is even and α satisfies Condition 2.4 (1); +• if d ⩾ 3, αδ(n) ∈ Z if, and only if, α satisfies Condition 2.4 (1). +Proof. First note that if n is odd, then +• if d = 2, αnα = αn+1 ̸= (−1)nα αn, therefore αδ(n) = αn ̸∈ Z; +• if d ⩾ 3, αδ(n) anti-commutes with all arrows since the products are 0 in B. +Therefore we may assume that n ⩾ 2 is an even integer and that αδ(n) ∈ Z. Set e = o(α). +Let β be an arrow ending at e with β ̸= α. Then, in B, +αδ(n)+d−1β = αδ(n) · αd−1β = αd−1β · αδ(n) = αd−1βαδ(n) +(the element αd−1β is in degree 2). Therefore we have an equality αδ(n)+d−1β = αd−1βαδ(n) between +two paths in the monomial algebra Λ +! +, so that both paths are zero. In particular, αδ(n)+d−1β contains +a subpath in σ, and since αd ̸∈ σ, we must have αd−1β ∈ σ. It follows that βαd−1 ̸∈ ρ. +Similarly, if β is an arrow that starts at e with β ̸= α, then αd−1β ̸∈ ρ. +Therefore α satisfies Condition 2.4 (1). +Conversely , assume that α satisfies Condition 2.4 (1). +Let β ̸= α be an arrow and let n ⩾ 2. By assumption, αd−1β ̸∈ ρ and βαd−1 ̸∈ ρ. It follows that +βαd−1 = 0 = αd−1β in Λ +! +(either in σ or not composable) and therefore that αδ(n)β = 0 = βαδ(n) +since δ(n) ⩾ δ(2) ⩾ d − 1. The path αδ(n) anticommutes with all elements of degree 1. +In particular, if d = 2 and n is even, then αn ∈ Z. + +12 +JAWAD, SNASHALL, AND TAILLEFER +Now assume that d ⩾ 3 and consider commutation with elements in B2. As a vector space, B2 is +generated by the paths p of length d such that p ∈ ρ. Let p = β1 · · · βd be a path in ρ. Since p has +degree 2 in B, products of elements in B with p in B and in Λ +! +are equal. +If p = αd, then clearly αδ(n)p = p αδ(n). +If p ̸= αd, set j = min {i | 1 ⩽ i ⩽ d, βi ̸= α} and k = max {i | 1 ⩽ i ⩽ d, βi ̸= α}. By assump- +tion, αd−1βj ̸∈ ρ and βkαd−1 ̸∈ ρ, therefore βjαd−1 = 0 and αd−1βk = 0 in Λ +! +. We have assumed that +n ⩾ 2, so δ(n) ⩾ d and therefore αδ(n)p = 0 = p αδ(n) in Λ +! +and in B, and αδ(n) anticommutes with +elements of degree 2. +Finally, we have proved that αδ(n) is in Z whenever d = 2 and n ⩾ 2 is even or d ⩾ 3 and +n ⩾ 2. +□ +For the next result, we need some more terminology for closed trails. Let n ⩾ 2 and let T = +α1 · · · αn be a closed trail in Q with ρT = {αi · · · αi+d−1 | 1 ⩽ i ⩽ n} ⊆ ρ. A subcycle of T is a cycle +of the form q = αi · · · αj with 1 ⩽ i ⩽ j ⩽ n and ℓ(q) < ℓ(T). We say that T has a repeated vertex if +T = α1 · · · αi−1vαi · · · αi+k−1vαi+k · · · αn for some i, k and vertex v such that the paths αi · · · αi+k−1 +and αi+k · · · αnα1 · · · αi−1 are non-trivial paths in KQ. +We make the following assumptions that we use in Proposition 2.11 and Lemma 2.13. The reason +for these specific assumptions becomes clear in the proof of Theorem 2.8. +(i) none of the αi are loops. +(ii) no subcycle of T satisfies the same assumptions as T (that is, there is no subcycle q of T of +length at least 2 with ρq ⊆ ρ). +Proposition 2.11. Let T = α1 · · · αn be a closed trail with n ⩾ 2, ρT ⊆ ρ and such that assumptions (i) +and (ii) hold. Let p be a path of length d such that dβ(p) = 0 if β ̸∈ {α1, . . . , αn} and which does not lie on +T. Then p ̸∈ ρ. +Proof. Let p = γ1 · · · γd with γi ∈ {α1, . . . , αn} for i = 1, . . . , d. The path p is a non-zero path in +KQ so t(γi) = o(γi+1) for all i. Suppose that γ1 = αj so p = αjγ2 · · · γd and γ2 ∈ {α1, . . . , αn} +with t(αj) = o(αj+1) = o(γ2). If T does not have a repeated vertex then necessarily αj+1 = γ2. +Inductively, p = αjαj+1 · · · αj+d−1 and hence p lies on the trail T. This contradicts our hypothesis. +Hence T has a repeated vertex. +Suppose that v is a repeated vertex so that T has a proper subpath q of length k with q = vqv +for some k; thus 2 ⩽ k ⩽ n − 1 since T does not have any loops and q is a closed trail. We claim +that k ⩾ d. Indeed, if we had k < d, then by Consequence 2.3 every path of length d that lies on q +would be in ρ, that is ρq ⊆ ρ, with ℓ(q) < ℓ(T). But this contradicts assumption (ii). Hence k ⩾ d. +Now suppose for contradiction that p ∈ ρ. As above, let αj be the first arrow in p. We know that +p does not lie on T and that T has a repeated vertex, so we may write +p = αj · · · αj+r−1γr+1 · · · γd +where r ⩾ 1, γr+1, . . . , γd ∈ {α1, . . . , αn}, t(αj+r−1) = o(αj+r) = o(γr+1) and γr+1 ̸= αj+r. Then +there is some t such that γr+1 = αt and t ̸≡ j + r (mod n). Moreover t(αt−1) = o(αt) = o(αj+r). +We may illustrate this as follows: +· +� · +· +· +αj+r−1� · +αj+r +�☎☎☎☎☎☎☎☎ +αt +� +... +· +� +· +· +αt+1 +� +· +αt−1 +�✿✿✿✿✿✿✿✿ +· +� +(We make no assumption as to whether γr+2 is or is not equal to αt+1.) We note that the path +αj+r · · · αt−1 · αt · · · αj+r−1 is a cyclic permutation of T and has length n. Moreover, from the previ- +ous part of this proof, both αj+r · · · αt−1 and αt · · · αj+r−1 are paths of length at least d. +Let S = αtαt+1 · · · αj+r−1. Then S is a closed path in KQ of length at least 2 and is a subcycle of +T. There is an overlap αj+r−d · · · αj · · · αj+r−1 with p so the subpath αj+r−d+1 · · · αj+r−1αt must also + +A COMBINATORIAL CHARACTERISATION OF (FG) +13 +be in ρ. Then we have an overlap of αj+r−d+1 · · · αj+r−1αt with αtαt+1 · · · αt+d−1; by Property 2.1, +all subpaths of length d of the path +αj+r−d+1 · · · αj+r−1αtαt+1 · · · αt+d−1 +must also be in ρ. Thus every path of length d that lies on S is in ρ. Hence ρS ⊆ ρ. +So S is a subcycle of T that satisfies the same assumptions as T, and this contradicts assump- +tion (ii). Hence p ̸∈ ρ. +□ +Remark 2.12. We keep the assumptions and notation of Proposition 2.11. Then T and all the paths +lying on T are in the basis B Λ +! +since none of their subpaths of length d are in σ; those of length δ(k) +for some k ⩾ 0 are in the basis BB. +Set Ti = αi · · · αnα1 · · · αi−1 so that Ti = αi−1 · · · α1αn · · · αi ∈ Λ +! +for all i. Then for any j ⩾ 1, we +have +T +j +iαk ̸= 0 ⇐⇒ k = i − 1 +αkT +j +i ̸= 0 ⇐⇒ k = i. +Lemma 2.13. Let T = α1 · · · αn be a closed trail with n ⩾ 2 and ρT ⊆ ρ that satisfies assumptions (i) and +(ii) and set zj = ∑n +i=1 T +j +i with nj = δ(u) for some integer u ⩾ 2 and with nj = u even if d = 2. Then +zj ∈ Z if, and only if, T satisfies Condition 2.4 (2). +Moreover, if T does not satisfy Condition 2.4 (2), then no element in B that is homogeneous with respect +to |·| and d (when viewed in Λ +! +) and that is a linear combination of non-trivial cycles that lie on T is in Z. +Proof. First assume that zj ∈ Z. Fix an integer i and let e = t(αi). Suppose for contradiction that +there is a path p of length d − 1 starting at e such that αip ∈ ρ and p ̸= αi+1 · · · αi+d−1. By Remark +2.12, at least one arrow in p is not in {α1, . . . , αn}, therefore p αi is not a subpath of any of the paths +that occur in zj (note that ℓ(zj) ⩾ δ(2) = d ⩾ ℓ(p)). The relation αipzj = zjαip in B becomes, in Λ +! +, +p αiT +j +i = zjp αi. +Since Λ +! +is monomial, it follows that p αiT +j +i contains a subpath in σ, that is, there is a subpath of +length d of Tj +i αip that is not in ρ. It cannot be a subpath of Tj +i αi because we have assumed that +ρT ⊆ ρ. Moreover, we have assumed that αip ∈ ρ. We also have αi−d+1 · · · αi−1αi ∈ ρT ⊆ ρ and ρ is +d-covering (Property 2.1), therefore every subpath of length d of αi−d+1 · · · αi−1αip is also in ρ and +so is every subpath of length d of Tj +i αip and we have obtained a contradiction. Therefore T satisfies +Condition 2.4 (2). +Conversely , assume that T satisfies Condition 2.4 (2). We prove that for all j ⩾ 1 such that +nj = δ(u) for some integer u ⩾ 2, and such that nj = u is even if d = 2, we have zj ∈ Z. +• First note that if d ⩾ 3 and +��zj +�� is odd, then zj anti-commutes with all arrows (the products +are 0 in B). Therefore assume that +��zj +�� is even and that d ⩾ 2, and let β be an arrow. +If β = αk for some k, then αkzj = αkT +j +k and zjαk = T +j +k+1αk using Remark 2.12, and these +paths are indeed equal, so that βzj = zjβ = (−1)|zj||β|zjβ. +If β ̸∈ {α1, . . . , αn} then β T +j +i = 0 = T +j +iβ for all i by assumption, therefore βzj = 0 = +(−1)|zj||β|zjβ. +• Now assume that d ⩾ 3 (and +��zj +�� is still even) and consider commutation with elements +in B2. As a vector space, B2 is generated by the paths p of length d such that p ∈ ρ. Let +p = β1 · · · βd be a path in ρ with βi ∈ Q1 for all i. Since p has degree 2 in B, products of +elements in B with p in B and in Λ +! +are equal. +If p lies on T, then p = αk · · · αk+d−1 for some k, and it is easy to check that pzj = zjp = +(−1)|zj||p|zjp (as for commutation with an arrow). +If p does not lie on T, then by Condition 2.4 (2), the first and last arrows in p are not in +{α1, . . . , αn}. Moreover, for all i, we have αiβ1 · · · βd−1 ̸∈ ρ; it follows that pzj = 0. Similarly, +for all i, we have β2 · · · βdαi ̸∈ ρ (since it ends with αi and is not in ρT), therefore zjp = 0. +Finally, pzj = zjp = (−1)|zj||p|zjp. + +14 +JAWAD, SNASHALL, AND TAILLEFER +We have proved that zj ∈ Z. +Now let z be an element in Z that is homogeneous with respect to |·| and d and that is a linear +combination of cycles that lie on T. Assume that ℓ(z) > 0; by (i) z is not a linear combination of +arrows so |z| ⩾ 2. Put z = ∑m +i=1 λici with λi ∈ K and the ci cycles in Qop that lie on T. Since z is +homogeneous with respect to |·| and d and the ci lie on T, the ci are cyclic permutations of c1. Up +to relabelling, we may write ci = T +j−1 +i +αi−1 · · · αi−s for some fixed s with 1 ⩽ s ⩽ n (and m = n). +We first consider the case where |z| is even. Then we must have αkz = zαk for all k, that is, +∑n +i=1 λiαkT +j−1 +i +αi−1 · · · αi−s = ∑n +i=1 λiT +j−1 +i +αi−1 · · · αi−sαk. Using Remark 2.12, this is equivalent to +λkαkT +j−1 +k +αk−1 · · · αk−s = λk+s+1T +j−1 +k+s+1αk+s · · · αk+1αk. +Therefore λk = 0 = λk+s+1 or k + s ≡ k (mod n) (that is, s = n), and λk = λk+1. +This is true for all k, so if z ̸= 0 then z = λ1zj with nj = ℓ(z) = δ(|z|). +We now consider the case where |z| is odd. +If d = 2 then the same reasoning as in the even case shows that λk+1 = (−1)|z|λk for all k with +1 ⩽ k ⩽ n − 1 and λ1 = (−1)|z|λn, therefore λ1 = (−1)n|z|λ1 = −λ1 (because nj = ℓ(z) = |z| is +odd hence n is odd) so that λk = 0 for all k and finally z = 0. +Now assume that d ⩾ 3. Since z ∈ Z, we have αk+d−1 · · · αkz = zαk+d−1 · · · αk for all k, that +is, ∑n +i=1 λiαk+d−1 · · · αkT +j−1 +i +αi−1 · · · αi−s = ∑n +i=1 λiT +j−1 +i +αi−1 · · · αi−sαk+d−1 · · · αk. Using Remark 2.12, +this is equivalent to +λkαk+d−1 · · · αkT +j−1 +k +αk−1 · · · αk−s = λk+s+dT +j−1 +k+s+dαk+s+d−1 · · · αk+dαk+d−1 · · · αk. +Therefore λk = 0 = λk+s+d or k + s + d − 1 ≡ k + d − 1 (mod n) (that is, s = n), and λk = λk+d. +When s = n, we have nj = ℓ(z) = δ(|z|) = |z| − 1 +2 +d + 1, therefore n and d are coprime. It +follows that if z ̸= 0, then all the λi are equal so that z = λ1zj. +We have proved that if z is a non-zero element in Z that is homogeneous with respect to |·| and +d and which is a linear combination of non-trivial cycles that lie on T, then if d = 2 we must have +|z| even and for all d ⩾ 2, z is then a non-zero scalar multiple of zj. Therefore zj is in Z and by the +first part of the proof, T satisfies Condition 2.4 (2). +□ +We have now all the tools we need for the proof of Theorem 2.8. +Proof of Theorem 2.8. The fact that (C2) implies (C1) is Theorem 2.7. The implication (C3) ⇒ (C4) is +clear and if, in addition, K is algebraically closed, then the implication (C1) ⇒ (C3) follows from +[7]. +We now prove that (C4) implies (C2). Suppose that (C2) does not hold, that is, Condition 2.4 does +not hold. Assume for contradiction that B is a finitely generated Z-module, generated by elements +g1, . . . , gt that are homogeneous with respect to both the grading |·| and the multi-grading d. +We first assume that Condition 2.4 (1) does not hold, so that there is a loop α that does not satisfy +this condition. Then for all j ⩾ 2, αδ(j) ̸∈ Z by Lemma 2.10. +Now consider αδ(k) ∈ B for some even integer k ⩾ 2. Then +dβ(αδ(k)) = +� +δ(k) +if β = α +0 +if β ̸= α +and +���αδ(k)��� = k. By assumption, and using the fact that αδ(k), g1, ..., gt are homogeneous with +respect to |·| and d, there exist elements u(k) +i +in Z, 1 ⩽ i ⩽ t, that are homogeneous with respect to +|·| and d, such that αδ(k) = ∑t +i=1 u(k) +i +gi. +Since αδ(k) is homogeneous with respect to |·| and d, we can assume that for all i we have +���u(k) +i +gi +��� = k and d(u(k) +i ) + d(gi) = d(u(k) +i +gi) = d(αδ(k)). If β ̸= α then dβ(u(k) +i ) + dβ(gi) = 0 so +dβ(u(k) +i ) = 0 and dβ(gi) = 0. It follows that u(k) +i +and gi are powers of α; since u(k) +i +∈ Z, we must + +A COMBINATORIAL CHARACTERISATION OF (FG) +15 +have +���u(k) +i +��� = 0 or 1 by assumption. If +���u(k) +i +��� = 1, then |gi| = k − 1 is odd and hence, in B, we have +u(k) +i +gi = 0. Therefore we may assume that u(k) +i +∈ Z0 = K. It follows that the sum contains only one +term and that gi is a (non-zero) scalar multiple of αδ(k) so that αδ(k) ∈ spanK {g1, . . . , gt}. +We have shown that +� +αδ(k) | k ⩾ 1, k even +� +⊆ spanK {g1, . . . , gt}. However, using the grading +|·|, we see that the αδ(k), k ⩾ 1, are linearly independent over K: we have reached a contradiction. +Therefore the Yoneda algebra E(Λ) = B is not finitely generated as a Z-module when Condi- +tion 2.4 (1) does not hold. +We now assume that Condition 2.4 (2) does not hold, so that there is a closed trail T = α1 · · · αn +with n ⩾ 2 and ρT ⊆ ρ that does not satisfy this condition. We can make the following assumptions +(and therefore use Proposition 2.11): +(i) none of the αi are loops. Indeed, if αi is a loop, then the paths αiαi+1 · · · αi+d−1 and αd +i are +in ρ and properly overlap, hence αd−1 +i +αi+1 is in ρ because ρ is d-covering, therefore α does +not satisfy Condition 2.4 (1), and in this case we already know that E(Λ) is not a finitely +generated Z-module. +(ii) no subcycle of T satisfies the same hypotheses as T (otherwise replace T with the shortest +such subcycle). +We have seen in Lemma 2.13 that no linear combination of non-trivial cycles that lie on T, that +is homogeneous with respect to |·| and d, is in Z. +Now consider zδ(k) = ∑n +i=1 T +δ(k) +i +∈ B for some even integer k ⩾ 2. Then +���zδ(k) +��� = nk and +dβ(zδ(k)) = +� +δ(k) +if β ∈ {α1, . . . , αn} +0 +if β ̸∈ {α1, . . . , αn} . +Since zδ(k) and the gi are homogeneous with respect to |·| and d, by assumption there exist +elements u(k) +i +in Z, 1 ⩽ i ⩽ t, that are homogeneous with respect to |·| and d, such that zδ(k) = +∑t +i=1 u(k) +i +gi. Note that Z ⊆ � +e∈Q0 e Λ +! +e, therefore the u(k) +i +are linear combinations of cycles. +Fix an integer j with 1 +⩽ +j +⩽ +n. +Then αj+d−1 · · · αjT +δ(k) +j += +αj+d−1 · · · αjzδ(k) += +∑t +i=1 αj+d−1 · · · αju(k) +i +gi. +Since αj+d−1 · · · αjT +δ(k) +j +is homogeneous with respect to |·| and d, we can assume that for all i we +have 2 + +���u(k) +i +��� + |gi| = +���αj+d−1 · · · αju(k) +i +gi +��� = +���αj+d−1 · · · αjzδ(j) +��� = nk + 2 and d(αj+d−1 · · · αj) + +d(u(k) +i ) + d(gi) = d(αj+d−1 · · · αju(k) +i +gi) = d(αj+d−1 · · · αjT +δ(k) +i +) = d(αj+d−1 · · · αj) + d(T +δ(k) +i +), and +therefore the only arrows that occur in u(k) +i +gi are the αj, 1 ⩽ j ⩽ n. Moreover, Proposition 2.11 +and the fact that the u(k) +i +are linear combinations of cycles show that the u(k) +i +must be linear com- +binations of cycles lying on T (otherwise, one at least of these cycles has a subpath of length d +that does not lie on T, hence that is in σ and this cycle vanishes in B and does not occur in u(k) +i ). +Our assumption shows that these cycles must be trivial (of length 0), and since Z0 = K we see +that u(k) +i +∈ K. Therefore αj+d−1 · · · αjT +δ(k) +j +is a linear combination of the αj+d−1 · · · αjgi, hence +� +αj+d−1 · · · αjT +δ(k) +j +| 1 ⩽ j ⩽ n, k ⩾ 1, k even +� +⊆ spanK +� +αj+d−1 · · · αjgi | 1 ⩽ j ⩽ n, 1 ⩽ i ⩽ t +� +. +The set +� +αj+d−1 · · · αJT +δ(k) +J +| 1 ⩽ j ⩽ n, k ⩾ 2, k even +� +is linearly independent over K (using the +grading |·|), therefore we have reached a contradiction. +Therefore the Yoneda algebra E(Λ) = B is not finitely generated as a Z-module when Condi- +tion 2.4 (2) does not hold. Hence (C4) ⇒ (C2). +□ +Remark 2.14. In the case where K is algebraically closed, we have extended the equivalence between +(C1) and (C3), already known for Koszul algebras from [7], to d-Koszul monomial algebras with +d ⩾ 3. In particular, we have the following corollary. +Corollary 2.15. Let Λ be a d-Koszul monomial algebra over an algebraically closed field with d ⩾ 2. +Assume that E(Λ) is a finitely generated Zgr(E(Λ))-module. Then the algebra Zgr(E(Λ)) is noetherian. + +16 +JAWAD, SNASHALL, AND TAILLEFER +3. EXTENSION TO (D, A)-STACKED MONOMIAL ALGEBRAS +3.1. Notation and properties of (D, A)-stacked monomial algebras +Let Λ = KQ/I be a monomial algebra with the length grading as before. Let D and A be integers +with D > A ⩾ 1. From [13, Definition 3.1], Λ is then a (D, A)-stacked monomial algebra if, for +any minimal projective right Λ-module resolution of Λ0, the n-th projective module is generated +in degree δA(n) where +δA(n) = + + + + + +n +if n = 0 or n = 1 +n +2 D +if n ⩾ 2 is even +n−1 +2 D + A +if n ⩾ 3 is odd. +When A = 1, we retrieve the definition of a D-Koszul algebra, so that a (D, 1)-stacked monomial +algebra is a D-Koszul monomial algebra. +It was shown in [13, Proposition 3.3] that if gldim Λ ⩾ 4 then A divides D; in particular, D ⩾ 2A. +If the global dimension of Λ is finite, then Condition (Fg), and in fact all the conditions (C1)–(C6) +stated in the Introduction, are satisfied by Λ. Therefore we shall assume throughout this section +that Λ is a (D, A)-stacked monomial algebra with gldim Λ ⩾ 4 and set d = D +A. We define +δ(n) = + + + + + + + + + +n +if n = 0 or n = 1 +n +2 d = δA(n) +A +if n ⩾ 2 is even +n − 1 +2 +d + 1 = δA(n) +A +if n ⩾ 3 is odd. +Definition 3.1. For A ⩾ 1, we define an A-path as a non-zero path p = α1 · · · αn where all the αi are +paths of length A (that is, αi ∈ QA for all i). An A-trail is an A-path in which all the αi are distinct. +An A-cycle is a closed A-path and finally an A-loop is an A-cycle of length A. +Given an A-path p as above, an A-subpath of p is an A-path of the form αi · · · αj with 1 ⩽ i ⩽ j ⩽ +n (note that not every A-path that is a subpath of p is an A-subpath of p). An A-subcycle of p is a +closed A-subpath of one of the non-zero A-paths αi · · · αnα1 · · · αi−1 with 1 ⩽ i ⩽ n. +We also define the A-length ℓA(p) of an A-path p = α1 · · · αn where the αi are paths of length A +as ℓA(p) = n, that is, ℓ(p) = AℓA(p). +We will need the following result from [13]. +Property 3.2. [13, Section 3] Let Λ = KQ/I be a finite dimensional monomial algebra. Then Λ is (D, A)- +stacked if, and only if, ρ = R2 has the following properties: +(1) every path in ρ is of length D; +(2) if R2 +2 ∈ R2 properly overlaps R2 +1 ∈ R2 with overlap R2 +1u, then ℓ(u) ⩾ A and there exists R2 +3 ∈ R2 +which properly overlaps R2 +1 with overlap R2 +1u′, ℓ(u′) = A and u′ is a prefix of u. +✤ +R2 +1 +✤ +✤ +R2 +3 +✤ +✤ +R2 +2 +✤ +� +u +� +� +u′ +� +Therefore ρ consists of paths of length D, and if Λ is (D, A)-stacked with gldim Λ ⩾ 4, we view +ρ as a set of A-paths of A-length d. +Example 3. We include first an example from [8] (Example 3.2). Let Λ = KQ/I where Q is the +quiver +· +α +� · +β +� +· +γ +� +· +δ +� +and the ideal I has minimal generating set ρ = {αβγδαβ, γδαβγδ}. Then Λ is a (6, 2)-stacked +monomial algebra. + +A COMBINATORIAL CHARACTERISATION OF (FG) +17 +The closed 2-trails are all the paths of length 4. +Example 4. Now we give an example where, as well as closed A-trails, there are A-loops. Let +Λ = KQ/I where Q is the quiver +· +γ4 +� · +β1 +� +γ1 +� +· +β2 +� · +α1 +� +· +γ3 +� +· +γ2 +� +· +α2 +� +and the ideal I has minimal generating set ρ = {(α1α2)2, (γ1γ2)(γ3γ4), (γ3γ4)(γ1γ2)}. Then Λ is a +(4, 2)-stacked monomial algebra. +The closed 2-trails are the paths of length 4 whose arrows are the γi and the 2-loops are α1α2 and +α2α1. +Example 5. Finally, we give an example in which an arrow, namely β2, occurs both in closed A-trails +and in A-loops. Let Λ = KQ/I where Q is the quiver +· +β3 +� +α2 +��������� +· +β4 +� · +β5 +� · +β6 +� · +β7 +��������� +· +α1 +� · +β2 +� +· +β1 +� +· +β9 +� +· +β8 +� +and the ideal I has minimal generating set +ρ = {(α1β2α2)2, (β1β2β3)(β4β5β6), (β4β5β6)(β7β8β9), (β7β8β9)(β1β2β3)}. +Then Λ is a (6, 3)-stacked monomial algebra. +The closed 3-trails are all the cycles of length 9 whose arrows are the βi and the 3-loops are +α1β2α2, α2α1β2 and β2α2α1. +We have the following consequences of Property 3.2. +Consequence 3.3. We keep the notation of Property 3.2, with D = dA. Then the length of u must be a +multiple of A, so that R2 +1u is an A-path, and every A-subpath of A-length d of R2 +1u is in ρ. Moreover, no +other subpath of length D of R2 +1u is in ρ. +Proof. Write ℓ(u) = qA + r with q ⩾ 1 and 0 ⩽ r < A. We prove the result by induction on q. +If q = 1, then the path R2 +2 ∈ ρ overlaps R2 +3 ∈ ρ with overlap R2 +3u3 for some path u3. If this +overlap is a proper overlap (that is, R2 +3 ̸= R2 +2), then ℓ(u) = ℓ(u′) + ℓ(u3) = A + ℓ(u3) so that +ℓ(u3) = (q − 1)A + r = r and 0 < r < A. Therefore by Property 3.2 we have a contradiction. +It follows that R2 +3 = R2 +2 and u = u′ has length A and that R2 +1 and R2 +3 are the only A-subpaths of +A-length d of R2 +1u and they are in ρ. Moreover, any other subpath of length D of R2 +1u is a proper +overlap of R2 +1 of length strictly smaller than D + A, which is impossible by Property 3.2. +Let q > 1 be such that ℓ(u) = qA + r with 0 ⩽ r < A and assume that the result is true for +any proper overlap of a path in ρ of length D + q′A + r′ with q′ < q and 0 ⩽ r′ < A. The path +R2 +2 ∈ ρ properly overlaps R2 +3 ∈ ρ with overlap R2 +3u3 for some path u3 with ℓ(u) = ℓ(u′) + ℓ(u3) = +A + ℓ(u3) so that ℓ(u3) = (q − 1)A + r and the overlap R2 +3u3 has length D + (q − 1)A + r. By +induction, ℓ(u3) is a multiple of A, therefore r = 0 and ℓ(u) is a multiple of A. Any A-subpath +of A-length d of R2 +1u is either R2 +1 or an A-subpath of A-length d of R2 +3u3. Again by induction, they +are all in ρ. Finally, a subpath of length D of R2 +1u which is not an A-subpath either is a subpath of +length D of R2 +3u3 that is not an A-subpath, therefore not in ρ by induction, or properly overlaps R2 +1 +with overlap R2 +1u′′′ with 0 < ℓ(u′′′) < A, which is impossible by Property 3.2. +□ +Consequence 3.4. Suppose that D = dA. Let n ⩾ 2 and let Rn +i be an element of Rn. Write Rn +i = +α1 · · · αδ(n) where each αi is a path of length A. Then for all i with 1 ⩽ i ⩽ δ(n) − d + 1, the path +αi · · · αi+d−1 is in ρ, that is, all the A-subpaths of A-length d of Rn +i are in ρ. Moreover, no other subpath of +Rn +i of length D is in ρ. + +18 +JAWAD, SNASHALL, AND TAILLEFER +Proof. The result is proved by induction. It is clear when n = 2. Moreover, if n = 3, since R3 +i ∈ R3 +is a maximal overlap of two elements in R2, it follows from Property 3.2 and using the notation +therein that R3 +i = R2 +1u′ = v′R2 +3 where v′ is the prefix of R2 +1 of length A. By Consequence 3.3, the +only subpaths of length D of R3 +i that are in ρ = R2 are R2 +1 and R2 +3. +Now let n ⩾ 4 and take Rn +i ∈ Rn. Then Rn +i is a maximal overlap of R2 +1 ∈ R2 with Rn−1 +2 +∈ Rn−1 +so that Rn +i = Rn−1 +2 +u for some path u. Write Rn−1 +2 += α1 · · · αδ(n−1) with ℓ(αi) = A for all i. By the +induction assumption, we have αi · · · αi+d−1 ∈ R2 for all i with i + d − 1 ⩽ δ(n − 1). In particular, +R2 +3 := αδ(n−1)−d+1 · · · αδ(n−1) is in R2. Since R2 +1 overlaps R2 +3 with overlap R2 +3u, by Property 3.2 we +have ℓ(u) = A and αδ(n−1)−d+2 · · · αδ(n−1)u = R2 +1 ∈ R2. Since Rn +i = Rn−1 +2 +u, we have proved the +first part of the result for Rn +i . +Now let p be another subpath of Rn +i of length D. We already know by induction that if p is a +subpath of Rn−1 +2 +, then p is not in R2. Therefore p is a subpath of R2 +3u which is neither R2 +3 nor R2 +1. By +Consequence 3.3, p is not in ρ. We have proved that p ̸∈ R2 and the induction step is complete. +□ +Consequence 3.5. Suppose that D = dA. Let T = α1 · · · αn be a closed A-trail in Q with αi ∈ QA for all +i and suppose that d ⩾ n + 1. Assume also that T is the prefix of an A-path in ρ and the suffix of an A-path +in ρ. Then all A-subpaths of A-length d of powers of the closed trail T are in ρ. +Proof. By assumption, there exist A-paths T′ and T′′ such that T′T ∈ ρ and TT′′ ∈ ρ. Since Λ is +finite dimensional, there is a path R2 ∈ ρ that lies on T, and ℓ(R2) = D = dA > ℓ(T) = nA. +Therefore R2 is a subpath of length D of TN = (α1 · · · αn)N for some N ⩾ 2. If R2 = Tm is a power +of T with m ⩾ 2 (and d = nm) then R2 overlaps itself with overlap T2m−1 and the result follows +using Consequence 3.3 (every A-subpath of A-length d of a power of T is an A-subpath of T2m−1). +Otherwise, TT′′ overlaps R2 or R2 overlaps T′T and we can use Consequence 3.3 again to prove +that R2 is an A-subpath of TN and then that every A-subpath of A-length d of the overlap is in ρ; +since every A-subpath of A-length d of a power of T is one of these, we obtain the result. +□ +3.2. Characterisations of (D, A)-stacked monomial algebras that satisfy (Fg) +We now give our combinatorial condition for (D, A)-stacked monomial algebras Λ. +Condition 3.6. +(1) Let c be an A-loop in QA. Write c = a1 · · · aA with ai ∈ Q1 for all i and cj = aj · · · aAa1 · · · aj−1 +for j ∈ {1, . . . , A}. Then there exists j such that cd +j ∈ ρ but there is no path in ρ of the form +cd−1 +j +β or βcd−1 +j +where β is a path of length A that is distinct from cj. +(2) Let T = α1 · · · αn be a closed A-trail in Q with n ⩾ 2 and αi ∈ QA for all i and such that +ρT := {α1 · · · αd, α2 · · · αdαd+1, . . . , αnα1 · · · αd−1} ⊆ ρ. Then there are no elements in ρ \ ρT +which begin or end with the path αi, for all i. +Remark 3.7. In part (1) of the condition, there is exactly one j such that cd +j ∈ ρ. Indeed, if cd +j and cd +k +were in ρ, they would overlap with an overlap of length at most D + A − 1, hence by Property 3.2 +we must have cd +j = cd +k and therefore j = k. +Remark 3.8. If A = 1 then Condition 3.6 is equivalent to Condition 2.4. +We first prove that this condition is sufficient for Λ to satisfy (Fg). +Theorem 3.9. Let Λ = KQ/I be a finite dimensional (D, A)-stacked monomial algebra. Assume that Λ +satisfies Condition 3.6. Then Λ satisfies (Fg). +Proof. The case D ⩾ 2 and A = 1 corresponds to d-Koszul monomial algebras (with D = d) and +is proved in Theorem 2.7. Therefore we assume that A > 1 so that necessarily D > 2. If gldim Λ +is finite then Λ satisfies (Fg) (and Condition 3.6 is empty), so we also assume that gldim Λ ⩾ 4 so +that D = dA. +The structure of this proof follows that of Theorem 2.7 by replacing each arrow in Q1 by a path +of length A in QA. We do not give all the details here, but indicate those places where we need to +provide additional arguments. +The first part of the proof is to show that the hypotheses of [13, Theorem 3.4] hold. + +A COMBINATORIAL CHARACTERISATION OF (FG) +19 +Let c1, . . . , cu be the A-loops in Q such that cd +i ∈ ρ for i = 1, . . . , u. (We remark that, in the +terminology of [13], these are precisely the closed paths in Q such that for each ci we have ci ̸= pri +i +for any path pi with ri ⩾ 2 and cd +i ∈ ρ. Firstly, cd +i ∈ ρ implies that ℓ(ci) = A. Then, if ci = pri +i for +some path pi with ri ⩾ 2, we have 1 ⩽ ℓ(pi) < A. Now pdri +i +is in ρ and pdri +i +overlaps itself with +overlap pdri+1 +i +, so there is a maximal overlap in R3 of length ⩽ D + ℓ(pi) < D + A. But this is a +contradiction since Λ is a (D, A)-stacked monomial algebra. So ci ̸= pri +i .) By Condition 3.6 (1), for +each i = 1, . . . , u, there are no elements in ρ of the form cd−1 +i +β or βcd−1 +i +where β is a path of length +A that is distinct from ci. +We need to show that there are no overlaps of cd +i with any element of ρ \ {cd +i }. If R ∈ ρ \ {cd +i } +and R overlaps cd +i , then, by Consequence 3.3, either R = cs +ib or R = bcs +i where 1 ⩽ s ⩽ d − 1 +and b is an A-path with ℓA(b) = d − s and that does not begin (respectively, end) with the path ci. +Suppose that R = cs +ib. Then R overlaps cd +i with overlap of length A(2d − s). By Consequence 3.3, +this is a maximal overlap since ci is not a prefix of b and thus gives an element R3 +1 ∈ R3. However, +ℓ(R3 +1) = D + A = (d + 1)A. Thus 2d − s = d + 1 and so s = d − 1. But then R = cd−1 +i +b and b is a +path of length A distinct from ci, which is a contradiction. The case R = bcs +i is similar. So there are +no overlaps of cd +i with any element of ρ \ {cd +i }. +Let Tu+1, . . . Tr be the distinct closed A-trails in Q with ℓA(Ti) > 1 such that the sets ρTi of +Condition 3.6 (2) are contained in ρ. For each i = u + 1, . . . , r, we write Ti = αi,1 · · · αi,mi, where the +αi,j are in QA so that ℓA(Ti) = mi > 1 and +ρTi = {αi,1 · · · αi,d, αi,2 · · · αi,d+1, . . . , αi,miαi,1 · · · αi,d−1} ⊆ ρ. +By Condition 3.6 (2), for each closed A-trail Ti (i = u + 1, . . . , r), there are no elements in ρ \ ρTi +which begin or end with the path αi,j, for all j = 1, . . . , mi. So no path αi,j of length A has overlaps +with any element in ρ \ ρTi. +The next step is to show that Λ satisfies (Fg1). Applying [13, Theorem 3.4], gives HH∗(Λ)/N ∼= +K[x1, . . . , xr]/⟨xaxb for a ̸= b⟩, where +• for i = 1, . . . , u, the vertices v1, . . . , vu are distinct and the element xi corresponding to the +A-loop ci is in degree 2 and is represented by the map P2 −→ Λ where for R2 ∈ R2, +o(R2) ⊗ t(R2) �→ +� +vi +if R2 = cd +i +0 +otherwise +• and for i = u + 1, . . . , r, the element xi corresponding to the closed A-trail Ti = αi,1 · · · αi,mi +is in degree 2µi such that µi = mi/ gcd(d, mi) and is represented by the map P2µi −→ Λ, +where for R2µi ∈ R2µi, +o(R2µi) ⊗ t(R2µi) �→ +� +o(Ti,k) +if R2µi = Td/ gcd(d,mi) +i,k +for all k = 1, . . . , mi +0 +otherwise. +Let H be the subring of HH∗(Λ) generated by Z(Λ) and {x1, . . . , xr}. As in Theorem 2.7, H is a +commutative Noetherian ring and so Λ satisfies (Fg1). +Now we show that Λ satisfies (Fg2). Again, we identify � +n⩾0 Rn with a basis of E(Λ). Set +N = max{3, |x1|, . . . , |xr|, |QA|}. We show that �N +n=0 Rn is a generating set for E(Λ) as a left H- +module and thus E(Λ) is finitely generated as a left H-module. +Let R ∈ Rn with n > N. Then ℓA(R) = δ(n) ⩾ 2d and we can write R = a1a2 · · · aδ(n) where +the ai are in QA. The proof now follows that of Theorem 2.7 by replacing each arrow by a path +of length A, and with extensive use of Consequences 3.3, 3.4 and 3.5, and Condition 3.6. Thus Λ +satisfies (Fg2) and we conclude that Λ has (Fg). +□ +Example 6. We return to Examples 3, 4 and 5. In all these examples, Condition 3.6 is satisfied and +therefore (Fg) holds for Λ. +For instance, in Example 3, the only closed 2-trails T such that ρT ⊆ ρ are αβγδ and γδαβ and, +in both cases, ρT = ρ. In Example 5, the closed 3-trails T such that ρT ⊆ ρ are those that start with +β1, β4 and β7, in all cases we have ρT = ρ \ +�(α1β2α2)2� +and (α1β2α2)2 does not start or end with +a βi. + +20 +JAWAD, SNASHALL, AND TAILLEFER +By [6, Theorem 2.5], it follows that Λ is Gorenstein in each case. Moreover, it was proved in [8] +that the algebra in Example 3 has injective dimension 2. +Our aim is now to prove the following theorem, and in particular the converse of Theorem 3.9. +Theorem 3.10. Let Λ be an indecomposable finite dimensional (D, A)-stacked monomial algebra. Suppose +that D ̸= 2A whenever A > 1. Consider the following statements: +(C1) Λ satisfies (Fg); +(C2) Condition 3.6 holds for Λ; +(C3) Zgr(E(Λ)) is noetherian and E(Λ) is a finitely generated Zgr(E(Λ))-module; +(C4) E(Λ) is finitely generated as a module over Zgr(E(Λ)). +Then (C4) implies (C2) which in turn implies (C1). +Moreover, if the field K is algebraically closed, then the four statements are equivalent. +We shall need, as in the d-Koszul case, a description of the Ext algebra of Λ. We give the details of +this in the Appendix, and we briefly describe it here. Since we have already proved Theorem 3.10 +when Λ is d-Koszul, we assume here that D > A > 1 and, in addition, that D ̸= 2A. +Let Γ be the quiver with the same vertices as Q and whose set of arrows corresponds to the set +of paths of length A in Q, that is, Γ1 = {α: i → j | there exists α ∈ QA, α: j → i}. Let +ρ +⊥ +be the +orthogonal of ρ for the bilinear form KΓd × K(QA)d → K defined on paths of length d in Γ and +A-paths of A-length d in Q by ⟨αd · · · α1, β1 · · · βd⟩ = 1 if α1 · · · αd = β1 · · · βd and 0 otherwise, +where the αi and βi are in QA. Set Λ +♮ += KΓ/J where J = ( ρ +⊥ ); it is a monomial algebra and the +ideal J has a minimal generating set σ given by all the paths αd · · · α1 such that the A-path α1 · · · αd +is not in ρ. +Let B = � +n⩾0 Bn be the algebra defined as follows: +• Bn = Λ +♮ +δ(n) +• for x ∈ Bn and y ∈ Bm, define x · y ∈ Bm+n by +x · y = + + + + + +0 +if n and m are odd; +0 +if n or m is equal to 1 and n ⩾ 1, m ⩾ 1; +xy +in Λ +♮ +otherwise. +Observe that if n or m is even and both are larger than 1, δ(n) + δ(m) = δ(n + m), so that the +algebra B is a graded K-algebra, generated in degrees 0, 1, 2 and 3. Note that this is also true of +E(Λ) by [12]. Moreover, we prove in Appendix A that the algebras E(Λ) and B are isomorphic, +generalising the description given in [11] when Λ is a d-Koszul algebra. This isomorphism uses +the assumption that D ̸= 2A. +There is a basis B Λ +♮ +of Λ +♮ +consisting of all paths p in Γ such that no path in σ is a subpath of p, +and basis BB of B contained in B Λ +♮ +consisting of all R +m +i for all m ⩾ 0 and all Rm +i ∈ Rm. +We now define several gradings, on Λ +♮ +and on B. +There is a natural grading on Λ +♮ +given by the length ℓ of paths. Note that if p is an A-path in +Q, then ℓ(p) = ℓA(p). The degree of a homogeneous element x in B will be denoted by |x|, so +x ∈ Λ +♮ +δ(|x|) or, in other terms, |x| = k if, and only if, ℓ(x) = δ(k). +The algebra Λ +♮ +is also multi-graded by NQ1: for each path p in Γ, we define an element d(p) = +(dα(p))α∈QA ∈ NQ1 as follows: write the A-path p in Q as p = α1 · · · αn where each αi is in QA; +• if ℓ(p) = 0, then d(p) = (0)α∈Q1 +• if ℓ(p) > 0, then dα(p) is the number of αi that are equal to α (it is 0 if none of the αi are +equal to α). +Note that even if α is a subpath of p, we can have dα(p) = 0 (if α is not one of the αi, that is, p = qαr +where q and r are paths in Q whose lengths are not multiples of A). +Since Λ +♮ +is monomial, the ideal J is homogeneous with respect to this multi-degree and therefore +Λ +♮ +is multi-graded. In B, if x and y are homogeneous and |x| or |y| is even with both degrees at +least 2, then dα(xy) = dα(x) + dα(y) but dα(xy) = 0 otherwise. + +A COMBINATORIAL CHARACTERISATION OF (FG) +21 +Let Z := Zgr(B) be the graded centre of B. As in the d-Koszul case, it is generated by elements z +that are homogeneous with respect to the grading |·| and the multi-degree and such that, for any +element y ∈ B that is homogeneous with respect to the grading |·|, we have zy = (−1)|y||z|yz. +Remark 3.11. Recall that B ∼= E(Λ) is generated in degrees 0, 1, 2 and 3 and that the product of an +element of degree 1 with any other element vanishes. Therefore when checking that an element +is in Z, we need to check that it is a linear combination of cycles and that it commutes or anti- +commutes with paths of degrees 2 and 3, that is, (non-zero) paths of length d and of length d + 1 +in Γ. +The proof of Theorem 3.10 relies on some preliminary results, namely Lemma 3.12, Proposi- +tion 3.13 and Lemma 3.15. We start with some comments on A-loops in QA. Let c be an A-loop in +QA. Since Λ is finite dimensional, there exists an integer N such that cN = 0 in Λ and therefore +there is some j such that cd +j ∈ ρ. To simplify notation and without loss of generality, write c = cj. +Then cd ∈ ρ, therefore cd ̸∈ σ and it follows that ck ̸= 0 in Λ +♮ +for all k ⩾ 0 and that cδ(k) ̸= 0 in B +for all k ⩾ 0. +Lemma 3.12. Let c be an A-loop in QA and let n ⩾ 2 be an integer. Then cδ(n) ∈ Z if, and only if, c +satisfies Condition 3.6 (1). +Proof. The proof is very similar to that of Lemma 2.10, using A-paths and Remark 3.11. +□ +We shall now consider part (2) of Condition 3.6. +Let T = α1 · · · αn be a closed A-trail in Q with αi ∈ QA for all i. Assume that n ⩾ 2 and that +ρT = {αi · · · αi+d−1 | 1 ⩽ i ⩽ n} ⊆ ρ. Then T and all the paths lying on T are in B Λ +♮ +(none of their +subpaths of length d are in σ); those of length δ(k) for some k ⩾ 0 are in BB. +In a similar way to Section 2.3, we make the following assumptions. +(i) none of the αi are A-loops. +(ii) no A-subcycle of T satisfies the same assumptions as T (that is, there is no A-subcycle q of +T of A-length at least 2, and ρq ⊆ ρ). +Proposition 3.13. Let T = α1 · · · αn be a closed A-trail with n ⩾ 2, ρT ⊆ ρ and such that assumptions (i) +and (ii) hold. Let p be an A-path of A-length d such that dβ(p) = 0 if β ∈ QA \ {α1, . . . , αn} and which is +not an A-subpath of a power of T. Then p ̸∈ ρ. +Proof. The proof is very similar to that of Proposition 2.11, replacing paths with A-paths and using +Consequence 3.5 in the proof that d > k. +□ +Remark 3.14. We keep the assumptions and notation of Proposition 3.13. Set Ti = αi · · · αnα1 · · · αi−1. +Then for any j ⩾ 1, we have +T +j +iαk ̸= 0 ⇐⇒ k = i − 1 +αkT +j +i ̸= 0 ⇐⇒ k = i. +Lemma 3.15. Let T = α1 · · · αn be a closed A-trail that satisfies assumptions (i) and (ii) and set zj = +∑n +i=1 T +j +i with nj = δ(u) for some u ⩾ 1. Then zj ∈ Z if, and only if, T satisfies Condition 3.6 (2). +Moreover, if T does not satisfy Condition 3.6 (2), then no element in B that is homogeneous with respect +to |·| and d (when viewed in Λ +♮ +) and that is a linear combination of non-trivial cycles lying on T is in Z. +Proof. The proof is very similar to that of Lemma 2.13, replacing paths with A-paths, again us- +ing Remark 3.11, replacing the d-covering property by Consequence 3.3 and Proposition 2.11 by +Proposition 3.13. Note also that for the proof of the last part, testing commutation with paths in +B2 gives s = n and λk = λk+d for all k and hence the result if |z| is odd; and if |z| is even, we +must use the fact that z commutes with elements in B3 in a similar way to obtain, in addition, that +λk = λk+d+1 for all k and hence that λk = λk+1 for all k. +□ +Proof of Theorem 3.10. We note first that if gldim Λ is finite then Λ satisfies (Fg) and Condition 3.6 +is empty. The implication (C2) ⇒ (C1) is Theorem 3.9. Again, the implication (C3) ⇒ (C4) is clear +and if, in addition, K is algebraically closed, then the implication (C1) ⇒ (C3) follows from [7]. It + +22 +JAWAD, SNASHALL, AND TAILLEFER +remains to prove that (C4) implies (C2). The proof is similar to that of Theorem 2.8, again replacing +paths with A-paths (we need not assume that the integers k are even). +□ +Remark 3.16. Suppose that K is algebraically closed. +We have now extended the equivalence +between (C1) and (C3), already known for Koszul algebras from [7], as well as d-Koszul monomial +algebras by Theorem 2.8, to (D, A)-stacked monomial algebras with D ̸= 2A whenever A > 1 . +In particular, we can extend Corollary 2.15 to (D, A)-stacked monomial algebras. +Corollary 3.17. Let Λ be a (D, A)-stacked monomial algebra over an algebraically closed field with D ̸= +2A whenever A > 1. Assume that E(Λ) is a finitely generated Zgr(E(Λ))-module. Then the algebra +Zgr(E(Λ)) is noetherian. +APPENDIX A. THE EXT ALGEBRA OF A (D, A)-STACKED MONOMIAL ALGEBRA +Leader and Snashall gave in [17] a presentation of the Yoneda algebra E(Λ) of a (D, A)-stacked +monomial algebra by quiver and relations. However, in our proof of Theorem 2.8 that (C4) implies +(C2) for d-Koszul monomial algebras, we used the description from [11, Sections 8 and 9] of E(Λ) +as an algebra contained, as a graded vector space, in the Koszul dual Λ +! +. +In this Appendix, we +generalise this description to (D, A)-stacked monomial algebras. +Throughout this section, Λ = KQ/I is a (D, A)-stacked monomial algebra with D = dA and +d ⩾ 2, where I an ideal generated by a set ρ of A-paths of A-length d = D +A. We view Λ as a quotient +of the tensor algebra: Λ = TΛ0(Λ1)/I. +All tensor products are taken over Λ0 and we write ⊗ for ⊗Λ0. The subspace S = I ∩ (Λ⊗D +1 +) = +span(ρ) of T = TΛ0(Λ1) is a Λ0-Λ0-submodule of Λ⊗D +1 +; it is finite dimensional over K. For an +element x ∈ T, write x for its image in Λ. Note that for 0 ⩽ i < D we have Λi = Λ⊗i +1 . +A.1. Generalised Koszul complex of S +Define spaces Hδ(n) ⊆ Tn +Λ0(Λ1) as follows: +H0 = Λ0, H1 = Λ1 and, for n ⩾ 2, Hδ(n) = +� +i+j=δ(n)−d +(Λ⊗i +A )⊗S⊗(Λ⊗j +A ). +For n ⩾ 0, let Pn be the right Λ-module defined by Pn = Hδ(n)⊗Λ; it is projective. +We have Hδ(1) = Λ1 = Hδ(0)⊗Λ1, Hδ(2) = S ⊆ Λ⊗D +1 += Hδ(1)⊗Λ⊗D−1 +1 +and, for any n ⩾ 3, +Hδ(n) ⊆ Hδ(n−1)⊗Λ⊗(δ(n)−δ(n−1)) +A +. Indeed, for any k ⩾ 1, +Hδ(2k+2) = H(k+1)d = +kd +� +j=0 +(Λ⊗(kd−j) +A +)⊗S⊗(Λ⊗j +A ) ⊆ +kd +� +j=d−1 +(Λ⊗(kd−j) +A +)⊗S⊗(Λ⊗j +A ) += +(k−1)d+1 +� +j=0 +(Λ⊗((k−1)d+1−j) +A +)⊗S⊗(Λ⊗j +A )⊗(Λ⊗(d−1) +A +) = Hδ(2k+1)⊗(Λ⊗(d−1) +A +) +Hδ(2k+1) = Hkd+1 = +kd+1 +� +j=0 +(Λ⊗((k−1)d+1−j) +A +)⊗S⊗(Λ⊗j +A ) ⊆ +kd+1 +� +j=A +(Λ⊗((k−1)d+1−j) +A +)⊗S⊗(Λ⊗j +A ) += +kd +� +j=0 +(Λ⊗((k−1)d−j) +A +)⊗S⊗(Λ⊗j +A )⊗ΛA = Hδ(2k)⊗ΛA +It follows that the maps F1 : Λ1⊗Λ → Λ0⊗Λ ∼= Λ, F2 : Λ⊗d +A ⊗Λ → Λ1⊗Λ and, for n ⩾ 3, +Fn : Λ⊗δ(n) +A +⊗Λ → Λ⊗δ(n−1) +A +⊗Λ defined by +F1(x1⊗λ) = x1λ +F2(x1⊗ · · · ⊗xD⊗λ) = x1⊗x2 · · · xDλ +Fn(y1⊗ · · · ⊗yδ(n)⊗λ) = y1⊗ · · · ⊗yδ(n−1)⊗yδ(n−1)+1 · · · yδ(n)λ, + +A COMBINATORIAL CHARACTERISATION OF (FG) +23 +where xi ∈ Λ1 and yi ∈ ΛA for all i, induce maps bn : Pn → Pn−1. More specifically, for all k ⩾ 1, +F2k+1(y1⊗ · · · ⊗ykd+1⊗λ) = y1⊗ · · · ⊗ykd⊗ykd+1λ if n = 2k + 1 is odd +F2k+2(y1⊗ · · · ⊗y(k+1)d⊗λ) = y1⊗ · · · ⊗ykd+1⊗ykd+2 · · · y(k+1)dλ if n = 2k + 2 is even. +Define also b0 : P0 = Λ0⊗Λ ∼= Λ → Λ0, which identifies with the natural projection. +Moreover, +for +n +⩾ +3 +we +have +Hδ(n+1) +⊆ +Hδ(n)⊗Λ⊗(δ(n+1)−δ(n)) +A +⊆ +Hδ(n−1)⊗Λ⊗(δ(n)−δ(n−1)) +A +⊗Λ⊗(δ(n+1)−δ(n)) +A += +Hδ(n−1)⊗Λ⊗d +A +and Hδ(n+1) +⊆ +Λ⊗δ(n−1) +A +⊗S, we +have Hδ(n+1) ⊆ +� +Λ⊗δ(n−1) +A +⊗S +� +∩ +� +Hδ(n−1)⊗Λ⊗d +A +� += Hδ(n−1)⊗S (all the spaces involved are +finitely generated and projective over Λ0) hence bn ◦ bn+1 = 0. It is easy to check that bn ◦ bn+1 = 0 +when n = 1, 2 or 3. +Therefore we have a complex (Pn, bn) of projective right Λ-modules. +Theorem A.1. Let Λ = KQ/I be a monomial algebra with I generated in degree D = dA with d ⩾ +2. Then Λ is (D, A)-stacked monomial if, and only if, (P•, b•) is a minimal projective right Λ-module +resolution of Λ0. +Proof. By construction, Pn is generated in degree δ(n). Therefore, if (P•, b•) is a minimal projective +right Λ-module resolution of Λ0, then Λ is a (D, A)-stacked monomial algebra. +Conversely, assume that Λ is a (D, A)-stacked monomial algebra. We already have a complex +(P•, b•) of right Λ-modules such that Pn is generated in degree δ(n). The beginning P1 +b1 +−→ P0 +b0 +−→ +Λ0 → 0 is exact. +We prove exactness at P2n+1 for n ⩾ 1 (the proof of exactness at P2n and at P1 is similar, without +the need for Consequence 3.3). +First note that b2n+2(P2n+2) is generated in degree δ(2n + 2) = (n + 1)D in P2n+1. +Let z = ∑i xnd+1,i⊗ · · · ⊗x1,i⊗λi be an element in Ker b2n+1 with xj,i ∈ ΛA and λi ∈ Λ +for all i, j. +Then ∑i xnd+1,i⊗ · · · ⊗x2,i⊗x1,iλi is in T⊗I. +It follows that λi ∈ � +k⩾D−A Λk and +that Ker b2n+1 is generated in degrees at least (n + 1)D. Therefore z can be rewritten as z = +∑i xnd+1,i⊗ · · · ⊗x1,i⊗yd−1,i · · · y1,iλ′ +i with yj,i ∈ ΛA and λ′ +i ∈ Λ for all i, j, with the yj,i right uni- +form and t(y1,i) ̸= t(y1,k) when i ̸= k. +Write λ′ +i = ∑l⩾0 λ′ +i,l with λ′ +i,l ∈ Λl for all i, l, and +λ′ +i,0 = t(y1,i). Then each of the ∑i xnd+1,i⊗ · · · ⊗x1,i⊗yd−1,i · · · y1,iλ′ +i,l is in Ker b2n+1 so in partic- +ular z′ := ∑i xnd+1,i⊗ · · · ⊗x1,i⊗yd−1,i · · · y1,i ∈ Ker b2n+1. +Consider z′′ := ∑i xnd+1,i⊗ · · · ⊗x1,i⊗yd−1,i⊗ · · · ⊗y1,i⊗t(y1,i) ∈ Λ⊗((n+1)d) +A +⊗Λ. We show that +z′′ ∈ P2n+2; this will imply that z′ = b2n+2(z′′) ∈ Im b2n+2. Since Λ is monomial and b2n+1(z′) = 0, +we may assume that that for all i, x1,iyd−1,i · · · y1,i is a path; since it is in I and has degree D, it is +in ρ = R2. By definition of P2n+1, we may assume that z is written so that for each i, xd,i · · · x1,i +is a path in ρ = R2. The path x1,iyd−1,i · · · y1,i properly overlaps xd,i · · · x1,i therefore, using Con- +sequence 3.3, it follows that for all k with 1 ⩽ k ⩽ d − 1 we have xk,i · · · x1,iyd−1,i · · · yk,i ∈ ρ +and hence z′′ ∈ �d−1 +k=1(Λ⊗(nd−k+1) +A +)⊗S⊗(Λ⊗(k−1) +A +)⊗Λ. Finally, using the fact that z ∈ P2n+1 = +Hnd+1⊗Λ, we get z′′ ∈ H(n+1)d⊗Λ = P2n+2. We have proved that (Ker b2n+1)(n+1)d ⊆ Im b2n+2. +Since Im b2n+2 is generated in degree (n + 1)d and Ker b2n+1 is generated in degrees at least +(n + 1)d, it follows that Ker b2n+1 ⊆ Im b2n+2 and finally that Ker b2n+1 = Im b2n+2. +Finally, since Im bn+1 is generated in degree δ(n + 1) and Pn is generated in degree δ(n) < +δ(n + 1), Im bn+1 ⊆ rPn for all n and therefore the resolution is minimal. +□ +A.2. The Ext algebra +A.2.1. Some duality results. We recall without proof some results stated in [2]. All modules are +finitely generated Λ0-Λ0-bimodules. All claims are easily checked for bimodules that are free +and finitely generated as left or as right Λ0-modules, and follow for arbitrary finitely generated +modules (since Λ0 is semisimple, all the Λ0-modules (left or right) are projective). +For any bimodules V and W, define V∗ = HomΛ0−(V, Λ0) and W +∗ += Hom−Λ0(W, Λ0); they are +Λ0-Λ0-bimodules, for the actions given for all e, e′ in Λ0, α ∈ V∗, β ∈ W +∗ +and v ∈ V, w ∈ W by: +(eαe′)(v) = α(ve)e′ and (eβe′)(w) = eβ(e′w). + +24 +JAWAD, SNASHALL, AND TAILLEFER +There are natural isomorphisms of Λ0-Λ0-bimodules Λ∗ +0 ∼= Λ0 ∼= Λ +∗ +0 which we view as identi- +fications. +There are also natural bimodule isomorphisms V ∼= (V∗) +∗ +and W ∼= ( W +∗ +)∗ of bimodules. +If V1 ⊆ V (respectively W1 ⊆ W), define V⊥ +1 += {α ∈ V∗ | α(V1) = 0} (respectively +W +⊥ +1 = +{β ∈ W +∗ +| β(W1) = 0}). If V1 (respectively W1) is a sub-bimodule, then they are sub-bimodules of +V∗ and W +∗ +respectively. +Fix a Λ0-Λ0-bimodule V. Then: +(i) if W is another Λ0-Λ0-bimodule, then there is an isomorphism of Λ0-Λ0-bimodules +ϕV,W : +V +∗ +⊗ W +∗ +→ +(W⊗V) +∗ +given for all α ∈ +V +∗ +, β ∈ +W +∗ +, v ∈ V and w ∈ W by +ϕV,W(α⊗β)(w⊗v) = α(β(w)v). There is a similar isomorphism with right duals, which +sends α⊗β to the map w⊗v �→ β(wα(v)); +(ii) if U and W are sub-bimodules of V, then +(U + W) +⊥ += +U +⊥ +∩ +W +⊥ +and (U + W)⊥ = U⊥ ∩ +W⊥; +(iii) if U is a sub-bimodule of V, then (V/U)∗ ∼= U⊥ and ( +∗ V/U) ∼= +U +⊥ +; +(iv) if W is a sub-bimodule of V, then for any idempotents ei, ej with (i, j) ∈ Q2 +0, we have +ei W +∗ +ej = ( +∗ ejWei); +(v) if U is a sub-bimodule of V, then for all i, j in Q0 we have dim ej U +⊥ +ei = dim eiVej − +dim eiUej = dim ejU⊥ei; +(vi) if U is a sub-bimodule of V, then under the identification of V with (V∗) +∗ +, we have +( +⊥ U⊥) +and under the identification of V with ( V +∗ +)∗, we have ( U +⊥ +)⊥; +(vii) if U is a sub-bimodule of V and W and Z are Λ0-Λ0-bimodules, there are bimodule iso- +morphisms +( +⊥ W⊗U⊗Z) ∼= Z +∗ ⊗ U +⊥ +⊗ W +∗ +and (W⊗U⊗Z)⊥ ∼= Z∗⊗U⊥⊗W∗. +A.2.2. Description of the Ext algebra. From the above, there is a natural isomorphism ( +∗ Λ⊗i +A ) ∼= +( Λ +∗ +A)⊗i, which we view as an identification. We also view S as a subspace of Λ⊗d +A (rather than +Λ⊗D +1 +). We may then consider +S +⊥ += +� +f ∈ ( Λ +∗ +A)⊗d | f(x) = 0 for all x ∈ S +� +. The dual algebra of +Λ is then Λ +♮ += TΛ0( Λ +∗ +A)/( S +⊥ ). It is a graded d-homogeneous algebra since +S +⊥ +is contained in +( Λ +∗ +A)⊗d, therefore Λ +♮ += � +n⩾0 Λ +♮ +n. +In terms of quivers, we have Λ0 = KQ0 and ΛA = KQA, the vector space whose basis is the set +QA of paths of length A in Q; moreover, Λ +∗ +A ∼= KQop +A using (iv). +Then Λ +♮ +is isomorphic to KΓ/( ρ +⊥ ) where Γ is the quiver with the same vertices as Q and whose +set of arrows is Γ1 = {α: i → j | there exists α ∈ QA, α: j → i} and where +ρ +⊥ +is the left orthogonal +of the set ρ viewed as a set of A-paths, for the bilinear form KΓd × K(QA)d → K defined on paths +of length d in Γ and A-paths of A-length d in Q by ⟨αd · · · α1, β1 · · · βd⟩ = 1 if α1 · · · αd = β1 · · · βd +and 0 otherwise, where the αi and βi are paths of length A. +The algebra KΓ/( ρ +⊥ ) is monomial, and the ideal ( ρ +⊥ ) has a minimal generating set σ given by +all the paths αd · · · α1 such that the A-path α1 · · · αd is not in ρ. In particular, if γ = αr . . . α1 is a path +in Γ, with αi ∈ QA for all i, then γ ̸∈ (σ) if, and only if, for each i with 1 ⩽ i ⩽ r − d + 1, the path +αi · · · αi+d−1 is in ρ (we use the fact that Λ is monomial here). +Lemma A.2. There is an isomorphism ( Λ +♮ +δ(n))∗ ∼= Hδ(n). +Proof. By definition, Λ +♮ +δ(n) = +( Λ +∗ +A)⊗δ(n) +∑ +δ(n)−d +i=0 +( Λ +∗ +A)⊗(δ(n)−d−i)⊗ S +⊥ ⊗( Λ +∗ +A)⊗i . Therefore we have +( Λ +♮ +δ(n))∗ ∼= +�δ(n)−d +∑ +i=0 +( Λ +∗ +A)⊗(δ(n)−d−i)⊗ S +⊥ ⊗( Λ +∗ +A)⊗i +�⊥ += +δ(n)−d +� +i=0 +� +( Λ +∗ +A)⊗(δ(n)−d−i)⊗ S +⊥ ⊗( Λ +∗ +A)⊗i�⊥ += +δ(n)−d +� +i=0 +� +(( Λ +∗ +A)⊗i) +∗⊗( S +⊥ ) +⊥⊗(( Λ +∗ +A)⊗(δ(n)−d−i)) +∗� + +A COMBINATORIAL CHARACTERISATION OF (FG) +25 +∼= +δ(n)−d +� +i=0 +� +Λ⊗i +A ⊗S⊗Λ⊗(δ(n)−d−i) +A +� += Hδ(n). +This isomorphism takes x = ∑ x1⊗ · · · ⊗xδ(n) ∈ Hδ(n) to the map gx : Λ +♮ +δ(n) → Λ0 defined by +gx(γδ(n)⊗ · · · ⊗γ1) = ∑ γδ(n)(γδ(n)−1(. . . (γ2(γ1(x1)x2) . . .)xδ(n)−1)xδ(n)) +where the xi are in ΛA and the γi are in Λ +∗ +A. +□ +Lemma A.3. There is an isomorphism ψ: Λ +♮ +δ(n) → HomΛ(Pn, Λ0) given by +ψ( f )(x1⊗ · · · ⊗xδ(n)⊗λ) = fδ(n)( fδ(n−1)(. . . ( f1(x1)) . . . xδ(n)−1)xδ(n))λ +where f = fδ(n)⊗ · · · ⊗ f1 ∈ Λ +♮ +δ(n) with fi ∈ Λ +∗ +1 for all i, xi ∈ Λ1 for all i and λ ∈ Λ. +Proof. The isomorphism ψ is the composition of the following isomorphisms: +• +Λ +♮ +δ(n) → Hom−Λ0(( Λ +♮ +δ(n))∗, Λ0), which sends f to the map [g �→ g( f )]; +• Hom−Λ0(( Λ +♮ +δ(n))∗, Λ0) → Hom−Λ0(Hδ(n), Λ0), which sends a map h to the map [x �→ +h(gx)], where gx is as in the proof of Lemma A.2; +• Hom−Λ0(Hδ(n), Λ0) → Hom−Λ(Hδ(n)⊗Λ, Λ0), which sends a map k to the map [x⊗λ �→ +k(x)λ]. +Applying these isomorphisms to f gives the expression in the statement. +□ +Let B be the vector space B = � +n⩾0 Bn where Bn = +Λ +♮ +δ(n). Define a multiplication on B as +follows: for x ∈ Bn and y ∈ Bm, set +x.y = + + + + + +0 +if n and m are odd; +0 +if n or m is equal to 1 and n ⩾ 1, m ⩾ 1; +xy +in Λ +♮ +otherwise. +The algebra B is a graded K-algebra generated in degrees 0, 1, 2 and 3. +We want to prove that E(Λ) ∼= B when A > 1 and D ̸= 2A. We first need a description of the +Yoneda product. +Proposition A.4. Let Λ be a (D, A)-stacked monomial algebra with A > 1, D = dA and d ⩾ 3. The +Yoneda product of fn ∈ HomΛ(Pn, Λ0) and fm ∈ HomΛ(Pm, Λ0) is given by +fn fm = + + + + + + + + + +0 if n and m are odd +0 if n ⩾ 1, m ⩾ 1 and n = 1 or m = 1 +∑ x1⊗ · · · ⊗xδ(n)+δ(m)⊗λ �→ fn(∑ fm(x1⊗ · · · ⊗xδ(m)⊗1)xδ(m)+1 +⊗xδ(m)+2⊗ · · · ⊗xδ(m)+δ(n)⊗λ) otherwise, +where the xi are all in ΛA. +Proof. If m and n are odd or if m ⩾ 1 or n ⩾ 1 is equal to 1 then, under the assumption that A > 1 +and D ̸= 2A, the Yoneda products vanish by [17, Theorem 3.4]. We now assume that m or n is even +and that m ̸= 1 and n ̸= 1. +Let σ: Λ0 → Λ be the natural inclusion. +Consider fm : Pm → Λ0; it lifts to f 0 +m = σ ◦ fm : Pm → Λ. We now define further liftings +f i +m : Pm+i → Pi for i ⩾ 1 as follows: +f 1 +m(x1⊗ · · · ⊗xδ(m+1)⊗λ) = f 0 +m(x1⊗ · · · ⊗xδ(m)⊗1)xδ(m)+1⊗xδ(m)+2 · · · xδ(m+1)λ +f i +m(y1⊗ · · · ⊗yδ(m+i)⊗λ) = f 0 +m(y1⊗ · · · ⊗yδ(m)⊗1)yδ(m)+1⊗ · · · ⊗yδ(m+i)⊗λ +if m or i ⩾ 2 is even +f i +m(y1⊗ · · · ⊗yδ(m+i)⊗λ) = f 0 +m(y1⊗ · · · ⊗yδ(m)⊗1)yδ(m)+1⊗ · · · ⊗yδ(m+i−1)+1 +⊗yδ(m+i−1)+2 · · · yδ(m+i)λ +if m and i ⩾ 2 are odd + +26 +JAWAD, SNASHALL, AND TAILLEFER +where the xi are in Λ1 and the yi are in ΛA. The proof that ( f i +m)i⩾0 is a family of liftings of fm, that +is, f i−1 ◦ bi+m = bi ◦ f i +m for all i ⩾ 1, is tedious but straightforward. +Finally, if n or m is even and n ⩾ 2, m ⩾ 2 ,then +fn fm(y1⊗ · · · ⊗yδ(m)+δ(n)⊗λ) = fn ◦ f n +m(y1⊗ · · · ⊗yδ(m+n)⊗λ) += fn( f 0 +m(y1⊗ · · · ⊗yδ(m)⊗1)yδ(m)+1⊗yδ(m)+2⊗ · · · ⊗yδ(m)+δ(n)⊗λ) += fn( fm(y1⊗ · · · ⊗yδ(m)⊗1)yδ(m)+1⊗yδ(m)+2⊗ · · · ⊗yδ(m)+δ(n)⊗λ). +□ +Theorem A.5. If Λ is a (D, A)-stacked monomial algebra, with A > 1, D = dA and d ⩾ 3 , then E(Λ) +is isomorphic to B as graded algebras. In particular, Extn +Λ(Λ0, Λ0) is isomorphic to Λ +♮ +δ(n). +Proof. We use the isomorphisms in Lemma A.3 and the cup-product described in Proposition A.4. +If f = fδ(n)⊗ · · · ⊗f1 ∈ Bn and g = gδ(m)⊗ · · · ⊗g1 ∈ Bm, where m or n is even and both are at least +2, then +ψ( f )ψ(g)(y1⊗ · · · ⊗yδ(m)+δ(n)⊗λ) = ψ( f ) +� +ψ(g)(y1⊗ · · · ⊗yδ(m)⊗1)yδ(m)+1⊗ · · · ⊗yδ(m+n)⊗λ +� += fδ(n)( fδ(n)−1(. . . ( f2( f1(ψ(g)(y1⊗ · · · ⊗yδ(m)⊗1)yδ(m)+1)yδ(m)+2) . . .)yδ′(m+n)−1)yδ(m+n))λ += fδ(n)( fδ(n)−1(. . . ( f2( f1(gδ(m)(. . . (g1(y1)) . . . yδ(m))yδ(m)+1)yδ(m)+2) . . .)yδ(m+n)−1)yδ(m+n))λ += ψ( f g)(y1⊗ · · · ⊗yδ(m)+δ(n)⊗λ) += ψ( f · g)(y1⊗ · · · ⊗yδ(m)+δ(n)⊗λ) +therefore ϕ( f · g) = ψ( f )ψ(g) and we have proved that ψ is an isomorphism of graded algebras. +□ +Remark A.6. Recall from Subsection 1.2 that the m-th projective in a minimal projective right Λ- +module resolution of Λ0 is Lm = � +Rm +i ∈Rm t(Rm +i )Λ. By Consequence 3.4, Rm +i ∈ Pm. We then have +an isomorphism Pm → Lm which is determined by Rm +i �→ t(Rm +i ) for all i. +As we mentioned in Subsection 1.2, the authors of [14] also gave a basis of E(Λ), namely the set +� +gm +i ∈ HomΛ(Lm, Λ0) | Rm +i ∈ Rm� +where gm +i (t(Rm +j )) = t(Rm +i ) if j = i and 0 otherwise. The element +gm +i corresponds to a map in HomΛ(Pm, Λ0) which we denote again by gm +i and that is defined by +gm +i (Rm +j ) = t(Rm +i ) if j = i and 0 otherwise. +We have isomorphisms KQA ∼= KΓ1 and KQA ∼= +Λ +∗ +A = +(KQA) +∗ +which combine to the iso- +morphism which associates to α ∈ Γ1 the linear form fα on KQA that sends β ∈ QA to t(α) if +β = α and to 0 otherwise. This extends to an isomorphism between the algebras KΓ/(σ) and Λ +♮ +that sends a path p of length n to the class of the linear map fp ∈ ( Λ +∗ +A)⊗n defined on A-paths by +fp(q) = t(p) if q = p and 0 otherwise. +Now consider Rm +i = α1 · · · αδ(m) with α ∈ QA for all i and R +m +i = αδ(m) · · · α1 in KΓ/(σ). It is easy +to check that gm +i = ψ +� +fαδ(m)⊗ · · · ⊗ fα1 +� +so that it corresponds, via the isomorphism above, to R +m +i . +Therefore we have a basis BB = +� +R +m +i | Rm +i ∈ Rm� +of B. +REFERENCES +[1] M.J. BARDZELL, The alternating syzygy behavior of monomial algebras, J. Algebra 188 (1997), p 69-89. +[2] A. BEILINSON, V. GINZBURG and W. SOERGEL, Koszul duality patterns in representation theory, JAMS 9 (1996), +p 473-527. +[3] R. BERGER, Koszulity for nonquadratic algebras, J. Algebra 239 (2001), p 705-734. +[4] V. DOTSENKO, V. GÉLINAS and P. TAMAROFF, Finite generation for Hochschild cohomology of Gorenstein +monomial algebras, (2019), arxiv:1909.00487v1. +[5] K. ERDMANN, Algebras with non-periodic bounded modules, J. Algebra 475 (2017), p 308-326. +[6] K. ERDMANN, M. HOLLOWAY, N. SNASHALL, Ø. SOLBERG and R. TAILLEFER, Support varieties for selfinjective +algebras, K Theory 33 (2004), p 67-87. +[7] K. ERDMANN and Ø. SOLBERG, Radical cube zero weakly symmetric algebras and support varieties, JPAA 215 +(2011), p 185-200. +[8] T. FURUYA and N. SNASHALL, Support varieties for modules over stacked monomial algebras, Comm. Alg. 39 +(2011), p 2926-2942. + +A COMBINATORIAL CHARACTERISATION OF (FG) +27 +[9] E.L. GREEN, D. HAPPEL and D. ZACHARIA, Projective resolutions over Artin algebras with zero relations, Illinois +J. Math. 29 (1985), p 180-190. +[10] E. L. GREEN and R. MARTINEZ-VILLA, Koszul and Yoneda algebras II, CMS Conf Proc Algebras and modules, II +(Geiranger, 1996) Amer. Math. Soc. 24 (1998), p 227-244. +[11] E. L. GREEN, E. N. MARCOS, R. MARTÍNEZ-VILLA and P. ZHANG, D-Koszul algebras, JPAA 193 (2004), p 141-162. +[12] E.L. GREEN and N. SNASHALL, Finite generation of Ext for a generalization of D-Koszul algebras, J. Algebra 295 +(2006), p 458-472. +[13] E. L. GREEN and N. SNASHALL, The Hochschild cohomology ring modulo nilpotence of a stacked monomial +algebra, Colloq. Math. 105 (2006), p 233-258. +[14] E. L. GREEN and D. ZACHARIA, The cohomology ring of a monomial algebra, Manus. Math. 85 (1994), p 11-23. +[15] R. JAWAD, Cohomology and finiteness conditions for generalisations of Koszul algebras, PhD, University of +Leicester (2019). +[16] J. KÜLSHAMMER, C. PSAROUDAKIS and Ø. SKARTSÆTERHAGEN, Derived invariance of support varieties, Proc. +Amer. Math. Soc. 147 (2019), p 1-14. +[17] J. LEADER and N. SNASHALL, The Ext Algebra and a New Generalization of D-Koszul Algebras, Quart. J. Math. 68 +(2016), p 433-458. +[18] A. POLISHCHUK and L. POSITSELSKI, Quadratic algebras, American Mathematical Society 37 (2005). +[19] C. PSAROUDAKIS, Ø. SKARTSÆTERHAGEN and Ø. SOLBERG, Gorenstein categories, singular equivalences and +finite generation of cohomology rings in recollements, Trans. Amer. Math. Soc. Ser. B 1 (2014), p 45-95. +[20] S. SCHROLL and N. SNASHALL, Hochschild cohomology and support varieties for tame Hecke algebras, Quart. J. +Math. 62 (2011), p 1017 1029. +[21] R. Y. SHARP, Steps in Commutative Algebra, LMS Student Texts 19, CUP, 1990. +[22] Ø. SKARTSAETERHAGEN, Singular equivalence and the (Fg) condition, J. Algebra 452 (2016), p 66-93. +[23] N. SNASHALL and Ø. SOLBERG, Support varieties and Hochschild cohomology rings, Proc. London Math. Soc. (3) +88 (2004), p 705-732. +[24] N. SNASHALL and R. TAILLEFER, Hochschild cohomology of socle deformations of a class of Koszul self-injective +algebras, Colloq. Math. 119 (2010), p 79-93. +[25] N. SNASHALL and R. TAILLEFER, The Hochschild cohomology ring of a class of special biserial algebras, J. Algebra +Appl. 9 (2010), p 73–122. +[26] S. WITHERSPOON, Varieties for modules of finite dimensional Hopf algebras, Geometric and topological aspects of the +representation theory of finite groups, Springer Proc. Math. Stat. 242 (2018), p 481-495. +RUAA JAWAD, TECHNICAL INSTRUCTORS TRAINING INSTITUTE, MIDDLE TECHNICAL UNIVERSITY, BAGHDAD, +IRAQ +Email address: ruaa.yousuf.jawad@mtu.edu.iq +NICOLE SNASHALL, SCHOOL OF COMPUTING AND MATHEMATICAL SCIENCES, UNIVERSITY OF LEICESTER, UNI- +VERSITY ROAD, LEICESTER LE1 7RH, UNITED KINGDOM +Email address: njs5@leicester.ac.uk +RACHEL TAILLEFER, UNIVERSITÉ CLERMONT AUVERGNE, CNRS, LMBP, F-63000 CLERMONT-FERRAND, FRANCE +Email address: rachel.taillefer@uca.fr + diff --git a/vNE5T4oBgHgl3EQfLw6U/content/tmp_files/load_file.txt b/vNE5T4oBgHgl3EQfLw6U/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..dae0e5eeb4d583c5ce09ac28ce80f109eaa7544b --- /dev/null +++ b/vNE5T4oBgHgl3EQfLw6U/content/tmp_files/load_file.txt @@ -0,0 +1,1599 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf,len=1598 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='05476v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='KT] 13 Jan 2023 A COMBINATORIAL CHARACTERISATION OF d-KOSZUL AND (D, A)-STACKED MONOMIAL ALGEBRAS THAT SATISFY (FG) RUAA JAWAD, NICOLE SNASHALL, AND RACHEL TAILLEFER ABSTRACT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Condition (Fg) was introduced in [6] to ensure that the theory of support varieties of a finite dimensional algebra, established by Snashall and Solberg, has some similar properties to that of a group algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' In this paper we give some easy to check combinatorial conditions that are equivalent to (Fg) for monomial d-Koszul algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' We then extend this to monomial (D, A)-stacked algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' We also extend the description of the Yoneda algebra of a d-Koszul algebra in [11] to (D, A)-stacked monomial algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' INTRODUCTION Let Λ be an indecomposable finite dimensional algebra over a field K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' In this Introduction, we shall assume that K is algebraically closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Support varieties for modules over Λ were introduced by Snashall and Solberg in [23], as a geometric tool to study the representation theory of Λ, using the Hochschild cohomology HH∗(Λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' It was then proved in [6] that many of the properties of support varieties for group algebras have analogues in this more general case, providing some finiteness conditions hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' These are now known as (Fg) and can be expressed in the following way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Let r be the Jacobson radical of Λ and let E(Λ) = Ext∗ Λ(Λ/r, Λ/r) be its Yoneda algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Then Condition (Fg) states that: (Fg1) there is a commutative noetherian graded subalgebra H of HH∗(Λ) with H0 = HH0(Λ) such that (Fg2) E(Λ) is a finitely generated H-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' In particular, it was shown in [6] that if Λ satisfies Condition (Fg), then Λ is necessarily Gorenstein, that the variety of a module is trivial if and only if the module has finite projective dimension, and periodic modules can be characterised up to projective summands as those whose support variety is a line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Moreover, the converse of the first result mentioned was proved for monomial algebras in [4], that is, a Gorenstein monomial algebra satisfies (Fg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Support varieties for group algebras have been very effective in the study of the representations of these algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Therefore Condition (Fg) has been much studied as it ensures a similarly useful theory of support varieties for finite dimensional algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' For instance, Condition (Fg) is invariant under various constructions, such as derived equivalence or singular equivalence of Morita type, see [16, 22, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Condition (Fg) has been studied or shown to hold for large families of algebras in [24, 26, 25, 5] among others, and support varieties have been studied for algebras that satisfy (Fg), see for instance [8, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Since Hochschild cohomology is generally very difficult to compute, Condition (Fg) can be diffi- cult to establish for a given algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' It is therefore useful to have necessary, sufficient or equivalent conditions for (Fg) to hold for a given algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' One such result was proved by Erdmann and Solberg in [7], where they showed that if (Fg) holds for Λ, then the graded centre Zgr(E(Λ)) of the Yoneda algebra is a noetherian algebra and E(Λ) is a finitely generated Zgr(E(Λ))-module;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' moreover, they proved that this is an equivalence when the algebra Λ is Koszul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' For monomial al- gebras, (Fg) was proved in [4] to be equivalent to the related condition that the A∞-centre Z∞(E(Λ)) is a noetherian algebra and E(Λ) is a finitely generated Z∞(E(Λ))-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' We note that if Λ has Date: 16th January 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' 2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' 16G20, 16E40, 16S37, 16E65, 16E30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' d-Koszul, Ext algebra, Hochschild cohomology, finiteness condition, (D, A)-stacked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Some of these results formed part of the first author’s PhD thesis at the University of Leicester, which was supported by The Higher Committee For Education Development in Iraq (HCED) Reference D1201116.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' 1 2 JAWAD, SNASHALL, AND TAILLEFER finite global dimension then both E(Λ) and HH∗(Λ) are finite dimensional as vector spaces, and Λ has (Fg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Thus we are particularly interested in algebras of infinite global dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' The aim of this paper is to prove that a number of conditions are equivalent to (Fg) for a large category of algebras, namely finite dimensional d-Koszul, and more generally (D, A)-stacked, monomial algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' This is motivated in particular by a result of the first author in her PhD thesis, who gave a sufficient and not difficult to check condition for d-Koszul monomial algebras to satisfy (Fg);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' this result is Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='7 in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Berger introduced d-Koszul algebras in [3] as a natural generalisation of Koszul algebras (which occur as 2-Koszul algebras).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' They are the algebras such that the n-th projective module in a min- imal projective resolution of Λ/r as a Λ-module is generated in a specific degree denoted by δ(n) (with δ(n) = n if d = 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Moreover, they were characterised in [11] as the algebras Λ that are d-homogeneous (that is, their ideal of relations can be generated by a set of homogeneous ele- ments of degree d) and such that E(Λ) is generated in degrees 0, 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' The (D, A)-stacked monomial algebras, where D ⩾ 2 and A ⩾ 1 are integers, were introduced by Green and Snashall in [13], and those of infinite global dimension were characterised by the same authors in [12] as the monomial algebras such that the n-th projective module in a minimal projective resolution of Λ/r as a Λ-module is generated in precisely one degree and such that E(Λ) is finitely generated (in which case E(Λ) is generated in degrees 0, 1, 2 and 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' In particular, when A = 1, a (D, 1)-stacked monomial algebra is D-Koszul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Thus (D, A)-stacked monomial algebras are natural generalisa- tions of d-Koszul and indeed Koszul monomial algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' In this paper, we consider Condition (Fg) for d-Koszul monomial algebras and more generally for (D, A)-stacked monomial algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' We introduce some combinatorial conditions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='4 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='6 that are easy to check in terms of a minimal set of relations for the algebra Λ, and we prove that they are equivalent to (Fg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' This gives a very practical way of checking whether a monomial d- Koszul or (D, A)-stacked algebra satisfies (Fg), because it is easy to check that a monomial algebra is d-Koszul using [11, Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='2] (recalled in Property 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='1) or (D, A)-stacked using [13, Section 3] (recalled in Property 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' To summarise, if Λ is a finite dimensional monomial K-algebra that is d-Koszul with d ⩾ 2 or (D, A)-stacked with D ̸= 2A whenever A > 1, then the following conditions are equivalent: (C1) Λ satisfies (Fg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' (C2) Λ satisfies some combinatorial conditions defined in Conditions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='4 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='6 (this follows from Theorems 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='8 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' (C3) Zgr(E(Λ)) is noetherian and E(Λ) is a finitely generated Zgr(E(Λ))-module (by [7] and Theorems 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='8 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' (C4) E(Λ) is a finitely generated Zgr(E(Λ))-module (again by Theorems 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='8 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' (C5) Z∞(E(Λ)) is noetherian and E(Λ) is a finitely generated Z∞(E(Λ))-module (by [4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' (C6) Λ is Gorenstein (by [4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' The paper is organised as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' In Section 1 we give some background on monomial algebras and the notion of overlaps, as well as on the Yoneda algebra and the Hochschild cohomology of a monomial algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Section 2 is devoted to the proof of the implications (C2)⇒(C1) and (C4)⇒(C2) for d-Koszul monomial algebras, which completes the equivalence of all the conditions above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' The first implication relies on a presentation of the Hochschild cohomology for (D, A)-stacked monomial algebras from [13] and the second one uses a description of the Yoneda algebra E(Λ) of a d-Koszul algebra Λ as a graded subspace of the Koszul dual algebra Λ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' from [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' In Section 3, we extend these results to (D, A)-stacked monomial algebras where D ̸= 2A whenever A > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Here again, we use a description of E(Λ) as a subspace of an analogue Λ ♮ of the Koszul dual of Λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' this description is detailed and proved in the Appendix, and is a generalisation of the corresponding result of [11] to (D, A)-stacked monomial algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' General assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Throughout the paper, Λ is an indecomposable finite dimensional algebra over a field K with char(K) ̸= 2 that is not necessarily algebraically closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Moreover, we assume that Λ = KQ/I where Q is a finite quiver (it has a finite number of vertices and arrows) and I is an admissible ideal in KQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' If Λ = KQ/I is also a monomial algebra then I is generated by a minimal set ρ of paths (monomials) and Λ is graded by the length of paths;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' we denote by ℓ(p) the length A COMBINATORIAL CHARACTERISATION OF (FG) 3 of a path p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Note that paths in any algebra given by quiver and relations are written from left to right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' For any j ⩾ 0, we shall denote by Qj the set of paths of length j in Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' In order to use the results in [13], we shall need to assume that gldim Λ ⩾ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' However, if Λ is a monomial algebra with finite global dimension, all the conditions (C1)–(C6) hold for Λ (we note that Condition (C2) is necessarily empty in this case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Therefore we do not lose any generality in making this assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' SOME BACKGROUND ON MONOMIAL ALGEBRAS AND THEIR COHOMOLOGY 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Overlaps Keeping the above assumptions, let Λ = KQ/I be a monomial algebra so that Λ = ⊕i⩾0Λi is a graded algebra with the length grading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' We denote by r = ⊕i⩾1Λi the radical of Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' An arrow α starts at the vertex o(α) and ends at the vertex t(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' If p = α1α2 · · · αn is a path with α1, α2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' , αn in Q1 then o(p) = o(α1) and t(p) = t(αn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' A path p is a prefix of a path q if there is some path p′ such that q = pp′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' if an arrow α is a prefix of q then we say that q begins with α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' A path p is a suffix of a path q if there is some path p′ such that q = p′p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' if an arrow α is a suffix of q then we say that q ends with α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' We use the concept of overlaps of [9] and [14] to describe the minimal projective resolution of Λ0 ∼= Λ/r over Λ, and to describe the minimal projective resolution of Λ over Λe, where Λe is the enveloping algebra Λop ⊗K Λ of Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' We recall the relevant definitions here using the notation of [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' (1) A path q overlaps a path p with overlap pu if there are paths u and v such that pu = vq and 1 ⩽ ℓ(u) < ℓ(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' We illustrate the definition with the following diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' ✤ p ✤ � v �✤ q ✤ � u � Note that we allow ℓ(v) = 0 here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' (2) A path q properly overlaps a path p with overlap pu if q overlaps p and ℓ(v) ⩾ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' (3) A path p has no overlaps with a path q if p does not properly overlap q and q does not properly overlap p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' We now define sets Rn recursively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Let R0 = Q0, the set of vertices of Q R1 = Q1, the set of arrows of Q R2 = ρ, the minimal generating set for I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' For n ⩾ 3, the construction is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' (1) For n ⩾ 3, we say that R2 ∈ R2 maximally overlaps Rn−1 ∈ Rn−1 with overlap Rn = Rn−1u if (a) Rn−1 = Rn−2p for some path p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' (b) R2 overlaps p with overlap pu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' (c) there is no element of R2 which overlaps p with overlap being a proper prefix of pu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' We may also say that Rn is a maximal overlap of R2 ∈ R2 with Rn−1 ∈ Rn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' The construction of Rn is illustrated in the following diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' ✤ Rn−2 ✤ ✤ Rn−1 ✤ � p � ✤ R2 ✤ � u � (2) For n ⩾ 3, the set Rn is defined to be the set of all overlaps Rn formed in this way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' We also recall from [14] that if Rn 1 p = Rn 2q, for Rn 1, Rn 2 ∈ Rn and paths p, q, then Rn 1 = Rn 2 and p = q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Any element Rn in Rn may be expressed uniquely as Rn−1 j aj and as bkRn−1 k for some Rn−1 j , Rn−1 k in Rn−1 and paths aj, bk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' We say that the elements Rn−1 j and Rn−1 k occur in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' 4 JAWAD, SNASHALL, AND TAILLEFER 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' The Ext algebra E(Λ) The Ext algebra E(Λ) is given by E(Λ) = Ext∗ Λ(Λ0, Λ0) = � n⩾0 Extn Λ(Λ0, Λ0) with the Yoneda product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' In the terminology of overlaps, the n-th projective module in a minimal projective Λ- resolution of Λ0 is � Rn∈Rn t(Rn)Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Then Extn Λ(Λ0, Λ0) has a basis indexed by Rn and E(Λ) has a basis indexed by � n⩾0 Rn (see [14, 9]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' We identify Rn i ∈ Rn with the corresponding element of Extn Λ(Λ0, Λ0), that is, with the map � Rn∈Rn t(Rn)Λ → Λ0 given by t(Rn)λ �→ � t(Rn i )λ + r if Rn = Rn i 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' The Hochschild cohomology ring HH∗(Λ) Let (P∗, ∂∗) be the minimal projective Λe-resolution of Λ from [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' We write ⊗ for ⊗K through- out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Then P n = � Rn∈Rn Λo(Rn) ⊗ t(Rn)Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' The maps are given as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' In odd degrees, if R2n+1 = R2n j aj = bkR2n k ∈ R2n+1 then ∂2n+1 : P2n+1 → P2n is given by o(R2n+1) ⊗ t(R2n+1) �→ o(R2n j ) ⊗ aj − bk ⊗ t(R2n k ) where the first tensor lies in the summand corresponding to R2n j and the second tensor lies in the summand corresponding to R2n k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' For even degrees, any element R2n in R2n may be expressed in the form pjR2n−1 j qj for some R2n−1 j ∈ R2n−1 and paths pj, qj with n ⩾ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Let R2n = p1R2n−1 1 q1 = · · · = prR2n−1 r qr be all expressions of R2n which contain some element of R2n−1 as a subpath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Then, for R2n ∈ R2n, the map ∂2n : P2n → P2n−1 is given by o(R2n) ⊗ t(R2n) �→ r ∑ j=1 pj ⊗ qj where the tensor pj ⊗ qj lies in the summand of P2n−1 corresponding to R2n−1 j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' If not specified, then it will always be clear from the context in which summand of a projective module our tensors lie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' The Hochschild cohomology ring HH∗(Λ) of Λ is given by HH∗(Λ) = Ext∗ Λe(Λ, Λ) = � n⩾0 Extn Λe(Λ, Λ) with the Yoneda product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' CHARACTERISATIONS OF d-KOSZUL MONOMIAL ALGEBRAS THAT SATISFY (FG) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Notation and properties of d-Koszul monomial algebras Let Λ = KQ/I be a monomial algebra, where Q is a finite quiver and I is an admissible ideal in KQ generated by a minimal set ρ of paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Recall that the algebra Λ = � i⩾0 Λi is graded by the length of paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' We can express Λ as a quotient Λ = TΛ0(Λ1)/I of the tensor algebra, where Λ/r ∼= Λ0 = KQ0 and Λ1 = KQ1 and I is an ideal generated by a minimal set ρ of monomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' The algebra Λ0 ∼= K|Q0| is isomorphic to a finite product of copies of the base field K;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' it is therefore a semisimple and commutative K-algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' We denote by ei the idempotent in Λ0 corresponding to the vertex i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Let d ⩾ 2 be an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' We assume that Λ is a d-Koszul algebra, that is, for any minimal projective right Λ-module resolution of Λ0, the n-th projective module is generated in degree δ(n) where δ(n) = � n 2 d if n is even n−1 2 d + 1 if n is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' It follows that Λ is d-homogeneous (that is, ρ consists of paths of length d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' A COMBINATORIAL CHARACTERISATION OF (FG) 5 The monomial d-Koszul algebras can be characterised as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Property 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' [11, Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='2] A finite dimensional d-homogeneous monomial algebra Λ = KQ/I is d-Koszul if, and only if, ρ is d-covering, that is, for any paths p, q and r in Q, (pq ∈ ρ, qr ∈ ρ, ℓ(q) ⩾ 1) ⇒ (all subpaths of pqr of length d are in ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Note that this condition is always satisfied if d = 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' it is indeed well known that all finite dimensional quadratic monomial algebras are Koszul, see [14] and [18, Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Let Λ = KQ/I where Q is the quiver γ3 �❃❃❃❃❃❃❃❃ γ2 � β � γ1 � α � and the ideal I has minimal generating set ρ = {α3, γ1γ2γ3, γ2γ3γ1, γ3γ1γ2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Then Λ is a 3-Koszul monomial algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' From now on, we assume that Λ = KQ/I is a finite dimensional d-Koszul monomial algebra with d ⩾ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' We have the following consequences of Property 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Consequence 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' [15, Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='13] Let Rn i be an element in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Then all subpaths of Rn i of length d are in ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' The result is proved by induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' It is clear when n = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Moreover, if n = 3, since R3 i ∈ R3 is a maximal overlap of two elements in R2, it follows from Property 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Now let n ⩾ 4 be an integer and take Rn i ∈ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Then Rn i is a maximal overlap of R2 1 ∈ R2 with Rn−1 2 ∈ Rn−1 so that Rn i = Rn−1 2 u for some path u, and Rn−1 2 is a maximal overlap of R2 3 ∈ R2 with Rn−2 4 ∈ Rn−2 so that Rn−1 2 = Rn−2 4 u′ for some path u′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' This can be illustrated as follows: ✤ Rn i ✤ ✤ Rn−2 4 ✤ ✤ Rn−1 2 ✤ ✤ R2 3 ✤ ✤ R2 1 ✤ � u′ � � u � Moreover, ℓ(u′u) = ℓ(Rn i ) − ℓ(Rn−2 4 ) = δ(n) − δ(n − 2) = d so u′u = R2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' By induction, every subpath of Rn−1 2 of length d is in ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Any other subpath of length d of Rn i is either u′u = R2 1 ∈ R2 or a proper subpath of R2 3u, therefore it is in ρ by Property 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' We have proved the induction step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' □ A trail in Q is a path T = α1 · · · αn with n ⩾ 1 such that the arrows αi are all distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' We say that the trail is closed when t(αn) = o(α1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' A path q is said to lie on the closed trail T if q is a subpath of Tm for some m ⩾ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' We say that two trails are distinct if neither lies on the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' We now have a second consequence of Property 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Consequence 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' [15, Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='14] Suppose that T = α1 · · · αn is a closed trail in Q and that d ⩾ n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Then all paths of length d that lie on the closed trail T are in ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Since Λ is finite dimensional, there is a path R2 ∈ ρ that lies on T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Now, ℓ(R2) = d and d ⩾ n + 1 so, without loss of generality, we may suppose that R2 = (α1α2 · · · αn)mα1α2 · · · αs for some 1 ⩽ s ⩽ n with d = nm + s and m ⩾ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Let p = (α1α2 · · · αn)m, q = α1α2 · · · αs and r = (αs+1 · · · αnα1 · · · αs)m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Then pq = R2 = qr and we can apply Property 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='1 so that all subpaths of pqr of length d are in ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Now, any path of length d that lies on the closed trail T is a subpath of pqr and hence is in ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' □ We now introduce Condition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Jawad showed in her PhD thesis [15] that this condition is sufficient for Λ to satisfy (Fg);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' we give a proof in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='7 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' 6 JAWAD, SNASHALL, AND TAILLEFER Condition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' [15, Theorems 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='11 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='15] (1) Let α be a loop in Q1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Then αd ∈ ρ but there is no path in ρ of the form αd−1β or βαd−1 where β is an arrow that is distinct from α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' (2) Let T = α1 · · · αn be a closed trail in Q with n > 1 and αi ∈ Q1 for all i and such that ρT := {α1 · · · αd, α2 · · · αdαd+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' , αnα1 · · · αd−1} ⊆ ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Then there are no elements in ρ \\ ρT which begin or end with the arrow αi, for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' If T = α1 · · · αn is a closed trail then the subscript i of αi is taken modulo n within the range 1 ⩽ i ⩽ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Thus ρT is the set of all paths of length d that lie on the closed trail T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Suppose that Condition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='4 is non-empty, that is, there is a loop or a closed trail with the given properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Then the description of the projective modules in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='2 using overlaps shows that Λ0 has infinite projective dimension as a Λ-module, and hence Λ has infinite global dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Condition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='4 is sufficient for Λ to satisfy (Fg) The proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='7 uses the description of the Hochschild cohomology ring modulo nil- potence of a (D, A)-stacked monomial algebra from [13, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' We recall the definition of a (D, A)-stacked monomial algebra in Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' The Hochschild cohomology ring modulo nilpotence is the quotient HH∗(Λ)/N where N is the ideal of HH∗(Λ) that is generated by the homogeneous nilpotent elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' It is well-known that HH∗(Λ) is a graded commutative ring, so, since char(K) ̸= 2, every homogeneous element of odd degree squares to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Moreover, N is the set of all nilpotent elements of HH∗(Λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Our calculations involving HH∗(Λ) use the minimal projective Λe-resolution (P∗, ∂∗) of Λ from [1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' see Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Noting that a d-Koszul monomial algebra is a (d, 1)-stacked monomial algebra (see [13]), we apply [13, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='4] in the special case where D = d and A = 1, and this simplifies the hypotheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Specifically, if there is a closed path C in Q with CD/A ∈ ρ then Cd ∈ ρ and it is immediate that C has length 1 and is necessarily a loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' [15, Theorems 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='11 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='15] Let Λ = KQ/I be a finite dimensional d-Koszul monomial algebra with d ⩾ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Assume that Λ satisfies Condition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Then Λ satisfies (Fg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' We keep the notation of Condition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Let α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' , αu be the loops in the quiver Q, and suppose that αi is a loop at the vertex vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Since Λ is a finite dimensional d-Koszul monomial algebra, αd i is necessarily in the minimal generating set ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' By Condition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='4 (1), for each i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' , u, there are no elements in ρ of the form αd−1 i β or βαd−1 i where β is an arrow that is distinct from αi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' We need to show that there are no overlaps of αd i with any element of ρ \\ {αd i }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' This is immediate if d = 2, so suppose that d ⩾ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' If R ∈ ρ \\ {αd i } and R overlaps αd i , then either R = αs ib or R = bαs i where 1 ⩽ s ⩽ d − 1 and b is a path of length d − s which does not begin (respectively, end) with the arrow αi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Suppose first that R = αs ib.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Then R overlaps αd i with overlap of length 2d − s as follows: � αd−s i � ✤ αd i ✤ ✤ R ✤ � b � This is a maximal overlap since αi is not the first arrow of b and thus gives an element R3 1 ∈ R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' However, ℓ(R3 1) = d + 1 since Λ is d-Koszul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Thus 2d − s = d + 1 and so s = d − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' But then R = αd−1 i b and b is an arrow distinct from αi, which contradicts our hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' The case where R = bαs i is similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' So there are no overlaps of αd i with any element of ρ \\ {αd i }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Moreover, as Λ is a finite dimensional monomial algebra, it follows that the vertices v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' , vu are distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Let Tu+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' , Tr be the distinct closed trails in Q such that all paths of length d that lie on these closed trails are contained in ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' For each i = u + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' , r, we write Ti = αi,1 · · · αi,mi, where the αi,j are arrows, and set ρTi = {αi,1 · · · αi,d, αi,2 · · · αi,d+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' , αi,miαi,1 · · · αi,d−1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' A COMBINATORIAL CHARACTERISATION OF (FG) 7 Then ρTi is contained in ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' By Condition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='4 (2), for each closed trail Ti (i = u + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' , r), there are no elements in ρ \\ ρTi which begin or end with the arrow αi,j, for all j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' , mi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' So no arrow αi,j has overlaps with any element in ρ \\ ρTi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' For i = u + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' , r, let Ti,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' , Ti,mi be defined by Ti,1 = Ti = αi,1αi,2 · · · αi,mi Ti,2 = αi,2αi,3 · · · αi,miαi,1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Ti,mi = αi,miαi,1 · · · αi,mi−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Then the paths Ti,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' , Ti,mi are all of length mi and lie on the closed path Ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' We now show that Λ satisfies (Fg1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' As noted above, Λ is a (d, 1)-stacked monomial algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Moreover, Condition (Fg) is always satisfied if the global dimension of Λ is finite, therefore we may assume that gldim Λ ⩾ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Hence we can apply [13, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='4], which gives HH∗(Λ)/N ∼= K[x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' , xr]/⟨xaxb for a ̸= b⟩, where for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' , u, the vertices v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' , vu are distinct and the element xi corresponding to the loop αi is in degree 2 and is represented by the map P2 −→ Λ where for R2 ∈ R2, o(R2) ⊗ t(R2) �→ � vi if R2 = αd i 0 otherwise and for i = u + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' , r, the element xi corresponding to the closed trail Ti = αi,1 · · · αi,mi is in degree 2µi such that µi = mi/ gcd(d, mi) and is represented by the map P2µi −→ Λ, where for R2µi ∈ R2µi, o(R2µi) ⊗ t(R2µi) �→ � o(Ti,k) if R2µi = Td/ gcd(d,mi) i,k for all k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' , mi 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Let H be the subring of HH∗(Λ) generated by Z(Λ) and {x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' , xr}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Since Z(Λ) = HH0(Λ) and HH∗(Λ) is graded commutative, it follows that H = Z(Λ)[x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' , xr]/⟨xaxb for a ̸= b⟩ and so H is a commutative ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Moreover, Z(Λ) is finite dimensional so is a commutative Noeth- erian ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Thus H is a Noetherian ring (see [21, Corollary 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Therefore Λ satisfies (Fg1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' The rest of this proof shows that Λ satisfies (Fg2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Following the discussion in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='2, we identify � n⩾0 Rn with a basis of E(Λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' The action of a homogeneous element x ∈ HHn(Λ) on E(Λ) is then given by left multiplication by ∑j Rn j where the sum is over all j such that x(o(Rn j ) ⊗ t(Rn j )) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Thus if xi ∈ HH2(Λ) corresponds to the loop αi, then the action of xi on E(Λ) is given by left multiplication by αd i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' And if xi in degree 2µi corresponds to the closed trail Ti, then the action of xi on E(Λ) is given by left multiplication by ∑mi k=1 Td/ gcd(d,mi) i,k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Set N = max{3, |x1|, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' , |xr|, |Q1|}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' We show that �N n=0 Rn is a generating set for E(Λ) as a left H-module and thus E(Λ) is finitely generated as a left H-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Let R ∈ Rn with n > N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Then ℓ(R) = δ(n) ⩾ 2d and we can write R = a1a2 · · · aδ(n) where the ai are in Q1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' From Consequence 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='2, all subpaths of R of length d are in ρ, so we may illustrate R with the following diagram: ✤ ✤ a1 ✤ ✤ a2 · · · ad ✤ ad+1 · · · ✤ ✤ aδ(n)−d+1 · · · aδ(n) Now, n > N ⩾ |Q1| so there is some repeated arrow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Choose j, k with k minimal and k ⩾ 1 such that aj is a repeated arrow, aj, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' , aj+k−1 are all distinct arrows and aj+k = aj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Write R = (a1 · · · aj−1)(aj · · · aj+k−1)(ajaj+k+1 · · · aδ(n)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' There are two cases to consider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Case (1): k = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Then aj = aj+1 and so aj is a loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' It follows that R = (a1 · · · aj−1)(ajaj)(aj+2 · · · aδ(n)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' 8 JAWAD, SNASHALL, AND TAILLEFER Suppose first that j ⩽ d − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Then j + d − 1 ⩽ δ(n) so from Consequence 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='2, a2 j aj+2 · · · aj+d−1 is in ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' But ad j ∈ ρ and we have already shown that there are no overlaps of ad j with any element of ρ \\ {ad j }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Thus aj = aj+2 = · · · = aj+d−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Inductively we see that R = (a1 · · · aj−1)aδ(n)−j+1 j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Similarly, a1 · · · aj−1ad−j+1 j is in ρ and d − j + 1 ⩾ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Again, there are no overlaps of ad j with any element of ρ \\ {ad j } so aj = a1 = · · · = aj−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Thus R = aδ(n) j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Now suppose that j ⩾ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Then j − d + 1 ⩾ 1, so by Consequence 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='2, aj−d+1 · · · aj−1aj is in ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' As there are no overlaps of ad j with any element of ρ \\ {ad j }, it follows that aj−d+1 = · · · = aj−1 = aj, and inductively R = aj+1 j (aj+2 · · · aδ(n)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Using Consequence 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='2 again, ad−1 j aj+2 is in ρ so aj = aj+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Inductively, we have R = aδ(n) j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Hence, for all j, R = aδ(n) j = � (ad j )(n/2) if n even (ad j )((n−1)/2)aj if n odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Let xi be the generator in H corresponding to the loop aj, so 1 ⩽ i ⩽ u and |xi| = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Then xi acts on E(Λ) as left multiplication by ad j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Hence R = � (xi)(n/2)o(aj) if n even (xi)((n−1)/2)aj if n odd with xi ∈ H, o(aj) ∈ R0 and aj ∈ R1, so that o(aj) and aj are in �N n=0 Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Case (2): k > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' We note by our choice of j, k that aj · · · aj+k−1 is a closed trail of length k, which we denote by T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Let ρT be the set of all paths of length d which lie on T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' The first step is to show that ρT is contained in ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' If d ⩾ k + 1, then this follows from Con- sequence 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' So, suppose that d ⩽ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Recall that R = (a1 · · · aj−1)(aj · · · aj+k−1)(ajaj+k+1 · · · aδ(n)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Then: ajaj+1 · · · aj+d−1, aj+1aj+2 · · · aj+d, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' aj+k−daj+k−d+1 · · · aj+k−1, aj+k−d+1aj+k−d+2 · · · aj+k−1aj are all paths of length d which are subpaths of R, and so, by Consequence 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='2, are in ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Now ajaj+1 · · · aj+d−1 overlaps aj+k−d+1aj+k−d+2 · · · aj+k−1aj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' So there is an element R2 1 ∈ ρ such that R2 1 maximally overlaps aj+k−d+1aj+k−d+2 · · · aj+k−1aj with maximal overlap of length d + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Then we have that R2 1 = aj+k−d+2aj+k−d+3 · · · aj+k−1ajaj+1 and this maximal overlap is � aj+k−d+1aj+k−d+2 · · · aj+k−1aj � aj+1 = aj+k−d+1R2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Continuing in this way, aj+1aj+2 · · · aj+d overlaps R2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' So there is an element R2 2 ∈ ρ such that R2 2 maximally overlaps R2 1 with maximal overlap of length d + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' So R2 2 = aj+k−d+3aj+k−d+4 · · · aj+k−1ajaj+1aj+2 and this maximal overlap is R2 1aj+2 = aj+k−d+2R2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Inductively, we see that every path of length d on the closed trail T is in ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Hence ρT is contained in ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' It follows from Condition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='4 (2), that there are no paths in ρ \\ ρT which begin or end with any of the arrows aj, aj+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' , aj+k−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Next we show that R can be written in the form R = p1Tqp2, where p1 is a suffix of T and p2 is a prefix of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' If d = 2 then ajaj+k+1 is a subpath of R of length 2 and hence is in ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' By Condition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='4 (2) ajaj+k+1 must be in ρT and so aj+k+1 = aj+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Then aj+k+1aj+k+2 = aj+1aj+k+2 and is a subpath of R A COMBINATORIAL CHARACTERISATION OF (FG) 9 of length 2, so we must have that aj+k+2 = aj+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Inductively, we see that R lies on the closed trail T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' So R = p1Tqp2, where p1 is a suffix of T and p2 is a prefix of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' So let d ⩾ 3, and suppose first that d ⩽ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Then aj+k−d+2 · · · aj+k−1ajaj+k+1 is a subpath of R of length d which begins with the arrow aj+k−d+2 ∈ {aj, aj+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' , aj+k−1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' So, by Consequence 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='2 and Condition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='4 (2), this path is in ρT and hence aj+k+1 = aj+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Inductively, we have aj+k+2 = aj+2, aj+k+3 = aj+3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Similarly, aj−1aj · · · aj+d−2 is a subpath of R of length d which ends with the arrow aj+d−2 ∈ {aj, aj+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' , aj+k−1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' So this path is in ρT and hence aj−1 = aj+k−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Inductively, we have aj−2 = aj+k−2, aj−3 = aj+k−3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' So we may write R = p1Tqp2, where p1 is a suffix of T and p2 is a prefix of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Now suppose that d ⩾ k + 1 (with d ⩾ 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' We consider j ⩽ d − 1 and j ⩾ d separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Let j ⩽ d − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Then j + d < δ(n), so aj+1aj+2 · · · aj+k−1ajaj+k+1 · · · aj+d is a subpath of R of length d and starts with the arrow aj+1 ∈ {aj, aj+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' , aj+k−1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' So by Consequence 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='2 and Condi- tion 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='4 (2), this path is in ρT and hence aj+k+1 = aj+1, aj+k+2 = aj+2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' So, inductively, we may write R = (a1 · · · aj−1)Tqp2, where p2 is a prefix of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Now a1a2 · · · aj−1 · · · ad is a subpath of R of length d and ends with the arrow ad ∈ {aj, aj+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' , aj+k−1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' So by Condition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='4 (2), this path is in ρT and hence aj−1 = aj+k−1, aj−2 = aj+k−2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Thus R = p1Tqp2, where p1 is a suffix of T and p2 is a prefix of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Finally, suppose j ⩾ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Then, we know that aj+k−d · · · aj−1aj · · · aj+k−1 is a subpath of R of length d and ends with the arrow aj+k−1 ∈ {aj, aj+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' , aj+k−1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' So by Con- sequence 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='2 and Condition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='4 (2), this path is in ρT and hence aj−1 = aj+k−1, aj−2 = aj+k−2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Also aj+k−d+2 · · · aj−1aj · · · aj+k+1 is a subpath of R of length d and starts with the arrow aj+k−d+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' But we have just shown that aj+k−d+2 ∈ {aj, aj+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' , aj+k−1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' So again, this path is in ρT and hence aj+k+1 = aj+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Inductively, aj+k+2 = aj+2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Thus R = p1Tqp2, where p1 is a suffix of T and p2 is a prefix of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' So, in all cases, R = p1Tqp2, where T = aj · · · aj+k−1, p1 is a suffix of T and p2 is a prefix of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Without loss of generality, relabel the trail T and write R = Tqp, where T = a1 · · · ak, p is a prefix of T, δ(n) = kq + ℓ(p), and we choose ℓ(p) in the range 1 ⩽ ℓ(p) ⩽ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Note that R has a repeated arrow so q ⩾ 1, and if q = 1 then ℓ(p) ⩾ 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' moreover if ℓ(p) = k then p = T and R = Tq+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Let xi be the generator in H corresponding to this closed trail T, so u + 1 ⩽ i ⩽ r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Let Ti,1 = T = a1a2 · · · ak;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Ti,2 = a2a3 · · · aka1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Ti,k = aka1 · · · ak−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' The action of xi on E(Λ) is left multiplication by Td/ gcd(d,k) i,1 + Td/ gcd(d,k) i,2 + · · · + Td/ gcd(d,k) i,k and |xi| = 2k/ gcd(d, k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Consequently, N ⩾ 2k/ gcd(d, k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Now R = Tqp with 1 ⩽ ℓ(p) ⩽ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Write q = d gcd(d,k)c + w with 0 ⩽ w ⩽ d gcd(d,k) − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Then R = � Td/ gcd(d,k)�c (Twp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Moreover, from the construction of R as a maximal overlap, we see that Twp is also constructed as a maximal overlap and so corresponds to a basis element of E(Λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' We have ℓ(Twp) = kw + ℓ(p) ⩽ k � d gcd(d,k) − 1 � + k = kd/ gcd(d, k) = δ(2k/ gcd(d, k)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' So Twp corresponds to a basis element of E(Λ) of degree at most 2k/ gcd(d, k), that is, Twp is in Rm for some m ⩽ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Let 2 ⩽ l ⩽ k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' we show that Td/ gcd(d,k) i,l (Twp) = 0 in E(Λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' We have Ti,l = alal+1 · · · aka1 · · · al−1, T = a1a2 · · · ak and p = a1a2 · · · aℓ(p) with 1 ⩽ ℓ(p) ⩽ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' If Td/ gcd(d,k) i,l (Twp) represents a non-zero element in E(Λ), then t(al−1) = o(a1) so that a1 · · · al−1 is a closed trail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' But l − 1 < k, so this contradicts the minimality of k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Hence Td/ gcd(d,k) i,l (Twp) = 0 in E(Λ) for 2 ⩽ l ⩽ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' 10 JAWAD, SNASHALL, AND TAILLEFER A similar argument also shows that � k ∑ l=1 Td/ gcd(d,k) i,l �c = k ∑ l=1 � Td/ gcd(d,k) i,l �c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Thus R = � Td/ gcd(d,k)�c (Twp) = k ∑ l=1 � Td/ gcd(d,k) i,l �c (Twp) = � k ∑ l=1 Td/ gcd(d,k) i,l �c (Twp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Hence R = xc i (Twp) with xi in H, and Twp ∈ Rm for some m ⩽ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Hence each R ∈ Rn with n > N can be written in the form hr for some h ∈ H and r ∈ �N n=0 Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' It follows that �N n=0 Rn is a generating set for E(Λ) as a left H-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Thus Λ satisfies (Fg2) and we conclude that Λ has (Fg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' □ Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' We return to Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Condition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='4 is satisfied: the only closed trails that are not loops are the cycles of length 3 (whose arrows are the γi);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' for all of these closed trails T, we have ρT = ρ \\ � α3� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Hence by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='7 the algebra Λ = KQ/I satisfies (Fg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Conditions equivalent to (Fg) for a d-Koszul monomial algebra Our aim is now to prove the converse, and more precisely, the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Let Λ be an indecomposable finite dimensional d-Koszul monomial K-algebra with d ⩾ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Consider the following statements: (C1) Λ satisfies (Fg);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' (C2) Condition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='4 holds for Λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' (C3) Zgr(E(Λ)) is noetherian and E(Λ) is a finitely generated Zgr(E(Λ))-module;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' (C4) E(Λ) is finitely generated as a module over Zgr(E(Λ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Then (C4) implies (C2) which in turn implies (C1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Moreover, if the field K is algebraically closed, then the four statements are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' We shall need the description of the Ext algebra E(Λ) from [11], which we recall here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Let Qop be the opposite quiver of Q, so that Qop 0 = Q0 and Qop 1 = {α: j → i | there is α: i → j in Q1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Now consider Λ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' = KQop/J with J = ( ρ ⊥ ), where the orthogonal is taken with respect to the natural bilinear form KQop d × KQd → K, that is, ⟨βd · · · β1, α1 · · · αd⟩ is equal to 1 if α1 · · · αd = β1 · · · βd and is equal to 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' (Recall that, for any n ⩾ 0, Qn denotes the set of paths of length n in Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=') If d = 2, set B = Λ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' and if d ⩾ 3 let B = � n⩾0 Bn be the algebra defined as follows: Bn = Λ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' δ(n) for x ∈ Bn and y ∈ Bm, define x · y ∈ Bn+m by x · y = � 0 if n and m are odd xy (in Λ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' ) if n or m is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Note that if n or m is even, δ(n) + δ(m) = δ(n + m), so that this defines a graded algebra B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Then by [10, 2, 11], the algebras E(Λ) and B are isomorphic (for d ⩾ 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Since Λ is monomial it is easy to see that the algebra Λ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' is d-homogeneous monomial and that the set σ of paths αd · · · α1 ∈ Qop d such that α1 · · · αd ∈ Qd is not in ρ is a minimal generating set for J consisting of paths of length d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' There is a basis B Λ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' of Λ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' consisting of all paths p in Qop such that no subpath of p is in σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' It follows from Consequence 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='2 that no subpath of length d of R n i is in σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Therefore R n i ∈ B Λ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' As we mentioned in Subsection 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='2, there is a basis of E(Λ) indexed by � n⩾0 Rn that corres- ponds, via the isomorphism with the algebra B, to the set of paths R n i for n ⩾ 0 and Rn i ∈ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' We then have an embedding of the basis � n⩾0 R n of B into B Λ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' where R n = � R n i | Rn i ∈ Rn� ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' denote by BB its image, which is a basis of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' A COMBINATORIAL CHARACTERISATION OF (FG) 11 We now define several gradings on Λ, Λ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' There are natural gradings on Λ and on Λ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' given by the lengths of paths;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' denote the length by ℓ for both algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' The degree of a homogeneous element x in B will be denoted by |x|, so x ∈ Λ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' δ(|x|) or, in other terms, |x| = k if, and only if, ℓ(x) = δ(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' The algebra Λ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' is also multi-graded by NQ1: for each path p in Qop, we define an element d(p) = (dα(p))α∈Q1 ∈ NQ1 as follows: if ℓ(p) = 0, then d(p) = (0)α∈Q1 if ℓ(p) > 0, then dα(p) is the number of occurrences of α in p (it is 0 if α does not occur in p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Since Λ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' is monomial, the ideal J is homogeneous with respect to this multi-degree and therefore Λ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' is multi-graded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' In B, if x and y are homogeneous and |x| or |y| is even, then dα(xy) = dα(x) + dα(y) but if both degrees are odd then dα(xy) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Let Z := Zgr(B) be the graded centre of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' It is generated as a subring of B by the homogeneous elements z ∈ B such that for all homogeneous y ∈ B, zy = (−1)|y||z|yz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Note that Z ⊂ � e∈Q0 eBe, therefore Z is generated by elements that are linear combinations of (non-zero) cycles in Qop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Moreover, it can be checked easily that the graded centre Z of B is generated by elements z that are homogeneous with respect to the grading |·| and the multi-degree d and such that, for any element y ∈ B that is homogeneous with respect to the grading |·|, we have zy = (−1)|y||z|yz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' If d = 2 then B ∼= E(Λ) is generated in degrees 0 and 1, so in order to check that an element of B is in Z, it is sufficient to check that it is a linear combination of cycles and that it commutes or anti-commutes with all arrows in Qop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' If d ⩾ 3, then B ∼= E(Λ) is generated in degrees 0, 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Therefore when checking that an element is in Z, we need to check that it is a linear combination of cycles and that it commutes or anti-commutes with paths of degrees 1 and 2, that is, arrows and (non-zero) paths of length d in Qop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' The proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='8 relies on some preliminary results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' These are Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='10, Propos- ition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='11 and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' For the first of these, we start with the following observation about loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Suppose α is a loop in Q1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Since Λ is finite dimensional, there is some integer N such that αN ∈ I;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' therefore αN has a subpath of length d that is in ρ and so αd ∈ ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Therefore αd ̸∈ σ and it follows that αj ̸= 0 in Λ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' for all j ⩾ 0 and that αδ(j) ̸= 0 in B for all j ⩾ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Let α be a loop in Q1 and let n ⩾ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Then if d = 2, αn ∈ Z if, and only if, n is even and α satisfies Condition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='4 (1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' if d ⩾ 3, αδ(n) ∈ Z if, and only if, α satisfies Condition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='4 (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' First note that if n is odd, then if d = 2, αnα = αn+1 ̸= (−1)nα αn, therefore αδ(n) = αn ̸∈ Z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' if d ⩾ 3, αδ(n) anti-commutes with all arrows since the products are 0 in B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Therefore we may assume that n ⩾ 2 is an even integer and that αδ(n) ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Set e = o(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Let β be an arrow ending at e with β ̸= α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Then, in B, αδ(n)+d−1β = αδ(n) · αd−1β = αd−1β · αδ(n) = αd−1βαδ(n) (the element αd−1β is in degree 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Therefore we have an equality αδ(n)+d−1β = αd−1βαδ(n) between two paths in the monomial algebra Λ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' , so that both paths are zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' In particular, αδ(n)+d−1β contains a subpath in σ, and since αd ̸∈ σ, we must have αd−1β ∈ σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' It follows that βαd−1 ̸∈ ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Similarly, if β is an arrow that starts at e with β ̸= α, then αd−1β ̸∈ ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Therefore α satisfies Condition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='4 (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Conversely , assume that α satisfies Condition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='4 (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Let β ̸= α be an arrow and let n ⩾ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' By assumption, αd−1β ̸∈ ρ and βαd−1 ̸∈ ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' It follows that βαd−1 = 0 = αd−1β in Λ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' (either in σ or not composable) and therefore that αδ(n)β = 0 = βαδ(n) since δ(n) ⩾ δ(2) ⩾ d − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' The path αδ(n) anticommutes with all elements of degree 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' In particular, if d = 2 and n is even, then αn ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' 12 JAWAD, SNASHALL, AND TAILLEFER Now assume that d ⩾ 3 and consider commutation with elements in B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' As a vector space, B2 is generated by the paths p of length d such that p ∈ ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Let p = β1 · · · βd be a path in ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Since p has degree 2 in B, products of elements in B with p in B and in Λ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' are equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' If p = αd, then clearly αδ(n)p = p αδ(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' If p ̸= αd, set j = min {i | 1 ⩽ i ⩽ d, βi ̸= α} and k = max {i | 1 ⩽ i ⩽ d, βi ̸= α}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' By assump- tion, αd−1βj ̸∈ ρ and βkαd−1 ̸∈ ρ, therefore βjαd−1 = 0 and αd−1βk = 0 in Λ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' We have assumed that n ⩾ 2, so δ(n) ⩾ d and therefore αδ(n)p = 0 = p αδ(n) in Λ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' and in B, and αδ(n) anticommutes with elements of degree 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Finally, we have proved that αδ(n) is in Z whenever d = 2 and n ⩾ 2 is even or d ⩾ 3 and n ⩾ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' □ For the next result, we need some more terminology for closed trails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Let n ⩾ 2 and let T = α1 · · · αn be a closed trail in Q with ρT = {αi · · · αi+d−1 | 1 ⩽ i ⩽ n} ⊆ ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' A subcycle of T is a cycle of the form q = αi · · · αj with 1 ⩽ i ⩽ j ⩽ n and ℓ(q) < ℓ(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' We say that T has a repeated vertex if T = α1 · · · αi−1vαi · · · αi+k−1vαi+k · · · αn for some i, k and vertex v such that the paths αi · · · αi+k−1 and αi+k · · · αnα1 · · · αi−1 are non-trivial paths in KQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' We make the following assumptions that we use in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='11 and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' The reason for these specific assumptions becomes clear in the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' (i) none of the αi are loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' (ii) no subcycle of T satisfies the same assumptions as T (that is, there is no subcycle q of T of length at least 2 with ρq ⊆ ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Let T = α1 · · · αn be a closed trail with n ⩾ 2, ρT ⊆ ρ and such that assumptions (i) and (ii) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Let p be a path of length d such that dβ(p) = 0 if β ̸∈ {α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' , αn} and which does not lie on T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Then p ̸∈ ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Let p = γ1 · · · γd with γi ∈ {α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' , αn} for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' , d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' The path p is a non-zero path in KQ so t(γi) = o(γi+1) for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Suppose that γ1 = αj so p = αjγ2 · · · γd and γ2 ∈ {α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' , αn} with t(αj) = o(αj+1) = o(γ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' If T does not have a repeated vertex then necessarily αj+1 = γ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Inductively, p = αjαj+1 · · · αj+d−1 and hence p lies on the trail T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' This contradicts our hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Hence T has a repeated vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Suppose that v is a repeated vertex so that T has a proper subpath q of length k with q = vqv for some k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' thus 2 ⩽ k ⩽ n − 1 since T does not have any loops and q is a closed trail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' We claim that k ⩾ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Indeed, if we had k < d, then by Consequence 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='3 every path of length d that lies on q would be in ρ, that is ρq ⊆ ρ, with ℓ(q) < ℓ(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' But this contradicts assumption (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Hence k ⩾ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Now suppose for contradiction that p ∈ ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' As above, let αj be the first arrow in p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' We know that p does not lie on T and that T has a repeated vertex, so we may write p = αj · · · αj+r−1γr+1 · · · γd where r ⩾ 1, γr+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' , γd ∈ {α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' , αn}, t(αj+r−1) = o(αj+r) = o(γr+1) and γr+1 ̸= αj+r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Then there is some t such that γr+1 = αt and t ̸≡ j + r (mod n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Moreover t(αt−1) = o(αt) = o(αj+r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' We may illustrate this as follows: � · αj+r−1� · αj+r �☎☎☎☎☎☎☎☎ αt � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' � αt+1 � αt−1 �✿✿✿✿✿✿✿✿ � (We make no assumption as to whether γr+2 is or is not equal to αt+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=') We note that the path αj+r · · · αt−1 · αt · · · αj+r−1 is a cyclic permutation of T and has length n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Moreover, from the previ- ous part of this proof, both αj+r · · · αt−1 and αt · · · αj+r−1 are paths of length at least d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Let S = αtαt+1 · · · αj+r−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Then S is a closed path in KQ of length at least 2 and is a subcycle of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' There is an overlap αj+r−d · · · αj · · · αj+r−1 with p so the subpath αj+r−d+1 · · · αj+r−1αt must also A COMBINATORIAL CHARACTERISATION OF (FG) 13 be in ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Then we have an overlap of αj+r−d+1 · · · αj+r−1αt with αtαt+1 · · · αt+d−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' by Property 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='1, all subpaths of length d of the path αj+r−d+1 · · · αj+r−1αtαt+1 · · · αt+d−1 must also be in ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Thus every path of length d that lies on S is in ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Hence ρS ⊆ ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' So S is a subcycle of T that satisfies the same assumptions as T, and this contradicts assump- tion (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Hence p ̸∈ ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' □ Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' We keep the assumptions and notation of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Then T and all the paths lying on T are in the basis B Λ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' since none of their subpaths of length d are in σ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' those of length δ(k) for some k ⩾ 0 are in the basis BB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Set Ti = αi · · · αnα1 · · · αi−1 so that Ti = αi−1 · · · α1αn · · · αi ∈ Λ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Then for any j ⩾ 1, we have T j iαk ̸= 0 ⇐⇒ k = i − 1 αkT j i ̸= 0 ⇐⇒ k = i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Let T = α1 · · · αn be a closed trail with n ⩾ 2 and ρT ⊆ ρ that satisfies assumptions (i) and (ii) and set zj = ∑n i=1 T j i with nj = δ(u) for some integer u ⩾ 2 and with nj = u even if d = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Then zj ∈ Z if, and only if, T satisfies Condition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='4 (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Moreover, if T does not satisfy Condition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='4 (2), then no element in B that is homogeneous with respect to |·| and d (when viewed in Λ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' ) and that is a linear combination of non-trivial cycles that lie on T is in Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' First assume that zj ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Fix an integer i and let e = t(αi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Suppose for contradiction that there is a path p of length d − 1 starting at e such that αip ∈ ρ and p ̸= αi+1 · · · αi+d−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' By Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='12, at least one arrow in p is not in {α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' , αn}, therefore p αi is not a subpath of any of the paths that occur in zj (note that ℓ(zj) ⩾ δ(2) = d ⩾ ℓ(p)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' The relation αipzj = zjαip in B becomes, in Λ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' , p αiT j i = zjp αi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Since Λ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' is monomial, it follows that p αiT j i contains a subpath in σ, that is, there is a subpath of length d of Tj i αip that is not in ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' It cannot be a subpath of Tj i αi because we have assumed that ρT ⊆ ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Moreover, we have assumed that αip ∈ ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' We also have αi−d+1 · · · αi−1αi ∈ ρT ⊆ ρ and ρ is d-covering (Property 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='1), therefore every subpath of length d of αi−d+1 · · · αi−1αip is also in ρ and so is every subpath of length d of Tj i αip and we have obtained a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Therefore T satisfies Condition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='4 (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Conversely , assume that T satisfies Condition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='4 (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' We prove that for all j ⩾ 1 such that nj = δ(u) for some integer u ⩾ 2, and such that nj = u is even if d = 2, we have zj ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' First note that if d ⩾ 3 and ��zj �� is odd, then zj anti-commutes with all arrows (the products are 0 in B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Therefore assume that ��zj �� is even and that d ⩾ 2, and let β be an arrow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' If β = αk for some k, then αkzj = αkT j k and zjαk = T j k+1αk using Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='12, and these paths are indeed equal, so that βzj = zjβ = (−1)|zj||β|zjβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' If β ̸∈ {α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' , αn} then β T j i = 0 = T j iβ for all i by assumption, therefore βzj = 0 = (−1)|zj||β|zjβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Now assume that d ⩾ 3 (and ��zj �� is still even) and consider commutation with elements in B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' As a vector space, B2 is generated by the paths p of length d such that p ∈ ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Let p = β1 · · · βd be a path in ρ with βi ∈ Q1 for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Since p has degree 2 in B, products of elements in B with p in B and in Λ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' are equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' If p lies on T, then p = αk · · · αk+d−1 for some k, and it is easy to check that pzj = zjp = (−1)|zj||p|zjp (as for commutation with an arrow).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' If p does not lie on T, then by Condition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='4 (2), the first and last arrows in p are not in {α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' , αn}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Moreover, for all i, we have αiβ1 · · · βd−1 ̸∈ ρ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' it follows that pzj = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Similarly, for all i, we have β2 · · · βdαi ̸∈ ρ (since it ends with αi and is not in ρT), therefore zjp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Finally, pzj = zjp = (−1)|zj||p|zjp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' 14 JAWAD, SNASHALL, AND TAILLEFER We have proved that zj ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Now let z be an element in Z that is homogeneous with respect to |·| and d and that is a linear combination of cycles that lie on T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Assume that ℓ(z) > 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' by (i) z is not a linear combination of arrows so |z| ⩾ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Put z = ∑m i=1 λici with λi ∈ K and the ci cycles in Qop that lie on T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Since z is homogeneous with respect to |·| and d and the ci lie on T, the ci are cyclic permutations of c1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Up to relabelling, we may write ci = T j−1 i αi−1 · · · αi−s for some fixed s with 1 ⩽ s ⩽ n (and m = n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' We first consider the case where |z| is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Then we must have αkz = zαk for all k, that is, ∑n i=1 λiαkT j−1 i αi−1 · · · αi−s = ∑n i=1 λiT j−1 i αi−1 · · · αi−sαk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Using Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='12, this is equivalent to λkαkT j−1 k αk−1 · · · αk−s = λk+s+1T j−1 k+s+1αk+s · · · αk+1αk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Therefore λk = 0 = λk+s+1 or k + s ≡ k (mod n) (that is, s = n), and λk = λk+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' This is true for all k, so if z ̸= 0 then z = λ1zj with nj = ℓ(z) = δ(|z|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' We now consider the case where |z| is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' If d = 2 then the same reasoning as in the even case shows that λk+1 = (−1)|z|λk for all k with 1 ⩽ k ⩽ n − 1 and λ1 = (−1)|z|λn, therefore λ1 = (−1)n|z|λ1 = −λ1 (because nj = ℓ(z) = |z| is odd hence n is odd) so that λk = 0 for all k and finally z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Now assume that d ⩾ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Since z ∈ Z, we have αk+d−1 · · · αkz = zαk+d−1 · · · αk for all k, that is, ∑n i=1 λiαk+d−1 · · · αkT j−1 i αi−1 · · · αi−s = ∑n i=1 λiT j−1 i αi−1 · · · αi−sαk+d−1 · · · αk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Using Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='12, this is equivalent to λkαk+d−1 · · · αkT j−1 k αk−1 · · · αk−s = λk+s+dT j−1 k+s+dαk+s+d−1 · · · αk+dαk+d−1 · · · αk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Therefore λk = 0 = λk+s+d or k + s + d − 1 ≡ k + d − 1 (mod n) (that is, s = n), and λk = λk+d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' When s = n, we have nj = ℓ(z) = δ(|z|) = |z| − 1 2 d + 1, therefore n and d are coprime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' It follows that if z ̸= 0, then all the λi are equal so that z = λ1zj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' We have proved that if z is a non-zero element in Z that is homogeneous with respect to |·| and d and which is a linear combination of non-trivial cycles that lie on T, then if d = 2 we must have |z| even and for all d ⩾ 2, z is then a non-zero scalar multiple of zj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Therefore zj is in Z and by the first part of the proof, T satisfies Condition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='4 (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' □ We have now all the tools we need for the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' The fact that (C2) implies (C1) is Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' The implication (C3) ⇒ (C4) is clear and if, in addition, K is algebraically closed, then the implication (C1) ⇒ (C3) follows from [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' We now prove that (C4) implies (C2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Suppose that (C2) does not hold, that is, Condition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='4 does not hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Assume for contradiction that B is a finitely generated Z-module, generated by elements g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' , gt that are homogeneous with respect to both the grading |·| and the multi-grading d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' We first assume that Condition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='4 (1) does not hold, so that there is a loop α that does not satisfy this condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Then for all j ⩾ 2, αδ(j) ̸∈ Z by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Now consider αδ(k) ∈ B for some even integer k ⩾ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Then dβ(αδ(k)) = � δ(k) if β = α 0 if β ̸= α and ���αδ(k)��� = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' By assumption, and using the fact that αδ(k), g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=', gt are homogeneous with respect to |·| and d, there exist elements u(k) i in Z, 1 ⩽ i ⩽ t, that are homogeneous with respect to |·| and d, such that αδ(k) = ∑t i=1 u(k) i gi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Since αδ(k) is homogeneous with respect to |·| and d, we can assume that for all i we have ���u(k) i gi ��� = k and d(u(k) i ) + d(gi) = d(u(k) i gi) = d(αδ(k)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' If β ̸= α then dβ(u(k) i ) + dβ(gi) = 0 so dβ(u(k) i ) = 0 and dβ(gi) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' It follows that u(k) i and gi are powers of α;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' since u(k) i ∈ Z, we must A COMBINATORIAL CHARACTERISATION OF (FG) 15 have ���u(k) i ��� = 0 or 1 by assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' If ���u(k) i ��� = 1, then |gi| = k − 1 is odd and hence, in B, we have u(k) i gi = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Therefore we may assume that u(k) i ∈ Z0 = K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' It follows that the sum contains only one term and that gi is a (non-zero) scalar multiple of αδ(k) so that αδ(k) ∈ spanK {g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' , gt}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' We have shown that � αδ(k) | k ⩾ 1, k even � ⊆ spanK {g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' , gt}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' However, using the grading |·|, we see that the αδ(k), k ⩾ 1, are linearly independent over K: we have reached a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Therefore the Yoneda algebra E(Λ) = B is not finitely generated as a Z-module when Condi- tion 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='4 (1) does not hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' We now assume that Condition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='4 (2) does not hold, so that there is a closed trail T = α1 · · · αn with n ⩾ 2 and ρT ⊆ ρ that does not satisfy this condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' We can make the following assumptions (and therefore use Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='11): (i) none of the αi are loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Indeed, if αi is a loop, then the paths αiαi+1 · · · αi+d−1 and αd i are in ρ and properly overlap, hence αd−1 i αi+1 is in ρ because ρ is d-covering, therefore α does not satisfy Condition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='4 (1), and in this case we already know that E(Λ) is not a finitely generated Z-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' (ii) no subcycle of T satisfies the same hypotheses as T (otherwise replace T with the shortest such subcycle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' We have seen in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='13 that no linear combination of non-trivial cycles that lie on T, that is homogeneous with respect to |·| and d, is in Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Now consider zδ(k) = ∑n i=1 T δ(k) i ∈ B for some even integer k ⩾ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Then ���zδ(k) ��� = nk and dβ(zδ(k)) = � δ(k) if β ∈ {α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' , αn} 0 if β ̸∈ {α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' , αn} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Since zδ(k) and the gi are homogeneous with respect to |·| and d, by assumption there exist elements u(k) i in Z, 1 ⩽ i ⩽ t, that are homogeneous with respect to |·| and d, such that zδ(k) = ∑t i=1 u(k) i gi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Note that Z ⊆ � e∈Q0 e Λ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' e, therefore the u(k) i are linear combinations of cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Fix an integer j with 1 ⩽ j ⩽ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Then αj+d−1 · · · αjT δ(k) j = αj+d−1 · · · αjzδ(k) = ∑t i=1 αj+d−1 · · · αju(k) i gi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Since αj+d−1 · · · αjT δ(k) j is homogeneous with respect to |·| and d, we can assume that for all i we have 2 + ���u(k) i ��� + |gi| = ���αj+d−1 · · · αju(k) i gi ��� = ���αj+d−1 · · · αjzδ(j) ��� = nk + 2 and d(αj+d−1 · · · αj) + d(u(k) i ) + d(gi) = d(αj+d−1 · · · αju(k) i gi) = d(αj+d−1 · · · αjT δ(k) i ) = d(αj+d−1 · · · αj) + d(T δ(k) i ), and therefore the only arrows that occur in u(k) i gi are the αj, 1 ⩽ j ⩽ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Moreover, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='11 and the fact that the u(k) i are linear combinations of cycles show that the u(k) i must be linear com- binations of cycles lying on T (otherwise, one at least of these cycles has a subpath of length d that does not lie on T, hence that is in σ and this cycle vanishes in B and does not occur in u(k) i ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Our assumption shows that these cycles must be trivial (of length 0), and since Z0 = K we see that u(k) i ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Therefore αj+d−1 · · · αjT δ(k) j is a linear combination of the αj+d−1 · · · αjgi, hence � αj+d−1 · · · αjT δ(k) j | 1 ⩽ j ⩽ n, k ⩾ 1, k even � ⊆ spanK � αj+d−1 · · · αjgi | 1 ⩽ j ⩽ n, 1 ⩽ i ⩽ t � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' The set � αj+d−1 · · · αJT δ(k) J | 1 ⩽ j ⩽ n, k ⩾ 2, k even � is linearly independent over K (using the grading |·|), therefore we have reached a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Therefore the Yoneda algebra E(Λ) = B is not finitely generated as a Z-module when Condi- tion 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='4 (2) does not hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Hence (C4) ⇒ (C2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' □ Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' In the case where K is algebraically closed, we have extended the equivalence between (C1) and (C3), already known for Koszul algebras from [7], to d-Koszul monomial algebras with d ⩾ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' In particular, we have the following corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Let Λ be a d-Koszul monomial algebra over an algebraically closed field with d ⩾ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Assume that E(Λ) is a finitely generated Zgr(E(Λ))-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Then the algebra Zgr(E(Λ)) is noetherian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' 16 JAWAD, SNASHALL, AND TAILLEFER 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' EXTENSION TO (D, A)-STACKED MONOMIAL ALGEBRAS 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Notation and properties of (D, A)-stacked monomial algebras Let Λ = KQ/I be a monomial algebra with the length grading as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Let D and A be integers with D > A ⩾ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' From [13, Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='1], Λ is then a (D, A)-stacked monomial algebra if, for any minimal projective right Λ-module resolution of Λ0, the n-th projective module is generated in degree δA(n) where δA(n) = \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 n if n = 0 or n = 1 n 2 D if n ⩾ 2 is even n−1 2 D + A if n ⩾ 3 is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' When A = 1, we retrieve the definition of a D-Koszul algebra, so that a (D, 1)-stacked monomial algebra is a D-Koszul monomial algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' It was shown in [13, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='3] that if gldim Λ ⩾ 4 then A divides D;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' in particular, D ⩾ 2A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' If the global dimension of Λ is finite, then Condition (Fg), and in fact all the conditions (C1)–(C6) stated in the Introduction, are satisfied by Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Therefore we shall assume throughout this section that Λ is a (D, A)-stacked monomial algebra with gldim Λ ⩾ 4 and set d = D A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' We define δ(n) = \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f3 n if n = 0 or n = 1 n 2 d = δA(n) A if n ⩾ 2 is even n − 1 2 d + 1 = δA(n) A if n ⩾ 3 is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' For A ⩾ 1, we define an A-path as a non-zero path p = α1 · · · αn where all the αi are paths of length A (that is, αi ∈ QA for all i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' An A-trail is an A-path in which all the αi are distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' An A-cycle is a closed A-path and finally an A-loop is an A-cycle of length A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Given an A-path p as above, an A-subpath of p is an A-path of the form αi · · · αj with 1 ⩽ i ⩽ j ⩽ n (note that not every A-path that is a subpath of p is an A-subpath of p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' An A-subcycle of p is a closed A-subpath of one of the non-zero A-paths αi · · · αnα1 · · · αi−1 with 1 ⩽ i ⩽ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' We also define the A-length ℓA(p) of an A-path p = α1 · · · αn where the αi are paths of length A as ℓA(p) = n, that is, ℓ(p) = AℓA(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' We will need the following result from [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Property 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' [13, Section 3] Let Λ = KQ/I be a finite dimensional monomial algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Then Λ is (D, A)- stacked if, and only if, ρ = R2 has the following properties: (1) every path in ρ is of length D;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' (2) if R2 2 ∈ R2 properly overlaps R2 1 ∈ R2 with overlap R2 1u, then ℓ(u) ⩾ A and there exists R2 3 ∈ R2 which properly overlaps R2 1 with overlap R2 1u′, ℓ(u′) = A and u′ is a prefix of u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' ✤ R2 1 ✤ ✤ R2 3 ✤ ✤ R2 2 ✤ � u � � u′ � Therefore ρ consists of paths of length D, and if Λ is (D, A)-stacked with gldim Λ ⩾ 4, we view ρ as a set of A-paths of A-length d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' We include first an example from [8] (Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Let Λ = KQ/I where Q is the quiver α � · β � γ � δ � and the ideal I has minimal generating set ρ = {αβγδαβ, γδαβγδ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Then Λ is a (6, 2)-stacked monomial algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' A COMBINATORIAL CHARACTERISATION OF (FG) 17 The closed 2-trails are all the paths of length 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Now we give an example where, as well as closed A-trails, there are A-loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Let Λ = KQ/I where Q is the quiver γ4 � · β1 � γ1 � β2 � · α1 � γ3 � γ2 � α2 � and the ideal I has minimal generating set ρ = {(α1α2)2, (γ1γ2)(γ3γ4), (γ3γ4)(γ1γ2)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Then Λ is a (4, 2)-stacked monomial algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' The closed 2-trails are the paths of length 4 whose arrows are the γi and the 2-loops are α1α2 and α2α1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Finally, we give an example in which an arrow, namely β2, occurs both in closed A-trails and in A-loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Let Λ = KQ/I where Q is the quiver β3 � α2 ��������� β4 � · β5 � · β6 � · β7 ��������� α1 � · β2 � β1 � β9 � β8 � and the ideal I has minimal generating set ρ = {(α1β2α2)2, (β1β2β3)(β4β5β6), (β4β5β6)(β7β8β9), (β7β8β9)(β1β2β3)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Then Λ is a (6, 3)-stacked monomial algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' The closed 3-trails are all the cycles of length 9 whose arrows are the βi and the 3-loops are α1β2α2, α2α1β2 and β2α2α1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' We have the following consequences of Property 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Consequence 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' We keep the notation of Property 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='2, with D = dA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Then the length of u must be a multiple of A, so that R2 1u is an A-path, and every A-subpath of A-length d of R2 1u is in ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Moreover, no other subpath of length D of R2 1u is in ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Write ℓ(u) = qA + r with q ⩾ 1 and 0 ⩽ r < A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' We prove the result by induction on q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' If q = 1, then the path R2 2 ∈ ρ overlaps R2 3 ∈ ρ with overlap R2 3u3 for some path u3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' If this overlap is a proper overlap (that is, R2 3 ̸= R2 2), then ℓ(u) = ℓ(u′) + ℓ(u3) = A + ℓ(u3) so that ℓ(u3) = (q − 1)A + r = r and 0 < r < A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Therefore by Property 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='2 we have a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' It follows that R2 3 = R2 2 and u = u′ has length A and that R2 1 and R2 3 are the only A-subpaths of A-length d of R2 1u and they are in ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Moreover, any other subpath of length D of R2 1u is a proper overlap of R2 1 of length strictly smaller than D + A, which is impossible by Property 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Let q > 1 be such that ℓ(u) = qA + r with 0 ⩽ r < A and assume that the result is true for any proper overlap of a path in ρ of length D + q′A + r′ with q′ < q and 0 ⩽ r′ < A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' The path R2 2 ∈ ρ properly overlaps R2 3 ∈ ρ with overlap R2 3u3 for some path u3 with ℓ(u) = ℓ(u′) + ℓ(u3) = A + ℓ(u3) so that ℓ(u3) = (q − 1)A + r and the overlap R2 3u3 has length D + (q − 1)A + r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' By induction, ℓ(u3) is a multiple of A, therefore r = 0 and ℓ(u) is a multiple of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Any A-subpath of A-length d of R2 1u is either R2 1 or an A-subpath of A-length d of R2 3u3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Again by induction, they are all in ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Finally, a subpath of length D of R2 1u which is not an A-subpath either is a subpath of length D of R2 3u3 that is not an A-subpath, therefore not in ρ by induction, or properly overlaps R2 1 with overlap R2 1u′′′ with 0 < ℓ(u′′′) < A, which is impossible by Property 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' □ Consequence 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Suppose that D = dA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Let n ⩾ 2 and let Rn i be an element of Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Write Rn i = α1 · · · αδ(n) where each αi is a path of length A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Then for all i with 1 ⩽ i ⩽ δ(n) − d + 1, the path αi · · · αi+d−1 is in ρ, that is, all the A-subpaths of A-length d of Rn i are in ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Moreover, no other subpath of Rn i of length D is in ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' 18 JAWAD, SNASHALL, AND TAILLEFER Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' The result is proved by induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' It is clear when n = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Moreover, if n = 3, since R3 i ∈ R3 is a maximal overlap of two elements in R2, it follows from Property 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='2 and using the notation therein that R3 i = R2 1u′ = v′R2 3 where v′ is the prefix of R2 1 of length A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' By Consequence 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='3, the only subpaths of length D of R3 i that are in ρ = R2 are R2 1 and R2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Now let n ⩾ 4 and take Rn i ∈ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Then Rn i is a maximal overlap of R2 1 ∈ R2 with Rn−1 2 ∈ Rn−1 so that Rn i = Rn−1 2 u for some path u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Write Rn−1 2 = α1 · · · αδ(n−1) with ℓ(αi) = A for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' By the induction assumption, we have αi · · · αi+d−1 ∈ R2 for all i with i + d − 1 ⩽ δ(n − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' In particular, R2 3 := αδ(n−1)−d+1 · · · αδ(n−1) is in R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Since R2 1 overlaps R2 3 with overlap R2 3u, by Property 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='2 we have ℓ(u) = A and αδ(n−1)−d+2 · · · αδ(n−1)u = R2 1 ∈ R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Since Rn i = Rn−1 2 u, we have proved the first part of the result for Rn i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Now let p be another subpath of Rn i of length D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' We already know by induction that if p is a subpath of Rn−1 2 , then p is not in R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Therefore p is a subpath of R2 3u which is neither R2 3 nor R2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' By Consequence 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='3, p is not in ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' We have proved that p ̸∈ R2 and the induction step is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' □ Consequence 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Suppose that D = dA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Let T = α1 · · · αn be a closed A-trail in Q with αi ∈ QA for all i and suppose that d ⩾ n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Assume also that T is the prefix of an A-path in ρ and the suffix of an A-path in ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Then all A-subpaths of A-length d of powers of the closed trail T are in ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' By assumption, there exist A-paths T′ and T′′ such that T′T ∈ ρ and TT′′ ∈ ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Since Λ is finite dimensional, there is a path R2 ∈ ρ that lies on T, and ℓ(R2) = D = dA > ℓ(T) = nA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Therefore R2 is a subpath of length D of TN = (α1 · · · αn)N for some N ⩾ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' If R2 = Tm is a power of T with m ⩾ 2 (and d = nm) then R2 overlaps itself with overlap T2m−1 and the result follows using Consequence 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='3 (every A-subpath of A-length d of a power of T is an A-subpath of T2m−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Otherwise, TT′′ overlaps R2 or R2 overlaps T′T and we can use Consequence 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='3 again to prove that R2 is an A-subpath of TN and then that every A-subpath of A-length d of the overlap is in ρ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' since every A-subpath of A-length d of a power of T is one of these, we obtain the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Characterisations of (D, A)-stacked monomial algebras that satisfy (Fg) We now give our combinatorial condition for (D, A)-stacked monomial algebras Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Condition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' (1) Let c be an A-loop in QA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Write c = a1 · · · aA with ai ∈ Q1 for all i and cj = aj · · · aAa1 · · · aj−1 for j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' , A}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Then there exists j such that cd j ∈ ρ but there is no path in ρ of the form cd−1 j β or βcd−1 j where β is a path of length A that is distinct from cj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' (2) Let T = α1 · · · αn be a closed A-trail in Q with n ⩾ 2 and αi ∈ QA for all i and such that ρT := {α1 · · · αd, α2 · · · αdαd+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' , αnα1 · · · αd−1} ⊆ ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Then there are no elements in ρ \\ ρT which begin or end with the path αi, for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' In part (1) of the condition, there is exactly one j such that cd j ∈ ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Indeed, if cd j and cd k were in ρ, they would overlap with an overlap of length at most D + A − 1, hence by Property 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='2 we must have cd j = cd k and therefore j = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' If A = 1 then Condition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='6 is equivalent to Condition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' We first prove that this condition is sufficient for Λ to satisfy (Fg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Let Λ = KQ/I be a finite dimensional (D, A)-stacked monomial algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Assume that Λ satisfies Condition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Then Λ satisfies (Fg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' The case D ⩾ 2 and A = 1 corresponds to d-Koszul monomial algebras (with D = d) and is proved in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Therefore we assume that A > 1 so that necessarily D > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' If gldim Λ is finite then Λ satisfies (Fg) (and Condition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='6 is empty), so we also assume that gldim Λ ⩾ 4 so that D = dA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' The structure of this proof follows that of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='7 by replacing each arrow in Q1 by a path of length A in QA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' We do not give all the details here, but indicate those places where we need to provide additional arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' The first part of the proof is to show that the hypotheses of [13, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='4] hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' A COMBINATORIAL CHARACTERISATION OF (FG) 19 Let c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' , cu be the A-loops in Q such that cd i ∈ ρ for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' , u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' (We remark that, in the terminology of [13], these are precisely the closed paths in Q such that for each ci we have ci ̸= pri i for any path pi with ri ⩾ 2 and cd i ∈ ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Firstly, cd i ∈ ρ implies that ℓ(ci) = A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Then, if ci = pri i for some path pi with ri ⩾ 2, we have 1 ⩽ ℓ(pi) < A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Now pdri i is in ρ and pdri i overlaps itself with overlap pdri+1 i , so there is a maximal overlap in R3 of length ⩽ D + ℓ(pi) < D + A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' But this is a contradiction since Λ is a (D, A)-stacked monomial algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' So ci ̸= pri i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=') By Condition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='6 (1), for each i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' , u, there are no elements in ρ of the form cd−1 i β or βcd−1 i where β is a path of length A that is distinct from ci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' We need to show that there are no overlaps of cd i with any element of ρ \\ {cd i }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' If R ∈ ρ \\ {cd i } and R overlaps cd i , then, by Consequence 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='3, either R = cs ib or R = bcs i where 1 ⩽ s ⩽ d − 1 and b is an A-path with ℓA(b) = d − s and that does not begin (respectively, end) with the path ci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Suppose that R = cs ib.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Then R overlaps cd i with overlap of length A(2d − s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' By Consequence 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='3, this is a maximal overlap since ci is not a prefix of b and thus gives an element R3 1 ∈ R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' However, ℓ(R3 1) = D + A = (d + 1)A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Thus 2d − s = d + 1 and so s = d − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' But then R = cd−1 i b and b is a path of length A distinct from ci, which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' The case R = bcs i is similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' So there are no overlaps of cd i with any element of ρ \\ {cd i }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Let Tu+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Tr be the distinct closed A-trails in Q with ℓA(Ti) > 1 such that the sets ρTi of Condition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='6 (2) are contained in ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' For each i = u + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' , r, we write Ti = αi,1 · · · αi,mi, where the αi,j are in QA so that ℓA(Ti) = mi > 1 and ρTi = {αi,1 · · · αi,d, αi,2 · · · αi,d+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' , αi,miαi,1 · · · αi,d−1} ⊆ ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' By Condition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='6 (2), for each closed A-trail Ti (i = u + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' , r), there are no elements in ρ \\ ρTi which begin or end with the path αi,j, for all j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' , mi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' So no path αi,j of length A has overlaps with any element in ρ \\ ρTi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' The next step is to show that Λ satisfies (Fg1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Applying [13, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='4], gives HH∗(Λ)/N ∼= K[x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' , xr]/⟨xaxb for a ̸= b⟩, where for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' , u, the vertices v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' , vu are distinct and the element xi corresponding to the A-loop ci is in degree 2 and is represented by the map P2 −→ Λ where for R2 ∈ R2, o(R2) ⊗ t(R2) �→ � vi if R2 = cd i 0 otherwise and for i = u + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' , r, the element xi corresponding to the closed A-trail Ti = αi,1 · · · αi,mi is in degree 2µi such that µi = mi/ gcd(d, mi) and is represented by the map P2µi −→ Λ, where for R2µi ∈ R2µi, o(R2µi) ⊗ t(R2µi) �→ � o(Ti,k) if R2µi = Td/ gcd(d,mi) i,k for all k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' , mi 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Let H be the subring of HH∗(Λ) generated by Z(Λ) and {x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' , xr}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' As in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='7, H is a commutative Noetherian ring and so Λ satisfies (Fg1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Now we show that Λ satisfies (Fg2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Again, we identify � n⩾0 Rn with a basis of E(Λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Set N = max{3, |x1|, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' , |xr|, |QA|}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' We show that �N n=0 Rn is a generating set for E(Λ) as a left H- module and thus E(Λ) is finitely generated as a left H-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Let R ∈ Rn with n > N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Then ℓA(R) = δ(n) ⩾ 2d and we can write R = a1a2 · · · aδ(n) where the ai are in QA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' The proof now follows that of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='7 by replacing each arrow by a path of length A, and with extensive use of Consequences 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='3, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='4 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='5, and Condition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Thus Λ satisfies (Fg2) and we conclude that Λ has (Fg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' □ Example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' We return to Examples 3, 4 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' In all these examples, Condition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='6 is satisfied and therefore (Fg) holds for Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' For instance, in Example 3, the only closed 2-trails T such that ρT ⊆ ρ are αβγδ and γδαβ and, in both cases, ρT = ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' In Example 5, the closed 3-trails T such that ρT ⊆ ρ are those that start with β1, β4 and β7, in all cases we have ρT = ρ \\ �(α1β2α2)2� and (α1β2α2)2 does not start or end with a βi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' 20 JAWAD, SNASHALL, AND TAILLEFER By [6, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='5], it follows that Λ is Gorenstein in each case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Moreover, it was proved in [8] that the algebra in Example 3 has injective dimension 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Our aim is now to prove the following theorem, and in particular the converse of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Let Λ be an indecomposable finite dimensional (D, A)-stacked monomial algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Suppose that D ̸= 2A whenever A > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Consider the following statements: (C1) Λ satisfies (Fg);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' (C2) Condition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='6 holds for Λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' (C3) Zgr(E(Λ)) is noetherian and E(Λ) is a finitely generated Zgr(E(Λ))-module;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' (C4) E(Λ) is finitely generated as a module over Zgr(E(Λ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Then (C4) implies (C2) which in turn implies (C1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Moreover, if the field K is algebraically closed, then the four statements are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' We shall need, as in the d-Koszul case, a description of the Ext algebra of Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' We give the details of this in the Appendix, and we briefly describe it here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Since we have already proved Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='10 when Λ is d-Koszul, we assume here that D > A > 1 and, in addition, that D ̸= 2A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Let Γ be the quiver with the same vertices as Q and whose set of arrows corresponds to the set of paths of length A in Q, that is, Γ1 = {α: i → j | there exists α ∈ QA, α: j → i}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Let ρ ⊥ be the orthogonal of ρ for the bilinear form KΓd × K(QA)d → K defined on paths of length d in Γ and A-paths of A-length d in Q by ⟨αd · · · α1, β1 · · · βd⟩ = 1 if α1 · · · αd = β1 · · · βd and 0 otherwise, where the αi and βi are in QA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Set Λ ♮ = KΓ/J where J = ( ρ ⊥ );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' it is a monomial algebra and the ideal J has a minimal generating set σ given by all the paths αd · · · α1 such that the A-path α1 · · · αd is not in ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Let B = � n⩾0 Bn be the algebra defined as follows: Bn = Λ ♮ δ(n) for x ∈ Bn and y ∈ Bm, define x · y ∈ Bm+n by x · y = \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 0 if n and m are odd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' 0 if n or m is equal to 1 and n ⩾ 1, m ⩾ 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' xy in Λ ♮ otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Observe that if n or m is even and both are larger than 1, δ(n) + δ(m) = δ(n + m), so that the algebra B is a graded K-algebra, generated in degrees 0, 1, 2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Note that this is also true of E(Λ) by [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Moreover, we prove in Appendix A that the algebras E(Λ) and B are isomorphic, generalising the description given in [11] when Λ is a d-Koszul algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' This isomorphism uses the assumption that D ̸= 2A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' There is a basis B Λ ♮ of Λ ♮ consisting of all paths p in Γ such that no path in σ is a subpath of p, and basis BB of B contained in B Λ ♮ consisting of all R m i for all m ⩾ 0 and all Rm i ∈ Rm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' We now define several gradings, on Λ ♮ and on B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' There is a natural grading on Λ ♮ given by the length ℓ of paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Note that if p is an A-path in Q, then ℓ(p) = ℓA(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' The degree of a homogeneous element x in B will be denoted by |x|, so x ∈ Λ ♮ δ(|x|) or, in other terms, |x| = k if, and only if, ℓ(x) = δ(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' The algebra Λ ♮ is also multi-graded by NQ1: for each path p in Γ, we define an element d(p) = (dα(p))α∈QA ∈ NQ1 as follows: write the A-path p in Q as p = α1 · · · αn where each αi is in QA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' if ℓ(p) = 0, then d(p) = (0)α∈Q1 if ℓ(p) > 0, then dα(p) is the number of αi that are equal to α (it is 0 if none of the αi are equal to α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Note that even if α is a subpath of p, we can have dα(p) = 0 (if α is not one of the αi, that is, p = qαr where q and r are paths in Q whose lengths are not multiples of A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Since Λ ♮ is monomial, the ideal J is homogeneous with respect to this multi-degree and therefore Λ ♮ is multi-graded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' In B, if x and y are homogeneous and |x| or |y| is even with both degrees at least 2, then dα(xy) = dα(x) + dα(y) but dα(xy) = 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' A COMBINATORIAL CHARACTERISATION OF (FG) 21 Let Z := Zgr(B) be the graded centre of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' As in the d-Koszul case, it is generated by elements z that are homogeneous with respect to the grading |·| and the multi-degree and such that, for any element y ∈ B that is homogeneous with respect to the grading |·|, we have zy = (−1)|y||z|yz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Recall that B ∼= E(Λ) is generated in degrees 0, 1, 2 and 3 and that the product of an element of degree 1 with any other element vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Therefore when checking that an element is in Z, we need to check that it is a linear combination of cycles and that it commutes or anti- commutes with paths of degrees 2 and 3, that is, (non-zero) paths of length d and of length d + 1 in Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' The proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='10 relies on some preliminary results, namely Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='12, Proposi- tion 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='13 and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' We start with some comments on A-loops in QA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Let c be an A-loop in QA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Since Λ is finite dimensional, there exists an integer N such that cN = 0 in Λ and therefore there is some j such that cd j ∈ ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' To simplify notation and without loss of generality, write c = cj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Then cd ∈ ρ, therefore cd ̸∈ σ and it follows that ck ̸= 0 in Λ ♮ for all k ⩾ 0 and that cδ(k) ̸= 0 in B for all k ⩾ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Let c be an A-loop in QA and let n ⩾ 2 be an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Then cδ(n) ∈ Z if, and only if, c satisfies Condition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='6 (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' The proof is very similar to that of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='10, using A-paths and Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' □ We shall now consider part (2) of Condition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Let T = α1 · · · αn be a closed A-trail in Q with αi ∈ QA for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Assume that n ⩾ 2 and that ρT = {αi · · · αi+d−1 | 1 ⩽ i ⩽ n} ⊆ ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Then T and all the paths lying on T are in B Λ ♮ (none of their subpaths of length d are in σ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' those of length δ(k) for some k ⩾ 0 are in BB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' In a similar way to Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='3, we make the following assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' (i) none of the αi are A-loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' (ii) no A-subcycle of T satisfies the same assumptions as T (that is, there is no A-subcycle q of T of A-length at least 2, and ρq ⊆ ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Let T = α1 · · · αn be a closed A-trail with n ⩾ 2, ρT ⊆ ρ and such that assumptions (i) and (ii) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Let p be an A-path of A-length d such that dβ(p) = 0 if β ∈ QA \\ {α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' , αn} and which is not an A-subpath of a power of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Then p ̸∈ ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' The proof is very similar to that of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='11, replacing paths with A-paths and using Consequence 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='5 in the proof that d > k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' □ Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' We keep the assumptions and notation of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Set Ti = αi · · · αnα1 · · · αi−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Then for any j ⩾ 1, we have T j iαk ̸= 0 ⇐⇒ k = i − 1 αkT j i ̸= 0 ⇐⇒ k = i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Let T = α1 · · · αn be a closed A-trail that satisfies assumptions (i) and (ii) and set zj = ∑n i=1 T j i with nj = δ(u) for some u ⩾ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Then zj ∈ Z if, and only if, T satisfies Condition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='6 (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Moreover, if T does not satisfy Condition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='6 (2), then no element in B that is homogeneous with respect to |·| and d (when viewed in Λ ♮ ) and that is a linear combination of non-trivial cycles lying on T is in Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' The proof is very similar to that of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='13, replacing paths with A-paths, again us- ing Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='11, replacing the d-covering property by Consequence 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='3 and Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='11 by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Note also that for the proof of the last part, testing commutation with paths in B2 gives s = n and λk = λk+d for all k and hence the result if |z| is odd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' and if |z| is even, we must use the fact that z commutes with elements in B3 in a similar way to obtain, in addition, that λk = λk+d+1 for all k and hence that λk = λk+1 for all k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' □ Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' We note first that if gldim Λ is finite then Λ satisfies (Fg) and Condition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='6 is empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' The implication (C2) ⇒ (C1) is Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Again, the implication (C3) ⇒ (C4) is clear and if, in addition, K is algebraically closed, then the implication (C1) ⇒ (C3) follows from [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' It 22 JAWAD, SNASHALL, AND TAILLEFER remains to prove that (C4) implies (C2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' The proof is similar to that of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='8, again replacing paths with A-paths (we need not assume that the integers k are even).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' □ Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Suppose that K is algebraically closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' We have now extended the equivalence between (C1) and (C3), already known for Koszul algebras from [7], as well as d-Koszul monomial algebras by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='8, to (D, A)-stacked monomial algebras with D ̸= 2A whenever A > 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' In particular, we can extend Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='15 to (D, A)-stacked monomial algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Let Λ be a (D, A)-stacked monomial algebra over an algebraically closed field with D ̸= 2A whenever A > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Assume that E(Λ) is a finitely generated Zgr(E(Λ))-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Then the algebra Zgr(E(Λ)) is noetherian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' APPENDIX A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' THE EXT ALGEBRA OF A (D, A)-STACKED MONOMIAL ALGEBRA Leader and Snashall gave in [17] a presentation of the Yoneda algebra E(Λ) of a (D, A)-stacked monomial algebra by quiver and relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' However, in our proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='8 that (C4) implies (C2) for d-Koszul monomial algebras, we used the description from [11, Sections 8 and 9] of E(Λ) as an algebra contained, as a graded vector space, in the Koszul dual Λ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' In this Appendix, we generalise this description to (D, A)-stacked monomial algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Throughout this section, Λ = KQ/I is a (D, A)-stacked monomial algebra with D = dA and d ⩾ 2, where I an ideal generated by a set ρ of A-paths of A-length d = D A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' We view Λ as a quotient of the tensor algebra: Λ = TΛ0(Λ1)/I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' All tensor products are taken over Λ0 and we write ⊗ for ⊗Λ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' The subspace S = I ∩ (Λ⊗D 1 ) = span(ρ) of T = TΛ0(Λ1) is a Λ0-Λ0-submodule of Λ⊗D 1 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' it is finite dimensional over K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' For an element x ∈ T, write x for its image in Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Note that for 0 ⩽ i < D we have Λi = Λ⊗i 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Generalised Koszul complex of S Define spaces Hδ(n) ⊆ Tn Λ0(Λ1) as follows: H0 = Λ0, H1 = Λ1 and, for n ⩾ 2, Hδ(n) = � i+j=δ(n)−d (Λ⊗i A )⊗S⊗(Λ⊗j A ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' For n ⩾ 0, let Pn be the right Λ-module defined by Pn = Hδ(n)⊗Λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' it is projective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' We have Hδ(1) = Λ1 = Hδ(0)⊗Λ1, Hδ(2) = S ⊆ Λ⊗D 1 = Hδ(1)⊗Λ⊗D−1 1 and, for any n ⩾ 3, Hδ(n) ⊆ Hδ(n−1)⊗Λ⊗(δ(n)−δ(n−1)) A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Indeed,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' for any k ⩾ 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Hδ(2k+2) = H(k+1)d = kd � j=0 (Λ⊗(kd−j) A )⊗S⊗(Λ⊗j A ) ⊆ kd � j=d−1 (Λ⊗(kd−j) A )⊗S⊗(Λ⊗j A ) = (k−1)d+1 � j=0 (Λ⊗((k−1)d+1−j) A )⊗S⊗(Λ⊗j A )⊗(Λ⊗(d−1) A ) = Hδ(2k+1)⊗(Λ⊗(d−1) A ) Hδ(2k+1) = Hkd+1 = kd+1 � j=0 (Λ⊗((k−1)d+1−j) A )⊗S⊗(Λ⊗j A ) ⊆ kd+1 � j=A (Λ⊗((k−1)d+1−j) A )⊗S⊗(Λ⊗j A ) = kd � j=0 (Λ⊗((k−1)d−j) A )⊗S⊗(Λ⊗j A )⊗ΛA = Hδ(2k)⊗ΛA It follows that the maps F1 : Λ1⊗Λ → Λ0⊗Λ ∼= Λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' F2 : Λ⊗d A ⊗Λ → Λ1⊗Λ and,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' for n ⩾ 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Fn : Λ⊗δ(n) A ⊗Λ → Λ⊗δ(n−1) A ⊗Λ defined by F1(x1⊗λ) = x1λ F2(x1⊗ · · · ⊗xD⊗λ) = x1⊗x2 · · · xDλ Fn(y1⊗ · · · ⊗yδ(n)⊗λ) = y1⊗ · · · ⊗yδ(n−1)⊗yδ(n−1)+1 · · · yδ(n)λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' A COMBINATORIAL CHARACTERISATION OF (FG) 23 where xi ∈ Λ1 and yi ∈ ΛA for all i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' induce maps bn : Pn → Pn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' More specifically, for all k ⩾ 1, F2k+1(y1⊗ · · · ⊗ykd+1⊗λ) = y1⊗ · · · ⊗ykd⊗ykd+1λ if n = 2k + 1 is odd F2k+2(y1⊗ · · · ⊗y(k+1)d⊗λ) = y1⊗ · · · ⊗ykd+1⊗ykd+2 · · · y(k+1)dλ if n = 2k + 2 is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Define also b0 : P0 = Λ0⊗Λ ∼= Λ → Λ0, which identifies with the natural projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Moreover, for n ⩾ 3 we have Hδ(n+1) ⊆ Hδ(n)⊗Λ⊗(δ(n+1)−δ(n)) A ⊆ Hδ(n−1)⊗Λ⊗(δ(n)−δ(n−1)) A ⊗Λ⊗(δ(n+1)−δ(n)) A = Hδ(n−1)⊗Λ⊗d A and Hδ(n+1) ⊆ Λ⊗δ(n−1) A ⊗S, we have Hδ(n+1) ⊆ � Λ⊗δ(n−1) A ⊗S � ∩ � Hδ(n−1)⊗Λ⊗d A � = Hδ(n−1)⊗S (all the spaces involved are finitely generated and projective over Λ0) hence bn ◦ bn+1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' It is easy to check that bn ◦ bn+1 = 0 when n = 1, 2 or 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Therefore we have a complex (Pn, bn) of projective right Λ-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Let Λ = KQ/I be a monomial algebra with I generated in degree D = dA with d ⩾ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Then Λ is (D, A)-stacked monomial if, and only if, (P•, b•) is a minimal projective right Λ-module resolution of Λ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' By construction, Pn is generated in degree δ(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Therefore, if (P•, b•) is a minimal projective right Λ-module resolution of Λ0, then Λ is a (D, A)-stacked monomial algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Conversely, assume that Λ is a (D, A)-stacked monomial algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' We already have a complex (P•, b•) of right Λ-modules such that Pn is generated in degree δ(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' The beginning P1 b1 −→ P0 b0 −→ Λ0 → 0 is exact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' We prove exactness at P2n+1 for n ⩾ 1 (the proof of exactness at P2n and at P1 is similar, without the need for Consequence 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' First note that b2n+2(P2n+2) is generated in degree δ(2n + 2) = (n + 1)D in P2n+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Let z = ∑i xnd+1,i⊗ · · · ⊗x1,i⊗λi be an element in Ker b2n+1 with xj,i ∈ ΛA and λi ∈ Λ for all i, j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Then ∑i xnd+1,i⊗ · · · ⊗x2,i⊗x1,iλi is in T⊗I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' It follows that λi ∈ � k⩾D−A Λk and that Ker b2n+1 is generated in degrees at least (n + 1)D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Therefore z can be rewritten as z = ∑i xnd+1,i⊗ · · · ⊗x1,i⊗yd−1,i · · · y1,iλ′ i with yj,i ∈ ΛA and λ′ i ∈ Λ for all i, j, with the yj,i right uni- form and t(y1,i) ̸= t(y1,k) when i ̸= k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Write λ′ i = ∑l⩾0 λ′ i,l with λ′ i,l ∈ Λl for all i, l, and λ′ i,0 = t(y1,i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Then each of the ∑i xnd+1,i⊗ · · · ⊗x1,i⊗yd−1,i · · · y1,iλ′ i,l is in Ker b2n+1 so in partic- ular z′ := ∑i xnd+1,i⊗ · · · ⊗x1,i⊗yd−1,i · · · y1,i ∈ Ker b2n+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Consider z′′ := ∑i xnd+1,i⊗ · · · ⊗x1,i⊗yd−1,i⊗ · · · ⊗y1,i⊗t(y1,i) ∈ Λ⊗((n+1)d) A ⊗Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' We show that z′′ ∈ P2n+2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' this will imply that z′ = b2n+2(z′′) ∈ Im b2n+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Since Λ is monomial and b2n+1(z′) = 0, we may assume that that for all i, x1,iyd−1,i · · · y1,i is a path;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' since it is in I and has degree D, it is in ρ = R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' By definition of P2n+1, we may assume that z is written so that for each i, xd,i · · · x1,i is a path in ρ = R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' The path x1,iyd−1,i · · · y1,i properly overlaps xd,i · · · x1,i therefore, using Con- sequence 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='3, it follows that for all k with 1 ⩽ k ⩽ d − 1 we have xk,i · · · x1,iyd−1,i · · · yk,i ∈ ρ and hence z′′ ∈ �d−1 k=1(Λ⊗(nd−k+1) A )⊗S⊗(Λ⊗(k−1) A )⊗Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Finally, using the fact that z ∈ P2n+1 = Hnd+1⊗Λ, we get z′′ ∈ H(n+1)d⊗Λ = P2n+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' We have proved that (Ker b2n+1)(n+1)d ⊆ Im b2n+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Since Im b2n+2 is generated in degree (n + 1)d and Ker b2n+1 is generated in degrees at least (n + 1)d, it follows that Ker b2n+1 ⊆ Im b2n+2 and finally that Ker b2n+1 = Im b2n+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Finally, since Im bn+1 is generated in degree δ(n + 1) and Pn is generated in degree δ(n) < δ(n + 1), Im bn+1 ⊆ rPn for all n and therefore the resolution is minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' □ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' The Ext algebra A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Some duality results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' We recall without proof some results stated in [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' All modules are finitely generated Λ0-Λ0-bimodules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' All claims are easily checked for bimodules that are free and finitely generated as left or as right Λ0-modules, and follow for arbitrary finitely generated modules (since Λ0 is semisimple, all the Λ0-modules (left or right) are projective).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' For any bimodules V and W, define V∗ = HomΛ0−(V, Λ0) and W ∗ = Hom−Λ0(W, Λ0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' they are Λ0-Λ0-bimodules, for the actions given for all e, e′ in Λ0, α ∈ V∗, β ∈ W ∗ and v ∈ V, w ∈ W by: (eαe′)(v) = α(ve)e′ and (eβe′)(w) = eβ(e′w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' 24 JAWAD, SNASHALL, AND TAILLEFER There are natural isomorphisms of Λ0-Λ0-bimodules Λ∗ 0 ∼= Λ0 ∼= Λ ∗ 0 which we view as identi- fications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' There are also natural bimodule isomorphisms V ∼= (V∗) ∗ and W ∼= ( W ∗ )∗ of bimodules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' If V1 ⊆ V (respectively W1 ⊆ W), define V⊥ 1 = {α ∈ V∗ | α(V1) = 0} (respectively W ⊥ 1 = {β ∈ W ∗ | β(W1) = 0}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' If V1 (respectively W1) is a sub-bimodule, then they are sub-bimodules of V∗ and W ∗ respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Fix a Λ0-Λ0-bimodule V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Then: (i) if W is another Λ0-Λ0-bimodule, then there is an isomorphism of Λ0-Λ0-bimodules ϕV,W : V ∗ ⊗ W ∗ → (W⊗V) ∗ given for all α ∈ V ∗ , β ∈ W ∗ , v ∈ V and w ∈ W by ϕV,W(α⊗β)(w⊗v) = α(β(w)v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' There is a similar isomorphism with right duals, which sends α⊗β to the map w⊗v �→ β(wα(v));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' (ii) if U and W are sub-bimodules of V, then (U + W) ⊥ = U ⊥ ∩ W ⊥ and (U + W)⊥ = U⊥ ∩ W⊥;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' (iii) if U is a sub-bimodule of V, then (V/U)∗ ∼= U⊥ and ( ∗ V/U) ∼= U ⊥ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' (iv) if W is a sub-bimodule of V, then for any idempotents ei, ej with (i, j) ∈ Q2 0, we have ei W ∗ ej = ( ∗ ejWei);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' (v) if U is a sub-bimodule of V, then for all i, j in Q0 we have dim ej U ⊥ ei = dim eiVej − dim eiUej = dim ejU⊥ei;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' (vi) if U is a sub-bimodule of V, then under the identification of V with (V∗) ∗ , we have ( ⊥ U⊥) and under the identification of V with ( V ∗ )∗, we have ( U ⊥ )⊥;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' (vii) if U is a sub-bimodule of V and W and Z are Λ0-Λ0-bimodules, there are bimodule iso- morphisms ( ⊥ W⊗U⊗Z) ∼= Z ∗ ⊗ U ⊥ ⊗ W ∗ and (W⊗U⊗Z)⊥ ∼= Z∗⊗U⊥⊗W∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Description of the Ext algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' From the above, there is a natural isomorphism ( ∗ Λ⊗i A ) ∼= ( Λ ∗ A)⊗i, which we view as an identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' We also view S as a subspace of Λ⊗d A (rather than Λ⊗D 1 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' We may then consider S ⊥ = � f ∈ ( Λ ∗ A)⊗d | f(x) = 0 for all x ∈ S � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' The dual algebra of Λ is then Λ ♮ = TΛ0( Λ ∗ A)/( S ⊥ ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' It is a graded d-homogeneous algebra since S ⊥ is contained in ( Λ ∗ A)⊗d, therefore Λ ♮ = � n⩾0 Λ ♮ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' In terms of quivers, we have Λ0 = KQ0 and ΛA = KQA, the vector space whose basis is the set QA of paths of length A in Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' moreover, Λ ∗ A ∼= KQop A using (iv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Then Λ ♮ is isomorphic to KΓ/( ρ ⊥ ) where Γ is the quiver with the same vertices as Q and whose set of arrows is Γ1 = {α: i → j | there exists α ∈ QA, α: j → i} and where ρ ⊥ is the left orthogonal of the set ρ viewed as a set of A-paths, for the bilinear form KΓd × K(QA)d → K defined on paths of length d in Γ and A-paths of A-length d in Q by ⟨αd · · · α1, β1 · · · βd⟩ = 1 if α1 · · · αd = β1 · · · βd and 0 otherwise, where the αi and βi are paths of length A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' The algebra KΓ/( ρ ⊥ ) is monomial, and the ideal ( ρ ⊥ ) has a minimal generating set σ given by all the paths αd · · · α1 such that the A-path α1 · · · αd is not in ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' In particular, if γ = αr .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' α1 is a path in Γ, with αi ∈ QA for all i, then γ ̸∈ (σ) if, and only if, for each i with 1 ⩽ i ⩽ r − d + 1, the path αi · · · αi+d−1 is in ρ (we use the fact that Λ is monomial here).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' There is an isomorphism ( Λ ♮ δ(n))∗ ∼= Hδ(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' By definition, Λ ♮ δ(n) = ( Λ ∗ A)⊗δ(n) ∑ δ(n)−d i=0 ( Λ ∗ A)⊗(δ(n)−d−i)⊗ S ⊥ ⊗( Λ ∗ A)⊗i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Therefore we have ( Λ ♮ δ(n))∗ ∼= �δ(n)−d ∑ i=0 ( Λ ∗ A)⊗(δ(n)−d−i)⊗ S ⊥ ⊗( Λ ∗ A)⊗i �⊥ = δ(n)−d � i=0 � ( Λ ∗ A)⊗(δ(n)−d−i)⊗ S ⊥ ⊗( Λ ∗ A)⊗i�⊥ = δ(n)−d � i=0 � (( Λ ∗ A)⊗i) ∗⊗( S ⊥ ) ⊥⊗(( Λ ∗ A)⊗(δ(n)−d−i)) ∗� A COMBINATORIAL CHARACTERISATION OF (FG) 25 ∼= δ(n)−d � i=0 � Λ⊗i A ⊗S⊗Λ⊗(δ(n)−d−i) A � = Hδ(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' This isomorphism takes x = ∑ x1⊗ · · · ⊗xδ(n) ∈ Hδ(n) to the map gx : Λ ♮ δ(n) → Λ0 defined by gx(γδ(n)⊗ · · · ⊗γ1) = ∑ γδ(n)(γδ(n)−1(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' (γ2(γ1(x1)x2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' )xδ(n)−1)xδ(n)) where the xi are in ΛA and the γi are in Λ ∗ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' □ Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' There is an isomorphism ψ: Λ ♮ δ(n) → HomΛ(Pn, Λ0) given by ψ( f )(x1⊗ · · · ⊗xδ(n)⊗λ) = fδ(n)( fδ(n−1)(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' ( f1(x1)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' xδ(n)−1)xδ(n))λ where f = fδ(n)⊗ · · · ⊗ f1 ∈ Λ ♮ δ(n) with fi ∈ Λ ∗ 1 for all i, xi ∈ Λ1 for all i and λ ∈ Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' The isomorphism ψ is the composition of the following isomorphisms: Λ ♮ δ(n) → Hom−Λ0(( Λ ♮ δ(n))∗, Λ0), which sends f to the map [g �→ g( f )];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Hom−Λ0(( Λ ♮ δ(n))∗, Λ0) → Hom−Λ0(Hδ(n), Λ0), which sends a map h to the map [x �→ h(gx)], where gx is as in the proof of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Hom−Λ0(Hδ(n), Λ0) → Hom−Λ(Hδ(n)⊗Λ, Λ0), which sends a map k to the map [x⊗λ �→ k(x)λ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Applying these isomorphisms to f gives the expression in the statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' □ Let B be the vector space B = � n⩾0 Bn where Bn = Λ ♮ δ(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Define a multiplication on B as follows: for x ∈ Bn and y ∈ Bm, set x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='y = \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 0 if n and m are odd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' 0 if n or m is equal to 1 and n ⩾ 1, m ⩾ 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' xy in Λ ♮ otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' The algebra B is a graded K-algebra generated in degrees 0, 1, 2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' We want to prove that E(Λ) ∼= B when A > 1 and D ̸= 2A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' We first need a description of the Yoneda product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Let Λ be a (D, A)-stacked monomial algebra with A > 1, D = dA and d ⩾ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' The Yoneda product of fn ∈ HomΛ(Pn, Λ0) and fm ∈ HomΛ(Pm, Λ0) is given by fn fm = \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f3 0 if n and m are odd 0 if n ⩾ 1, m ⩾ 1 and n = 1 or m = 1 ∑ x1⊗ · · · ⊗xδ(n)+δ(m)⊗λ �→ fn(∑ fm(x1⊗ · · · ⊗xδ(m)⊗1)xδ(m)+1 ⊗xδ(m)+2⊗ · · · ⊗xδ(m)+δ(n)⊗λ) otherwise, where the xi are all in ΛA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' If m and n are odd or if m ⩾ 1 or n ⩾ 1 is equal to 1 then, under the assumption that A > 1 and D ̸= 2A, the Yoneda products vanish by [17, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' We now assume that m or n is even and that m ̸= 1 and n ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Let σ: Λ0 → Λ be the natural inclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Consider fm : Pm → Λ0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' it lifts to f 0 m = σ ◦ fm : Pm → Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' We now define further liftings f i m : Pm+i → Pi for i ⩾ 1 as follows: f 1 m(x1⊗ · · · ⊗xδ(m+1)⊗λ) = f 0 m(x1⊗ · · · ⊗xδ(m)⊗1)xδ(m)+1⊗xδ(m)+2 · · · xδ(m+1)λ f i m(y1⊗ · · · ⊗yδ(m+i)⊗λ) = f 0 m(y1⊗ · · · ⊗yδ(m)⊗1)yδ(m)+1⊗ · · · ⊗yδ(m+i)⊗λ if m or i ⩾ 2 is even f i m(y1⊗ · · · ⊗yδ(m+i)⊗λ) = f 0 m(y1⊗ · · · ⊗yδ(m)⊗1)yδ(m)+1⊗ · · · ⊗yδ(m+i−1)+1 ⊗yδ(m+i−1)+2 · · · yδ(m+i)λ if m and i ⩾ 2 are odd 26 JAWAD, SNASHALL, AND TAILLEFER where the xi are in Λ1 and the yi are in ΛA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' The proof that ( f i m)i⩾0 is a family of liftings of fm, that is, f i−1 ◦ bi+m = bi ◦ f i m for all i ⩾ 1, is tedious but straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Finally, if n or m is even and n ⩾ 2, m ⩾ 2 ,then fn fm(y1⊗ · · · ⊗yδ(m)+δ(n)⊗λ) = fn ◦ f n m(y1⊗ · · · ⊗yδ(m+n)⊗λ) = fn( f 0 m(y1⊗ · · · ⊗yδ(m)⊗1)yδ(m)+1⊗yδ(m)+2⊗ · · · ⊗yδ(m)+δ(n)⊗λ) = fn( fm(y1⊗ · · · ⊗yδ(m)⊗1)yδ(m)+1⊗yδ(m)+2⊗ · · · ⊗yδ(m)+δ(n)⊗λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' □ Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' If Λ is a (D, A)-stacked monomial algebra, with A > 1, D = dA and d ⩾ 3 , then E(Λ) is isomorphic to B as graded algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' In particular, Extn Λ(Λ0, Λ0) is isomorphic to Λ ♮ δ(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' We use the isomorphisms in Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='3 and the cup-product described in Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' If f = fδ(n)⊗ · · · ⊗f1 ∈ Bn and g = gδ(m)⊗ · · · ⊗g1 ∈ Bm, where m or n is even and both are at least 2, then ψ( f )ψ(g)(y1⊗ · · · ⊗yδ(m)+δ(n)⊗λ) = ψ( f ) � ψ(g)(y1⊗ · · · ⊗yδ(m)⊗1)yδ(m)+1⊗ · · · ⊗yδ(m+n)⊗λ � = fδ(n)( fδ(n)−1(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' ( f2( f1(ψ(g)(y1⊗ · · · ⊗yδ(m)⊗1)yδ(m)+1)yδ(m)+2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' )yδ′(m+n)−1)yδ(m+n))λ = fδ(n)( fδ(n)−1(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' ( f2( f1(gδ(m)(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' (g1(y1)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' yδ(m))yδ(m)+1)yδ(m)+2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' )yδ(m+n)−1)yδ(m+n))λ = ψ( f g)(y1⊗ · · · ⊗yδ(m)+δ(n)⊗λ) = ψ( f · g)(y1⊗ · · · ⊗yδ(m)+δ(n)⊗λ) therefore ϕ( f · g) = ψ( f )ψ(g) and we have proved that ψ is an isomorphism of graded algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' □ Remark A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Recall from Subsection 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='2 that the m-th projective in a minimal projective right Λ- module resolution of Λ0 is Lm = � Rm i ∈Rm t(Rm i )Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' By Consequence 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='4, Rm i ∈ Pm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' We then have an isomorphism Pm → Lm which is determined by Rm i �→ t(Rm i ) for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' As we mentioned in Subsection 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='2, the authors of [14] also gave a basis of E(Λ), namely the set � gm i ∈ HomΛ(Lm, Λ0) | Rm i ∈ Rm� where gm i (t(Rm j )) = t(Rm i ) if j = i and 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' The element gm i corresponds to a map in HomΛ(Pm, Λ0) which we denote again by gm i and that is defined by gm i (Rm j ) = t(Rm i ) if j = i and 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' We have isomorphisms KQA ∼= KΓ1 and KQA ∼= Λ ∗ A = (KQA) ∗ which combine to the iso- morphism which associates to α ∈ Γ1 the linear form fα on KQA that sends β ∈ QA to t(α) if β = α and to 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' This extends to an isomorphism between the algebras KΓ/(σ) and Λ ♮ that sends a path p of length n to the class of the linear map fp ∈ ( Λ ∗ A)⊗n defined on A-paths by fp(q) = t(p) if q = p and 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Now consider Rm i = α1 · · · αδ(m) with α ∈ QA for all i and R m i = αδ(m) · · · α1 in KΓ/(σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' It is easy to check that gm i = ψ � fαδ(m)⊗ · · · ⊗ fα1 � so that it corresponds, via the isomorphism above, to R m i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Therefore we have a basis BB = � R m i | Rm i ∈ Rm� of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' REFERENCES [1] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' BARDZELL, The alternating syzygy behavior of monomial algebras, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' Algebra 188 (1997), p 69-89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' [2] A.' metadata={'source': 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(2018), p 481-495.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content=' RUAA JAWAD, TECHNICAL INSTRUCTORS TRAINING INSTITUTE, MIDDLE TECHNICAL UNIVERSITY, BAGHDAD, IRAQ Email address: ruaa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='yousuf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='jawad@mtu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE5T4oBgHgl3EQfLw6U/content/2301.05476v1.pdf'} +page_content='iq NICOLE SNASHALL, SCHOOL OF COMPUTING AND MATHEMATICAL SCIENCES, UNIVERSITY OF LEICESTER, UNI- VERSITY ROAD, LEICESTER LE1 7RH, UNITED KINGDOM Email address: 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0000000000000000000000000000000000000000..c70760830ab3d3c59d4a6594f292aa8c7781f21a --- /dev/null +++ b/w9E0T4oBgHgl3EQfcQA2/content/tmp_files/2301.02359v1.pdf.txt @@ -0,0 +1,1941 @@ +CHARM: Composing Heterogeneous AcceleRators for Matrix +Multiply on Versal ACAP Architecture +Jinming Zhuang +University of Pittsburgh +USA +jinming.zhuang@pitt.edu +Jason Lau +University of California, +Los Angeles, USA +lau@cs.ucla.edu +Hanchen Ye +University of Illinois at +Urbana-Champaign, USA +hanchen8@illinois.edu +Zhuoping Yang +University of Pittsburgh +USA +zhuoping.yang@pitt.edu +Yubo Du +University of Pittsburgh +USA +yubo.du@pitt.edu +Jack Lo +Advanced Micro +Devices Inc., USA +jack.lo@amd.com +Kristof Denolf +Advanced Micro +Devices Inc., USA +kristof.denolf@amd.com +Stephen +Neuendorffer +Advanced Micro +Devices Inc., USA +stephen.neuendorffer@amd.com +Alex Jones +University of Pittsburgh +USA +akjones@pitt.edu +Jingtong Hu +University of Pittsburgh +USA +jthu@pitt.edu +Deming Chen +University of Illinois at +Urbana-Champaign, USA +dchen@illinois.edu +Jason Cong +University of California, +Los Angeles, USA +cong@cs.ucla.edu +Peipei Zhou +University of Pittsburgh +USA +peipei.zhou@pitt.edu +ABSTRACT +Dense matrix multiply (MM) serves as one of the most heavily +used kernels in deep learning applications. To cope with the high +computation demands of these applications, heterogeneous archi- +tectures featuring both FPGA and dedicated ASIC accelerators have +emerged as promising platforms. For example, the AMD/Xilinx +Versal ACAP architecture combines general-purpose CPU cores +and programmable logic with AI Engine processors optimized for +AI/ML. An array of 400 AI Engine processors executing at 1 GHz +can provide up to 6.4 TFLOPs performance for 32-bit floating-point +(fp32) data. However, machine learning models often contain both +large and small MM operations. While large MM operations can +be parallelized efficiently across many cores, small MM operations +typically cannot. We observe that executing some small MM lay- +ers from the BERT natural language processing model on a large, +monolithic MM accelerator in Versal ACAP achieved less than 5% +of the theoretical peak performance. Therefore, one key question +arises: How can we design accelerators to fully use the abundant +computation resources under limited communication bandwidth for +end-to-end applications with multiple MM layers of diverse sizes? +We identify the biggest system throughput bottleneck result- +ing from the mismatch of massive computation resources of one +monolithic accelerator and the various MM layers of small sizes in +the application. To resolve this problem, we propose the CHARM +framework to compose multiple diverse MM accelerator archi- +tectures working concurrently towards different layers within one +application. CHARM includes analytical models which guide de- +sign space exploration to determine accelerator partitions and layer +scheduling. To facilitate the system designs, CHARM automatically +generates code, enabling thorough onboard design verification. We +deploy the CHARM framework on four different deep learning +FPGA ’23, February 12–14, 2023, Monterey, CA, USA +© 2023 Copyright held by the owner/author(s). +This is the author’s version of the work. It is posted here for your personal use. Not +for redistribution. The definitive Version of Record was published in Proceedings of the +2023 ACM/SIGDA International Symposium on Field Programmable Gate Arrays (FPGA +’23), February 12–14, 2023, Monterey, CA, USA, https://doi.org/10.1145/3543622.3573210. +applications, including BERT, ViT, NCF, MLP, on the AMD/Xilinx +Versal ACAP VCK190 evaluation board. Our experiments show +that we achieve 1.46 TFLOPs, 1.61 TFLOPs, 1.74 TFLOPs, and 2.94 +TFLOPs inference throughput for BERT, ViT, NCF, MLP, respec- +tively, which obtain 5.29×, 32.51×, 1.00× and 1.00× throughput +gains compared to one monolithic accelerator. +CCS CONCEPTS +• Computer systems organization → Heterogeneous (hybrid) +systems; • Hardware → Hardware-software codesign. +KEYWORDS +Heterogeneous Architecture, Domain-Specific Accelerator, Versal +ACAP, Mapping Framework, Matrix-Multiply, Deep Learning +ACM Reference Format: +Jinming Zhuang, Jason Lau, Hanchen Ye, Zhuoping Yang, Yubo Du, Jack +Lo, Kristof Denolf, Stephen Neuendorffer, Alex Jones, Jingtong Hu, Deming +Chen, Jason Cong, Peipei Zhou. 2023. CHARM: Composing Heterogeneous +Accelerators for Matrix Multiply on Versal ACAP Architecture. In Proceed- +ings of the 2023 ACM/SIGDA International Symposium on Field Programmable +Gate Arrays (FPGA ’23), February 12–14, 2023, Monterey, CA, USA. ACM, +New York, NY, USA, 12 pages. https://doi.org/10.1145/3543622.3573210 +1 +INTRODUCTION +Dense matrix multiply (MM) serves as one of the most heavily used +kernels in many deep learning workloads, including BERT [1] for +natural language processing, NCF [2] for recommendations, ViT [3] +for vision classification, and MLP [4] for multilayer perceptron +classification or regression. According to profiling results from +Google [5], dense matrix multiply tasks occupied 90% of Neural +Network (NN) inference workload in Google’s data center in 2017. +The increasing complexity of these applications leads to extreme +demands for computation and data movement. +According to [6, 7, 8, 9], the off-chip bandwidth has been a bot- +tleneck for both the performance and energy efficiency of a system +and a common trend on current platforms is that the off-chip band- +width does not scale as fast as the computation resources. Therefore, +arXiv:2301.02359v1 [cs.AR] 6 Jan 2023 + +FPGA ’23, February 12–14, 2023, Monterey, CA, USA +Jinming Zhuang et al. +One Monolithic Acc +8 Duplicated Accs +10-1 +104 +103 +102 +101 +100 +GFLOPs +64 128 256 512 1K 1.5K 2K 3K +Square Matrix Size +X +X +X +X +A +D +C +B +Large data reuse +Compared to A, less data reuse due +to resource partition +2821.14 GFLOPs +692.96 GFLOPs +7.20 GFLOPs +0.41 GFLOPs +Shape mismatch, waste on +both computation and bandwidth +Compared to B, less waste +as the size of Accs decreases +Figure 1: Throughput of square MM under different sizes. +One Monolithic Acc 0123 +5 +4 +6 +7 +Two Diverse Accs +6 +7 +0 +4 +1 2 3 +5 +Acc0 +Acc1 +Region B +Region better than C +Region A +Region A +Figure 2: Execution timeline of one monolithic MM design +vs. two diverse MM accs design for BERT on VCK190. +the first research question arises: How to sustain the faster scaling +computation with the slower scaling off-chip bandwidth? +A common solution is to increase data reuse by allocating more +on-chip storage within an accelerator (acc). As shown in asymptotic +analysis in [9], the total off-chip communication volume in MM +scales as O( 1 +√ +𝑀 ) where M is the on-chip tile size. If we increase the +tile size, we can reduce the total communication volume, therefore +reducing the pressure on the off-chip bandwidth. +In this work, we target on the AMD/Xilinx Versal ACAP ar- +chitecture [10], which combines general-purpose CPU cores and +programmable logic (PL) with AI Engine processors (AIE) optimized +for AI/ML computation. For example, we implemented an MM ac- +celerator on an AMD/Xilinx VCK190 board using 384 AIEs and over +80% on-chip URAM and BRAM resources. The red line in Figure 1 +illustrates the performance of this accelerator. This design operates +on a native tile size of 1536×128×1024 and achieves 2.8 TFLOPs +throughput when carrying a tiled execution of a large square MM +(point A). However, when simply mapping different sizes of MM +to such a design, the performance decreases significantly as the +square MM size drops below 512, since each tile is padded to the +native tile size of the accelerator. For instance, at point B, the per- +formance of such a monolithic design goes to 0.41 GFLOPs, which +is 6880× lower than point A. Although padding is a common and +simple approach to implementing small MM operations on a large +accelerator, padding can waste both computation and bandwidth. +An alternative to padding is implementing multiple accelerators +with smaller native tile sizes, potentially executing different tasks on +each accelerator in parallel [11]. We apply this approach using eight +independent accelerators with a native tile size of 256×128×256, +illustrated by the blue dash line in Figure 1. For small square MM +operations with size 64, this approach achieves 7.2 GFLOPS at point +C, approximately 17× speedup compared to point B. +However, the smaller accelerator size also means less data reuse +for large MM, with total throughput almost saturation when the +operation size is larger than 256. When the MM size is 3072 (point +D), the total throughput from eight duplicate accs is 4.08× smaller +than point A in one monolithic design. +These experiments expose two conflicting design goals. Firstly, +we want to implement large MM operations with sufficient data +reuse to achieve the highest possible performance on the devices. +Secondly, we want to implement small MM operations while min- +imizing computation and communication overheads. Neither of +these simple designs seems able to achieve these design goals si- +multaneously. Therefore, the second research question arises: How +to trade-off between the two design goals for real-world, end-to-end +applications where MM layers with large and small sizes coexist? +To illustrate how these conflicting design goals can affect the +performance of practical machine learning models, we consider +BERT [1] as a representative workload containing MM layers with +both large and small sizes. In a transformer layer of BERT, there are a +total of 8 types of MM kernels where Kernels 0-5 are large MMs and +Kernels 6 and 7 are batch dots, i.e., small MMs. The detailed shapes +can be referred to Table 5. Take Kernel 5 and Kernel 6 as examples, +Kernel 5 is an MM with the shape 3072×1024×4096, Kernel 6 is a +batch dot with the shape 96×512×512×64, which means there are +96 small independent MMs sized at 512×512×64. +As shown in Figure 2, when using one monolithic MM accelera- +tor, Kernels 0-5 consume 92% of the total BERT MM computation +operations and 12% of the total MM acc time. In contrast, Kernels +6-7 consume 8% of the total operations but take 88% of the total +MM acc time. For Kernels 0-5, they lie in Region A (a region that +performs similarly to Point A in Figure 1), where the throughput of +acc is more than 2082 GFLOPS. For Kernel 6-7, they lie in Region B, +where the throughput of acc is only 23.6 GFLOPS. Given there is +a large portion of acc execution underutilized in the timeline, the +overall MM acc throughput is only 276 GFLOPS. Can we achieve a +design for BERT that lies in region A, i.e., good for large MMs, and +also in a region better than point C, i.e., good for small MMs with +less or no waste computation/bandwidth? +Our answer is “Yes". The key idea is to allocate more portion +of the resources to accs dedicated to computing larger MMs and a +smaller portion of the resources to other accs to compute smaller +MMs at the same time, as shown in Figure 2 where a two-diverse +accs system is illustrated. To achieve our design goals, we need to +solve these new challenges. First, we need to achieve high compu- +tation utilization for every single acc, i.e., use the smaller acc(s) to +reduce the waste for small MMs and use the larger acc(s) to max- +imize the data reuse for large MMs. Second, to maximize overall +utilization while maintaining high throughput and low latency, we +need to carefully overlap the execution time for these accs by co- +optimizing workload and resource partitioning. Third, to facilitate +the design space explorations (DSE), we need analytical models +to optimize the overall throughput under resource and bandwidth +constraints. Fourth, to reduce the programming efforts for the sys- +tem implementation, we need automatic code generation. Fifth, +to resolve the dependency of the kernels within the application +graph when running multiple accs we need an accelerator runtime +to schedule kernels from different tasks onto the accs. +To answer the research questions, we propose the CHARM archi- +tecture and its corresponding automation framework, the CHARM +framework. Our contributions are summarized below: + +CHARM: Composing Heterogeneous AcceleRators for Matrix Multiply on Versal ACAP Architecture +FPGA ’23, February 12–14, 2023, Monterey, CA, USA +• CHARM Systematical Design Methodology on Versal: To +achieve high computation and communication efficiency of each +acc, in Section 4, we propose a thorough design methodology +on Versal heterogeneous platform. We further provide automatic +CHARM DSE (CDSE) to find the optimized single acc configu- +ration. To the best of our knowledge, this is the first work that +provides a detailed analysis of the systematical data movement +and computation on Versal. +• CHARM Architecture and Framework: To achieve the de- +sign goals of good performance for MMs with both small and +large sizes in an application, in Section 5, we propose the CHARM +architecture and the CHARM framework to find the optimized +design. In the CHARM framework, there are several modules. +First, on top of CDSE, we propose the CHARM diverse accel- +erator composer (CDAC), which features a sort-based two-step +search algorithm to find an optimized CHARM design in the +polynomial time complexity instead of exponential time com- +plexity. Furthermore, to automate the system implementation, +CHARM automatic code generation (CACG) is proposed to gen- +erate source code files for AIEs, PL, and host CPU. Lastly, the +CHARM runtime system (CRTS) is launched in the host CPU +that schedules different kernels to the accs for optimizing both +task latency and overall system throughput. +• We deploy the CHARM framework to accelerate four applica- +tions on VCK190 in Section 6. Our on-board experiments demon- +strate that CHARM achieves 1.46 TFLOPs, 1.61 TFLOPs, 1.74 +TFLOPs, and 2.94 TFLOPs inference throughput for BERT, ViT, +NCF, MLP, respectively, which obtain 5.29×, 32.51×, 1.00×, and +1.00×, throughput gains compared to one monolithic accelerator. +• White-Box Open-Source Tools for Versal. While AMD pro- +vides users a block-box IP for NN applications called DPU [11], +we open-sourced our tools completely as a white-box with a de- +tailed step-by-step guide to reproduce all of the results presented +in this paper and for the other users to learn and leverage in +their end-to-end systems. (https://github.com/arc-research- +lab/CHARM) +2 +PRIOR WORK +To achieve high throughput and energy efficiency, NN accelerators +usually employ a large number of processing elements (PE) and +share a similar memory hierarchy. That is, while the big bulk of data +is stored in the off-chip memory, there are multiple levels of on-chip +buffers, including the local memory attached to each PE and global +shared memory, to further reduce the costly data movement from/to +off-chip memory. Several works contribute to NN accelerators by +discussing the data reuse opportunities, computation parallelism, +and the choice of dataflow. +However, many of the prior works apply a one-size-fits-all mono- +lithic design that cannot efficiently handle layers with huge dif- +ferences in shapes and sizes (Eyeriss [12, 13], ShiDiannao [14], +NPU [15, 16, 17] and others [18, 19, 20, 21]). AutoSA [22] is a +polyhedral-based compilation framework that generates mono- +lithic systolic array designs for dense matrices. Sextans and Ser- +pens [23, 24] are general-purpose monolithic accelerators for sparse +matrices. [25, 26] analyze layout and pipeline efficiency. Other +works like AMD DPU [11], Mocha [27] explore task-level paral- +lelism by allocating multiple duplicate accs on the device without +Table 1: Comparison with prior works. +Prior +Works +One +Mono +Multi +Duplicate +Multi +Diverse +Workload +Assignment +Specializa +-tion for Acc +Eyeriss etc. +[12]-[26] +✓ +× +× +× +× +DPU etc. +[11, 27] +✓ +✓ +× +× +× +DNN Expl. +etc. [28, 29] +✓ +✓ +✓ +× +× +Herald [32] +✓ +✓ +✓ +✓ +× +CHARM +(Ours) +✓ +✓ +✓ +✓ +✓ +specializing each acc. DNNBuilder [28] designs a dedicated acc for +each layer according to the number of operations within the layer. +DNNExplorer [29] enhances DNNBuilder by combining dedicated +accs for the first several layers and a monolithic acc for the rest of +the layers. While it employs multiple accelerators, it lacks a com- +prehensive exploration of workload assignments. TETRIS [30] and +TANGRAM [31] propose multiple dataflow optimizations within +and across the NN layers to improve performance and energy effi- +ciency. Although they offer diverse accelerator designs, they lack +the DSE and workload assignment for high overall throughput. Her- +ald [32] proposes an architecture with multiple diverse accelerators +and explores the workload assignment and resource partition. Still, +they choose several existing acc designs from their candidate pool, +e.g., ShiDiannao [14], NVDLA [33] without doing DSE for each +acc. FPCA [34] and CHARM'12 [35] propose a fully pipelined and +dynamically composable coarse-grained reconfigurable architec- +ture and compose loosely coupled accelerators for different kernels +within an application via permutation network, which costs high +in chip area. +In conclusion, we summarize the differences between our work +and prior works in Table 1. Our work is capable of choosing the de- +sign from one monolithic, multiple duplicates, and multiple diverse +accelerators, and each accelerator is a specialized design considering +the different workload assignments, dataflow, and data parallelism +strategies that are covered by our DSE. +3 +VERSAL ACAP ARCHITECTURE +OVERVIEW +In this section, we first introduce the system architecture of AMD/ +Xilinx Versal ACAP architecture in Section 3.1 and then the memory +model of AIE array in Section 3.2. +3.1 +Versal ACAP Architecture +Figure 3 illustrates the overall architecture of the VCK190 [36] +board and highlights the AIE array on the top. The VCK190 board +features (1) the first-generation AIE architecture, which has 8 × 50 +1 GHz 7-way VLIW processors supporting vector operations up to +1024 bits [37], (2) ARM processors to run Linux and general-purpose +applications, and (3) PL to design application-specific hardware +with Digital Signal Processors (DSP) available for integration. The +AI engine cores and ARM CPUs can be programmed with C/C++ +code, while PL can be programmed using both RTL and C/C++ code +using High-Level Synthesis (HLS) [38, 39, 40, 41, 42, 43]. These three + +FPGA ’23, February 12–14, 2023, Monterey, CA, USA +Jinming Zhuang et al. + +Processor +System +(ARM) +AIE +Core +Interface +Switch +AIE +Core +Switch +AIE +Core +Switch +AIE +Core +Switch +AIE +Core +Switch +AIE +Core +Switch +Switch +Interface +Switch +Interface +Switch +… +… +… +… +… +… +NOC +DRAM +AI Engine Array +Configuration +Reconfigurable Hardware +Instruction-Based Processors +AXI to DDR +Shared Memory +AXI Stream +Programmable Logic +(FPGA) +Figure 3: Versal ACAP architecture. +components are integrated with I/O peripherals, such as PCIe and +DRAM controllers, into a heterogeneous SoC with a Network-on- +Chip (NoC). The VCK190 board is equipped with one DDR4-DIMM +off-chip memory with a 25.6 GB/s peak bandwidth. +3.2 +AIE Memory Model +Each AIE processor tile contains 32 KB of data and is capable of shar- +ing data with the adjacent AIEs in four directions (AIE↔neighbor +AIE). In addition to local memory shared with adjacent tiles, each +AIE tile also connects to an AXI-Stream (AXIS) switch network, +which enables non-local communication between AIE processors +(AIE↔non-local) and communication with the PL through the +PLIOs in the 39 interface tiles (PL↔AIEs). The VCK190 board +provides 1.2 TB/s (PL↔AIEs) / 0.9 TB/s (AIEs↔PL) bandwidth be- +tween PL and AIEs, which is 46× more than the bandwidth between +DDR4 and PL. The AXIS switches support both circuit-switched +and packet-switched connections between ports. Circuit-switched +connections provide dedicated, deterministic communication and +support broadcast, where data from a single input channel is trans- +mitted to multiple output channels simultaneously. Packet-switched +connections allow data from an input channel to be dynamically +routed to different destinations based on a destination header at the +start of each packet. This enables data flows to be time-multiplexed +on a single routing path. One situation in which we can use packet- +switched connections happens when the computation-to-communi- +cation (CTC) ratio of an AIE is more than one. During the compu- +tation of AIE 0, the port assigned to this AIE is idle and thus can +be used to transfer data to another AIE, say AIE 1, by assigning a +different header that matches the destination ID of AIE 1. +4 +CHARM SINGLE ACCELERATOR DESIGN +In this section, we describe the dataflow and mapping strategy +for a single MM acc using hundreds AIEs in Section 4.1. Then, in +1 +// Off-Chip <-> On-Chip Time Loop +2 +for(int i.0=0;i.0