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Let ρ∈𝒮⁢(ℋ)𝜌𝒮ℋ\rho\in\mathcal{S}(\mathcal{H})italic_ρ ∈ caligraphic_S ( caligraphic_H ) and σ∈𝒫⁢(ℋ)𝜎𝒫ℋ\sigma\in\mathcal{P}(\mathcal{H})italic_σ ∈ caligraphic_P ( caligraphic_H ). The sandwiched Rényi

\otimes\bar{\omega}_{A}(\alpha))italic_I start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_R : italic_A ) start_POSTSUBSCRIPT italic_ρ end_POSTSUBSCRIPT = italic_D start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_ρ start_POSTSUBSCRIPT italic_R italic_A end_POSTSUBSCRIPT ∥ over¯ start_ARG italic_σ end_ARG start_POSTSUBSCRIPT italic_R end_POSTSUBSCRIPT ( italic_α ) ⊗ over¯ start_ARG italic_ω end_ARG start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_α ) ). By definition and the monotonicity of Dα∗superscriptsubscript𝐷𝛼D_{\alpha}^{*}italic_D start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT with respect to the Rényi parameter (Proposition 4 (i)), we have

Monotonicity in Rényi parameter MDSFT2013on ; Beigi2013sandwiched ; MosonyiOgawa2017strong : if 0≤α≤β0𝛼𝛽0\leq\alpha\leq\beta0 ≤ italic_α ≤ italic_β, then

conditional Rényi entropy MDSFT2013on and the log-Euclidean conditional Rényi entropy CHDH2020non of order α∈(0,+∞)∖{1}𝛼01\alpha\in(0,+\infty)\setminus\{1\}italic_α ∈ ( 0 , + ∞ ) ∖ { 1 } for a

Monotonicity in σ𝜎\sigmaitalic_σ MDSFT2013on ; MosonyiOgawa2017strong : if σ′≥σsuperscript𝜎′𝜎\sigma^{\prime}\geq\sigmaitalic_σ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ≥ italic_σ, then Dα(t)⁢(ρ∥σ′)≤Dα(t)⁢(ρ∥σ)superscriptsubscript𝐷𝛼𝑡conditional𝜌superscript𝜎′superscriptsubscript𝐷𝛼𝑡conditional𝜌𝜎D_{\alpha}^{(t)}(\rho\|\sigma^{\prime})\leq D_{\alpha}^{(t)}(\rho\|\sigma)italic_D start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT ( italic_ρ ∥ italic_σ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) ≤ italic_D start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT ( italic_ρ ∥ italic_σ ),

B

\text{with $\eta=\zeta$}]{\text{ Algorithm 4.1}}\mathcal{D}_{\alpha}^{\eta}.caligraphic_D start_ARROW underwith the optimal ζ start_ARROW start_OVERACCENT italic_α - Process end_OVERACCENT → end_ARROW end_ARROW caligraphic_D start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT start_ARROW underwith z determined by () start_ARROW overRandomly pick z data points → end_ARROW end_ARROW caligraphic_D start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_z end_POSTSUPERSCRIPT start_ARROW underwith η=ζ start_ARROW over Algorithm 4.1 → end_ARROW end_ARROW caligraphic_D start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_η end_POSTSUPERSCRIPT .

count the number of data points in the ζ𝜁\zetaitalic_ζ-vicinity of ξd(i)subscriptsuperscript𝜉𝑖d\xi^{(i)}_{\rm d}italic_ξ start_POSTSUPERSCRIPT ( italic_i ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_d end_POSTSUBSCRIPT and save it to set 𝒟αsubscript𝒟𝛼\mathcal{D}_{\alpha}caligraphic_D start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT if

We used three representative test systems, i.e., IEEE 6, 39, and- 118-bus systems [1]. The problem sizes of the p-OPFs for these test systems are provided in Table 2 (recall that n𝑛nitalic_n and r𝑟ritalic_r are the numbers of decision and uncertain variables respectively, and m𝑚mitalic_m is the number of constraints that are contributed by one data point). In this numerical experiment, α=1%𝛼percent1\alpha=1\%italic_α = 1 % is considered. Table 2 also tabulates the sizes of sets 𝒟𝒟\mathcal{D}caligraphic_D, 𝒟αsubscript𝒟𝛼\mathcal{D}_{\alpha}caligraphic_D start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT, 𝒟αzsuperscriptsubscript𝒟𝛼𝑧\mathcal{D}_{\alpha}^{z}caligraphic_D start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_z end_POSTSUPERSCRIPT, and 𝒟αηsuperscriptsubscript𝒟𝛼𝜂\mathcal{D}_{\alpha}^{\eta}caligraphic_D start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_η end_POSTSUPERSCRIPT (recall that they are the original historical data set, the data set after the α𝛼\alphaitalic_α-process, the input data sets of D-DA(𝒟αzsuperscriptsubscript𝒟𝛼𝑧\mathcal{D}_{\alpha}^{z}caligraphic_D start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_z end_POSTSUPERSCRIPT) and D-DA(𝒟αηsuperscriptsubscript𝒟𝛼𝜂\mathcal{D}_{\alpha}^{\eta}caligraphic_D start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_η end_POSTSUPERSCRIPT), respectively) for each case.

The value of ζ𝜁\zetaitalic_ζ for each case listed in Table 2 are the values that make the resulting probability distribution smoothest. The case of IEEE 39-bus system is given in Figure 6 as an example for elaborating this idea. It can be observed that, with an ζ𝜁\zetaitalic_ζ at around 0.12, we obtain a smooth probability distribution. Both bigger and smaller ζ𝜁\zetaitalic_ζ will result in less smooth distribution.

A pictorial interpretation of the above definition is given in Figure 4. In Definition 3.2, the selection of the bandwidth ζ𝜁\zetaitalic_ζ influences the accuracy of the estimated joint probability ℙ⁢[ξ=ξd(j)]ℙdelimited-[]𝜉subscriptsuperscript𝜉𝑗d\mathbb{P}[\xi=\xi^{(j)}_{\rm d}]blackboard_P [ italic_ξ = italic_ξ start_POSTSUPERSCRIPT ( italic_j ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_d end_POSTSUBSCRIPT ]. The optimal value of ζ𝜁\zetaitalic_ζ varies from case to case. Fortunately, there exist various methods, such as the plug-in [9] and cross validation [19] selectors, that one can use for determining the optimal ζ𝜁\zetaitalic_ζ for a specific case. A numberical example of selecting the best ζ𝜁\zetaitalic_ζ is provided in Subsection 5.2.

C

As such, it is well-suited for the task of estimating the prior and the obtained density estimate is proper and of the desired mixture type.

The rest of the paper is organized as follows. In Section 2 we introduce the class of minimum distance estimators and identify conditions on the distance function that guarantees the existence and uniqueness of the minimizer. The theoretical guarantees in terms of density estimation and regret are presented in Theorem 2 and Theorem 3 therein. The proof sketches of these results are presented in Section 3 and Section 4 respectively. In Section 5 we present an algorithm for computing minimum-distance estimators in the one-dimensional setting and study their numerical performance in empirical Bayes estimation with both simulated and real datasets. In Section 6 we mention our theoretical results in a multi-dimensional setting. For all the other related details of proofs, see Appendix.

In this section we analyze the performances of the empirical Bayes estimators based on the minimum-H2superscript𝐻2H^{2}italic_H start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT, the minimum-χ2superscript𝜒2\chi^{2}italic_χ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT, and the minimum-KL divergence estimator (i.e., the NPMLE). We compare them against the Robbins estimator and also draw comparisons among their individual performances. Unlike the Robbins estimator, the minimum-distance based estimators do not admit a closed form solution.

Assumption 3 and Assumption 4 are identical to Assumption 1 and Assumption 2 respectively. The loss functions such as the Kullback-Leibler divergence, squared Hellinger distance, Chi-squared divergence satisfy the above assumptions, similarly as in the one-dimensional case. We have the following results.

As a concrete example, we consider a simple uniform prior and compare the numerical performance of Robbins and three prototypical examples of minimum-distance estimators of G𝐺Gitalic_G, with respect to the Kullback-Leibler (KL) divergence (i.e., the NPMLE), the Hellinger distance, and the χ2superscript𝜒2\chi^{2}italic_χ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT-divergence, respectively (see Section 2.1 for the formal definitions). As evident in

D

ϕ9⁢({r})⁢(1)−ϕ9⁢({s})⁢(1)(mod9)annotatedsubscriptitalic-ϕ9𝑟1subscriptitalic-ϕ9𝑠1pmod9\phi_{9}(\{r\})(1)-\phi_{9}(\{s\})(1)\pmod{9}italic_ϕ start_POSTSUBSCRIPT 9 end_POSTSUBSCRIPT ( { italic_r } ) ( 1 ) - italic_ϕ start_POSTSUBSCRIPT 9 end_POSTSUBSCRIPT ( { italic_s } ) ( 1 ) start_MODIFIER ( roman_mod start_ARG 9 end_ARG ) end_MODIFIER

The authors declare that the data supporting the findings of this study are available within the paper, its source code is available on https://github.com/anthonydoyon/Ramanujan-s-tau-and-MT-elts.

The authors thank Rob Pollack for answering many of their questions during the preparation of the article. The authors are also indebted to the anonymous referees for helpful comments and suggestions on earlier versions of the article. This article forms part of the master thesis of the first named author at Université Laval, who was supported by an NSERC and a FRQNT graduate scholarships. The research of the second named author is supported by the NSERC Discovery Grants Program RGPIN-2020-04259 and RGPAS-2020-00096.

The computations in [DL21] show that the λ𝜆\lambdaitalic_λ-invariants of the Mazur–Tate elements attached to ΔΔ\Deltaroman_Δ at p∈{5,7}𝑝57p\in\{5,7\}italic_p ∈ { 5 , 7 } satisfy exactly the same formula given in Theorem B for small n𝑛nitalic_n. We prove that this formula, in fact, holds for all n𝑛nitalic_n. As we have already alluded to above, multiplicity one modulo p𝑝pitalic_p results are not available in our current setting. Instead, we shall take a computational approach. Another theoretical issue we face is that the Fontaine–Laffaille condition (which requires that the weight of the modular form to be smaller than the prime p𝑝pitalic_p) does not hold in our current setting. Consequently, the canonical period of Vatsal [Vat99], which behaves well under congruences, is not available. We overcome this by considering cohomological periods instead, following the approach taken in [PW11].

The calculations in the following are based on the methods discussed in [PS11] and [Ste07, §8], which are carried out on SageMath. The code of our calculations can be found in [Doy22, lambda.sagews].

A

“ρk⁢(ω)=αk⁢(ω)subscript𝜌𝑘𝜔subscript𝛼𝑘𝜔\rho_{k}(\omega)=\alpha_{k}(\omega)italic_ρ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_ω ) = italic_α start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_ω )” holds for X=M⁢(𝒜)𝑋𝑀𝒜X=M(\mathcal{A})italic_X = italic_M ( caligraphic_A ) the

the integral homology of the Milnor fiber H1⁢(F,ℤ)subscript𝐻1𝐹ℤH_{1}(F,\mathbb{Z})italic_H start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_F , blackboard_Z ) has

F⟶M⁢(𝒜)⟶𝐹𝑀𝒜F\longrightarrow M(\mathcal{A})italic_F ⟶ italic_M ( caligraphic_A ) which is the Milnor fiber of the cone of 𝒜𝒜\mathcal{A}caligraphic_A.

of the Milnor fiber F𝒜subscript𝐹𝒜F_{\mathcal{A}}italic_F start_POSTSUBSCRIPT caligraphic_A end_POSTSUBSCRIPT for

The basic strategy to study the Milnor fiber F𝒜subscript𝐹𝒜F_{\mathcal{A}}italic_F start_POSTSUBSCRIPT caligraphic_A end_POSTSUBSCRIPT of a

B

}})roman_ext ( italic_f ) ∈ italic_C start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_rad end_POSTSUBSCRIPT ( over¯ start_ARG blackboard_B start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT end_ARG ) with tr⁡(ext⁡(f))=ftrext𝑓𝑓\operatorname{tr}(\operatorname{ext}(f))=froman_tr ( roman_ext ( italic_f ) ) = italic_f for all f∈C∞⁢([0,r2])𝑓superscript𝐶0superscript𝑟2f\in C^{\infty}([0,r^{2}])italic_f ∈ italic_C start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT ( [ 0 , italic_r start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ] ). It follows from Theorem 1.1 that the operators trtr\operatorname{tr}roman_tr and extext\operatorname{ext}roman_ext are bounded and thus have unique bounded linear extensions to the closure of their respective domains which are inverses of each other. In the same way, an isomorphism is also established for homogeneous Sobolev norms on ℝdsuperscriptℝ𝑑\mathbb{R}^{d}blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT.

Corotational maps are a special case of so-called equivariant maps between rotationally symmetric manifolds, see e.g., [12, p. 109, Definition 8.1] for a general definition. As such, they appear as critical points to certain geometric action functionals, most prominently in the wave maps equation and in Yang-Mills equations. Equivalence results for Sobolev norms of corotational maps then allow to link the function spaces for the reduced equations of the radial profile to the spaces that underlie the general equations. Radial lemmas that characterize the Sobolev norm of F𝐹Fitalic_F in terms of its radial profile f𝑓fitalic_f were formulated in [13] on ℝdsuperscriptℝ𝑑\mathbb{R}^{d}blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT when k=d2𝑘𝑑2k=\frac{d}{2}italic_k = divide start_ARG italic_d end_ARG start_ARG 2 end_ARG in the context of wave maps and lately in [7] with the restriction 0≤k<d2+20𝑘𝑑220\leq k<\frac{d}{2}+20 ≤ italic_k < divide start_ARG italic_d end_ARG start_ARG 2 end_ARG + 2 in the context of Yang-Mills equations. However, it is expected in [7, Remark A.4] that an equivalence result also holds for spaces of higher order. Since a complete charactersiation of Sobolev norms for corotational maps is of current interest in the study of stability of blowup in nonlinear wave equations, we provide the full result in the following theorem.

In this paper, we provide in Theorems 1.1 and 1.2 the missing characterization of radial Sobolev spaces. We also apply our results in Theorem 1.3 to describe Sobolev norms of corotational maps.

“≳greater-than-or-equivalent-to\gtrsim≳”: For the converse inequality, we start off with Theorem 1.1 and use the second part of Lemma 2.3 to get the estimate

Together with Theorem 1.1 and the previous lemmas, we are able to determine Sobolev norms of corotational maps.

D

\right).roman_dimGr start_POSTSUPERSCRIPT italic_P end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_H start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT ( italic_X , blackboard_Q ) = roman_dim italic_H start_POSTSUPERSCRIPT italic_m - italic_i - ( italic_a - italic_r ) end_POSTSUPERSCRIPT ( italic_Y , start_FLOATSUPERSCRIPT fraktur_p end_FLOATSUPERSCRIPT caligraphic_H start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ( italic_R italic_f start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT blackboard_Q start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT [ italic_a - italic_r ] ) ) .

with ℋi𝔭⁢(R⁢f∗⁢ℚX)superscriptsuperscriptℋ𝑖𝔭𝑅subscript𝑓subscriptℚ𝑋{{}^{\mathfrak{p}}{\mathcal{H}}^{i}}(Rf_{*}{\mathbb{Q}}_{X})start_FLOATSUPERSCRIPT fraktur_p end_FLOATSUPERSCRIPT caligraphic_H start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ( italic_R italic_f start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT blackboard_Q start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT ) semisimple perverse sheaves on Y𝑌Yitalic_Y; we say that a closed Z⊂Y𝑍𝑌Z\subset Yitalic_Z ⊂ italic_Y is a support of f𝑓fitalic_f if it is a support of a simple summand of some ℋi𝔭⁢(R⁢f∗⁢ℚX)superscriptsuperscriptℋ𝑖𝔭𝑅subscript𝑓subscriptℚ𝑋{{}^{\mathfrak{p}}{\mathcal{H}}^{i}}(Rf_{*}{\mathbb{Q}}_{X})start_FLOATSUPERSCRIPT fraktur_p end_FLOATSUPERSCRIPT caligraphic_H start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ( italic_R italic_f start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT blackboard_Q start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT ). A particularly interesting case is that f𝑓fitalic_f has full support, that is, Y𝑌Yitalic_Y is the only support of f𝑓fitalic_f; in this case the cohomology of any closed fiber of f𝑓fitalic_f is governed by the nonsingular fibers.

The decomposition for the parabolic Hitchin map h^parsuperscript^ℎpar\widehat{h}^{\mathrm{par}}over^ start_ARG italic_h end_ARG start_POSTSUPERSCRIPT roman_par end_POSTSUPERSCRIPT has full support, i.e., any non-trivial simple perverse summand of

Since a perverse sheaf, as a complex of constructible sheaves, is concentrated in degrees [−b,0]𝑏0[-b,0][ - italic_b , 0 ], it only has non-trivial cohomology in degrees ≥−babsent𝑏\geq-b≥ - italic_b. In particular, the right-hand side of (2) is non-trivial only if

Since c1⁢(L⁢(ξ))subscript𝑐1𝐿𝜉c_{1}(L(\xi))italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_L ( italic_ξ ) ) automatically has strong perversity 2222, showing it has strong perversity 1111 is equivalent to showing that the induced morphism of perverse cohomology sheaves

C

((a)) Planewave scattering by a homogeneous obstacle, for k=2⁢π𝑘2𝜋k=2\piitalic_k = 2 italic_π and η=4𝜂4\eta=4italic_η = 4 in ΩΩ\Omegaroman_Ω. Left: Real part of the total field u𝑢uitalic_u solution of the transmission problem (96); nearly-singular volume integration arising in near-field evaluation of the representation formula Eq. 95 is handled seamlessly by VDIM per Remark 3.1 Right: Logarithm in base ten of the absolute pointwise error in the Lippmann-Schwinger equation solution for interpolation degrees n=1,2,3𝑛123n=1,2,3italic_n = 1 , 2 , 3 and 4444 over a fixed mesh of ΩΩ\Omegaroman_Ω with h≈0.09ℎ0.09h\approx 0.09italic_h ≈ 0.09. The maximum poinwise errors in those figures are around 3.0⋅10−3⋅3.0superscript1033.0\cdot 10^{-3}3.0 ⋅ 10 start_POSTSUPERSCRIPT - 3 end_POSTSUPERSCRIPT, 2.3⋅10−5⋅2.3superscript1052.3\cdot 10^{-5}2.3 ⋅ 10 start_POSTSUPERSCRIPT - 5 end_POSTSUPERSCRIPT, 8.8⋅10−6⋅8.8superscript1068.8\cdot 10^{-6}8.8 ⋅ 10 start_POSTSUPERSCRIPT - 6 end_POSTSUPERSCRIPT, and 9.2⋅10−7⋅9.2superscript1079.2\cdot 10^{-7}9.2 ⋅ 10 start_POSTSUPERSCRIPT - 7 end_POSTSUPERSCRIPT, respectively.

that can itself be recast into a second-kind boundary integral equation [48]. By solving the boundary integral equation (and employing an exponentially convergent Nyström method, based on the Martensen-Kussmaul quadrature rule [24, Sec. 3.5] for kernels with logarithmic singularities to do so), we obtain a highly accurate reference solution that serves as the benchmark to quantify the error in the solution of the volume integral equation for this piecewise constant refractive index medium. Figure 6(a) displays the real part of the total field u𝑢uitalic_u in this example as well as the pointwise errors within ΩΩ\Omegaroman_Ω in the volume integral equation solutions obtained utilizing the proposed method for interpolation degrees n=1,2,3𝑛123n=1,2,3italic_n = 1 , 2 , 3 and 4444 for a fixed mesh with h≈0.09ℎ0.09h\approx 0.09italic_h ≈ 0.09. The resulting linear system was solved via GMRES with a tolerance of 10−7superscript10710^{-7}10 start_POSTSUPERSCRIPT - 7 end_POSTSUPERSCRIPT, which was achieved in approximately the same number of iterations (∼340similar-toabsent340\sim 340∼ 340) in all the examples presented in that figure.

To assess the accuracy of such a Nyström method for the Lippmann-Schwinger equation, we first focus on a specific scenario where η𝜂\etaitalic_η maintains a constant value of 4444 within ΩΩ\Omegaroman_Ω. In this case, the total field can be determined by solving the transmission problem

((b)) Gaussian beam scattering by an inhomogeneous obstacle, for k=3⁢π𝑘3𝜋k=3\piitalic_k = 3 italic_π and a variable refractive index. Left: Piecewise smooth refractive index η𝜂\etaitalic_η generated from a skull x-ray image. Right: Real part of the total field, resulting from the scattering of a Gaussian beam by the skull-like inhomogeneous obstacle, obtained from an approximate Lippmann-Schwinger equation solution.

We consider, finally, a challenging scattering problem involving a piecewise-smooth refractive index η𝜂\etaitalic_η generated from an x-ray image of a human skull, seen in the left panel of Figure 6(b). The right panel of that figure shows the real part of the total field resulting from the scattering of a Gaussian beam that impinges on the skull obstacle from the top-left corner. The Lippmann-Schwinger integral equation solution displayed in that figure, was obtained using the proposed method with n=2𝑛2n=2italic_n = 2. The scatterer diameter is about 9⁢λ9𝜆9\lambda9 italic_λ, where the wavelength is λ=2⁢π/k=2/3𝜆2𝜋𝑘23\lambda=2\pi/k=2/3italic_λ = 2 italic_π / italic_k = 2 / 3 in this example. GMRES convergence was achieved after 753753753753 iterations for a relative error tolerance of 10−7superscript10710^{-7}10 start_POSTSUPERSCRIPT - 7 end_POSTSUPERSCRIPT. In order to generate a reference solution to roughly estimate the error, the same problem was solved over the same mesh (h=0.11ℎ0.11h=0.11italic_h = 0.11) but with n=4𝑛4n=4italic_n = 4. The relative error, measured on a rectangular curve enclosing the obstacle, is approximately 10−4superscript10410^{-4}10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT.

C

Later, Newton et al., (2021) modified that procedure by re-weighting both the individual likelihood terms and the log-prior.

When run with sparsity-inducing priors like the SSL, repeatedly solving the weighted MAP problem produces exactly sparse samples, which are not representative of the absolutely continuous posterior distribution.

In fact, with slight modifications, our proof of Theorem 2 can be used to establish the posterior contraction rate for the sparse Gaussian graphical model with SSL priors; see Section S2 in the Supplemental Materials.

Then, we repeatedly solve the the randomized optimization problem in Equation (8) to obtain 500 bootstrap re-samples of (B,Ω).𝐵Ω(B,\Omega).( italic_B , roman_Ω ) .

They further showed that the distribution of their re-weighted and re-centered Bayesian bootstrap samples contracts with the same rate as the true posterior towards the true parameters.

A

\infty&\text{ else }.\end{cases}roman_FW ( italic_z ) = { start_ROW start_CELL - ( ∫ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT italic_b ( italic_z ( italic_t ) ) italic_d italic_z ( italic_t ) - divide start_ARG 1 end_ARG start_ARG 2 end_ARG ∫ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT italic_b start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_z ( italic_t ) ) italic_d italic_t ) + divide start_ARG 1 end_ARG start_ARG 2 end_ARG ∫ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT ( italic_z start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_t ) ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_d italic_t end_CELL start_CELL if italic_z ∈ caligraphic_W start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 , 2 end_POSTSUPERSCRIPT end_CELL end_ROW start_ROW start_CELL ∞ end_CELL start_CELL else . end_CELL end_ROW

Small noise large deviations for arbitrary Gaussian measures on Banach space are well known. In [1], for instance, it is shown that the Freidlin-Wentzell rate function for a general Gaussian measure μ0subscript𝜇0\mu_{0}italic_μ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT with Cameron-Martin space ℋμ0subscriptℋsubscript𝜇0\mathcal{H}_{\mu_{0}}caligraphic_H start_POSTSUBSCRIPT italic_μ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT is 12∥⋅∥μ02\frac{1}{2}\|\cdot\|_{\mu_{0}}^{2}divide start_ARG 1 end_ARG start_ARG 2 end_ARG ∥ ⋅ ∥ start_POSTSUBSCRIPT italic_μ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT. In particular, the small noise rate function for ε⁢B⁢(t)𝜀𝐵𝑡\varepsilon B(t)italic_ε italic_B ( italic_t ) is

The Friedlin-Wentzell function arises as the small noise large deviations rate function of a Markov diffusion process [9, 15] . The Friedlin-Wentzell theory is also used in pathwise analyses of metastable behavior, with the large deviations rate function begin used to determine the time-scale of ‘tunneling’ behavior in the small noise setting [26]. Metastable phenomena emerge outside the small noise setting, and therefore the relationship between the Onsager-Machlup and Friedlin-Wentzell theories is of significance and interest. The convergence of the Onsager-Machlup to Freidlin-Wentzell for SDEs, specifically, was studied in [19] and [12]. This paper establishes the relationship between them for measures that are equivalent to Gaussian measures on Banach spaces.

We begin this section with a motivating example for small noise large deviations for Gaussian measures in finite dimensions.

The proof of Theorem 1.1 boils down to two “tilting” lemmas - one for Onsager-Machlup in Corollary 2.6 and one for large deviations in Lemma 3.2. In large deviations analysis, the exponential tilting principle is a mechanism for transferring large deviations principles from a given sequence of measures to a sequence of equivalent measures. Lemma 3.2 proves a slightly generalized version of this principle that applies in the setting of measures equivalent to a Gaussian. In Corollary 2.6, we present the Onsager-Machlup function for measures equivalent to Gaussians, and represents a type of tilting. Our proof uses these two results to establish the main Theorem 1.1.

C

This study thus supports conjectures from biology stating that the optimization of certain mechanical properties, in particular the bending and torsional rigidities, are a driving force in the development of plant morphology [34, 35].

The authors would like to thank the livMatS Cluster of Excellence (Living, Adaptive and Energy-autonomous Materials Systems) and, in particular, the members Olga Speck and Thomas Speck, from the Plant Biomechanics Group Freiburg, for sharing their knowledge on Plant Biomechanics. The authors would also like to thank Thomas Speck for providing the picture in Fig. 2, Laura Melas for useful discussions on the topic and the anonymous referees for their careful reading.

We refer the interested reader to the in-depth monograph [10] for a comprehensive introduction and deep analysis of the topic.

For a comprehensive introduction on the topic, we refer the interested reader to the monographs of Antman [4, 5] and to [18, 33], for further discussions about the history of the subject.

Figure 2. Cross-sections of liana Condylocarpon Guianense in the non-self-supporting old ontogenetic stage after attachment to a support. The secondary xylem is marked with (1) and the cortex is marked with (2). ©  Plant Biomechanics Group Freiburg, annotated and used with permission.

A

Throughout, when 𝐋𝐋{\mathbf{L}}bold_L is an F𝐹Fitalic_F-stable Levi subgroup of 𝐇𝐇{\mathbf{H}}bold_H, we use R𝐋𝐇superscriptsubscript𝑅𝐋𝐇{R_{\mathbf{L}}^{\mathbf{H}}}italic_R start_POSTSUBSCRIPT bold_L end_POSTSUBSCRIPT start_POSTSUPERSCRIPT bold_H end_POSTSUPERSCRIPT

b=bG⁢(𝐋,λ)𝑏subscript𝑏𝐺𝐋𝜆b=b_{G}({\mathbf{L}},\lambda)italic_b = italic_b start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( bold_L , italic_λ ) for some e𝑒eitalic_e-split Levi subgroup 𝐋𝐋{\mathbf{L}}bold_L of 𝐆𝐆{\mathbf{G}}bold_G, proper

𝐇𝐇{\mathbf{H}}bold_H under a Frobenius morphism F:𝐇→𝐇:𝐹→𝐇𝐇F\colon{\mathbf{H}}\rightarrow{\mathbf{H}}italic_F : bold_H → bold_H endowing 𝐇𝐇{\mathbf{H}}bold_H with

𝐏𝐏{\mathbf{P}}bold_P of 𝐇𝐇{\mathbf{H}}bold_H containing 𝐋𝐋{\mathbf{L}}bold_L as a Levi complement. In our situation, since

Throughout, when 𝐋𝐋{\mathbf{L}}bold_L is an F𝐹Fitalic_F-stable Levi subgroup of 𝐇𝐇{\mathbf{H}}bold_H, we use R𝐋𝐇superscriptsubscript𝑅𝐋𝐇{R_{\mathbf{L}}^{\mathbf{H}}}italic_R start_POSTSUBSCRIPT bold_L end_POSTSUBSCRIPT start_POSTSUPERSCRIPT bold_H end_POSTSUPERSCRIPT

C

0})}}.italic_C start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ≤ divide start_ARG italic_V end_ARG start_ARG square-root start_ARG - italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_ARG end_ARG ≤ italic_C , | italic_D italic_V | ≤ divide start_ARG italic_C end_ARG start_ARG square-root start_ARG roman_log ( - italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) end_ARG end_ARG .

To estimate this, note that by Corollary 2.11 (cylindrical estimate) in the region {V≥L⁢|t|/log⁡|t|}𝑉𝐿𝑡𝑡\{V\geq L\sqrt{|t|/\log|t|}\}{ italic_V ≥ italic_L square-root start_ARG | italic_t | / roman_log | italic_t | end_ARG } we have

Together with the estimates (3.2) and (3.74), taking into account again the fact that η=1+o⁢(1)𝜂1𝑜1\eta=1+o(1)italic_η = 1 + italic_o ( 1 ) thanks to (3.76), this proves the proposition.

Together with Corollary 2.11 (cylindrical estimate) and the already established estimates this implies

Hence, by standard interior estimates for perturbations of the heat equation, taking also into account that we have good control for the differentiated coefficients thanks to Corollary 2.11 (cylindrical estimate), we obtain

D

This vector can be seen as the score vector of an auxiliary parametric model associated with (h1,1⁢(p),…,hs,d⁢(s)⁢(p))subscriptℎ11𝑝…subscriptℎ𝑠𝑑𝑠𝑝\big{(}h_{1,1}(p),\ldots,h_{s,d(s)}(p)\big{)}( italic_h start_POSTSUBSCRIPT 1 , 1 end_POSTSUBSCRIPT ( italic_p ) , … , italic_h start_POSTSUBSCRIPT italic_s , italic_d ( italic_s ) end_POSTSUBSCRIPT ( italic_p ) ) modelling an alternative

Consider the χ2superscript𝜒2\chi^{2}italic_χ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT-type test statistic for the GoF

3.4.  χ2superscript𝜒2\chi^{2}italic_χ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT-type test statistic and a selection rule for d⁢(s)𝑑𝑠d(s)italic_d ( italic_s ) for testing Gaussianity

With these notations, the data driven GoF χ2superscript𝜒2\chi^{2}italic_χ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT-type test statistic

With these notations, our data driven GoF χ2superscript𝜒2\chi^{2}italic_χ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT-type test statistic

A

Looking more closely at specific genres, we have already noticed that most audio samples labeled as Blues are misclassified as unknown in the open set recognition task, as shown in Figure 3.

Overall, the plots in Figure 4 indicate how distributed and less tightly defined the UUCs are than KKCs. In part this is to be expected because the activation vectors visualized are optimal representations learned from audio to separate the samples of each KKC. Pop and Hip-hop are significantly overlapping, and are also distributed over Classical, Jazz and Metal clusters. However, even some of the UUCs are more tightly distributed than others. For example, the Classical samples in plot (e) of Figure 4 are noticeably nearer to each other than those for Blues. A genre that can be succinctly defined in terms of the features in its audio is likely to have higher precision, even if belonging to the set of UUCs.

Looking more closely at specific genres, we have already noticed that most audio samples labeled as Blues are misclassified as unknown in the open set recognition task, as shown in Figure 3.

Table 2: This table contains the data sources as well as the number of samples for the training, evaluation, and test sets for each experimental setup. The size of the training and evaluation sets are based only on the audio samples in the KKCs defined in Table 3 respectively.

This is consistent with research found in [25] that there are manually mislabeled audio samples between Blues and Country. In addition, there are mislabelled data points for the Metal genre label. Plot (a) in Figure 4 illustrates a number of the outliers for different classes, such as the Blues activation vectors near the Metal cluster. The combination of limited data points labelled Blues, outliers, and highly distributed samples could be resulting in probabilities of all KKCs, thus resulting in all Blues samples being labeled as unknown.

D

[9]) Let 𝒜⊆B⁢(ℋ)𝒜𝐵ℋ{\mathcal{A}}\subseteq B({\mathcal{H}})caligraphic_A ⊆ italic_B ( caligraphic_H ) be a von

Neumann algebra and let τ:𝒜→𝒜:𝜏→𝒜𝒜\tau:{\mathcal{A}}\to{\mathcal{A}}italic_τ : caligraphic_A → caligraphic_A be a

Let 𝒜𝒜{\mathcal{A}}caligraphic_A be a von Neumann algebra and let τ:𝒜→𝒜:𝜏→𝒜𝒜\tau:{\mathcal{A}}\to{\mathcal{A}}italic_τ : caligraphic_A → caligraphic_A be a normal UCP map. Suppose the group generated

Let 𝒜⊆B⁢(ℋ)𝒜𝐵ℋ{\mathcal{A}}\subseteq B({\mathcal{H}})caligraphic_A ⊆ italic_B ( caligraphic_H ) be a von Neumann algebra and let τ:𝒜→𝒜:𝜏→𝒜𝒜\tau:{\mathcal{A}}\to{\mathcal{A}}italic_τ : caligraphic_A → caligraphic_A be a normal unital completely positive map. Let

Neumann algebra and let τ:𝒜→𝒜:𝜏→𝒜𝒜\tau:{\mathcal{A}}\to{\mathcal{A}}italic_τ : caligraphic_A → caligraphic_A be a

A

A face F⊆P𝐹𝑃F\subseteq Pitalic_F ⊆ italic_P is the set of points in the polytope P𝑃Pitalic_P that maximizes a linear functional ϕ:ℝd→ℝ:italic-ϕ→superscriptℝ𝑑ℝ\phi:{\mathbb{R}}^{d}\to{\mathbb{R}}italic_ϕ : blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT → blackboard_R.

For a cosmological polytope PGsubscript𝑃𝐺P_{G}italic_P start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT and a vertex w∈V⁢(G)𝑤𝑉𝐺w\in V(G)italic_w ∈ italic_V ( italic_G ) it holds that fPG∖w⁢(t)subscript𝑓subscript𝑃𝐺𝑤𝑡f_{P_{G\setminus w}}(t)italic_f start_POSTSUBSCRIPT italic_P start_POSTSUBSCRIPT italic_G ∖ italic_w end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_t ) equals the “upper f𝑓fitalic_f-vector” of the vertex face Fwsubscript𝐹𝑤F_{w}italic_F start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT in PGsubscript𝑃𝐺P_{G}italic_P start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT, that is the f𝑓fitalic_f-vector of faces containing Fwsubscript𝐹𝑤F_{w}italic_F start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT.

This implies that the upper f𝑓fitalic_f-vector of Fwsubscript𝐹𝑤F_{w}italic_F start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT in PGsubscript𝑃𝐺P_{G}italic_P start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT equals the (shifted) f𝑓fitalic_f-vector of PG∖wsubscript𝑃𝐺𝑤P_{G\setminus w}italic_P start_POSTSUBSCRIPT italic_G ∖ italic_w end_POSTSUBSCRIPT.∎

Each polytope has finitely many faces and their numbers are counted in the f𝑓fitalic_f-vector f⁢(P)𝑓𝑃f(P)italic_f ( italic_P ) of P𝑃Pitalic_P which is

For a polytope P𝑃Pitalic_P, with f𝑓fitalic_f-vector (f−1,…,fdimP)subscript𝑓1…subscript𝑓dimension𝑃(f_{-1},\ldots,f_{\dim P})( italic_f start_POSTSUBSCRIPT - 1 end_POSTSUBSCRIPT , … , italic_f start_POSTSUBSCRIPT roman_dim italic_P end_POSTSUBSCRIPT ) we will define its f𝑓fitalic_f-polynomial fP⁢(t)subscript𝑓𝑃𝑡f_{P}(t)italic_f start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT ( italic_t ) to be

C

Given vector bundles 𝒟𝒟\mathcal{D}caligraphic_D and ℱℱ\mathcal{F}caligraphic_F on X𝑋Xitalic_X such that the minimum slope in HN⁢(𝒟)HN𝒟\mathrm{HN}(\mathcal{D})roman_HN ( caligraphic_D ) is greater than or equal to the maximum slope in HN⁢(ℱ)HNℱ\mathrm{HN}(\mathcal{F})roman_HN ( caligraphic_F ), every extension of ℱℱ\mathcal{F}caligraphic_F by 𝒟𝒟\mathcal{D}caligraphic_D splits.

Let 𝒟𝒟\mathcal{D}caligraphic_D and ℰℰ\mathcal{E}caligraphic_E be vector bundles on X𝑋Xitalic_X of rank n𝑛nitalic_n.

Given vector bundles 𝒟𝒟\mathcal{D}caligraphic_D, ℰℰ\mathcal{E}caligraphic_E and ℱℱ\mathcal{F}caligraphic_F on X𝑋Xitalic_X, we define

Let 𝒟𝒟\mathcal{D}caligraphic_D, ℰℰ\mathcal{E}caligraphic_E and ℱℱ\mathcal{F}caligraphic_F be vector bundles on X𝑋Xitalic_X. We write the HN decomposition of ℱℱ\mathcal{F}caligraphic_F as

Let 𝒟𝒟\mathcal{D}caligraphic_D, ℰℰ\mathcal{E}caligraphic_E and ℱℱ\mathcal{F}caligraphic_F be vector bundles on X𝑋Xitalic_X such that there exists a short exact sequence

D

\right)}\leq C∥ italic_R start_POSTSUBSCRIPT italic_g , italic_λ end_POSTSUBSCRIPT ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( 0 , italic_T ; italic_H start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ( italic_M ) ) end_POSTSUBSCRIPT ≤ italic_C for some constant C>0𝐶0C>0italic_C > 0 independent of λ𝜆\lambdaitalic_λ.

Using ρ=φ+i⁢ψ,𝜌𝜑𝑖𝜓\rho=\varphi+i\psi,italic_ρ = italic_φ + italic_i italic_ψ , and φ=λ2⁢β2⁢t+λ⁢x1𝜑superscript𝜆2superscript𝛽2𝑡𝜆subscript𝑥1\varphi=\lambda^{2}\beta^{2}t+\lambda x_{1}italic_φ = italic_λ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_β start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_t + italic_λ italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT, we get

\lVert F_{\lambda}\rVert_{L^{2}(M_{T})},∥ italic_R start_POSTSUBSCRIPT italic_d , italic_λ end_POSTSUBSCRIPT ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( 0 , italic_T ; italic_H start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ( italic_M ) ) end_POSTSUBSCRIPT ≤ italic_C ∥ italic_F start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_M start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT ) end_POSTSUBSCRIPT , for some constant C>0𝐶0C>0italic_C > 0 not depending on λ𝜆\lambdaitalic_λ. To construct these solutions, we first start with the construction of ψ𝜓\psiitalic_ψ and Tdsubscript𝑇𝑑T_{d}italic_T start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT following the arguments used in Theorem 3.1. Denote by ρ:=φ−i⁢ψ,assign𝜌𝜑𝑖𝜓\rho:=\varphi-i\psi,italic_ρ := italic_φ - italic_i italic_ψ , then one can check that the conjugated operator

operator ℒρ:=e−ρ⁢ℒA,q⁢eρassignsubscriptℒ𝜌superscript𝑒𝜌subscriptℒ𝐴𝑞superscript𝑒𝜌\mathcal{L}_{\rho}:=e^{-\rho}\mathcal{L}_{A,q}e^{\rho}caligraphic_L start_POSTSUBSCRIPT italic_ρ end_POSTSUBSCRIPT := italic_e start_POSTSUPERSCRIPT - italic_ρ end_POSTSUPERSCRIPT caligraphic_L start_POSTSUBSCRIPT italic_A , italic_q end_POSTSUBSCRIPT italic_e start_POSTSUPERSCRIPT italic_ρ end_POSTSUPERSCRIPT will have the following expression

Following [29], if we denote ρ:=φ+i⁢ψ,assign𝜌𝜑𝑖𝜓\rho:=\varphi+i\psi,italic_ρ := italic_φ + italic_i italic_ψ , then simple calculations show that the conjugated

D

{2}}{2})italic_J start_POSTSUBSCRIPT roman_BS end_POSTSUBSCRIPT ( italic_a ) = 4 roman_inf start_POSTSUBSCRIPT italic_ρ ≥ 0 end_POSTSUBSCRIPT ( italic_F ( italic_ρ ) + divide start_ARG 1 + italic_a start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG 2 italic_a end_ARG - divide start_ARG italic_π start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG 2 end_ARG ). This relation follows from Proposition 6 in [28] and the relation J⁢(a)=14⁢JBS⁢(a)𝐽𝑎14subscript𝐽BS𝑎J(a)=\frac{1}{4}J_{\rm BS}(a)italic_J ( italic_a ) = divide start_ARG 1 end_ARG start_ARG 4 end_ARG italic_J start_POSTSUBSCRIPT roman_BS end_POSTSUBSCRIPT ( italic_a ). However it does not seem easy to use this relation to connect the expansions of these two functions, so we treat them separately.

represent the functions F⁢(ρ),G⁢(ρ)𝐹𝜌𝐺𝜌F(\rho),G(\rho)italic_F ( italic_ρ ) , italic_G ( italic_ρ ) as follows:

for 0<ρ≤10𝜌10<\rho\leq 10 < italic_ρ ≤ 1 and π−λ⁢(ρ)=i⁢h⁢(1/ρ)𝜋𝜆𝜌𝑖ℎ1𝜌\pi-\lambda(\rho)=i\sqrt{h(1/\rho)}italic_π - italic_λ ( italic_ρ ) = italic_i square-root start_ARG italic_h ( 1 / italic_ρ ) end_ARG for

The functions κ⁢(ρ),λ⁢(ρ)𝜅𝜌𝜆𝜌\kappa(\rho),\lambda(\rho)italic_κ ( italic_ρ ) , italic_λ ( italic_ρ ) are related to the function

The mathematics of the series expansions for the functions J⁢(a),F⁢(ρ),G⁢(ρ)𝐽𝑎𝐹𝜌𝐺𝜌J(a),F(\rho),G(\rho)italic_J ( italic_a ) , italic_F ( italic_ρ ) , italic_G ( italic_ρ )

C

To implement FE-WGF we also need to specify the reference measure π0subscript𝜋0\pi_{0}italic_π start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, a Gaussian with mean and variance given by the empirical mean and variance of Y𝑌Yitalic_Y, and the time step γ=10−3𝛾superscript103\gamma=10^{-3}italic_γ = 10 start_POSTSUPERSCRIPT - 3 end_POSTSUPERSCRIPT.

replacement m=N=103𝑚𝑁superscript103m=N=10^{3}italic_m = italic_N = 10 start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT times from it at each iteration of both SMC-EMS and FE-WGF.

We consider different particle sizes (from 102superscript10210^{2}10 start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT to 104superscript10410^{4}10 start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT) and compare reconstruction accuracy and total runtime of SMC-EMS, FE-WGF and DKDE

Figure 1: Average accuracy and runtime for FE-WGF, SMC-EMS and DKDE with number of particles N𝑁Nitalic_N

To compare the two algorithms we consider N=103𝑁superscript103N=10^{3}italic_N = 10 start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT and we assume that μ𝜇\muitalic_μ is

B

\forall m,n,\omega_{u},\omega_{v}.roman_min start_POSTSUBSCRIPT over~ start_ARG italic_x end_ARG end_POSTSUBSCRIPT italic_L ( over~ start_ARG italic_x end_ARG ) roman_s . roman_t . | italic_y start_POSTSUBSCRIPT italic_ω start_POSTSUBSCRIPT italic_u end_POSTSUBSCRIPT , italic_ω start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_m , italic_n end_POSTSUPERSCRIPT | ≥ | italic_c start_POSTSUBSCRIPT italic_u , italic_v , italic_ω start_POSTSUBSCRIPT italic_u end_POSTSUBSCRIPT , italic_ω start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT end_POSTSUBSCRIPT over~ start_ARG italic_x end_ARG start_POSTSUBSCRIPT italic_u , italic_v end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_m , italic_n end_POSTSUPERSCRIPT | , ∀ italic_m , italic_n , italic_ω start_POSTSUBSCRIPT italic_u end_POSTSUBSCRIPT , italic_ω start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT .

Figure 4 shows the solution spaces for (11) and (12). The solution space is limited to the two isolated points for (11), while it is extended to the line connecting the two points for (12).

Figure 2 summarizes sign retrieval. It is formulated as an optimization problem with specific constraints in the DCT domain, and it is solved using deep neural networks (DNNs) [34]. We first compute an initial image from the magnitude information as the input for our solver. The solver, which corresponds to the gray region in Fig. 2, consists of two building blocks; the first is a DNN whose input and output are images, and the second is the projection onto convex set (POCS) operator [35] for enforcing the constraints imposed by the DCT magnitudes. Taking the initial image as the input, the solver alternates between DNN inference and POCS, similar to iterative methods in numerical optimization [36]. The initial image is gradually refined while this alternation is repeated many times. It is expected that, after a sufficient number of iterations, a restored image can be obtained that has mostly correct signs in the DCT domain. We finally obtain the restored sign information from the restored image.

We derive our solution method for (12) and (14) from the framework of POCS [35]. Note that (14) describes the training process of the DNN over the training images X𝑋Xitalic_X, while (12) describes the inference process for a target image x~~𝑥\tilde{x}over~ start_ARG italic_x end_ARG using the pre-trained DNN ϕθsubscriptitalic-ϕ𝜃\phi_{\theta}italic_ϕ start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT.

Figure 3 shows a geometrical interpretation for the convex relaxation in the DFT domain; the solution spaces for (7) and (9) are illustrated in (a) and (b), respectively. The solution is limited to the circle in (7), while the solution space is extended to the disk (the area inside the circle) in (9). Note that the solution space is non-convex for the circle, but it is convex for the disk. Therefore, the solution to (9) can be efficiently computed by a convex optimization algorithm.

A

In Section 3, we show some analytic results about an integral operator needed in the proof of Theorem 1.2.1.

Then, by (4.11) and the relation between V𝑉Vitalic_V and Vℝsubscript𝑉ℝV_{\mathbb{R}}italic_V start_POSTSUBSCRIPT blackboard_R end_POSTSUBSCRIPT, ΦssubscriptΦ𝑠\Phi_{s}roman_Φ start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT satisfies the following property

Besides the relation between Theorem 1.2.1 and Goldman and Mondello’s results, Theorem 1.2.1 can also be viewed as a variant of a seminal result due to Tian [Tian_1987aa, Theorem 2].

In Section 3, we show some analytic results about an integral operator needed in the proof of Theorem 1.2.1.

In Section 4, we explain the relation between Theorem 1.2.1 and Tian’s result and finish the proof of it.

D

\end{array}\right.\text{ on }\mathbb{R}^{d}.{ start_ARRAY start_ROW start_CELL divide start_ARG 1 end_ARG start_ARG 2 end_ARG roman_Δ italic_φ - italic_φ = - | italic_x | start_POSTSUPERSCRIPT - italic_α end_POSTSUPERSCRIPT over¯ start_ARG italic_φ end_ARG italic_ψ ≤ 0 , end_CELL end_ROW start_ROW start_CELL divide start_ARG italic_κ end_ARG start_ARG 2 end_ARG roman_Δ italic_ψ - 2 italic_ψ = - divide start_ARG 1 end_ARG start_ARG 2 end_ARG | italic_x | start_POSTSUPERSCRIPT - italic_α end_POSTSUPERSCRIPT italic_φ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≤ 0 , end_CELL end_ROW end_ARRAY on blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT .

which is (4.8). The estimate (4.7) is treated similarly (even simpler) as for (4.8). The proof is complete.

In particular, 𝕎⁢(φ,ψ)≤𝕎⁢(|φ|∗,|ψ|∗)𝕎𝜑𝜓𝕎superscript𝜑superscript𝜓\mathbb{W}(\varphi,\psi)\leq\mathbb{W}(|\varphi|^{*},|\psi|^{*})blackboard_W ( italic_φ , italic_ψ ) ≤ blackboard_W ( | italic_φ | start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT , | italic_ψ | start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ) or (|φ|∗,|ψ|∗)superscript𝜑superscript𝜓(|\varphi|^{*},|\psi|^{*})( | italic_φ | start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT , | italic_ψ | start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ) is also an optimizer for CGNsubscript𝐶GNC_{\operatorname*{GN}}italic_C start_POSTSUBSCRIPT roman_GN end_POSTSUBSCRIPT. Thus we can assume, without loss of generality, that φ𝜑\varphiitalic_φ and ψ𝜓\psiitalic_ψ are non-negative, radially symmetric, and radially decreasing. It follows that

On the other hand, we have |χ2⁢R⁢(r)|≲1less-than-or-similar-tosubscript𝜒2𝑅𝑟1|\chi_{2R}(r)|\lesssim 1| italic_χ start_POSTSUBSCRIPT 2 italic_R end_POSTSUBSCRIPT ( italic_r ) | ≲ 1 (by (5.12)). Therefore, (5.14) holds provided R>0𝑅0R>0italic_R > 0 is taken sufficiently large. The proof is complete.

By the maximum principle (see e.g., [24, Theorem 3.5]), both φ𝜑\varphiitalic_φ and ψ𝜓\psiitalic_ψ are positive. The proof is complete.

D

Indeed, this number of rows is controlled by the vanishing of the local cohomology modules at certain degrees, including the control of the saturation index of the homogeneous ideal I⁢(f)𝐼𝑓I(f)italic_I ( italic_f ) generated by general polynomials f0,…,ftsubscript𝑓0…subscript𝑓𝑡f_{0},\ldots,f_{t}italic_f start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , italic_f start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT of degrees α0,…,αtsubscript𝛼0…subscript𝛼𝑡\alpha_{0},\ldots,\alpha_{t}italic_α start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , italic_α start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT. In the case of hybrid elimination matrices, the situation is similar with the difference that now one considers the homogeneous ideal generated by f0,…,ftsubscript𝑓0…subscript𝑓𝑡f_{0},\ldots,f_{t}italic_f start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , italic_f start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT and their toric Sylvester forms, whose saturation index is smaller than the one of I⁢(f)𝐼𝑓I(f)italic_I ( italic_f ).

We remark that the number of columns of hybrid elimination matrices may increase fast when the number of equations is large compared to the dimension of the ground projective toric variety. Further work is needed to analyze if some toric Sylvester forms can be avoided or combined to gain in efficiency. A more practical approach for

We conclude this section with a comment on the computational impact of the hybrid elimination matrices obtained in Theorem 5.4. Indeed, these matrices are intended for solving overdetermined 0-dimensional polynomial systems via eigenvalue and eigenvector computations, applicable over projective spaces, multi-projective spaces, or more broadly, smooth projective toric varieties that are σ𝜎\sigmaitalic_σ-positive for a given maximal cone σ𝜎\sigmaitalic_σ. In comparison with the more classical Macaulay-type matrices, hybrid elimination matrices are more compact. In particular, these matrices have a smaller number of rows, which is a key ingredient with respect to computational complexity.

Hybrid elimination matrices already appeared in the existing literature in various contexts and they are known to be more compact than Macaulay matrices. In order to be a little more concrete, we provide some specific examples in Table 1 to show explicitly the gain in terms of the number of rows.

The first property is related to resultant theory (see e.g. [cattani1997residuesresultants, gkz1994]) whilst the second is used for solving 00-dimensional polynomial systems (see e.g. [bender2021toric, emirismourrain]). In this paper, we introduce a new family of elimination matrices by adding to a classical Macaulay-block matrix in some degree α∈Cl⁡(XΣ)𝛼Clsubscript𝑋Σ\alpha\in\operatorname{Cl}(X_{\Sigma})italic_α ∈ roman_Cl ( italic_X start_POSTSUBSCRIPT roman_Σ end_POSTSUBSCRIPT ), a block-matrix built from the toric Sylvester forms of degree α𝛼\alphaitalic_α (see Definition 5.1). We call these matrices hybrid elimination matrices and prove their main properties in Theorem 5.1. Compared with the more classical Macaulay matrices, this new family yields more compact matrices that can still be used for solving 0-dimensional polynomial systems. In addition, we also prove that the construction of hybrid elimination matrices can be extended to polynomial systems defined by more than n+1𝑛1n+1italic_n + 1 polynomials (see Theorem 5.3).

C

\mathfrak{m}\mathcal{R}_{A}^{[r,1)}caligraphic_R start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT start_POSTSUPERSCRIPT [ italic_r , 1 ) end_POSTSUPERSCRIPT ⊗ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT italic_A / fraktur_m = caligraphic_R start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT start_POSTSUPERSCRIPT [ italic_r , 1 ) end_POSTSUPERSCRIPT / fraktur_m caligraphic_R start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT start_POSTSUPERSCRIPT [ italic_r , 1 ) end_POSTSUPERSCRIPT is complete. Since A𝐴Aitalic_A is Noetherian the ideal 𝔪⁢ℛA[r,1)𝔪superscriptsubscriptℛ𝐴𝑟1\mathfrak{m}\mathcal{R}_{A}^{[r,1)}fraktur_m caligraphic_R start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT start_POSTSUPERSCRIPT [ italic_r , 1 ) end_POSTSUPERSCRIPT is a finitely generated submodule of a coadmissible module hence itself coadmissible. By [ST03, Lemma 3.6] the quotient in question is complete.

When studying (φL,ΓL)subscript𝜑𝐿subscriptΓ𝐿(\varphi_{L},\Gamma_{L})( italic_φ start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT , roman_Γ start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT )-modules over the Robba ring ℛKsubscriptℛ𝐾\mathcal{R}_{K}caligraphic_R start_POSTSUBSCRIPT italic_K end_POSTSUBSCRIPT it turns out that they admit a so-called model over some half-open interval [r,1),𝑟1[r,1),[ italic_r , 1 ) , meaning that it arises as a base extension from a module over ℛK[r,1).superscriptsubscriptℛ𝐾𝑟1\mathcal{R}_{K}^{[r,1)}.caligraphic_R start_POSTSUBSCRIPT italic_K end_POSTSUBSCRIPT start_POSTSUPERSCRIPT [ italic_r , 1 ) end_POSTSUPERSCRIPT . When working with families of such modules, we enforce the existence of such a model, which in turn allows us to view the modules (more precisely their models) as vector bundles on Sp⁡(A)×K𝔹[r,1),subscript𝐾Sp𝐴superscript𝔹𝑟1\operatorname{Sp}(A)\times_{K}\mathbb{B}^{[r,1)},roman_Sp ( italic_A ) × start_POSTSUBSCRIPT italic_K end_POSTSUBSCRIPT blackboard_B start_POSTSUPERSCRIPT [ italic_r , 1 ) end_POSTSUPERSCRIPT , where Sp⁡(A)Sp𝐴\operatorname{Sp}(A)roman_Sp ( italic_A ) denotes the rigid analytic variety attached to A𝐴Aitalic_A. A suitable frame work to do so is the theory of coadmissible modules in the sense of Schneider and Teitelbaum.

The definition of coherent sheaves so far is too restrictive because we can not change the lower boundary of a given half-open interval [r,1).𝑟1[r,1).[ italic_r , 1 ) . Recall that a collection of subsets of a topological space is called locally finite if every point admits an open neighbourhood intersecting only finitely many members of the collection.

The goal of this section is to prove that for any analytic (φL,ΓL)subscript𝜑𝐿subscriptΓ𝐿(\varphi_{L},\Gamma_{L})( italic_φ start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT , roman_Γ start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT )-module M𝑀Mitalic_M over ℛAsubscriptℛ𝐴\mathcal{R}_{A}caligraphic_R start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT the cohomology groups HφL,Zi⁢(M)subscriptsuperscript𝐻𝑖subscript𝜑𝐿𝑍𝑀H^{i}_{\varphi_{L},Z}(M)italic_H start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_φ start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT , italic_Z end_POSTSUBSCRIPT ( italic_M ) are finitely generated over A𝐴Aitalic_A. We follow the strategy of [Bel21] using the result from the previous chapter regarding the Z𝑍Zitalic_Z action on the kernel of ψ𝜓\psiitalic_ψ to arrive at a situation that allows us to apply results from [KL16]. This approach differs from [KPX14] who first prove the finiteness of the Iwasawa cohomology of M𝑀Mitalic_M and compare it to the (φ,Γ)𝜑Γ(\varphi,\Gamma)( italic_φ , roman_Γ )-cohomology of the cyclotomic deformation of M.𝑀M.italic_M . The finiteness of the cohomology of M𝑀Mitalic_M is obtained in [KPX14] as a corollary by writing M𝑀Mitalic_M as a base change of its deformation. Similar arguments to the ones of Bellovin already appear in [KP18]. We denote by 𝐃p⁢e⁢r⁢fb⁢(A),𝐃p⁢e⁢r⁢f−⁢(A),𝐃p⁢e⁢r⁢f[a,b]⁢(A)subscriptsuperscript𝐃𝑏𝑝𝑒𝑟𝑓𝐴subscriptsuperscript𝐃𝑝𝑒𝑟𝑓𝐴superscriptsubscript𝐃𝑝𝑒𝑟𝑓𝑎𝑏𝐴\mathbf{D}^{b}_{perf}(A),\mathbf{D}^{-}_{perf}(A),\mathbf{D}_{perf}^{[a,b]}(A)bold_D start_POSTSUPERSCRIPT italic_b end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p italic_e italic_r italic_f end_POSTSUBSCRIPT ( italic_A ) , bold_D start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p italic_e italic_r italic_f end_POSTSUBSCRIPT ( italic_A ) , bold_D start_POSTSUBSCRIPT italic_p italic_e italic_r italic_f end_POSTSUBSCRIPT start_POSTSUPERSCRIPT [ italic_a , italic_b ] end_POSTSUPERSCRIPT ( italic_A ) the full subcategory of the derived category consisting of objects which are quasi-isomorphic to bounded (resp. bounded above, resp. concentrated in degree [a,b]𝑎𝑏[a,b][ italic_a , italic_b ]) complexes of finite projective A𝐴Aitalic_A-modules.

For the full analytic Herr cohomology we obtain a variant of 3.3.12. Because the cohomology is defined by taking the invariants of the (φL,Z)subscript𝜑𝐿𝑍(\varphi_{L},Z)( italic_φ start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT , italic_Z )-cohomology we can not formulate similar perfectness results (outside of the case e<p−1𝑒𝑝1e<p-1italic_e < italic_p - 1) and we only obtain a base change result in the flat case.

B

\mu}(\Phi<-t)=c_{2}t^{-\alpha}+o(t^{-\alpha}),over^ start_ARG italic_μ end_ARG ( roman_Φ > italic_t ) = italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_t start_POSTSUPERSCRIPT - italic_α end_POSTSUPERSCRIPT + italic_o ( italic_t start_POSTSUPERSCRIPT - italic_α end_POSTSUPERSCRIPT ) and over^ start_ARG italic_μ end_ARG ( roman_Φ < - italic_t ) = italic_c start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_t start_POSTSUPERSCRIPT - italic_α end_POSTSUPERSCRIPT + italic_o ( italic_t start_POSTSUPERSCRIPT - italic_α end_POSTSUPERSCRIPT ) ,

In certain settings, limit theorems can be deduced from properties of the induced observable ΦΦ\Phiroman_Φ. In particular, it is proved in Theorems 1.1 and 1.2 of [Gou04] that, precisely in our setting 555The assumptions of [Gou04, Theorems 1.1 and 1.2]

The last part of our argument is focused on obtaining the limit theorems stated in Theorem E. When β=0𝛽0\beta=0italic_β = 0 the decay of correlations is exponential and the result follows from [You99]. Similarly, after having established Proposition 2.2 and Corollary 2.8, the case that only one of ℓ1,ℓ2subscriptℓ1subscriptℓ2\ell_{1},\ell_{2}roman_ℓ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , roman_ℓ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT is positive implies that there is only one intermittent fixed point, and thus essentially reduces to the argument given in [Gou04, Theorem 1.3] for the LSV map. We only therefore need to consider the case that both ℓ1,ℓ2>0subscriptℓ1subscriptℓ20\ell_{1},\ell_{2}>0roman_ℓ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , roman_ℓ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT > 0, which implies in particular that β∈(0,1)𝛽01\beta\in(0,1)italic_β ∈ ( 0 , 1 ).

We will argue that in each case of Theorem E, the induced observable ΦΦ\Phiroman_Φ satisfies one of the above.

Our final result concerns a number of limit theorems for maps g∈𝔉𝑔𝔉g\in\mathfrak{F}italic_g ∈ fraktur_F, which depend on the parameters of the map and, in some cases, also on some additional regularity conditions. These are arguably some of the most interesting results of the paper, and those in which the existence of two indifferent fixed points, instead of just one, really comes into play, giving rise to quite a complex scenario of possibilities.

A

We present the details in Section 2 below. The proof of Theorem 1.2(b) follows from a closely related argument. We prove Theorem 1.2(c) in Section 3.

If G𝐺Gitalic_G is not defined over the prime field F𝐹Fitalic_F then the argument above needs modification. The trouble is that the polynomials that define G𝐺Gitalic_G as a closed subgroup of SLn⁢(k)subscriptSL𝑛𝑘{\rm SL}_{n}(k)roman_SL start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_k ) may involve parameters from k𝑘kitalic_k, so (∗)s,tsubscript𝑠𝑡(*)_{s,t}( ∗ ) start_POSTSUBSCRIPT italic_s , italic_t end_POSTSUBSCRIPT may fail to translate into an honest sentence. We get around this by using a trick from [13] (cf. also the discussion after Theorem 1.1 of [12]): first we replace these parameters with variables, then we quantify over all possible values of these variables. This amounts to quantifying over all linear algebraic groups of bounded complexity in an appropriate sense. In [13, Sec. 3.2] this idea is formulated using the language of Hopf algebras; here we give a more concrete description.

J. Wilson gives some other first-order properties of groups that hold for finite groups but not for arbitrary groups [18]; see [19], [5]. One can use the methods of this paper to prove that these properties hold for linear algebraic groups over an algebraically closed or pseudo-finite field. Likewise, Theorem 1.2 holds for definable groups in the sense of [15, introduction] over an algebraically closed or pseudo-finite field k𝑘kitalic_k; in particular, this includes group schemes of finite type over such k𝑘kitalic_k. I’m grateful to Lenstra and Tiersma for these observations.

An infinite field k𝑘kitalic_k is pseudo-finite if it is a model of the theory of finite fields: that is, a sentence is true for k𝑘kitalic_k if it is true for every finite field. A nonprincipal ultraproduct of an infinite collection of finite fields is pseudo-finite. For instance, if 𝒰𝒰{\mathcal{U}}caligraphic_U is a nonprincipal ultrafilter on the set of all primes then the ultraproduct ∏𝒰𝔽psubscriptproduct𝒰subscript𝔽𝑝\prod_{\mathcal{U}}{\mathbb{F}}_{p}∏ start_POSTSUBSCRIPT caligraphic_U end_POSTSUBSCRIPT blackboard_F start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT is a pseudo-finite field of characteristic 0, and many of the infinite subfields of 𝔽p¯¯subscript𝔽𝑝\overline{{\mathbb{F}}_{p}}over¯ start_ARG blackboard_F start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT end_ARG are pseudo-finite fields of characteristic p𝑝pitalic_p. See [3] and [4, (5.1)] for more details and examples.

It is natural to ask which other groups have the Honda property. Pride has given an example of a one-relator group with torsion which does not have the Honda property [16, Result (C), p488]. In this note we extend Theorem 1.1 to certain linear algebraic groups. By a linear algebraic group over a ring k𝑘kitalic_k we mean a smooth closed subgroup scheme of GLnsubscriptGL𝑛{\rm GL}_{n}roman_GL start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT for some n∈ℕ𝑛ℕn\in{\mathbb{N}}italic_n ∈ blackboard_N; if k𝑘kitalic_k is a field then these are just linear algebraic groups in the usual sense [1], and they correspond to the affine algebraic groups of finite type by [14, Cor. 4.10].

B

Making use of (5.1) and proceeding as in [9, Section 4], we obtain the ellipticity of −𝒜H⁢(v0)superscript𝒜Hsubscript𝑣0-\mathcal{A}^{\mathrm{H}}(v_{0})- caligraphic_A start_POSTSUPERSCRIPT roman_H end_POSTSUPERSCRIPT ( italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ).

Comparing this definition of Hibler’s operator to the one presented in [9, Section 3], we observe that the sign is changed and 1/ρice⁢h1subscript𝜌iceℎ\nicefrac{{1}}{{{\rho}_{\mathrm{ice}}h}}/ start_ARG 1 end_ARG start_ARG italic_ρ start_POSTSUBSCRIPT roman_ice end_POSTSUBSCRIPT italic_h end_ARG is included in the definition for the sake of consistency with the hydrostatic Stokes operator and to ease notation.

Making use of (5.1) and proceeding as in [9, Section 4], we obtain the ellipticity of −𝒜H⁢(v0)superscript𝒜Hsubscript𝑣0-\mathcal{A}^{\mathrm{H}}(v_{0})- caligraphic_A start_POSTSUPERSCRIPT roman_H end_POSTSUPERSCRIPT ( italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ).

Subsection 5.1 for the ellipticity properties of the negative Hibler operator −𝒜H⁢(v0)superscript𝒜Hsubscript𝑣0-\mathcal{A}^{\mathrm{H}}(v_{0})- caligraphic_A start_POSTSUPERSCRIPT roman_H end_POSTSUPERSCRIPT ( italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) and using (5.1) in conjunction with the fact

For the notion of strong ellipticity and parameter ellipticity, we refer e. g. to [13, Definition 5.1].

D

\right\rangle)^{{\dagger}}=\left\langle{q}\right|.( | italic_q ⟩ ) start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT = ⟨ italic_p | ( | italic_p ⟩ ) start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT = ⟨ italic_q | .

The Wigner functional is recovered from the characteristic functional via the inverse symplectic Grassmann functional Fourier transform.

The characteristic functional is the symplectic Grassmann functional Fourier transform of the Wigner functional with respect to both q𝑞qitalic_q and p𝑝pitalic_p. It reads

We can also interpret (72) as orthogonality conditions for exponential functionals. It allows us to define Grassmann functional Fourier integrals. The Grassmann functional Fourier transform of an arbitrary Grassmann functional W⁢[q]𝑊delimited-[]𝑞W[q]italic_W [ italic_q ] and its associated inverse are thus represented by

When the inverse Fourier transform is applied to the result of the Fourier transform, it reproduces the original functional.

C

,\quad 0\leq m\leq 3\,.roman_sup start_POSTSUBSCRIPT italic_t ∈ [ 0 , italic_T ] end_POSTSUBSCRIPT ∥ ( start_ARG start_ROW start_CELL over¯ start_ARG italic_κ end_ARG end_CELL end_ROW start_ROW start_CELL italic_ξ end_CELL end_ROW end_ARG ) ∥ start_POSTSUBSCRIPT over˙ start_ARG italic_H end_ARG start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT × over˙ start_ARG italic_H end_ARG start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ≤ italic_c ∥ ( start_ARG start_ROW start_CELL over¯ start_ARG italic_κ end_ARG start_POSTSUPERSCRIPT roman_in end_POSTSUPERSCRIPT end_CELL end_ROW start_ROW start_CELL italic_ξ start_POSTSUPERSCRIPT roman_in end_POSTSUPERSCRIPT end_CELL end_ROW end_ARG ) ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT × italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ≤ italic_c italic_ε , 0 ≤ italic_m ≤ 3 .

We will thus consider equation (7) as an equation holding along the normal direction of the fiber only:

Given θ|t=0evaluated-at𝜃𝑡0\theta\big{|}_{t=0}italic_θ | start_POSTSUBSCRIPT italic_t = 0 end_POSTSUBSCRIPT and (κ¯,ξ)¯𝜅𝜉(\overline{\kappa},\xi)( over¯ start_ARG italic_κ end_ARG , italic_ξ ) solving (1.2), we may recover θ⁢(s,t)𝜃𝑠𝑡\theta(s,t)italic_θ ( italic_s , italic_t ) via

We next prove the following lemma regarding the elliptic equation (15) for the tension τ¯¯𝜏\overline{\tau}over¯ start_ARG italic_τ end_ARG.

Using the equation (18) for θ˙˙𝜃\dot{\theta}over˙ start_ARG italic_θ end_ARG in the frame evolution (19), we thus have

D

We will start by introducing some notation concerning both finite and infinite sequences of zero’s and one’s, as some proofs make use of the Cantor set. We proceed by introducing the so-called Menger order of a point and giving an easy observation to use later.

If the set of all cut points of X𝑋Xitalic_X is uncountable, we can use Theorem 14. Hence assume that there are uncountably many local cut points which are not cut points and we will find three witnessing maps for X𝑋Xitalic_X in three steps.

Although it is possible to prove the results for Peano continua with uncountably many local cut points using very similar approach, we have decided to make use of a semiconjugacy with the circle in this case. Thus in Lemma 17 we prove that every Peano continuum X𝑋Xitalic_X with uncountably many local cut points is semiconjugated with the circle and in Lemma 16 we give some properties of such semiconjugacies. Lifting three contractions on the circle through the semiconjugacy to self-maps of X𝑋Xitalic_X gives three maps witnessing that X𝑋Xitalic_X is a topological fractal, as it is shown in the proof of Theorem 18, which closes the section.

The backward implication is trivial, so we are left to prove the forward one. Suppose that every Peano continuum which is a topological fractal contains a self-regenerating fractal as a subset with nonempty interior and let X𝑋Xitalic_X be an arbitrary Peano continuum. We may assume that X𝑋Xitalic_X is nondegenerate.

Afterwards, we prove Lemma 13 stating that every Peano continuum X𝑋Xitalic_X with uncountably many cut points is of a specific structure that enables us to define self-maps of X𝑋Xitalic_X naturally. In Theorem 14 we prove that we can find two such self-maps of X𝑋Xitalic_X that witness that X𝑋Xitalic_X is a topological fractal.

D

52967312788185296731278818\frac{529673}{1278818}divide start_ARG 529673 end_ARG start_ARG 1278818 end_ARG

1354498327650913544983276509\frac{1354498}{3276509}divide start_ARG 1354498 end_ARG start_ARG 3276509 end_ARG

1010\frac{1}{0}divide start_ARG 1 end_ARG start_ARG 0 end_ARG, 0101\frac{0}{1}divide start_ARG 0 end_ARG start_ARG 1 end_ARG, and 1111\frac{1}{1}divide start_ARG 1 end_ARG start_ARG 1 end_ARG, respectively.

0101\frac{0}{1}divide start_ARG 0 end_ARG start_ARG 1 end_ARG1313\frac{1}{3}divide start_ARG 1 end_ARG start_ARG 3 end_ARG3838\frac{3}{8}divide start_ARG 3 end_ARG start_ARG 8 end_ARG513513\frac{5}{13}divide start_ARG 5 end_ARG start_ARG 13 end_ARG2525\frac{2}{5}divide start_ARG 2 end_ARG start_ARG 5 end_ARG12291229\frac{12}{29}divide start_ARG 12 end_ARG start_ARG 29 end_ARG1212\frac{1}{2}divide start_ARG 1 end_ARG start_ARG 2 end_ARG

for the triples (01,25,12)012512(\frac{0}{1},\frac{2}{5},\frac{1}{2})( divide start_ARG 0 end_ARG start_ARG 1 end_ARG , divide start_ARG 2 end_ARG start_ARG 5 end_ARG , divide start_ARG 1 end_ARG start_ARG 2 end_ARG ) and

A

≥2⁢σ⁢∑j|vj|2.absent2𝜎subscript𝑗superscriptsubscript𝑣𝑗2\geq 2\sigma\sum_{j}|v_{j}|^{2}.≥ 2 italic_σ ∑ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT | italic_v start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT .

We state the main propositions here, so that all the consecutive sections are independent of each other and only refer to the results and notation of this section.

Like in the linear setting, the weak bound (1.10) implies almost everywhere convergence of |a|𝑎|a|| italic_a |. However, unlike in the linear setting where Stein’s principle [Ste61] applies, the reverse implication is not known to be true. Furthermore, the convergence of arg⁡(a)𝑎\arg(a)roman_arg ( italic_a ) is not known to us even for the Cantor group model considered in [MTT02], for which (1.10) is proved.

We will need differential equations for the absolute value and the argument of E𝐸Eitalic_E that are consequences of (2.2). We have

So the only possible value of k𝑘kitalic_k for which in (7.6) does not hold is either 1111 or 15151515 but not both, so we are done.

D

Starting from this section, we shall extend the graphical scale from O⁢(log⁡t)𝑂𝑡O(\sqrt{\log t})italic_O ( square-root start_ARG roman_log italic_t end_ARG ) to O⁢(t)𝑂𝑡O(\sqrt{t})italic_O ( square-root start_ARG italic_t end_ARG ). One of the main ingredients of the extension is the pseudolocality theorem, which says that if we have two hypersurfaces that are close to each other in a large domain, then under the MCF or RMCF they will still be close to each other for a short but definite amount of time. This allows us to extend the graphical scale of the RMCF to exponentially growing radius for a short time. However, the derivative estimates of the graph deteriorate as time goes on. To address this problem, we use the regularization property of the Ornstein-Uhlenbeck operator to show that within the graphical radius of order O⁢(t)𝑂𝑡O(\sqrt{t})italic_O ( square-root start_ARG italic_t end_ARG ), various derivative estimates of the graph actually improve as time goes on.

The following pseudolocality argument was first studied by Ecker-Huisken in [EH91]. Later Ilmanen-Neves-Schulze [INS19, Section 9] gave a simple proof using White’s regularity theorem [Whi05]. There they study the MCF of graphs over a hyperplane, and we study the MCF of graphs over a cylinder.

To study the asymptotics of the RMCF approaching the limiting cylinder, we need to study the evolution of graphical function.

In this section, we explore the application of the main results to rotational graphs. The MCF of rotational graphs was studied in [AAG95], where the authors utilized a Liouville-Sturm theorem to prove that all singularities of the MCF of rotational graphs are isolated. Consequently, firecracker singularities cannot appear.

The paper is organized as follows. In Section 2, we give a preliminary on the eigenfunctions of the L𝐿Litalic_L-operator. Of particular importance for later proofs is our interpretation of the eigenfunction yisubscript𝑦𝑖y_{i}italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT. In Section 3, we show how to extend the graphical scale from O⁢(log⁡t)𝑂𝑡O(\sqrt{\log t})italic_O ( square-root start_ARG roman_log italic_t end_ARG ) to O⁢(t)𝑂𝑡O(\sqrt{t})italic_O ( square-root start_ARG italic_t end_ARG ) for the rescaled mean curvature flows with cylindrical singularities. In Section 4, we study the local dynamics of the RMCF near the cylindrical shrinker and prove that the linear part of the equation approximates very well the local dynamics. In Section 5, we prove normal form Theorem 1.3 and 1.4. In Section 6, we prove that nondegenerate singularities are isolated (Theorem 1.6). In Section 7 and 8, we prove the genericity and stability of nondegenerate singularities (Theorem 1.9 and 1.10). In Section 10, we give the proof of the global theorem 1.12. All other theorems mentioned above will be proved in Section 9 as an application of Theorem 1.9 and 1.10.

A

1osuperscript1o1^{\rm o}1 start_POSTSUPERSCRIPT roman_o end_POSTSUPERSCRIPT. If ‖F⁢(x)‖≤cnorm𝐹𝑥𝑐||F(x)||\leq c| | italic_F ( italic_x ) | | ≤ italic_c for all x∈X𝑥𝑋x\in Xitalic_x ∈ italic_X then ∫XF⁢(x)⁢𝑑ϕ≤csubscript𝑋𝐹𝑥differential-ditalic-ϕ𝑐\int_{X}F(x)d\phi\leq c∫ start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT italic_F ( italic_x ) italic_d italic_ϕ ≤ italic_c.

The strong estimate on 𝔟⁢(x,η)𝔟𝑥𝜂\mathfrak{b}(x,\eta)fraktur_b ( italic_x , italic_η ) required in definition 3.5, 2osuperscript2o2^{\rm o}2 start_POSTSUPERSCRIPT roman_o end_POSTSUPERSCRIPT, is not needed in the item 2osuperscript2o2^{\rm o}2 start_POSTSUPERSCRIPT roman_o end_POSTSUPERSCRIPT of definition 5.5 because of an almost canonical choice of coordinate systems in the tangent spaces along the orbit (and the stabilizer-invariance of the symbols).

2osuperscript2o2^{\rm o}2 start_POSTSUPERSCRIPT roman_o end_POSTSUPERSCRIPT σ∗↦ℱ∗maps-tosuperscript𝜎superscriptℱ\sigma^{*}\mapsto{\cal F}^{*}italic_σ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ↦ caligraphic_F start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT.

2osuperscript2o2^{\rm o}2 start_POSTSUPERSCRIPT roman_o end_POSTSUPERSCRIPT Self-adjoint operators on ungraded spaces have index in the K1superscript𝐾1K^{1}italic_K start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT-groups.

2osuperscript2o2^{\rm o}2 start_POSTSUPERSCRIPT roman_o end_POSTSUPERSCRIPT. The integral is additive, multiplicative, and

D

Furthermore, using the second-kind formulations in the trace spaces has the disadvantage that the inner products on H±1/2⁢(Γ)superscript𝐻plus-or-minus12ΓH^{\pm 1/2}(\Gamma)italic_H start_POSTSUPERSCRIPT ± 1 / 2 end_POSTSUPERSCRIPT ( roman_Γ ) are non-local and non-trivial to evaluate; even if the basis functions ϕNsubscriptitalic-ϕ𝑁\phi_{N}italic_ϕ start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT and ψNsubscript𝜓𝑁\psi_{N}italic_ψ start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT in (1.4) have supports only on a subset of ΓΓ\Gammaroman_Γ, (A⁢ϕN,ψN)ℋsubscript𝐴subscriptitalic-ϕ𝑁subscript𝜓𝑁ℋ(A\phi_{N},\psi_{N})_{{\cal H}}( italic_A italic_ϕ start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT , italic_ψ start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT caligraphic_H end_POSTSUBSCRIPT is an integral over all of ΓΓ\Gammaroman_Γ, and the calculation of the Galerkin matrix in this case is impractical.

1.3 The rationale for using second-kind BIEs posed in L2⁢(Γ)superscript𝐿2Γ{L^{2}(\Gamma)}italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( roman_Γ )

For the second-kind BIEs, an attractive alternative to working in the trace spaces is to work in L2⁢(Γ)superscript𝐿2Γ{L^{2}(\Gamma)}italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( roman_Γ ).

An alternative definition of second kind BIEs is that, in addition to the unknown function appearing outside the integrand as well as inside, the BIO is Fredholm of index zero (i.e., the Fredholm alternative applies to the BIE); see, e.g., [4, §1.1.4].

The disadvantage of using second-kind BIEs in L2⁢(Γ)superscript𝐿2Γ{L^{2}(\Gamma)}italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( roman_Γ ) is that convergence of the Galerkin method is harder to establish when

B

4.1. Effect of b0subscript𝑏0b_{0}italic_b start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT and the connection to the conventional ADF test

by a linear model (the solid green line in Figure 4.1), which is invariant to b0subscript𝑏0b_{0}italic_b start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT. (Under the parametrisation

We now illustrate how the values of the initial condition b0subscript𝑏0b_{0}italic_b start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT

as b0subscript𝑏0b_{0}italic_b start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT grows, is similarly manifest in the critical values given

Figure 4.1 depicts how a change in b0subscript𝑏0b_{0}italic_b start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT

D

{L}^{p})^{2d}( - bold_Δ ) start_POSTSUPERSCRIPT - 1 / 2 end_POSTSUPERSCRIPT bold_∇ : caligraphic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT → ( caligraphic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT 2 italic_d end_POSTSUPERSCRIPT is a bounded operator for any p∈(1,∞)𝑝1p\in(1,\infty)italic_p ∈ ( 1 , ∞ ) with norm independent of ℏPlanck-constant-over-2-pi\hbarroman_ℏ.

One of the important tools in our analysis will be the use of a quantum analogue of the convolution, actually already introduced in [66], that we present in this section.

One of the important property of this definition for the semiclassical convolution that was proved in [66] is the fact that the analogue of Young’s convolution inequality holds. It also allows to better understand when the semiclassical convolution is well defined.

To get the quantum analogue of the convolution by a Gaussian, we use the above defined semiclassical convolution. We define

As was already introduced by Werner in [66], we define the semiclassical convolution of a phase space function f𝑓fitalic_f and a bounded operator 𝝆𝝆\boldsymbol{\rho}bold_italic_ρ as the following operator-valued integral444See Remark 4.1 for the more rigorous meaning of this integral.

A

We note in passing that there are non-trivial solutions to the Fermat cubic over ℚ⁢(2)ℚ2\mathbb{Q}(\sqrt{2})blackboard_Q ( square-root start_ARG 2 end_ARG ) (see [22, page 184]).

This proof fails for p=13𝑝13p=13italic_p = 13 in the case that ord𝔓⁡(b)=1subscriptord𝔓𝑏1\operatorname{ord}_{\mathfrak{P}}(b)=1roman_ord start_POSTSUBSCRIPT fraktur_P end_POSTSUBSCRIPT ( italic_b ) = 1, and we treat this case separately in the rest of the section.

We can thus conclude that if P∈C⁢(L)𝑃𝐶𝐿P\in C(L)italic_P ∈ italic_C ( italic_L ) then P=∞𝑃P=\inftyitalic_P = ∞. This completes the proof of Lemma 6.1 and therefore Theorem 6.1.

We are very grateful to Samir Siksek for the lengthy discussions and ideas that resulted in this proof.

𝒩0={𝔓if we are in case (i) of Lemma 3.1𝔓4if we are in case (ii) of Lemma 3.1𝔓5if ⁢p=13⁢ and ⁢ord𝔓⁡(b)=1.subscript𝒩0cases𝔓if we are in case (i) of Lemma 3.1superscript𝔓4if we are in case (ii) of Lemma 3.1superscript𝔓5if 𝑝13 and subscriptord𝔓𝑏1\mathcal{N}_{0}=\begin{cases}\mathfrak{P}&\text{if we are in case (i) of Lemma%

C

}F(X)=F(\overline{X}^{*}).∃ over¯ start_ARG italic_X end_ARG start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ∈ caligraphic_C start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT : roman_min start_POSTSUBSCRIPT italic_X ∈ blackboard_R start_POSTSUPERSCRIPT italic_n × italic_d start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_F ( italic_X ) = italic_F ( over¯ start_ARG italic_X end_ARG start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ) .

_{y}}blackboard_E ∥ bold_Y start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT - over¯ start_ARG bold_Y end_ARG start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≤ italic_δ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT in all iterations. So, we can estimate noise and bias for the gradient with respect to 𝐘𝐘\mathbf{Y}bold_Y for each inner loop. The proof of the following lemma is presented in Appendix D.

Note, in the inner iterations we have gradient process with respect to 𝐘𝐘\mathbf{Y}bold_Y. Uniting this assumption and results of the previous part we obtain the following result.

Uniting the results of Lemmas 4.3, 4.4 and Theorem 3.4, we obtain the following result for the convergence of Algorithm 3.

Uniting obtained above results for convergences by 𝐗𝐗\mathbf{X}bold_X and 𝐘𝐘\mathbf{Y}bold_Y, we obtain the result of the theorem.

B

We study the Finsler norm on 𝔛Θ⁢(Γ,𝖦)subscript𝔛ΘΓ𝖦\mathfrak{X}_{\Theta}(\Gamma,\mathsf{G})fraktur_X start_POSTSUBSCRIPT roman_Θ end_POSTSUBSCRIPT ( roman_Γ , sansserif_G ) following the same approach, namely, by finding a general dynamical construction inspired by [GKL21], and then pulling back this construction to spaces of Anosov representations. Observe, however, that in this case we need a more complicated expression than what’s available in [GKL21] because we cannot assume that the entropy is constant.

Even though the Local Length Spectrum Rigidity Theorem is a geometric statement, the proofs of (1.5) and (1.6) can be abstracted to a more general dynamical framework inspired from [BCLS15, Section 3]. We develop this general dynamical framework in detail in Sections 2 and 3, as well as the specific statements needed for the construction of an asymmetric distance and a Finsler norm in that setting. As we explain, these general constructions can then be applied not only to the space 𝔐𝔐\mathfrak{M}fraktur_M as in Guillarmou-Knieper-Lefeuvre, but also to other geometric settings, such as spaces of Anosov representations. We expect that this can be applicable in many more geometric contexts.

The proofs of our main results follow closely the approach by Guillarmou-Knieper-Lefeuvre [GKL21], which is based on work of Knieper [Kni95] and Bridgeman-Canary-Labourie-Sambarino [BCLS15]. In [GKL21], the authors work with the space 𝔐𝔐\mathfrak{M}fraktur_M of isometry classes of negatively curved, entropy one Riemannian metrics on a closed manifold M𝑀Mitalic_M. For g∈𝔐𝑔𝔐g\in\mathfrak{M}italic_g ∈ fraktur_M and an isotopy class c𝑐citalic_c of closed curves in M𝑀Mitalic_M, one may define Lg⁢(c)subscript𝐿𝑔𝑐L_{g}(c)italic_L start_POSTSUBSCRIPT italic_g end_POSTSUBSCRIPT ( italic_c ) as we did when g𝑔gitalic_g was a point in Teichmüller space. Guillarmou-Knieper-Lefeuvre define

We may summarize the above discussion by saying that the results of this paper are obtained by adapting the corresponding constructions in [GKL21] to the context of Anosov representations: we can rely on the Thermodynamical Formalism, on which part of the constructions in [GKL21] are based, using the work of Sambarino [Sam14b] and Bridgeman-Canary-Labourie-Sambarino [BCLS15], and the local rigidity statement needed in [GKL21] is replaced here by rigidity statements for Anosov representations from [BCLS15]. One of the strong points of our approach is to find a suitable general setup where both contexts can be encompassed, and which might prove useful for other geometric situations.

In Section 2 we discuss the dynamical setup, and in Section 3 we construct the asymmetric metric and the corresponding Finsler norm in this general setting. In Section 4 we recall the definition and main examples of interest of Anosov representations. In Section 5 we recall Sambarino’s Reparametrizing Theorem. In Section 6 we pull back the construction of Section 3 to spaces of Anosov representations and also discuss the renormalized length spectrum rigidity in general. In Sections 7 and 8 we specify the discussion to Hitchin representations, as well as some components of Benoist and positive representations. In Appendix A we discuss in detail the link between the rigidity statement (1.7) and Bonahon’s Rigidity Intersection Theorem.

C

Second, note that the lattice of subspaces of 𝔽q3superscriptsubscript𝔽𝑞3\mathbb{F}_{q}^{3}blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT consists of a minimal element, a maximal element, and elements corresponding to one-dimensional and two-dimensional subspaces. The one-dimensional subspaces are naturally in bijection with points of the finite projective plane ℙ2⁢(𝔽q)superscriptℙ2subscript𝔽𝑞\mathbb{P}^{2}(\mathbb{F}_{q})blackboard_P start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ), and the two-dimensional subspaces are naturally in bijection with lines in this plane. We can generalize this by considering finite projective planes which do not necessarily arise from vector spaces over finite fields. Since these are exactly the finite projective planes that don’t satisfy Desargues’s theorem, they are called non-Desarguesian planes. The smallest examples have 91919191 points and 91919191 lines.

First, note that the lattice of subspaces of 𝔽q2superscriptsubscript𝔽𝑞2\mathbb{F}_{q}^{2}blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT consists of a minimal element (the zero-dimensional subspace), a maximal element (the two-dimensional subspace), and q+1𝑞1q+1italic_q + 1 incomparable elements (the one-dimensional subspaces). We can generalize this by taking the lattice consisting of a minimal element, a maximal element, and q+1𝑞1q+1italic_q + 1 incomparable elements, where q𝑞qitalic_q is not necessarily a prime power.

We form a lattice from a finite projective plane by choosing the set of elements of the lattice to consist of one minimal element, one maximal element, the set of points of the plane, and the set of lines of the plane, with a partial ordering defined so that the minimal element is ≤\leq≤ every other element, the maximal element is ≥\geq≥ every other element, a point is ≤\leq≤ a line if the point lies on the line, and every other pair of elements is incomparable.

A lattice consisting of one minimal element, one maximal element, the set of points in P𝑃Pitalic_P, and the set of lines in P𝑃Pitalic_P, for P𝑃Pitalic_P a non-Desarguesian finite projective plane, with ≤\leq≤ defined as in Example 3.4.

First note that every object of this category lies in ℒℒ\mathcal{L}caligraphic_L where ℒℒ\mathcal{L}caligraphic_L is generated by the one-element set, because joins correspond to unions of sets and every set is the union of one-element sets.

B

_{\Lambda}^{b}(\Lambda)italic_m start_POSTSUBSCRIPT roman_Λ end_POSTSUBSCRIPT : roman_Sh start_POSTSUPERSCRIPT italic_b end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_Λ end_POSTSUBSCRIPT ( italic_M ) → start_OPFUNCTION italic_μ roman_sh end_OPFUNCTION start_POSTSUBSCRIPT roman_Λ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_b end_POSTSUPERSCRIPT ( roman_Λ )

The structure of spherical adjunctions has appeared in a number of previous works and leads to interesting applications in homological mirror symmetry [AbAurouxHMS, Nadspherical, Gammagespherical, Jeffs]. However, there are still important problems that remain. Namely, the cap functor ∩\cap∩ is only defined in the setting of a symplectic Landau-Ginzburg model instead of a general Weinstein pair, since it is completely not clear whether there are enough Lagrangian submanifolds asymptotic to a general Legendrian stop [Ganatra-Pardon-Shende3]. This means that even though the cup functor ∪\cup∪ is proved to be spherical as long as the stop is swappable [SylvanOrlov], it is difficult to characterize the adjoint functors geometrically.

is a spherical functor, and over the category of compact objects of sheaves, we will show that the left adjoint of the microlocalization

(even though we do not explicitly show compatibility of the Serre functor, we believe that it is basically clear from the definition in [KPSsphericalCY]).

Though the doubling functor wΛsubscript𝑤Λw_{\Lambda}italic_w start_POSTSUBSCRIPT roman_Λ end_POSTSUBSCRIPT in our current setting has appeared already in works of Nadler–Shende [NadShen, Section 7], we do not know how to prove further adjunction properties. Our main result crucially relies on the fact that the doubling functor will induce both the left adjoint and the right adjoint of the microlocalization functor, which does not seem obvious at the first place. For this reason, we will provide two different approaches generalizing the work of Guillermou [Gui, Section 13–15], and in particular providing an explicit local formula of the doubling functor as opposed to [NadShen].

C

It could be interesting to consider a modified Fourier spectrum, following the modified Fourier dimension defined in [5], but we do not pursue this here. We will be more focused on the Fourier spectrum of measures and many of our results for sets would also hold for the modified or compact variants considered by [5]. We leave details to the reader.

After establishing some fundamental theory, we move towards applications of the Fourier spectrum, especially concerning sumsets, convolutions, distance sets and certain random sets. A rough heuristic which emerges is that when the Fourier spectrum is not (the restriction of) an affine function, it provides more information than the Hausdorff and Fourier dimension on their own and this leads to new estimates in various contexts. For example, the Sobolev dimension of a measure increases under convolution with itself if and only if the Fourier spectrum is not (the restriction of) a linear function (Corollary 6.3) and when the Fourier spectrum of a measure is not an affine function, it provides better estimates for the Hausdorff dimension of the distance set of its support than the Hausdorff and Fourier dimension provide on their own via Mattila integrals (Theorem 7.1). As a result we solve the distance set problem for sets satisfying certain Fourier analytic conditions. A simple special case shows that if μ𝜇\muitalic_μ is a finite Borel measure on ℝdsuperscriptℝ𝑑\mathbb{R}^{d}blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT with ∫|μ^|4<∞superscript^𝜇4\int|\widehat{\mu}|^{4}<\infty∫ | over^ start_ARG italic_μ end_ARG | start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT < ∞, then the distance set of the support of μ𝜇\muitalic_μ has positive Lebesgue measure (Corollary 7.4). We note that the exponent 4 is sharp. We also use the Fourier spectrum to give conditions ensuring a measure is ‘Sobolev improving’ (Corollary 6.8), to give estimates for the Hausdorff dimension of certain random constructions where the Fourier dimension alone provides only trivial estimates (Corollary 8.3), and to provide a one line proof (and extension) of a well-known connection between moment analysis and Fourier dimension in random settings (Lemma 8.1).

It could be interesting to consider a modified Fourier spectrum, following the modified Fourier dimension defined in [5], but we do not pursue this here. We will be more focused on the Fourier spectrum of measures and many of our results for sets would also hold for the modified or compact variants considered by [5]. We leave details to the reader.

The idea to introduce a continuum of dimensions in-between a given pair of ‘fractal dimensions’ is part of a growing programme sometimes referred to as ‘dimension interpolation’. Previous examples include the Assouad spectrum which lives in-between the (upper) box-counting and Assouad dimensions [9] and the intermediate dimensions [8] which live in-between the Hausdorff and box-counting dimensions. The Fourier spectrum is of a rather different flavour since the aforementioned notions are defined via coverings. Despite their recent inception, the Assouad spectrum and intermediate dimensions are proving useful tools in a growing range of (often unexpected) areas, for example, in quasi-conformal mapping theory [3] and in analysis of spherical maximal functions [19]. We believe this will also be the case for the Fourier spectrum.

Finally, we hope that our use of ‘Fourier spectrum’ does not cause confusion with other uses of the phrase in the literature, for example [2]. In a previous draft of the paper we opted for the ‘Fourier dimension spectrum’ to avoid this issue, but in practice this name was too cumbersome and was always shortened. Further, we prefer to align the Fourier spectrum philosophically with the Assouad spectrum and hence the general programme of dimension interpolation.

D

\overline{\theta}(t)\theta(s).italic_α ( italic_t ) = italic_t italic_s start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT + over¯ start_ARG italic_θ end_ARG ( italic_t ) italic_s and italic_β ( italic_t ) = italic_t italic_θ ( italic_s start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ) + over¯ start_ARG italic_θ end_ARG ( italic_t ) italic_θ ( italic_s ) .

t}(x,y)+4|\theta(t)|C^{\prime}}{2}≤ divide start_ARG over^ start_ARG italic_d end_ARG start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ( italic_x , italic_o ) + over^ start_ARG italic_d end_ARG start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ( italic_o , italic_y ) - over^ start_ARG italic_d end_ARG start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ( italic_x , italic_y ) + 4 | italic_θ ( italic_t ) | italic_C start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_ARG start_ARG 2 end_ARG

By the definition of θ¯¯𝜃\overline{\theta}over¯ start_ARG italic_θ end_ARG, for all t𝑡titalic_t we get

where dtθ¯superscriptsubscript𝑑𝑡¯𝜃d_{t}^{\overline{\theta}}italic_d start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT over¯ start_ARG italic_θ end_ARG end_POSTSUPERSCRIPT is the pseudo metric given by Proposition 4.1 applied to θ¯¯𝜃\overline{\theta}over¯ start_ARG italic_θ end_ARG. Since h⁢(dtθ¯)=1ℎsuperscriptsubscript𝑑𝑡¯𝜃1h(d_{t}^{\overline{\theta}})=1italic_h ( italic_d start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT over¯ start_ARG italic_θ end_ARG end_POSTSUPERSCRIPT ) = 1 by Remark 4.7 (1), from the definition of θ𝜃\thetaitalic_θ we deduce that

{\prime}(t_{0})+\theta(s)=0italic_s start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT over¯ start_ARG italic_θ end_ARG start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) + italic_s = italic_θ ( italic_s start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ) over¯ start_ARG italic_θ end_ARG start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) + italic_θ ( italic_s ) = 0 from which it follows that s∗⁢θ¯⁢(s)=s⁢θ¯⁢(s∗)subscript𝑠∗¯𝜃𝑠𝑠¯𝜃subscript𝑠∗s_{\ast}\overline{\theta}(s)=s\overline{\theta}(s_{\ast})italic_s start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT over¯ start_ARG italic_θ end_ARG ( italic_s ) = italic_s over¯ start_ARG italic_θ end_ARG ( italic_s start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ). Again this is impossible due to Remark 4.7 (2) and the proof is complete.

B

We assume that clients and MPC servers have access to a shared randomness source, e.g., by agreeing on a PRNG seed.

Such a configuration has been widely employed in MPC protocols [7, 57, 35, 98, 79] and in ML systems [127, 12] to optimize communication.

While linear operations like additions can be calculated for “free” in most MPC protocols, non-linear operations such as multiplications require some form of communication between the MPC servers.

This is possible in MPC with a bit-to-arithmetic conversion protocol Π𝖡𝗂𝗍𝖠subscriptΠ𝖡𝗂𝗍𝖠\Pi_{\sf BitA}roman_Π start_POSTSUBSCRIPT sansserif_BitA end_POSTSUBSCRIPT [98, 108, 2, 79], which computes the arithmetic shares of 𝖻∈ℤ2𝖻subscriptℤ2\mathsf{b}\in\mathbb{Z}_{2}sansserif_b ∈ blackboard_Z start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT from its Boolean sharing.

As MPC protocols typically induce significant overhead in terms of (inter-server) communication, we propose optimizations for secure aggregation that support any “linear” quantization scheme, including 1-bit quantization schemes that uses preprocessing like random rotations [127] and Kashin’s representation [32].

A

_{\mu-1}\otimes\cdots\otimes I_{1}.italic_K = italic_A start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ⊕ italic_A start_POSTSUBSCRIPT italic_d - 1 end_POSTSUBSCRIPT ⊕ ⋯ ⊕ italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = ∑ start_POSTSUBSCRIPT italic_μ = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT italic_A start_POSTSUBSCRIPT ⊗ italic_μ end_POSTSUBSCRIPT , italic_A start_POSTSUBSCRIPT ⊗ italic_μ end_POSTSUBSCRIPT = italic_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ⊗ ⋯ ⊗ italic_I start_POSTSUBSCRIPT italic_μ + 1 end_POSTSUBSCRIPT ⊗ italic_A start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT ⊗ italic_I start_POSTSUBSCRIPT italic_μ - 1 end_POSTSUBSCRIPT ⊗ ⋯ ⊗ italic_I start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT .

1}\ldots j_{d}}\prod_{\mu=1}^{d}\ell_{i_{\mu}j_{\mu}}^{\mu},italic_w start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT … italic_i start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT end_POSTSUBSCRIPT = ∑ start_POSTSUBSCRIPT italic_j start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ⋯ ∑ start_POSTSUBSCRIPT italic_j start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT italic_v start_POSTSUBSCRIPT italic_j start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT … italic_j start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT end_POSTSUBSCRIPT ∏ start_POSTSUBSCRIPT italic_μ = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT roman_ℓ start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT italic_j start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_μ end_POSTSUPERSCRIPT ,

Here and throughout the paper d∈ℕ𝑑ℕd\in\mathbb{N}italic_d ∈ blackboard_N, μ=1,…,d𝜇1…𝑑\mu=1,\ldots,ditalic_μ = 1 , … , italic_d,

Here and in the next tableaux φℓ,jsubscript𝜑ℓ𝑗\varphi_{\ell,j}italic_φ start_POSTSUBSCRIPT roman_ℓ , italic_j end_POSTSUBSCRIPT denotes

Let Lμ∈ℂnμ×nμsubscript𝐿𝜇superscriptℂsubscript𝑛𝜇subscript𝑛𝜇L_{\mu}\in\mathbb{C}^{n_{\mu}\times n_{\mu}}italic_L start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT ∈ blackboard_C start_POSTSUPERSCRIPT italic_n start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT × italic_n start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT end_POSTSUPERSCRIPT be matrices, with μ=1,…,d𝜇1…𝑑\mu=1,\ldots,ditalic_μ = 1 , … , italic_d,

B

⟨a,b,c∣b2⁢a2⁢c2⁢a⁢b⁢c⟩.inner-product𝑎𝑏𝑐superscript𝑏2superscript𝑎2superscript𝑐2𝑎𝑏𝑐\langle a,b,c\mid b^{2}a^{2}c^{2}abc\rangle\,.⟨ italic_a , italic_b , italic_c ∣ italic_b start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_a start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_c start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_a italic_b italic_c ⟩ .

The proposition has one more consequence which is very useful in practice, because it depends only on combinatorial information. By the girth of a graph we mean the combinatorial length of the shortest non-trivial loop.

The presentation complex X𝑋Xitalic_X provides an example with ρ+⁢(X)=σ+⁢(X)<0subscript𝜌𝑋subscript𝜎𝑋0\rho_{+}(X)=\sigma_{+}(X)<0italic_ρ start_POSTSUBSCRIPT + end_POSTSUBSCRIPT ( italic_X ) = italic_σ start_POSTSUBSCRIPT + end_POSTSUBSCRIPT ( italic_X ) < 0. To see this, note that the link of the unique vertex of X𝑋Xitalic_X is the 1-skeleton of a triangular prism, and in particular has girth equal to 3. The face has length 9 and so

The link of the vertex is the complete bipartite graph K3,3subscript𝐾33K_{3,3}italic_K start_POSTSUBSCRIPT 3 , 3 end_POSTSUBSCRIPT [39, Example 2.1], which has girth 4, so

Let X𝑋Xitalic_X be the presentation complex associated to the presentation ⟨a,b∣b⁢a⁢b−1⁢a−2,b⟩inner-product𝑎𝑏𝑏𝑎superscript𝑏1superscript𝑎2𝑏\langle a,b\mid bab^{-1}a^{-2},b\rangle⟨ italic_a , italic_b ∣ italic_b italic_a italic_b start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_a start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT , italic_b ⟩. Since this is a balanced presentation of the trivial group, it follows from the Hurewicz and Whitehead theorems that X𝑋Xitalic_X is contractible. But the link of the unique vertex of X𝑋Xitalic_X is the complete graph on 4 generators, so X𝑋Xitalic_X is irreducible by Remark 4.2.

C

Let G𝐺Gitalic_G be a group and H𝐻Hitalic_H be a finitely generated subgroup. Then e⁢(G,H)≤e~⁢(G,H)𝑒𝐺𝐻~𝑒𝐺𝐻e(G,H)\leq\tilde{e}(G,H)italic_e ( italic_G , italic_H ) ≤ over~ start_ARG italic_e end_ARG ( italic_G , italic_H ).

Equality is certainly possible, but not true in general. A simple counterexample is presented in [36, p. 32], which involves a one-sided essential simple closed curve on a non-orientable surface. We finally conclude with the following, which demonstrates the common thread between the two competing definitions.

Λ⁢(v)={ℓ:ℓ is a simple loop through Δ with o⁢(ℓ)=t⁢(ℓ)=v}Λ𝑣conditional-setℓℓ is a simple loop through Δ with o⁢(ℓ)=t⁢(ℓ)=v\Lambda(v)=\{\ell:\textrm{$\ell$ is a simple loop through $\Delta$ with $o(%

Intuitively, Λ′⁢(v)superscriptΛ′𝑣\Lambda^{\prime}(v)roman_Λ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_v ) is the set of “lollipop loops” based at v𝑣vitalic_v, which are formed by attaching a simple loop to a simple path and its inverse.

Π⁢(v,v′)={p:p is a simple path through Δ with o⁢(p)=v, t⁢(p)=v′}Π𝑣superscript𝑣′conditional-set𝑝p is a simple path through Δ with o⁢(p)=v, t⁢(p)=v′\Pi(v,v^{\prime})=\{p:\textrm{$p$ is a simple path through $\Delta$ with $o(p)%

A

)^{-1}\mathcal{S}(\boldsymbol{Y}^{*,j},p^{j})\Delta p,roman_Δ bold_italic_Y start_POSTSUBSCRIPT italic_E italic_u end_POSTSUBSCRIPT = - caligraphic_K ( bold_italic_Y start_POSTSUPERSCRIPT ∗ , italic_j end_POSTSUPERSCRIPT , italic_p start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT caligraphic_S ( bold_italic_Y start_POSTSUPERSCRIPT ∗ , italic_j end_POSTSUPERSCRIPT , italic_p start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT ) roman_Δ italic_p ,

Δ⁢𝒀E⁢u=−𝒦⁢(𝒀∗,j,pj)−1⁢𝒮⁢(𝒀∗,j,pj)⁢Δ⁢p,Δsubscript𝒀𝐸𝑢𝒦superscriptsuperscript𝒀𝑗superscript𝑝𝑗1𝒮superscript𝒀𝑗superscript𝑝𝑗Δ𝑝\displaystyle\Delta\boldsymbol{Y}_{Eu}=-\mathcal{K}(\boldsymbol{Y}^{*,j},p^{j}%

Δ⁢𝒀N⁢e=−𝒦⁢(𝒀E⁢u∗,j+1,pj+1)−1⁢𝑻⁢(𝒀E⁢u∗,j+1,pj+1),Δsubscript𝒀𝑁𝑒𝒦superscriptsubscriptsuperscript𝒀𝑗1𝐸𝑢superscript𝑝𝑗11𝑻subscriptsuperscript𝒀𝑗1𝐸𝑢superscript𝑝𝑗1\displaystyle\Delta\boldsymbol{Y}_{Ne}=-\mathcal{K}(\boldsymbol{Y}^{*,j+1}_{Eu%

𝒀E⁢u∗,j+1=𝒀∗,j+Δ⁢𝒀E⁢u,subscriptsuperscript𝒀𝑗1𝐸𝑢superscript𝒀𝑗Δsubscript𝒀𝐸𝑢\displaystyle\boldsymbol{Y}^{*,j+1}_{Eu}=\boldsymbol{Y}^{*,j}+\Delta%

𝒀N⁢e∗,j+1=𝒀E⁢u∗,j+1+Δ⁢𝒀N⁢e.subscriptsuperscript𝒀𝑗1𝑁𝑒subscriptsuperscript𝒀𝑗1𝐸𝑢Δsubscript𝒀𝑁𝑒\displaystyle\boldsymbol{Y}^{*,j+1}_{Ne}=\boldsymbol{Y}^{*,j+1}_{Eu}+\Delta%

C

The boundary of ♢ψsubscript♢𝜓\diamondsuit_{\psi}♢ start_POSTSUBSCRIPT italic_ψ end_POSTSUBSCRIPT joins consecutive conifold points (x𝒪⁢(m),y𝒪⁢(m))subscript𝑥𝒪𝑚subscript𝑦𝒪𝑚(x_{\mathcal{O}(m)},y_{\mathcal{O}(m)})( italic_x start_POSTSUBSCRIPT caligraphic_O ( italic_m ) end_POSTSUBSCRIPT , italic_y start_POSTSUBSCRIPT caligraphic_O ( italic_m ) end_POSTSUBSCRIPT ), m∈ℤ𝑚ℤm\in\mathds{Z}italic_m ∈ blackboard_Z, so that ♢ψsubscript♢𝜓\diamondsuit_{\psi}♢ start_POSTSUBSCRIPT italic_ψ end_POSTSUBSCRIPT borders all translates of ΔψsubscriptΔ𝜓\Delta_{\psi}roman_Δ start_POSTSUBSCRIPT italic_ψ end_POSTSUBSCRIPT, while ΔψsubscriptΔ𝜓\Delta_{\psi}roman_Δ start_POSTSUBSCRIPT italic_ψ end_POSTSUBSCRIPT borders three large volume regions.

The initial rays (6.29) emanate from vertices of the tessellation by construction, and they enter different regions depending on ψ𝜓\psiitalic_ψ.

|Z⁢(γ)|𝑍𝛾|Z(\gamma)|| italic_Z ( italic_γ ) | must lie along active rays ℛψ⁢(γ)subscriptℛ𝜓𝛾\mathcal{R}_{\psi}(\gamma)caligraphic_R start_POSTSUBSCRIPT italic_ψ end_POSTSUBSCRIPT ( italic_γ ), for a suitable value of ψ𝜓\psiitalic_ψ determined by the initial value of the flow. Since vertices in the attractor flow tree have to

Depending on the phase ψ𝜓\psiitalic_ψ, we find that 𝒟ψΠsuperscriptsubscript𝒟𝜓Π\mathcal{D}_{\psi}^{\Pi}caligraphic_D start_POSTSUBSCRIPT italic_ψ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT roman_Π end_POSTSUPERSCRIPT has the following structure,

The outcome of this analysis in §C.3 is that the initial rays of the exact scattering diagram are262626To lighten the notation, we omit the ψ𝜓\psiitalic_ψ-dependence of the rays.

A

Note that (2.8) also appeared as [3, Entry 1.7.12], (2.9) also appeared in [3, Entries 1.7.11 and 4.2.15]. A typo has been corrected for (2.17). The identities (2.8) and (2.9) are usually referred as the Göllnitz–Gordon identities.

Setting z=q−2𝑧superscript𝑞2z=q^{-2}italic_z = italic_q start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT, q−1superscript𝑞1q^{-1}italic_q start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT and 1 in (3.15), we obtain (3.4), (3.6) and (3.7), respectively. But one cannot get (3.5) and (3.8) by specializing the choice of z𝑧zitalic_z. These two identities were also known in the literature. For example, Uncu [24, Corollary 3.8] gave some double sum representations of the Göllnitz–Gordon identities (see (2.8) and (2.9)), which prove (3.5) and (3.8).

For convenience, we label the examples in Zagier’s list from 1 to 11 according to their order in [28, Table 2]. We are able to prove nine of them. Namely, Examples 1–4, 6–9 and 11. It should be mentioned that seven of Zagier’s examples have been discussed (explicitly or implicitly) in the literature. See Table 1 for known cases and references which discuss them. Note that for Example 2, there are five choices for the vector part B𝐵Bitalic_B. Lee’s arguments [14] and the work of Vlasenko and Zwegers [25] are applicable for three of them. The remaining two cases can be justified by double sum variants of the Göllnitz–Gordon identities (see (2.8) and (2.9)). We will give unified proofs which apply to all these five cases. As for Example 10, Vlasenko and Zwegers [25] only provided conjectural identities, and the modularity of this example were later confirmed by Cherednik and Feigin [6] via the nilpotent double affine Hecke algebras. We will state new identities for Example 10 which is different from but equivalent to that of Vlasenko and Zwegers [25]. The only example which remains open is Example 5. We will state conjectural identities for it.

As seen from Table 1, it appears that this is the first time to state explicit identities for Examples 5, 7, 8 and 9. Moreover, it seems to be the first time to give proofs for Examples 7–9. It is worth mentioning that many of Zagier’s examples can be reduced to some known single sum Rogers–Ramanujan type identities from Slater’s list [22]. The reduction processes vary for different examples. For most of the examples, we achieve it by summing over one of the indexes first. For Example 8 we will use an integral method to find new expressions for the sum sides, and then either eliminate one of the summation indexes or follow the techniques in the author’s work [26].

For Example 8 we will provide two different proofs. For both proofs we will use an integral method to rewrite the sum sides as integrals of some infinite products. This method was applied by Rosengren [20] to prove some conjectural identities of Kanade and Russell [13]. Later it has been applied in several works. For example, it was used by Chern [7] and the author [26] to prove a conjecture of Andrews and Uncu [4]. It was also utilized by Mc Laughlin [15] and Cao and Wang [8] in finding some new multi-sum Rogers–Ramanujan type identities.

D

Let ChRsubscriptCh𝑅\operatorname{Ch}_{R}roman_Ch start_POSTSUBSCRIPT italic_R end_POSTSUBSCRIPT be the category of chain complexes over a fixed commutative ring R𝑅Ritalic_R with its standard symmetric monoidal structure given by tensor product of chain complexes. The reader should keep the case where R𝑅Ritalic_R is the Novikov ring in mind, since that is the only one that we will use.

We warn the reader that the set of n𝑛nitalic_n-cubes of the tensor product ℋ∙⁢(H→1;H2,i)⊗ℋ∙⁢(H→2;H)tensor-productsubscriptℋ∙superscript→𝐻1superscript𝐻2𝑖subscriptℋ∙superscript→𝐻2𝐻\mathcal{H}_{\bullet}(\vec{H}^{1};H^{2,i})\otimes\mathcal{H}_{\bullet}(\vec{H}%

Let ChRsubscriptCh𝑅\operatorname{Ch}_{R}roman_Ch start_POSTSUBSCRIPT italic_R end_POSTSUBSCRIPT be the category of chain complexes over a fixed commutative ring R𝑅Ritalic_R with its standard symmetric monoidal structure given by tensor product of chain complexes. The reader should keep the case where R𝑅Ritalic_R is the Novikov ring in mind, since that is the only one that we will use.

The projection map from the tensor product over the ground ring to the tensor product over the group ring of Aut⁡(x→)Aut→𝑥\operatorname{Aut}(\vec{x})roman_Aut ( over→ start_ARG italic_x end_ARG )

If we do not write anything under a tensor product sign it means that it is the tensor product of chain complexes over R𝑅Ritalic_R.

D

A partition π={π1,…,πk}𝜋subscript𝜋1…subscript𝜋𝑘\pi=\{\pi_{1},\ldots,\pi_{k}\}italic_π = { italic_π start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_π start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT } of a linearly ordered set

X:=j!⁢(A)assign𝑋subscript𝑗𝐴X:=j_{!}(A)italic_X := italic_j start_POSTSUBSCRIPT ! end_POSTSUBSCRIPT ( italic_A ), where X1subscript𝑋1X_{1}italic_X start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT is the set of all noncrossing partitions,

sizes of the layers, so that the set An={𝐦→𝐧}subscript𝐴𝑛→𝐦𝐧A_{n}=\{\mathbf{m}\to\mathbf{n}\}italic_A start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT = { bold_m → bold_n } is

𝐧={1,2,…,n}𝐧12…𝑛\mathbf{n}=\{1,2,\ldots,n\}bold_n = { 1 , 2 , … , italic_n } is called noncrossing where there are no

Let Ansubscript𝐴𝑛A_{n}italic_A start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT denote the set of noncrossing partitions of 𝐧𝐧\mathbf{n}bold_n.

C

\{s_{k}\right\}}\neq\varnothing.italic_g start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_W start_POSTSUBSCRIPT italic_S ∖ { italic_s start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT } end_POSTSUBSCRIPT ∩ … ∩ italic_g start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_W start_POSTSUBSCRIPT italic_S ∖ { italic_s start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT } end_POSTSUBSCRIPT ≠ ∅ .

Left multiplication on cosets induces an action of W𝑊Witalic_W on Σ⁢(W,S)Σ𝑊𝑆\Sigma(W,S)roman_Σ ( italic_W , italic_S ).

In the setting of Lemma 2.2, any Coxeter generator s∈S𝑠𝑆s\in Sitalic_s ∈ italic_S of W𝑊Witalic_W acts by (−1)1(-1)( - 1 ) on the standard apartment class of Σ⁢(W,S)Σ𝑊𝑆\Sigma(W,S)roman_Σ ( italic_W , italic_S ),

The standard apartment class of Σ⁢(W,S)Σ𝑊𝑆\Sigma(W,S)roman_Σ ( italic_W , italic_S ) is the generator

Left multiplication on cosets induces an action of 𝒢⁢(𝕂)𝒢𝕂\mathcal{G}(\mathbb{K})caligraphic_G ( blackboard_K ) on Δ⁢(𝒢⁢(𝕂))Δ𝒢𝕂\Delta(\mathcal{G}(\mathbb{K}))roman_Δ ( caligraphic_G ( blackboard_K ) ).

A

Our contribution lies in not only offering a possible strategy for solving Questions 1 and 2, which bridges ergodic theory and topological dynamics of random dynamical systems in a more elegant manner, but also providing new ideas and methods for the construction of invariant measures (including the constructing of the certain margin) of random dynamical systems, which is different from the classical approach for topological entropy[Mis75, Wal82, Kif01].

In this section, we first introduce some powerful tools in subsection 3.1 and then give the proof of Theorem 1.1 in subsection 3.2.

In this subsection, we introduce the notion of metric mean dimension with potential and deduce some elementary properties.

In this subsection, we introduce the notion of (upper) measure-theoretical metric mean dimension and give some equivalent characterizations for it.

The rest of this paper is organized as follows. In section 2, we introduce the notions of random metric mean dimension both in topological and measure-theoretical situation, and deduce some elementary properties. In section 3, we develop some new tools to prove Theorem 1.1.

D

It appears that all the known examples of double symplectic groupoids are particular cases of the previous result. In most examples, the relevant transitive Courant algebroid is of the form 𝕋H⁢X0⊕𝔤direct-sumsubscript𝕋𝐻subscript𝑋0𝔤\mathbb{T}_{H}X_{0}\oplus\mathfrak{g}blackboard_T start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT italic_X start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ⊕ fraktur_g where 𝔤𝔤\mathfrak{g}fraktur_g is a quadratic Lie algebra. The general way of showing that an integrable Dirac structure in a transitive Courant algebroid actually integrates to a 2-shifted lagrangian groupoid is by means of Van Est Theorem [5] as we shall discuss in [4]. However, in practice one can often find the desired integration by ad hoc considerations.

Symplectic groupoids are fundamental tools in Poisson geometry, being the global counterparts of Poisson structures [57, 20]. Not every Poisson manifold can be associated with a symplectic groupoid, only those that satisfy the necessary and sufficient conditions of [19] admit such a corresponding global object. Interestingly, some Poisson structures that are compatible with a group(oid) structure such as those defined on Poisson-Lie groups give rise to symplectic groupoids that are equipped with an additional Lie groupoid structure which makes them into double symplectic groupoids [39]. Some of the most interesting applications of double symplectic groupoids are the following. In the case of a Poisson-Lie group, the presence of an additional groupoid multiplication on its symplectic groupoid is responsible for the existence of a monoidal structure on the category of its hamiltonian spaces, such a monoidal structure is even braided monoidal in some situations [58]; similar properties should hold for more general double symplectic groupoids by using one of the groupoid multiplications to define the monoidal structure. Another remarkable aspect of double symplectic groupoids is their relevance to the AKSZ formalism [2]. The infinitesimal counterpart of a double symplectic groupoid is a Lie bialgebroid [43] and the doubles of Lie bialgebroids are Courant algebroids [34]. This means that double symplectic groupoids induce integrations of Courant algebroids [44, 21] and so they can be regarded as classical phase spaces associated with the Courant sigma model [48]. It was pointed out in [55] (after [53]) that this viewpoint could lead to a full integration of Lie bialgebroids; conversely, we expect that the explicit double symplectic groupoids we construct can be used to better understand the Courant sigma model in special circumstances, see [17].

The construction of double symplectic groupoids has proceeded so far on a case by case basis without a general principle behind. In this work we describe a way of constructing double symplectic groupoids that can be applied to most cases of interest, motivated by the ideas in [39, 51, 50]. Instead of attempting to integrate a Lie bialgebroid (A,B)𝐴𝐵(A,B)( italic_A , italic_B ) by stages, first to a Poisson groupoid [42] and then to a double symplectic groupoid [55], we propose to look at both A𝐴Aitalic_A and B𝐵Bitalic_B as Dirac structures and integrate them according to their interpretation as infinitesimal 2-shifted lagrangians [47]; these 2-shifted lagrangians can then be used to produce a double symplectic groupoid following the outline of [39]. Under mild assumptions, our approach allows us to integrate a Lie bialgebroid whose associated Courant algebroid is transitive to a double symplectic groupoid, thus giving a unified explanation for most of the known examples such as those coming from dynamical Poisson groupoids [29] or moduli spaces of flat bundles [39, 51]. We can also use this construction to produce new families of double symplectic groupoids in two interesting contexts: (1) in generalised Kähler geometry, where the relevant Lie bialgebroids are holomorphic and induce exact Courant algebroids [26] and (2) in Lie theory, where we can give integrations in the algebraic category of the configuration Poisson groupoids of flags [38], these applications will be given in upcoming works.

The double symplectic groupoid that we get by applying Theorem 4.10 to a Lie bialgebroid over a point is the original construction of [39] since the 2-forms associated with a lagrangian subgroup of a Lie group equipped with a quadratic Lie algebra are zero. According to [50, 51], this double symplectic groupoid is an example of a decorated moduli space of flat G𝐺Gitalic_G-bundles. The double symplectic groupoids that appear in [3] are examples of this construction, see the example below.

By applying the Lie functor to a double Lie groupoid in one direction we get an LA-groupoid: a groupoid in the category of Lie algebroids [40]. In the case of a double quasi-symplectic groupoid, we get a multiplicative Dirac structure.

C

A mechanism (or channel) 𝖪:𝒳→𝒫⁢(𝒵):𝖪→𝒳𝒫𝒵{\mathsf{K}}:{\mathcal{X}}\to{\mathcal{P}}({\mathcal{Z}})sansserif_K : caligraphic_X → caligraphic_P ( caligraphic_Z ) is specified by a collection of distributions {𝖪(⋅|x)∈𝒫(𝒵):x∈𝒳}\{{\mathsf{K}}(\cdot|x)\in{\mathcal{P}}({\mathcal{Z}}):x\in{\mathcal{X}}\}{ sansserif_K ( ⋅ | italic_x ) ∈ caligraphic_P ( caligraphic_Z ) : italic_x ∈ caligraphic_X }. Given such mechanism 𝖪𝖪{\mathsf{K}}sansserif_K and P∈𝒫⁢(𝒳)𝑃𝒫𝒳P\in{\mathcal{P}}({\mathcal{X}})italic_P ∈ caligraphic_P ( caligraphic_X ), we denote by P⁢𝖪𝑃𝖪P{\mathsf{K}}italic_P sansserif_K the output distribution of 𝖪𝖪{\mathsf{K}}sansserif_K when the input is distributed according to P𝑃Pitalic_P, given by P⁢𝖪⁢(A)≔∫P⁢(d⁢x)⁢𝖪⁢(A|x)≔𝑃𝖪𝐴𝑃d𝑥𝖪conditional𝐴𝑥P{\mathsf{K}}(A)\coloneqq\int P(\text{d}x){\mathsf{K}}(A|x)italic_P sansserif_K ( italic_A ) ≔ ∫ italic_P ( d italic_x ) sansserif_K ( italic_A | italic_x ) for A⊂𝒵𝐴𝒵A\subset{\mathcal{Z}}italic_A ⊂ caligraphic_Z. We use 𝔼P⁢[⋅]subscript𝔼𝑃delimited-[]⋅{\mathbb{E}}_{P}[\cdot]blackboard_E start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT [ ⋅ ] to write the expectation with respect to P𝑃Pitalic_P and [n]delimited-[]𝑛[n][ italic_n ] for an integer n≥1𝑛1n\geq 1italic_n ≥ 1 to denote {1,…,n}1…𝑛\{1,\dots,n\}{ 1 , … , italic_n }.

In this section, we use the results presented in the previous section to examine several statistical problems under LDP constraint, including minimax estimation risks in Sections IV-A to IV-D and sample complexity of hypothesis testing in Section IV-E. In all these applications, we allow our mechanisms to be sequentially interactive.

In this section, we give basic definitions of f𝑓fitalic_f-divergence, contraction coefficients, and LDP mechanisms.

Let 𝒬εsubscript𝒬𝜀{\mathcal{Q}}_{\varepsilon}caligraphic_Q start_POSTSUBSCRIPT italic_ε end_POSTSUBSCRIPT be the collection of all ε𝜀\varepsilonitalic_ε-LDP mechanisms 𝖪𝖪{\mathsf{K}}sansserif_K. It can be shown that LDP mechanisms can be equivalently defined in terms of the hockey-stick divergence:

Therefore, binary randomized response mechanism satisfies the inequality in (8) with equality. In addition to the tightness of Theorem 1, this implies that the binary randomized response mechanism has the largest contraction coefficient among all ε𝜀\varepsilonitalic_ε-LDP mechanisms. Considering k𝑘kitalic_k-ary randomized response mechanisms, it is therefore expected that the contraction coefficients decrease as k𝑘kitalic_k increases, as proved next.

B

(ln⁡(T)).99superscript𝑇.99(\ln(T))^{.99}( roman_ln ( italic_T ) ) start_POSTSUPERSCRIPT .99 end_POSTSUPERSCRIPT

probability. In the Haar case we use either ℋTsubscriptℋ𝑇\mathcal{H}_{T}caligraphic_H start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT above, and 𝒦Tsubscript𝒦𝑇\mathcal{K}_{T}caligraphic_K start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT === [(ln⁡(T)).99]delimited-[]superscript𝑇.99[(\ln(T))^{.99}][ ( roman_ln ( italic_T ) ) start_POSTSUPERSCRIPT .99 end_POSTSUPERSCRIPT ]. Refer to Table 1.

(ln⁡(T)).99superscript𝑇.99(\ln(T))^{.99}( roman_ln ( italic_T ) ) start_POSTSUPERSCRIPT .99 end_POSTSUPERSCRIPT

log2(T).99\log_{2}(T)^{.99}roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_T ) start_POSTSUPERSCRIPT .99 end_POSTSUPERSCRIPT −-- 3333

(ln⁡(T)).99superscript𝑇.99(\ln(T))^{.99}( roman_ln ( italic_T ) ) start_POSTSUPERSCRIPT .99 end_POSTSUPERSCRIPT

B

The assertion follows formally from 2.15, the fact that sheafification preserves small colimits (in particular, all coequalizers) and the fact that sheafification of multiplicative presheaves preserves countable products (Proposition 2.14).

In this section, our goal is to prove A. We begin by summarizing some of the necessary background on the homotopy theory of ∞\infty∞-topoi (following [Lur09, § 6.5]) that we will need and fix some notation along the way.

Here, we will only discuss some examples and constructions that will be useful to us in Section 3.2.

We include two applications of A; the necessary notations and background are explained before the statements of the corollaries.

In order to prove A, we will use the notion of multiplicative presheaves (Definition 2.5) and also some additional preparations. An important observation is the following proposition which shows that certain naturally occuring presheaves are multiplicative.

A

{k(\varphi(a))/k}.roman_Φ start_POSTSUBSCRIPT italic_r , italic_B end_POSTSUBSCRIPT ( italic_φ ( [ italic_a ] ) ) = [ italic_k ( italic_a ) : italic_k ( italic_φ ( italic_a ) ) ] { italic_φ ( italic_a ) , … , italic_φ ( italic_a ) } start_POSTSUBSCRIPT italic_k ( italic_φ ( italic_a ) ) / italic_k end_POSTSUBSCRIPT .

In [Gaz15] the author constructed an integral descending (a priori infinite) filtration {Fr⁢(A)}r≥0subscriptsuperscript𝐹𝑟𝐴𝑟0\{F^{r}(A)\}_{r\geq 0}{ italic_F start_POSTSUPERSCRIPT italic_r end_POSTSUPERSCRIPT ( italic_A ) } start_POSTSUBSCRIPT italic_r ≥ 0 end_POSTSUBSCRIPT of CH0⁡(A)subscriptCH0𝐴\operatorname{CH}_{0}(A)roman_CH start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_A ) for an abelian variety A𝐴Aitalic_A, which agrees with the Beauville-Bloch filtration Gr⁢(A)superscript𝐺𝑟𝐴G^{r}(A)italic_G start_POSTSUPERSCRIPT italic_r end_POSTSUPERSCRIPT ( italic_A ) after ⊗ℚtensor-productabsentℚ\otimes\mathbb{Q}⊗ blackboard_Q. The distinctive property of this new construction is that the successive quotients Fr/Fr+1superscript𝐹𝑟superscript𝐹𝑟1F^{r}/F^{r+1}italic_F start_POSTSUPERSCRIPT italic_r end_POSTSUPERSCRIPT / italic_F start_POSTSUPERSCRIPT italic_r + 1 end_POSTSUPERSCRIPT have an interpretation as a Somekawa K𝐾Kitalic_K-group. We note that it has been shown by Kahn and Yamazaki (cf. [KY13, (1.1)]) that these K𝐾Kitalic_K-groups have the correct motivic interpretation as motivic cohomology groups111In fact, they are ExtExt\operatorname{Ext}roman_Ext groups in the category of effective motivic complexes of Voevodsky.. Thus, to the extend to which the category ℳ⁢ℳℳℳ\mathcal{MM}caligraphic_M caligraphic_M has been constructed, the above constructions provide good candidates for the Bloch-Beilinson filtration.

The equality of these two symbols follows by the projection formula of the Somekawa K𝐾Kitalic_K-group (cf. [GL21, 2.5]). Namely,

(cf. [RS00, Conjecture 3.5.4], [CT95, 1.4 (d),(e),(f)]) The group F2⁢(X)superscript𝐹2𝑋F^{2}(X)italic_F start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_X ) is the direct sum of a divisible group and a finite group.

We note that the proof of section 1 is not immediate. It requires various properties of the Somekawa K𝐾Kitalic_K-group and a careful comparison of the two constructions in [RS00] and [Gaz15], which are a priori very different. Lastly, section 1 improves in this special case the equality Fr⁢(A)⊗ℚ=Gr⁢(A)⊗ℚtensor-productsuperscript𝐹𝑟𝐴ℚtensor-productsuperscript𝐺𝑟𝐴ℚF^{r}(A)\otimes\mathbb{Q}=G^{r}(A)\otimes\mathbb{Q}italic_F start_POSTSUPERSCRIPT italic_r end_POSTSUPERSCRIPT ( italic_A ) ⊗ blackboard_Q = italic_G start_POSTSUPERSCRIPT italic_r end_POSTSUPERSCRIPT ( italic_A ) ⊗ blackboard_Q shown in [Gaz15, Corollary 4.4]. It is not unreasonable to expect that the filtration {Fr⁢(A)}r≥0subscriptsuperscript𝐹𝑟𝐴𝑟0\{F^{r}(A)\}_{r\geq 0}{ italic_F start_POSTSUPERSCRIPT italic_r end_POSTSUPERSCRIPT ( italic_A ) } start_POSTSUBSCRIPT italic_r ≥ 0 end_POSTSUBSCRIPT agrees integrally with the Beauville-Bloch filtration and an integral vanishing Gd+1⁢(A)=0superscript𝐺𝑑1𝐴0G^{d+1}(A)=0italic_G start_POSTSUPERSCRIPT italic_d + 1 end_POSTSUPERSCRIPT ( italic_A ) = 0 holds (cf. subsection 2.3).

B

In Sec. 4.4, we compare our proposed centrality measure to other edge centralities for these three empirical case studies.

Despite these challenges, we utilized our approximation theory for perturbed diffusion-kernel VNE to study the entropic importance of edges via three case studies with empirical networks to investigate structural/dynamical mechanisms that can systematically influence the entropic importance of edges. In Sec. 4.1, we studied a network encoding voting similarity in the 117th U.S. Senate and studied the importance of interparty edges (i.e., those that connect across the two large-scale communities that result from party polarization). In Sec. 4.2, we studied a multimodal transportation network and studied the importance of edges representing metro lines as opposed to those representing roads. In Sec. 4.3, we studied a multiplex brain network and studied the importance of interlayer edges that couple network layers versus intralayer edges that connect nodes within a particular layer. In each of these case studies, we found that the edges that are deemed to be the most important drastically change by considering different diffusion timescales (i.e., different values of the parameter β𝛽\betaitalic_β).

Observe in the right-most column of Fig. 2(A) for β=1𝛽1\beta=1italic_β = 1 that the top-ranked edges are the interparty edges that connect Senators in different political parties. Interestingly, for β=10−1𝛽superscript101\beta=10^{-1}italic_β = 10 start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT (center column) the interparty edges are the ones with the lowest rankings, and for β=10−2𝛽superscript102\beta=10^{-2}italic_β = 10 start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT (left column) the rankings of interparty and intraparty edges are similar.

Similar to our study in Sec. 4.1 (where we examined the rankings of interparty edges), we will now pay particular attention to the rankings R~p⁢qsubscript~𝑅𝑝𝑞\tilde{R}_{pq}over~ start_ARG italic_R end_ARG start_POSTSUBSCRIPT italic_p italic_q end_POSTSUBSCRIPT of interlayer edges. We ask the following: When do the interlayer edges have the top rankings?

In our first experiment, we study the importance of edges according to Algorithm 2 for a network that encodes voting similarity among U.S. Senators, and we will compare the rankings of interparty edges that connect Senators in different parties to the rankings of intraparty edges between Senators in the same party. More generally, we ask the following: which edges are most important in a graph that contains community structure, the edges between communities or the ones inside of communities? As we shall show, the answer will depend sensitively on the timescale of the dynamics considered.

D

In broad terms, quasi-coherent complexes correspond to complexes ℱ∙subscriptℱ∙\mathscr{F}_{\bullet}script_F start_POSTSUBSCRIPT ∙ end_POSTSUBSCRIPT of presheaves of 𝒪Xsubscript𝒪𝑋\mathscr{O}_{X}script_O start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT-modules whose homology presheaves ℋn⁢(ℱ∙)subscriptℋ𝑛subscriptℱ∙\mathscr{H}_{n}(\mathscr{F}_{\bullet})script_H start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( script_F start_POSTSUBSCRIPT ∙ end_POSTSUBSCRIPT ) are quasi-coherent.

Resolutions of this sort were used by Olsson in [Ols] to study quasi-coherent sheaves on Artin stacks.

The following is given by [Pri, Theorems LABEL:stacks-lshfthm, LABEL:stacks-hshfthm and LABEL:stacks-dgshfthm]:

As in §4, inverse images of quasi-coherent complexes are easy to compute, while derived direct images are more complicated — see [Pri, §LABEL:stacks-directsn].

To understand how these are related to complexes of quasi-coherent sheaves on schemes, see [Pri, Remarks LABEL:stacks-hcartrks].

D

The following construction follows Reznikov [24]. We recall it for the reader’s convenience and because it is of prime importance in what follows. In this section M𝑀Mitalic_M may be any 2⁢n2𝑛2n2 italic_n-dimensional Riemannian manifold whose Reznikov 2-form is non-degenerate.

Let ΣΣ\Sigmaroman_Σ be an oriented totally geodesic 3-dimensional submanifold of an oriented hyperbolic 6-manifold. The quantum homology of the Reznikov lift LΣsubscript𝐿ΣL_{\Sigma}italic_L start_POSTSUBSCRIPT roman_Σ end_POSTSUBSCRIPT is

Let ΣΣ\Sigmaroman_Σ be an n𝑛nitalic_n-dimensional submanifold of M𝑀Mitalic_M and consider the submanifold

If ΣΣ\Sigmaroman_Σ is an oriented totally geodesic submanifold of an oriented hyperbolic 6-manifold M𝑀Mitalic_M and LΣsubscript𝐿ΣL_{\Sigma}italic_L start_POSTSUBSCRIPT roman_Σ end_POSTSUBSCRIPT denotes the Reznikov Lagrangian lift in the twistor space of M𝑀Mitalic_M then

If ΣΣ\Sigmaroman_Σ is an oriented totally geodesic submanifold of an oriented hyperbolic 6-manifold M𝑀Mitalic_M and LΣsubscript𝐿ΣL_{\Sigma}italic_L start_POSTSUBSCRIPT roman_Σ end_POSTSUBSCRIPT denotes the Reznikov Lagrangian lift in the twistor space of M𝑀Mitalic_M then

B

K}(\gamma)}))=0roman_Φ start_POSTSUBSCRIPT italic_K end_POSTSUBSCRIPT ( fraktur_h ( italic_γ start_POSTSUPERSCRIPT italic_ν start_POSTSUBSCRIPT italic_K end_POSTSUBSCRIPT ( italic_γ ) end_POSTSUPERSCRIPT ) , fraktur_h ( italic_γ start_POSTSUPERSCRIPT italic_ν start_POSTSUBSCRIPT italic_K end_POSTSUBSCRIPT ( italic_γ ) end_POSTSUPERSCRIPT ) ) = 0.

Taking γ′=γsuperscript𝛾′𝛾\gamma^{\prime}=\gammaitalic_γ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = italic_γ in Theorem 2.3, we get the aforementioned characterization of s.c.c.’s:

(cf. Remark 5.4). Composing the map 𝔥𝒞subscript𝔥𝒞{\mathfrak{h}}_{\mathscr{C}}fraktur_h start_POSTSUBSCRIPT script_C end_POSTSUBSCRIPT with the pairing Φ𝒞subscriptΦ𝒞\Phi_{\mathscr{C}}roman_Φ start_POSTSUBSCRIPT script_C end_POSTSUBSCRIPT, we get a continuous pairing

characterization of this property in terms of the homology of finite unramified coverings of Sg,nsubscript𝑆𝑔𝑛S_{g,n}italic_S start_POSTSUBSCRIPT italic_g , italic_n end_POSTSUBSCRIPT.

Combining this remark with Theorem 2.3 and its corollaries, we get the following characterization of simple

D

The calculations to this point allow the following interesting description of the bar construction on a free n𝑛nitalic_n-disk algebra. Let 𝒱𝒱\mathcal{V}caligraphic_V be a symmetric monoidal ∞infinity\operatorname{\infty}∞-category which is ⊗tensor-product\otimes⊗-presentable.

For 𝒱𝒱\mathcal{V}caligraphic_V a symmetric monoidal ∞infinity\operatorname{\infty}∞-category which is ⊗tensor-product\otimes⊗-presentable, there is an equivalence

For any object V∈𝒱𝑉𝒱V\in\mathcal{V}italic_V ∈ caligraphic_V in a symmetric monoidal ∞\infty∞-category which is ⊗tensor-product\otimes⊗-presentable, there is an natural equivalence

For each symmetric monoidal ∞\infty∞-category 𝒱𝒱\mathcal{V}caligraphic_V which is ⊗tensor-product\otimes⊗-presentable, there is a natural equivalence in 𝒱𝒱\mathcal{V}caligraphic_V:

For an associative algebra A𝐴Aitalic_A in a symmetric monoidal ∞infinity\operatorname{\infty}∞-category 𝒱𝒱\mathcal{V}caligraphic_V which is ⊗tensor-product\otimes⊗-presentable, there is an equivalence

B

Rε=R∪{ε}subscript𝑅𝜀𝑅𝜀R_{\varepsilon}=R\cup\{\varepsilon\}italic_R start_POSTSUBSCRIPT italic_ε end_POSTSUBSCRIPT = italic_R ∪ { italic_ε } with

A(k)=(E⊕𝒯k⊗GT⊕(𝒯k⊗GT)2)⊕(𝒯k⊗GT)3)⊗𝒯kA(k)=(E\oplus{\cal T}_{k}\otimes G^{T}\oplus({\cal T}_{k}\otimes G^{T})^{2})%

ε=−∞𝜀\varepsilon=-\inftyitalic_ε = - ∞. The operations ⊕direct-sum\oplus⊕ and ⊗tensor-product\otimes⊗

⟨Rε,⊕,⊗⟩subscript𝑅𝜀direct-sumtensor-product\langle R_{\varepsilon},\oplus,\otimes\rangle⟨ italic_R start_POSTSUBSCRIPT italic_ε end_POSTSUBSCRIPT , ⊕ , ⊗ ⟩, where

111122223333444455556666(a) Network schemeG=(ε00εεεεεε0εεεεε00εεεεεε0εεεεε0εεεεεε)𝐺𝜀00𝜀𝜀𝜀𝜀𝜀𝜀0𝜀𝜀𝜀𝜀𝜀00𝜀𝜀𝜀𝜀𝜀𝜀0𝜀𝜀𝜀𝜀𝜀0𝜀𝜀𝜀𝜀𝜀𝜀G=\left(\begin{array}[]{cccccc}\varepsilon&0&0&\varepsilon&\varepsilon&%

B

Rule (3) describes the two cases of an in-flow: either a tree with multiple branches leading to the same root (parallel branches) or a path (conveyor belt).

An important additional constraint in the notation is that each point can only occur at most once.333

One of the most useful features of linear notation is that when restricted to permutations it is identical to orbit-cycle form. Also, by looking for parentheses we can easily spot the existence of nontrivial permutations even in large examples.

It is important to note that this is the only dynamics in systems where the only action is the unidirectional passage of discrete time.111This is in contrast to more general discrete dynamical systems which are generated by several such transformations on a finite set X𝑋Xitalic_X, as studied in transformation semigroup theory.

The notation w𝑤witalic_w thus clearly determines a unique well-defined transformation f=ℐ⁢(w)𝑓ℐ𝑤f=\mathcal{I}(w)italic_f = caligraphic_I ( italic_w ) on {1,…,n}1…𝑛\{1,\ldots,n\}{ 1 , … , italic_n } to itself, since each element p𝑝pitalic_p appears at most once in w𝑤witalic_w. The transformation is total (i.e, not partial, but fully defined) by Equation 6. Moreover, every f𝑓fitalic_f can be written in this attractor-cycle notation in a canonical form.

A

Let K𝐾Kitalic_K be the set of all finite ordered n𝑛nitalic_n-partite n𝑛nitalic_n-uniform hypergraphs and let K∗={A:A⊆B∈K}superscript𝐾conditional-set𝐴𝐴𝐵𝐾K^{*}=\{A:A\subseteq B\in K\}italic_K start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT = { italic_A : italic_A ⊆ italic_B ∈ italic_K } be the hereditary closure of K𝐾Kitalic_K.

Then K𝐾Kitalic_K and K∗superscript𝐾K^{*}italic_K start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT are Ramsey classes.

In addition, if K𝐾Kitalic_K has the (A,B)𝐴𝐵(A,B)( italic_A , italic_B )-Ramsey property for every A,B∈K𝐴𝐵𝐾A,B\in Kitalic_A , italic_B ∈ italic_K, then we say that K𝐾Kitalic_K has Ramsey property (or it is a Ramsey class).

We say that K𝐾Kitalic_K has the (A,B)𝐴𝐵(A,B)( italic_A , italic_B )-Ramsey property if for every k∈ω𝑘𝜔k\in\omegaitalic_k ∈ italic_ω there is C∈K𝐶𝐾C\in Kitalic_C ∈ italic_K such that C→(B)kA→𝐶subscriptsuperscript𝐵𝐴𝑘C\to(B)^{A}_{k}italic_C → ( italic_B ) start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT.

Then both K𝐾Kitalic_K and K∗superscript𝐾K^{*}italic_K start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT have Ramsey property.

D

By Proposition 4.1, Br1⁡(X)=Br⁡(ℚ)subscriptBr1𝑋Brℚ\operatorname{Br}_{1}(X)=\operatorname{Br}({\mathbb{Q}})roman_Br start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_X ) = roman_Br ( blackboard_Q ) and therefore there is no algebraic Brauer-Manin obstruction to weak approximation on X𝑋Xitalic_X. By (1),

(Br(X)/Br1(X))ℓ∞=Br(X)ℓ∞/Br1(X)ℓ∞.(\operatorname{Br}(X)/\operatorname{Br}_{1}(X))_{\ell^{\infty}}=\operatorname{%

Br(X)/Br(ℚ)=Br(E×E)/Br1(E×E)=Br(E×E)3/Br1(E×E)3=(ℤ/3)η≅ℤ/3\operatorname{Br}(X)/\operatorname{Br}(\mathbb{Q})=\operatorname{Br}(E\times E%

Br(X)/Br(ℚ)=Br(X)3/Br(ℚ)3=Br(E×E)3/Br1(E×E)3≅ℤ/3.\operatorname{Br}(X)/\operatorname{Br}({\mathbb{Q}})=\operatorname{Br}(X)_{3}/%

Br(X)/Br(ℚ)=Br(X)3/Br1(X)3=Br(Ec×Ec)3/Br1(Ec×Ec)3.\operatorname{Br}(X)/\operatorname{Br}({\mathbb{Q}})=\operatorname{Br}(X)_{3}/%

D

Conversely, all the above cases of r𝑟ritalic_r define a Demushkin group with the appropriate invariants.

The proof may be found in Appendix B, and is obtained by showing that k𝑘kitalic_k being formally real is in all cases incompatible with K𝐾Kitalic_K being Demushkin. Because the proof proceeds by using the ‘norm-combinatorics’ machinery developed in the next section, the reader is recommended to leave the verification of this Proposition to the end. One may reasonably ask whether a group theoretic proof can be given of Proposition 13, for example by proving that Gk⁢(2)subscript𝐺𝑘2G_{k}(2)italic_G start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( 2 ) has to be Demushkin. It is possible to identify several potential structures of Gk⁢(2)subscript𝐺𝑘2G_{k}(2)italic_G start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( 2 ) by (i) using the arithmetic information (existence of orderings/valuations) to identify the potential Witt rings of k𝑘kitalic_k and (ii) referring to Tables 5.2 and 5.3 of [JW89], which give the isomorphism type of Gk⁢(2)subscript𝐺𝑘2G_{k}(2)italic_G start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( 2 ) for each such Witt ring. However, in all such cases the authors considered, the abstract group Gk⁢(2)subscript𝐺𝑘2G_{k}(2)italic_G start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( 2 ) obtained does occur as a quotient of GK⁢(2)subscript𝐺𝐾2G_{K}(2)italic_G start_POSTSUBSCRIPT italic_K end_POSTSUBSCRIPT ( 2 ), suggesting that Case B cannot be straightforwardly ruled out by purely group-theoretic arguments. It seems that some form of direct appeal to the field structure is necessary.

Given k=K∩ℚ¯𝑘𝐾¯ℚk=K\cap\overline{\mathbb{Q}}italic_k = italic_K ∩ over¯ start_ARG blackboard_Q end_ARG, one of the following cases holds:

Details and proofs of all cases are given in [Lab67], with the three cases described in the Introduction.

Conversely, all the above cases of r𝑟ritalic_r define a Demushkin group with the appropriate invariants.

C

\varepsilon/4blackboard_P { italic_Z start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( roman_Φ start_POSTSUBSCRIPT ( italic_ε / 2 ) ( italic_F + italic_h italic_G ) end_POSTSUBSCRIPT | start_POSTSUBSCRIPT [ italic_S ( italic_K ) ] end_POSTSUBSCRIPT ) ≠ ∅ } ≥ italic_ε / 4.

given h∈[0,ε2]ℎ0superscript𝜀2h\in[0,\varepsilon^{2}]italic_h ∈ [ 0 , italic_ε start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ] we have almost surely

h∈[0,ε2/2]ℎ0superscript𝜀22h\in[0,\varepsilon^{2}/2]italic_h ∈ [ 0 , italic_ε start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 ].

For 0≤h≤ε2/20ℎsuperscript𝜀220\leq h\leq\varepsilon^{2}/20 ≤ italic_h ≤ italic_ε start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2, we have

Suppose 0≤h≤ε2/20ℎsuperscript𝜀220\leq h\leq\varepsilon^{2}/20 ≤ italic_h ≤ italic_ε start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2.

C

Considering i.i.d. offspring (and hence independent of the parent’s position), and i.i.d. displacement, Harris [20] conjectured that the distribution of the descendants’ position of the nth generation approaches Gaussian distribution. This conjecture was proved by [32], [23]. Its extended generation-dependent versions, where offspring and displacement distribution are dependent on generation n𝑛nitalic_n, were proved by [12], [24], to mention a few. Problems where offspring and displacement are dependent of parents’ positions are studied in adhoc approaches though (e.g. [34]). Concerning the deviation from the expectation, [9] proved that for any δ>0𝛿0\delta>0italic_δ > 0 the number of particles locating beyond μ−δ𝜇𝛿\mu-\deltaitalic_μ - italic_δ is zero almost surely, μ𝜇\muitalic_μ denoting the scaled expectation. In another direction of estimating the rareness of the large deviation, extremum is well studied (e.g. [16] [29], [5], [13] [22], [1] [2] ). [16] showed tightness for Mn−E⁢Mnsubscript𝑀𝑛𝐸subscript𝑀𝑛M_{n}-EM_{n}italic_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT - italic_E italic_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT, where Mnsubscript𝑀𝑛M_{n}italic_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT is minimum of n⁢t⁢h𝑛𝑡ℎnthitalic_n italic_t italic_h generation. [29] gave probability bound for the deviation of Mnsubscript𝑀𝑛M_{n}italic_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT, i.e. P⁢r⁢(Mn−M⁢e⁢dn>x)<e−δ⁢x𝑃𝑟subscript𝑀𝑛𝑀𝑒subscript𝑑𝑛𝑥superscript𝑒𝛿𝑥Pr(M_{n}-Med_{n}>x)<e^{-\delta x}italic_P italic_r ( italic_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT - italic_M italic_e italic_d start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT > italic_x ) < italic_e start_POSTSUPERSCRIPT - italic_δ italic_x end_POSTSUPERSCRIPT, where M⁢e⁢dn𝑀𝑒subscript𝑑𝑛Med_{n}italic_M italic_e italic_d start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT is the median of Mnsubscript𝑀𝑛M_{n}italic_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT. So far probability bound for deviation ϵitalic-ϵ\epsilonitalic_ϵ of appropriate scale (ϵ>0italic-ϵ0\epsilon>0italic_ϵ > 0) is established. Little is known as to how sparse the population is at a distance of O⁢(n)𝑂𝑛O(\sqrt{n})italic_O ( square-root start_ARG italic_n end_ARG ) far from the mean in terms of concentration inequalities.

• In CSP (constraints satisfaction problem) (e.g. K-SAT [26]), enumerating the whole problem instances can be formulated as a branching random walk. In this BRW the forest has only one tree because every tree is the same; misubscript𝑚𝑖m_{i}italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT is not random given birth place, and in addition branching factor is large (say (2⁢n)ksuperscript2𝑛𝑘(2n)^{k}( 2 italic_n ) start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT). Because of dependence of branching and position, the BRW can not be reduced to random walk; (2.1) can not be reduced to (2.2). The concentration inequalities for this BRW are developed in a separate paper ([27]).

• As far as the stochastic behavior at generation n𝑛nitalic_n is concerned, random walk can be viewed as a special BRW (with branching factor = 1), or random walk is a special BRW where birth-rate is independent of birth-place.

Throughout we consider BRW on 𝐍𝐍\mathbf{N}bold_N for notational simplicity, though the argument applies to 𝐑𝐑\mathbf{R}bold_R. We start by a new setting for BRW, which is more general in three respects. First, in our framework both subcritical and supercritical BRW are treated equally. While traditionally subcritical BRW is considered trivial because a branching random walk process almost surely ends with zero population when n𝑛nitalic_n is large, we see that the probability space for the survival BRW processes is well defined with infinitely large number of ancestors. Second, the underlying random walks between siblings are not assumed to be independent. Third, branching factor (birth rate) is not assumed identical across generations and siblings. The only major requirement is the independence between birth-rate and birth-place.

In the literature, a branching random walk on the real line is described as follows (see, for example, [11] [8] [9]). In generation zero, an initial particle at the origin on the real line 𝐑𝐑\mathbf{R}bold_R. It splits into a random number of child particles who form generation one. The children’s displacements, relative to their parent, correspond to a point process on 𝐑𝐑\mathbf{R}bold_R. The children in turn split too to form the second generation, and so on. If the average split number (branching factor) is greater than one, with positive probability the number of the descendants grows exponentially through generations. Current BRW studies typically address models where the offspring’s behavior is independent of that of their previous generation (e.g. [8], [9] [10], [22], [19]). The law of large numbers and central limit theorem type results about the distribution of position are established, under some conditions of independence (for instance, i.i.d of branching and walking). Yet unlike random walk, Chernoff bound is not known so far even in the case of i.i.d aforementioned, while the minimal (and maximal) is studied by many.

C

\nolimits RF)( roman_sing italic_R italic_D : roman_sing italic_R italic_D : roman_sing italic_R italic_F ).

We evaluate the left side of (2) first. From right triangle R⁢F⁢G𝑅𝐹𝐺RFGitalic_R italic_F italic_G, we have

Now we must replace R⁢F𝑅𝐹RFitalic_R italic_F with R⁢C𝑅𝐶RCitalic_R italic_C in the integral of the left-hand side of (3) in order to have the single unknown R⁢C𝑅𝐶RCitalic_R italic_C on both sides. Since R⁢F=C⁢F−R⁢C𝑅𝐹𝐶𝐹𝑅𝐶RF=CF-RCitalic_R italic_F = italic_C italic_F - italic_R italic_C, we have

Conversion of the homogeneous coordinates is more complicated. We first need the coordinates of the vertices of the triangle of reference in the embedded space. We then find the equations of the three planes that pass through the edges of the reference triangle and the center of the sphere. Next we translate each plane in the direction of its normal (positive is in the direction of the remaining vertex of the reference triangle) by the amount of the corresponding homogeneous coordinate. These planes will intersect at a point, which we can project onto the sphere from its center.

Now let us deal with the integral on the right-hand side of (2). We no longer have the luxury of a right triangle. But we do know that

A

\omega+51].[ 7 italic_ω + 24 , 48 italic_ω - 83 , - 72 italic_ω + 117 ] and [ 3 italic_ω + 14 , - 12 italic_ω + 49 , - 24 italic_ω + 51 ] .

 23⋅113⋅31⋅X3⋅superscript23superscript11331superscript𝑋3\displaystyle\;2^{3}\cdot 11^{3}\cdot 31\cdot X^{3}2 start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ⋅ 11 start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ⋅ 31 ⋅ italic_X start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT

X3+(−1520.81864577885822782322+358.629756234205144714067⁢i)⁢X2superscript𝑋31520.81864577885822782322358.629756234205144714067𝑖superscript𝑋2\displaystyle\,X^{3}+(-1520.81864577885822782322+358.629756234205144714067i)X^%

192⋅592⋅X4+(1381745663216332313130⁢ωr−3547293859211493542130)⁢X3⋅superscript192superscript592superscript𝑋41381745663216332313130subscript𝜔𝑟3547293859211493542130superscript𝑋3\displaystyle 19^{2}\cdot 59^{2}\cdot X^{4}+(1381745663216332313130\omega_{r}-%

{2}italic_X start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT + ( - 1520.81864577885822782322 + 358.629756234205144714067 italic_i ) italic_X start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT

B

In [2] Elder introduced the loop shortening and basepoint loop shortening properties as natural generalisations of the falsification by fellow traveller property (FFTP). Where FFTP gives a simple way to check if a word is a geodesic, each of the loop shortening properties gives a somewhat simple way to check if a word represents the identity in the group.

The properties considered in this article require us to interpret the Cayley graph ΓΓ\Gammaroman_Γ as a metric space, which we denote Γ~~Γ\tilde{\Gamma}over~ start_ARG roman_Γ end_ARG. Edges are then considered to be segments of length 1 between the vertices that they join and the points in this metric space are the vertices of ΓΓ\Gammaroman_Γ as well the interior points of the edges. Assuming ΓΓ\Gammaroman_Γ is a connected graph, any two points u,v𝑢𝑣u,vitalic_u , italic_v in Γ~~Γ\tilde{\Gamma}over~ start_ARG roman_Γ end_ARG are joined by at least one continuous path. We define the distance d⁢(u,v)𝑑𝑢𝑣d(u,v)italic_d ( italic_u , italic_v ) between u𝑢uitalic_u and v𝑣vitalic_v to be the length of the shortest such path. This is sometimes called the path metric. Since the elements of the underlying group G𝐺Gitalic_G are precisely the vertices of ΓΓ\Gammaroman_Γ, these are also points in the metric space Γ~~Γ\tilde{\Gamma}over~ start_ARG roman_Γ end_ARG. In particular the identity 1111 of G𝐺Gitalic_G is a point in Γ~~Γ\tilde{\Gamma}over~ start_ARG roman_Γ end_ARG. For convenience, we write |z|=d⁢(1,z)𝑧𝑑1𝑧|z|=d(1,z)| italic_z | = italic_d ( 1 , italic_z ) for z∈Γ~𝑧~Γz\in\tilde{\Gamma}italic_z ∈ over~ start_ARG roman_Γ end_ARG. For g∈G𝑔𝐺g\in Gitalic_g ∈ italic_G, the value |g|𝑔|g|| italic_g | can alternatively be seen as the length of the shortest word representing g𝑔gitalic_g over the generating set. By the symmetry of the Cayley graph, we have d⁢(g1,g2)=d⁢|g1−1⁢g2|𝑑subscript𝑔1subscript𝑔2𝑑superscriptsubscript𝑔11subscript𝑔2d(g_{1},g_{2})=d|g_{1}^{-1}g_{2}|italic_d ( italic_g start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_g start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) = italic_d | italic_g start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_g start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT | for g1,g2∈Gsubscript𝑔1subscript𝑔2𝐺g_{1},g_{2}\in Gitalic_g start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_g start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ∈ italic_G.

Strictly speaking the paths in Γ~~Γ\tilde{\Gamma}over~ start_ARG roman_Γ end_ARG are continuous objects, however since we will only be interested in paths that are in some sense minimal, we can safely assume that all paths considered in this article are made up of a sequence of joined edges, rather than occasionally backtracking in the middle of an edge. The paths can then be defined by the starting point and the word over the generating set representing the path. The Cayley graphs that we consider have no double edges, so can alternatively describe paths between vertices in Γ~~Γ\tilde{\Gamma}over~ start_ARG roman_Γ end_ARG by the sequence of adjacent vertices they pass through. The restriction that a path lies inside the ball B⁢(r)𝐵𝑟B(r)italic_B ( italic_r ) is equivalent to these vertices all lying inside B⁢(r)𝐵𝑟B(r)italic_B ( italic_r ) and no two adjacent vertices in the path both lying on S⁢p⁢h⁢(r)𝑆𝑝ℎ𝑟Sph(r)italic_S italic_p italic_h ( italic_r ), as then the edge between them would be outside B⁢(r)𝐵𝑟B(r)italic_B ( italic_r ). For our purposes this distinction is not important as the Cayley graph ΓΓ\Gammaroman_Γ that we consider is bipartite, so no two adjacent vertices could simultaneously lie on S⁢p⁢h⁢(r)𝑆𝑝ℎ𝑟Sph(r)italic_S italic_p italic_h ( italic_r ).

Note that the only difference between these two properties is that for the basepoint loop shortening property, the initial loop is around a basepoint which the shorter loop has to pass through, whereas for the loop shortening property no such restriction is imposed. Hence, it is clear that

We define these loop shortening properties below. For convenience we describe the loops only by the vertices that they pass through. As discussed earlier this uniquely defines the loops in our cases as each pair of adjacent vertices in the graph we consider is joined by only one edge. Even without this assumption the definition would make sense if a loop of vertices is considered to mean a sequence of vertices where each is adjacent to the previous, and the final vertex in the sequence is the same as the starting vertex.

D

Note that Corollary 1.6 follows from Theorem 1.5, since any periodic orbit is a preperiodic one and any singularity σ𝜎\sigmaitalic_σ is accumulated by regular orbits, it cannot present ind⁡(σ)<ind⁡𝒥ind𝜎ind𝒥\operatorname{ind}(\sigma)<\operatorname{ind}{\mathcal{J}}roman_ind ( italic_σ ) < roman_ind caligraphic_J. Indeed, X∈𝔛1⁢(M)𝑋superscript𝔛1𝑀X\in\mathfrak{X}^{1}(M)italic_X ∈ fraktur_X start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ( italic_M ), 𝒥𝒥\mathcal{J}caligraphic_J is a continuous field of quadratic forms and X𝑋Xitalic_X is 𝒥𝒥\mathcal{J}caligraphic_J- monotonic over any compact invariant nonsingular set ΓΓ\Gammaroman_Γ.⋄⋄\diamond⋄

Now, Birkhoff’s ergodic theorem and Corollary 2.8 imply that the Lyapunov exponents on E𝐸Eitalic_E are negative and the sectional Lyapunov exponents are positive, in a total probability subset of ΛΛ\Lambdaroman_Λ.

Before to proceed the proofs, we need to recall some results related to the theory of the Extended Linear Poincaré Flow from Gan, Li and Wen [16].

To proceed the demonstration of Theorem 1.7, we recall some definitions which are necessary from now.

The definitions concerning the quadratic forms are given in the next section, precisely in Definition 6 and remarks just after this.

C

\gamma\cdot\sum_{\sigma\in R_{k^{*}}}diam(J_{\sigma}).italic_d italic_i italic_a italic_m ( italic_B ) ≥ italic_ρ start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT ( { italic_J start_POSTSUBSCRIPT italic_σ ∗ italic_i end_POSTSUBSCRIPT : italic_σ ∈ italic_R start_POSTSUBSCRIPT italic_k start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT , italic_i = 1 , 2 , … italic_m } ) ≥ italic_γ ⋅ ∑ start_POSTSUBSCRIPT italic_σ ∈ italic_R start_POSTSUBSCRIPT italic_k start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_d italic_i italic_a italic_m ( italic_J start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ) .

Let 𝒳𝒳\mathcal{X}caligraphic_X be a collection of compact subsets of X𝑋Xitalic_X and let 𝒞⁢(𝒳)𝒞𝒳\mathcal{C}(\mathcal{X})caligraphic_C ( caligraphic_X ) denote the collection of all compressions on 𝒳𝒳\mathcal{X}caligraphic_X.

Let 𝒳𝒳\mathcal{X}caligraphic_X, D𝐷Ditalic_D, M𝑀Mitalic_M, and ℱℱ\mathcal{F}caligraphic_F be as in section §5.3.

In this section, we fix an m𝑚mitalic_m-ary tree D𝐷Ditalic_D, a parameterization space M𝑀Mitalic_M, a collection 𝒳𝒳\mathcal{X}caligraphic_X of compact subsets of a metric space X𝑋Xitalic_X, and a marking ℱℱ\mathcal{F}caligraphic_F of M𝑀Mitalic_M into 𝒞⁢(𝒳)m𝒞superscript𝒳𝑚\mathcal{C}(\mathcal{X})^{m}caligraphic_C ( caligraphic_X ) start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT.

Let D𝐷Ditalic_D be an m𝑚mitalic_m-ary tree, let 𝒳𝒳\mathcal{X}caligraphic_X be a collection of compact subsets of X𝑋Xitalic_X, and let 𝒞⁢(𝒳)𝒞𝒳\mathcal{C}(\mathcal{X})caligraphic_C ( caligraphic_X ) denote the collection of all compressions on 𝒳𝒳\mathcal{X}caligraphic_X.

B

Note the inverse relationship between the Lipschitz constant L𝐿Litalic_L and the step-size 1/L1𝐿1/L1 / italic_L, which is characteristic of results that rely on a Lipschitz property of the gradient for non-asymptotic analysis.

A comparison of dsgd with gradient descent and variants of gradient descent using several different norms.

The Stochastic Variance Reduced Gradient (SVRG) algorithm combines features of deterministic and stochastic gradient descent, alternating between full gradient calculations and sgd iterations [Johnson and Zhang, 2013]. Notably, it was shown that SVRG for non-convex functions requires fewer gradient evaluations on average compared to RSG [Allen-Zhu and Hazan, 2016] [Reddi et al., 2016]. The step-sizes follow a 1/L1𝐿1/L1 / italic_L rule, and the variance assumptions are weaker compared to RSG. For machine learning on a large scale, distributed and decentralized algorithms become of interest. Decentralized sgd was analyzed in [Lian et al., 2017], leading to a 1/L1𝐿1/L1 / italic_L-type result for this setting.

In [Ghadimi and Lan, 2013] it was proved that the expected squared-norm of the returned gradient tends to zero at rate of 1/T1𝑇1/\sqrt{T}1 / square-root start_ARG italic_T end_ARG.

Most practical algorithms in machine learning are stochastic variants of gradient descent. The Randomized Stochastic Gradient (RSG) algorithm is one such example [Ghadimi and Lan, 2013]. In RSG, a stochastic gradient update is run for T𝑇Titalic_T steps, and then a random iterate is returned.

D

\mathbb{C}}([0,1])\widehat{\otimes}_{\pi}X^{*})^{*}=L(L_{\infty}([0,1]),X^{**})( italic_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT blackboard_C end_POSTSUPERSCRIPT ( [ 0 , 1 ] ) over^ start_ARG ⊗ end_ARG start_POSTSUBSCRIPT italic_ε end_POSTSUBSCRIPT italic_X ) start_POSTSUPERSCRIPT ∗ ∗ end_POSTSUPERSCRIPT = ( italic_L start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT blackboard_C end_POSTSUPERSCRIPT ( [ 0 , 1 ] ) over^ start_ARG ⊗ end_ARG start_POSTSUBSCRIPT italic_π end_POSTSUBSCRIPT italic_X start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT = italic_L ( italic_L start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT ( [ 0 , 1 ] ) , italic_X start_POSTSUPERSCRIPT ∗ ∗ end_POSTSUPERSCRIPT ) fails the SD2P too (c.f. e.g. [14, Proposition 2.14]). However, L∞ℂ⁢([0,1])superscriptsubscript𝐿ℂ01L_{\infty}^{\mathbb{C}}([0,1])italic_L start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT blackboard_C end_POSTSUPERSCRIPT ( [ 0 , 1 ] ) has the SD2P because L∞ℂ⁢([0,1])superscriptsubscript𝐿ℂ01L_{\infty}^{\mathbb{C}}([0,1])italic_L start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT blackboard_C end_POSTSUPERSCRIPT ( [ 0 , 1 ] ) has the Daugavet property (see e.g. [21, Section 2, Examples (b)]) and Daugavet spaces have SD2P (c.f. e.g. [1, Theorem 4.4]).

x2:=(0,12,−12,i2,−i2,1,1,1,1,1)assignsubscript𝑥201212𝑖2𝑖211111x_{2}:=(0,\frac{1}{2},-\frac{1}{2},\frac{i}{2},-\frac{i}{2},1,1,1,1,1)italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT := ( 0 , divide start_ARG 1 end_ARG start_ARG 2 end_ARG , - divide start_ARG 1 end_ARG start_ARG 2 end_ARG , divide start_ARG italic_i end_ARG start_ARG 2 end_ARG , - divide start_ARG italic_i end_ARG start_ARG 2 end_ARG , 1 , 1 , 1 , 1 , 1 ). Then L1⁢⊗^ε⁢Xsubscript𝐿1subscript^tensor-product𝜀𝑋L_{1}\widehat{\otimes}_{\varepsilon}Xitalic_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT over^ start_ARG ⊗ end_ARG start_POSTSUBSCRIPT italic_ε end_POSTSUBSCRIPT italic_X identifies with the space of 10101010-tuples (f1,f2,…,f10)subscript𝑓1subscript𝑓2…subscript𝑓10(f_{1},f_{2},\ldots,f_{10})( italic_f start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_f start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , … , italic_f start_POSTSUBSCRIPT 10 end_POSTSUBSCRIPT ), with fi∈L1subscript𝑓𝑖subscript𝐿1f_{i}\in L_{1}italic_f start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∈ italic_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT, of the form g1⊗x1+g2⊗x2tensor-productsubscript𝑔1subscript𝑥1tensor-productsubscript𝑔2subscript𝑥2g_{1}\otimes x_{1}+g_{2}\otimes x_{2}italic_g start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ⊗ italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_g start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ⊗ italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT and with the norm ‖f‖:=max1⩽k⩽10⁡‖fk‖assignnorm𝑓subscript1𝑘10normsubscript𝑓𝑘\|f\|:=\max\limits_{1\leqslant k\leqslant 10}\|f_{k}\|∥ italic_f ∥ := roman_max start_POSTSUBSCRIPT 1 ⩽ italic_k ⩽ 10 end_POSTSUBSCRIPT ∥ italic_f start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∥ (this identification comes from the identification L1⁢⊗^ε⁢ℓ∞10=ℓ∞10⁢(L1)subscript𝐿1subscript^tensor-product𝜀superscriptsubscriptℓ10superscriptsubscriptℓ10subscript𝐿1L_{1}\widehat{\otimes}_{\varepsilon}\ell_{\infty}^{10}=\ell_{\infty}^{10}(L_{1})italic_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT over^ start_ARG ⊗ end_ARG start_POSTSUBSCRIPT italic_ε end_POSTSUBSCRIPT roman_ℓ start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 10 end_POSTSUPERSCRIPT = roman_ℓ start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 10 end_POSTSUPERSCRIPT ( italic_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) [20, Chapter 3.2] and by the well known fact that the injective tensor product respects subspaces isometrically [20, Proposition 3.2]).

Given two Banach spaces X𝑋Xitalic_X and Y𝑌Yitalic_Y we will denote by L⁢(X,Y)𝐿𝑋𝑌L(X,Y)italic_L ( italic_X , italic_Y ) (resp. K⁢(X,Y)𝐾𝑋𝑌K(X,Y)italic_K ( italic_X , italic_Y )) the space of all linear and bounded (resp. compact linear) operators from X𝑋Xitalic_X to Y𝑌Yitalic_Y, and we will denote by X⁢⊗^π⁢Y𝑋subscript^tensor-product𝜋𝑌X\widehat{\otimes}_{\pi}Yitalic_X over^ start_ARG ⊗ end_ARG start_POSTSUBSCRIPT italic_π end_POSTSUBSCRIPT italic_Y and X⁢⊗^ε⁢Y𝑋subscript^tensor-product𝜀𝑌X\widehat{\otimes}_{\varepsilon}Yitalic_X over^ start_ARG ⊗ end_ARG start_POSTSUBSCRIPT italic_ε end_POSTSUBSCRIPT italic_Y the projective and the injective tensor product of X𝑋Xitalic_X and Y𝑌Yitalic_Y. See [20] for a detailed treatment of the tensor product theory.

For the notation section we consider real or complex Banach spaces. Given a Banach space X𝑋Xitalic_X, we denote by BXsubscript𝐵𝑋B_{X}italic_B start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT and SXsubscript𝑆𝑋S_{X}italic_S start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT the closed unit ball and the closed unit sphere of X𝑋Xitalic_X. We also denote by X∗superscript𝑋X^{*}italic_X start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT the topological dual of X𝑋Xitalic_X.

This work was supported by MCIN/AEI/10.13039/501100011033: grant PID2021-122126NB-C31, Junta de Andalucía: grant FQM-0185, by Fundación Séneca: ACyT Región de Murcia: grant 21955/PI/22 and by Generalitat Valenciana: grant CIGE/2022/97.

D

Define uhℍsuperscriptsubscript𝑢ℎℍu_{h}^{\mathbb{H}}italic_u start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT start_POSTSUPERSCRIPT blackboard_H end_POSTSUPERSCRIPT by (3) and let uhm⁢s,j=T⁢(I−Pj)⁢λHjsubscriptsuperscript𝑢𝑚𝑠𝑗ℎ𝑇𝐼superscript𝑃𝑗superscriptsubscript𝜆𝐻𝑗u^{ms,j}_{h}=T(I-P^{j})\lambda_{H}^{j}italic_u start_POSTSUPERSCRIPT italic_m italic_s , italic_j end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT = italic_T ( italic_I - italic_P start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT ) italic_λ start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT, where λHjsuperscriptsubscript𝜆𝐻𝑗\lambda_{H}^{j}italic_λ start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT is as in (15). Then

|uhm⁢s−uhm⁢s,j|H𝒜1⁢(𝒯H)2≤|uhm⁢s−u^hm⁢s,j|H𝒜1⁢(𝒯H)2≤(cγ⁢j)2⁢(9⁢α)2⁢e−j−2(1+9⁢α)⁢|T⁢IH⁢λhm⁢s|H𝒜1⁢(𝒯H)2.superscriptsubscriptsuperscriptsubscript𝑢ℎ𝑚𝑠superscriptsubscript𝑢ℎ𝑚𝑠𝑗superscriptsubscript𝐻𝒜1subscript𝒯𝐻2superscriptsubscriptsuperscriptsubscript𝑢ℎ𝑚𝑠subscriptsuperscript^𝑢𝑚𝑠𝑗ℎsuperscriptsubscript𝐻𝒜1subscript𝒯𝐻2superscriptsubscript𝑐𝛾𝑗2superscript9𝛼2superscript𝑒𝑗219𝛼superscriptsubscript𝑇subscript𝐼𝐻superscriptsubscript𝜆ℎ𝑚𝑠superscriptsubscript𝐻𝒜1subscript𝒯𝐻2|u_{h}^{ms}-u_{h}^{ms,j}|_{H_{\mathcal{A}}^{1}({\mathcal{T}_{H}})}^{2}\leq|u_{%

|uhℍ−uhm⁢s|H𝒜1⁢(Ω)≤cP,L⁢H⁢‖g‖Lρ2⁢(Ω),subscriptsuperscriptsubscript𝑢ℎℍsuperscriptsubscript𝑢ℎ𝑚𝑠superscriptsubscript𝐻𝒜1Ωsubscript𝑐𝑃𝐿𝐻subscriptnorm𝑔subscriptsuperscript𝐿2𝜌Ω|u_{h}^{\mathbb{H}}-u_{h}^{ms}|_{H_{\mathcal{A}}^{1}(\Omega)}\leq c_{P,L}H\|g%

|uhℍ−um⁢s,j|H𝒜1⁢(𝒯H)≤H⁢{cP,L+[γ2⁢κ⁢(1+log⁡(H/h))]1/2⁢cγ⁢j2⁢9⁢α⁢e−([j−22⁢(1+9⁢α)−log(cP,G/H))}⁢‖g‖Lρ2⁢(Ω).|u_{h}^{\mathbb{H}}-u^{ms,j}|_{H_{\mathcal{A}}^{1}({\mathcal{T}_{H}})}\leq H%

|T⁢ψ~h|H𝒜1⁢(𝒯H)2≤9⁢α⁢e−j−22⁢(1+9⁢α)⁢∑K∈𝒯H|T⁢ψ~h|H𝒜1⁢(𝒯j+1⁢(K))⁢|T⁢PK⁢νh|H𝒜1⁢(𝒯H)≤9⁢α⁢e−j−22⁢(1+9⁢α)⁢cγ⁢j⁢|T⁢ψ~h|H𝒜1⁢(𝒯H)⁢(∑K∈𝒯H|T⁢PK⁢νh|H𝒜1⁢(𝒯H)2)1/2.superscriptsubscript𝑇subscript~𝜓ℎsuperscriptsubscript𝐻𝒜1subscript𝒯𝐻29𝛼superscript𝑒𝑗2219𝛼subscript𝐾subscript𝒯𝐻subscript𝑇subscript~𝜓ℎsuperscriptsubscript𝐻𝒜1subscript𝒯𝑗1𝐾subscript𝑇superscript𝑃𝐾subscript𝜈ℎsuperscriptsubscript𝐻𝒜1subscript𝒯𝐻9𝛼superscript𝑒𝑗2219𝛼subscript𝑐𝛾𝑗subscript𝑇subscript~𝜓ℎsuperscriptsubscript𝐻𝒜1subscript𝒯𝐻superscriptsubscript𝐾subscript𝒯𝐻superscriptsubscript𝑇superscript𝑃𝐾subscript𝜈ℎsuperscriptsubscript𝐻𝒜1subscript𝒯𝐻212|T\tilde{\psi}_{h}|_{H_{\mathcal{A}}^{1}({\mathcal{T}_{H}})}^{2}\leq 9\alpha e%

C

[37, 38, 43], the change-point model to a latent transition model [8, 17, 12, 44, 6, 18], and the treatments to educational items, such as formative assessment, practice, intervention.

We first assume that the change-point and response models are completely known. We obtain the optimal solution under Markovian change-point models via dynamic programming approach, which however does not admit an explicit assignment rule and whose computation can be demanding. Thus, we propose an alternative, flexible procedure, and we establish its asymptotic optimality for a large class of (not necessarily Markovian) change-point models. Further, we extend the proposed procedure to the setup where the models have unknown parameters with a given joint prior distribution. For future work, it would be interesting to study the acceleration of multiple changes, which in the educational setup corresponds to the mastery of multiple skills [1, 47],

we consider the memoryless model with K=3𝐾3K=3italic_K = 3 treatments, as in the previous subsection, but now both the response model and the change-point models have unknown parameters with a given prior distribution.

These references consider multiple latent attributes (i.e., skills) and estimate the model parameters based on data from many users. In the present work, we assume that both models have been calibrated offline, and we focus on the real-time instruction of a single attribute to a single user. Note that the online instruction is individualized since the developed models consider subject-specific variables that capture the characteristics of users. This separation of offline model estimation and online decision-making is realistic and practical, as

Until Section 7, we assume that both the response and the change-point models are completely specified. This may be considered a realistic assumption when we have access to a large amount of historical data and the offline estimation has negligible variance. In Section 7, we consider the case that both models have unknown parameters, which are random with a given joint prior distribution.

C

_{\lambda}(f)\sigma_{\lambda}[z,t]^{*}\big{)}|\lambda|^{n}\,d\lambda\,.italic_f [ italic_z , italic_t ] = divide start_ARG 1 end_ARG start_ARG ( 2 italic_π ) start_POSTSUPERSCRIPT italic_n + 1 end_POSTSUPERSCRIPT end_ARG ∫ start_POSTSUBSCRIPT blackboard_R end_POSTSUBSCRIPT roman_tr ( italic_σ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ( italic_f ) italic_σ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT [ italic_z , italic_t ] start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ) | italic_λ | start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT italic_d italic_λ .

have that F∈Hol⁡(𝒰)𝐹Hol𝒰F\in\operatorname{Hol}({\mathcal{U}})italic_F ∈ roman_Hol ( caligraphic_U ).

𝒟(m)={F∈Hol(𝒰):\displaystyle{\mathcal{D}}_{(m)}=\Big{\{}F\in\operatorname{Hol}({\mathcal{U}}):caligraphic_D start_POSTSUBSCRIPT ( italic_m ) end_POSTSUBSCRIPT = { italic_F ∈ roman_Hol ( caligraphic_U ) :

ℋm={F∈Hol⁡(𝒰¯):∂ζαF∈H2,|α|≤m}.subscriptℋ𝑚conditional-set𝐹Hol¯𝒰formulae-sequencesuperscriptsubscript𝜁𝛼𝐹superscript𝐻2𝛼𝑚{\mathcal{H}}_{m}=\big{\{}F\in\operatorname{Hol}(\overline{\mathcal{U}}):\,%

For F∈Hol⁡(𝒰)𝐹Hol𝒰F\in\operatorname{Hol}({\mathcal{U}})italic_F ∈ roman_Hol ( caligraphic_U ), recalling (4), we write

D

Here, we introduce several common definitions and statements that will be used in the rest of the paper.

In Appendix 6.1 we briefly outline the construction of the polyhedron Ppsubscript𝑃𝑝P_{p}italic_P start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT in [5]. In Appendix 6.2 we give an implicit proof why there must exist (some) polyhedron Qpsubscript𝑄𝑝Q_{p}italic_Q start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT suitable for our purposes in place of Ppsubscript𝑃𝑝P_{p}italic_P start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT.

The anonymous referees (for both this paper and [2]) asked for more explanations on the structure of the polyhedron Pm⊂Fm⁢(ℝ2)subscript𝑃𝑚subscript𝐹𝑚superscriptℝ2P_{m}\subset F_{m}(\mathbb{R}^{2})italic_P start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ⊂ italic_F start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ). We encourage the interested readers to read [5] (having in mind the case d=2𝑑2d=2italic_d = 2 there for simplicity), but we also outline the main idea under the construction of Pmsubscript𝑃𝑚P_{m}italic_P start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT there.

a prime power. In Section 2.2 we recall some common facts about the transversality of equivariant maps of polyhedra. The reader who is comfortable with these notions may skip this section. In Section 2.3 we list the properties of a certain polyhedron Ppsubscript𝑃𝑝P_{p}italic_P start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT (constructed in [5]) which is going to be used in the subsequent proofs; we also show how this polyhedron is important for the Borsuk–Ulam type result needed to complete the proof scheme in Section 2.1.

We also recommend the textbooks [12, 14, 15, 6] for general notions of PL topology. There are many definitions of a polyhedron, equivalent to each other. For our purposes it is sufficient to consider a polyhedron as a subset of the Euclidean space with certain structure.

D

If ΓΓ\Gammaroman_Γ is an LCA group, we say that K⊂Γ𝐾ΓK\subset\Gammaitalic_K ⊂ roman_Γ is a Kronecker set if for every continuous f:K→𝒮1:𝑓→𝐾superscript𝒮1f:K\to\mathcal{S}^{1}italic_f : italic_K → caligraphic_S start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT and all ε>0𝜀0\varepsilon>0italic_ε > 0, there is a χ∈Γ^𝜒^Γ\chi\in\widehat{\Gamma}italic_χ ∈ over^ start_ARG roman_Γ end_ARG such that |f⁢(x)−χ⁢(x)|<ε𝑓𝑥𝜒𝑥𝜀|f(x)-\chi(x)|<\varepsilon| italic_f ( italic_x ) - italic_χ ( italic_x ) | < italic_ε for all x∈K𝑥𝐾x\in Kitalic_x ∈ italic_K. For a fixed q∈ℕ𝑞ℕq\in\mathbb{N}italic_q ∈ blackboard_N, we say that K𝐾Kitalic_K is a Kqsubscript𝐾𝑞K_{q}italic_K start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT-set if this only holds for all continuous functions f:K→Zq:𝑓→𝐾subscript𝑍𝑞f:K\to Z_{q}italic_f : italic_K → italic_Z start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT. See Chapter 5 of [24] for exposition.

The main argument in the proof of Proposition 1.9 will use the following corollary, so we prove it now. The remainder of this section will be dedicated to the proof of Lemma 6.3.

The language of Cayley graphs is useful for describing how recurrence properties of special sets can be transferred between groups. The main result in this section is Lemma 5.6, which will be used to prove Lemma 6.7.

Section 8 constructs the special measures on G^^𝐺\widehat{G}over^ start_ARG italic_G end_ARG introduced in §3, proving Proposition 1.9.

After outlining some previous constructions in the direction of Theorem 1.3, we introduce a special class of measures in Definition 3.1. Lemma 3.4 asserts that these measures exist, and will be proved in §8. We prove Proposition 1.9 at the end of this section.

D

O⁢(1T+dT)𝑂1𝑇𝑑𝑇O\bigg{(}\frac{1}{\sqrt{T}}+\frac{d}{\sqrt{T}}\bigg{)}italic_O ( divide start_ARG 1 end_ARG start_ARG square-root start_ARG italic_T end_ARG end_ARG + divide start_ARG italic_d end_ARG start_ARG square-root start_ARG italic_T end_ARG end_ARG )

Li & Orabona (2020) presented a high probability analysis for Delayed AdaGrad algorithm with momentum in the smooth nonconvex setting.

Assumption 5.1 is commonly considered when studying high probability bounds (Li & Orabona, 2020). It is weaker than Assumption B2 in Li & Orabona (2020): for the case when ∇f⁢(𝐱,ξ)−∇f⁢(𝐱)∇𝑓𝐱𝜉∇𝑓𝐱\nabla f(\mathbf{x},\xi)-\nabla f(\mathbf{x})∇ italic_f ( bold_x , italic_ξ ) - ∇ italic_f ( bold_x ) is a standard Gaussian vector, σ2superscript𝜎2\sigma^{2}italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT defined in Li & Orabona (2020) is of order O⁢(d)𝑂𝑑O(d)italic_O ( italic_d ), while σ2=O⁢(1)superscript𝜎2𝑂1\sigma^{2}=O(1)italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = italic_O ( 1 ) in our definition.

      Li & Orabona (2020)333To be precise, Li & Orabona (2020) studies a delayed AdaGrad algorithm with momentum.

Alacaoglu et al. (2020) proposed a new framework to derive data-dependent regret bounds with a constant momentum parameter in various settings.

C

  (b) If w2subscript𝑤2w_{2}italic_w start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT is an end of some m′∈M′superscript𝑚′superscript𝑀′m^{\prime}\in M^{\prime}italic_m start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ italic_M start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT, and a cycle C𝐶Citalic_C such that v2⁢w2∈E⁢(C)subscript𝑣2subscript𝑤2𝐸𝐶v_{2}w_{2}\in E(C)italic_v start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_w start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ∈ italic_E ( italic_C ) exists in J−v2⁢w1⁢F⁢x2⁢w2𝐽subscript𝑣2subscript𝑤1𝐹subscript𝑥2subscript𝑤2J-v_{2}w_{1}Fx_{2}w_{2}italic_J - italic_v start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_w start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_F italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_w start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT, then mark v2⁢w2subscript𝑣2subscript𝑤2v_{2}w_{2}italic_v start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_w start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT and let J:=J−v2⁢w1assign𝐽𝐽subscript𝑣2subscript𝑤1J:=J-v_{2}w_{1}italic_J := italic_J - italic_v start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_w start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT.

(5) Repeat (2) - (4) until in (4) all m∈M′∪M′′𝑚superscript𝑀′superscript𝑀′′m\in M^{\prime}\cup M^{\prime\prime}italic_m ∈ italic_M start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∪ italic_M start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT are chosen.

(7) Repeat (2) - (6) until in (6) all m∈M′𝑚superscript𝑀′m\in M^{\prime}italic_m ∈ italic_M start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT are chosen.

(9) Repeat (2) - (8) until in (8) all m∈M′𝑚superscript𝑀′m\in M^{\prime}italic_m ∈ italic_M start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT are chosen.

(3) Repeat (2) until in (2) all m∈M′∪M′′𝑚superscript𝑀′superscript𝑀′′m\in M^{\prime}\cup M^{\prime\prime}italic_m ∈ italic_M start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∪ italic_M start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT are chosen.

B

},W|_{[s,t]\times\Omega};\mathbb{R}^{d})).roman_span ( caligraphic_S ( blackboard_P , blackboard_F start_POSTSUBSCRIPT italic_S end_POSTSUBSCRIPT , italic_W ; blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT ) ) ⊆ roman_span ( caligraphic_S ( blackboard_P , blackboard_F start_POSTSUBSCRIPT [ italic_S , italic_s ] ∪ [ italic_t , italic_T ] end_POSTSUBSCRIPT , italic_W | start_POSTSUBSCRIPT [ italic_s , italic_t ] × roman_Ω end_POSTSUBSCRIPT ; blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT ) ) .

The proof of Lemma A.3 is almost literally identical to the proofs of Proposition 4.2 and Proposition 4.4 in Kruse [35] and therefore omitted.

The next result, Lemma A.9, proves that if a sequence of integrals converges weakly and has uniformly bounded Skorohod integrals, then the limit is Skorohod-integrable and the sequence of Skorohod integrals of the sequence converges weakly.

In this appendix we provide sufficient results to rewrite Itô integrals as Skorohod integrals and then to write the sum of these as a single Skorohod integral.

The following result, Proposition A.7, shows how to change the domain of integration for Skorohod integrals.

D

If X𝑋Xitalic_X is a finite quandle, then the number of X𝑋Xitalic_X-colorings of D𝐷Ditalic_D is an invariant of link L𝐿Litalic_L [14].

The map C:𝒜⁢(D)⊔R⁢(D)→X:𝐶→square-union𝒜𝐷𝑅𝐷𝑋C:\mathcal{A}(D)\sqcup R(D)\to Xitalic_C : caligraphic_A ( italic_D ) ⊔ italic_R ( italic_D ) → italic_X is an X𝑋Xitalic_X-shadow coloring of D𝐷Ditalic_D [Definition.4.3 [7]], as shown in

Figure 1 for C|𝒜⁢(D):𝒜⁢(D)→X:evaluated-at𝐶𝒜𝐷→𝒜𝐷𝑋C|_{\mathcal{A}(D)}:\mathcal{A}(D)\to Xitalic_C | start_POSTSUBSCRIPT caligraphic_A ( italic_D ) end_POSTSUBSCRIPT : caligraphic_A ( italic_D ) → italic_X and in Figure 2 for C|R⁢(D):R⁢(D)→X:evaluated-at𝐶𝑅𝐷→𝑅𝐷𝑋C|_{R(D)}:R(D)\to Xitalic_C | start_POSTSUBSCRIPT italic_R ( italic_D ) end_POSTSUBSCRIPT : italic_R ( italic_D ) → italic_X.

Let R⁢(D)𝑅𝐷R(D)italic_R ( italic_D ) be the set of regions of the underlying immersed curves of 𝒜⁢(D)𝒜𝐷\mathcal{A}(D)caligraphic_A ( italic_D ).

xx\mathrm{x}roman_x be a crossing of the diagram D𝐷Ditalic_D, and 𝒜⁢(D)𝒜𝐷\mathcal{A}(D)caligraphic_A ( italic_D ) be the set of arcs of D𝐷Ditalic_D.

C

𝐁⊂U+𝐁superscript𝑈\mathbf{B}\subset U^{+}bold_B ⊂ italic_U start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT be the canonical basis of U+superscript𝑈U^{+}italic_U start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT

up to isomorphism and ℂ×superscriptℂ\mathbb{C}^{\times}blackboard_C start_POSTSUPERSCRIPT × end_POSTSUPERSCRIPT-equivariant twist in the following way

The basis {B~⁢(𝐚)∣𝐚∈KPn⁢(β)}conditional-set~𝐵𝐚𝐚subscriptKP𝑛𝛽\{\widetilde{B}(\mathbf{a})\mid\mathbf{a}\in\mathrm{KP}_{n}(\beta)\}{ over~ start_ARG italic_B end_ARG ( bold_a ) ∣ bold_a ∈ roman_KP start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_β ) } of (𝒜n)βsubscriptsubscript𝒜𝑛𝛽(\mathscr{A}_{n})_{\beta}( script_A start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_β end_POSTSUBSCRIPT is characterized by the following

(more precisely, of a certain localization of a subalgebra of its restricted dual) thanks to [KKKO18]. So it is natural to

It is characterized up to sign as the set of elements b∈U+𝑏superscript𝑈b\in U^{+}italic_b ∈ italic_U start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT

D

1049,1051,1061,1063,1069,1087,1091,1093,1097,1103,1109,11171049105110611063106910871091109310971103110911171049,1051,1061,1063,1069,1087,1091,1093,1097,1103,1109,11171049 , 1051 , 1061 , 1063 , 1069 , 1087 , 1091 , 1093 , 1097 , 1103 , 1109 , 1117,

In this paper, using both the sieve of Eratosthenes and Bertrand postulate, we obtain prime numbers in a faster way by considering [pi+1,pi2+4⁢pi+3]subscript𝑝𝑖1superscriptsubscript𝑝𝑖24subscript𝑝𝑖3[p_{i}+1,p_{i}^{2}+4p_{i}+3][ italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT + 1 , italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + 4 italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT + 3 ] as consecutive intervals where p1,p2,…subscript𝑝1subscript𝑝2…p_{1},p_{2},\ldotsitalic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_p start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , … denotes the primes 2, 3, …numbered in increasing order, i≥2𝑖2i\geq 2italic_i ≥ 2. It is a tight bound in the sense that if we increase the width of the interval still further, then the algorithm will not work, in general. We present two more methods to show the fastness of our method. In all the three methods, we obtain primes in each successive intervals of the form [pi+1,2⁢pj]subscript𝑝𝑖12subscript𝑝𝑗[p_{i}+1,2p_{j}][ italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT + 1 , 2 italic_p start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ] and existence of prime(s) in each such intervals is ensured by Bertrand postulate, j≥i≥2𝑗𝑖2j\geq i\geq 2italic_j ≥ italic_i ≥ 2. In the first method, we consider successive intervals [pi+1,2⁢pi]subscript𝑝𝑖12subscript𝑝𝑖[p_{i}+1,2p_{i}][ italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT + 1 , 2 italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ], i≥2𝑖2i\geq 2italic_i ≥ 2. In the second method, successive intervals [pi+1,pi2]subscript𝑝𝑖1superscriptsubscript𝑝𝑖2[p_{i}+1,p_{i}^{2}][ italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT + 1 , italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ] are considered, i≥2𝑖2i\geq 2italic_i ≥ 2. In the third method, we consider [pi+1,pi2+4⁢pi+3]subscript𝑝𝑖1superscriptsubscript𝑝𝑖24subscript𝑝𝑖3[p_{i}+1,p_{i}^{2}+4p_{i}+3][ italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT + 1 , italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + 4 italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT + 3 ] as consecutive intervals. The author feels that this development is going to revolutionise development in Mathematics, especially in Cryptography, Number theory, Signal processing and Computational Mathematics.

1123,1129,1151,1153,1163,1171,1181,1187,1193,1201,1213,12171123112911511153116311711181118711931201121312171123,1129,1151,1153,1163,1171,1181,1187,1193,1201,1213,12171123 , 1129 , 1151 , 1153 , 1163 , 1171 , 1181 , 1187 , 1193 , 1201 , 1213 , 1217,

The problem of computing π⁢(x)𝜋𝑥\pi(x)italic_π ( italic_x ), the number of primes less than or equal to x𝑥xitalic_x is one of the oldest problem in Mathematics, x∈ℕ𝑥ℕx\in\mathbb{N}italic_x ∈ blackboard_N. For a very long time, the sieve of Eratosthenes has been the practical way to compute π⁢(x)𝜋𝑥\pi(x)italic_π ( italic_x ) despite its time complexity. Legendre [8] observed a combinatorial formula, known as Legendre sum, for the number of primes p𝑝pitalic_p for which x12<p≤xsuperscript𝑥12𝑝𝑥x^{\frac{1}{2}}<p\leq xitalic_x start_POSTSUPERSCRIPT divide start_ARG 1 end_ARG start_ARG 2 end_ARG end_POSTSUPERSCRIPT < italic_p ≤ italic_x. Since then, a large number of writers have suggested variants and improvements of the formula. During 1870 to 1885, astronomer Meissel [10] developed practical combinatorial method to compute π⁢(x)𝜋𝑥\pi(x)italic_π ( italic_x ) and in 1959, Lehmer [8] extended and simplified Meissel’s method. In 1985, the Meissel-Lehmer method was used to compute several values of π⁢(x)𝜋𝑥\pi(x)italic_π ( italic_x ) up to x=4.1016𝑥superscript4.1016x=4.10^{16}italic_x = 4.10 start_POSTSUPERSCRIPT 16 end_POSTSUPERSCRIPT [7] and in 1996, Deleglise and Rivat [4] developed a modified form of the Meissel-Lehmer method saving much computation.

Even though the third method generates primes in a faster way and is the best, we present the other two to highlight their differences. After generating, it is easy to check, from the listing, whether any given number is prime or not. We also present Riemann zeta function [6] related to prime generation.

B

To bolster customer participation, several government authorities and regulators have launched websites and programs to educate customers on the benefits of retail choice in energy markets [13, 14, 15, 16]. Recently, a number of start-up companies emerged to capitalize on the opportunities of retail choice in energy markets by providing assistance services and online tools to automatically determine the best energy plans for customers and to offer personalized selection advice. These assistance services and online tools are integral to the success of retail choice in energy markets by automating the confusing and complex decision-making processes of energy plan selections. In the future, household PVs, batteries, and smart appliances will optimize their usage and performance in conjunction with energy plan selection. Therefore, we anticipate the importance of proper decision-making processes for energy plan selection in energy markets with retail choice.

Uncertain Future Information: To decide the best energy plan that may last for a long period, one has to estimate the future usage and fluctuation of energy tariffs. It is difficult to predict the uncertain information accurately to make the best decisions. For example, energy tariffs may depend on global energy markets. If there are rooftop PVs, their performance is conditional on unpredictable weather. The various sources of uncertainty complicate the decision-making processes of energy plan selection.

There are several challenging research questions arisen in the decision-making processes for energy plan selection:

To bolster customer participation, several government authorities and regulators have launched websites and programs to educate customers on the benefits of retail choice in energy markets [13, 14, 15, 16]. Recently, a number of start-up companies emerged to capitalize on the opportunities of retail choice in energy markets by providing assistance services and online tools to automatically determine the best energy plans for customers and to offer personalized selection advice. These assistance services and online tools are integral to the success of retail choice in energy markets by automating the confusing and complex decision-making processes of energy plan selections. In the future, household PVs, batteries, and smart appliances will optimize their usage and performance in conjunction with energy plan selection. Therefore, we anticipate the importance of proper decision-making processes for energy plan selection in energy markets with retail choice.

Assurance of Online Decision-Making: The energy plan selection decision-making processes are determined over time when new information is gradually revealed (e.g., the present demand and updated tariffs). A sequential decision-making process with a sequence of gradually revealed events is called online decision-making. The average customers are reluctant to switch to a new energy plan unless there is absolute assurance provided to their selection decisions. The online decision-making processes should incorporate proper bounds on the optimality of decisions as a metric of confidence for customers.

B

\mathscr{O}_{X})∐ start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT roman_Spec italic_D start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT roman_Γ ( italic_U start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT , italic_σ start_POSTSUBSCRIPT ∙ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT script_O start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT ), with a similar replacement for each Spec⁢D∗⁢O⁢(XΔn)Specsuperscript𝐷𝑂superscript𝑋superscriptΔ𝑛\mathrm{Spec}\,D^{*}O(X^{\Delta^{n}})roman_Spec italic_D start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT italic_O ( italic_X start_POSTSUPERSCRIPT roman_Δ start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT ) based on affine covers of each Xnsubscript𝑋𝑛X_{n}italic_X start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT.

For a linear algebraic group G𝐺Gitalic_G acting on an affine scheme Y𝑌Yitalic_Y, we can apply the theorem to the quotient stack [Y/G]delimited-[]𝑌𝐺[Y/G][ italic_Y / italic_G ], in which case we just recover the description of Example 1.16. Explicitly,

Given a smooth affine group scheme G𝐺Gitalic_G acting on an affine scheme Y𝑌Yitalic_Y, we can form a simplicial scheme X∙subscript𝑋∙X_{\bullet}italic_X start_POSTSUBSCRIPT ∙ end_POSTSUBSCRIPT as the nerve of the associated groupoid (with Xn≅Y×Gnsubscript𝑋𝑛𝑌superscript𝐺𝑛X_{n}\cong Y\times G^{n}italic_X start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ≅ italic_Y × italic_G start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT). Functions on this simplicial scheme form a cosimplicial algebra O⁢(X)𝑂𝑋O(X)italic_O ( italic_X ), and then by

Given a linear algebraic group G𝐺Gitalic_G acting on an affine scheme Y𝑌Yitalic_Y, the calculation of [Pri4, Example LABEL:poisson-DstarBG] implies that for the associated groupoid R𝑅Ritalic_R-scheme (Y×G⇒Y)⇒𝑌𝐺𝑌(Y\times G\Rightarrow Y)( italic_Y × italic_G ⇒ italic_Y ), we have

For an arbitrary linear algebraic group G𝐺Gitalic_G, we can consider the stack B⁢G𝐵𝐺BGitalic_B italic_G parametrising G𝐺Gitalic_G-torsors. A similar argument to the previous example characterises (D∗⁢B⁢G)⁢(B)subscript𝐷𝐵𝐺𝐵(D_{*}BG)(B)( italic_D start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT italic_B italic_G ) ( italic_B )

A

Relating the permanent and the determinant, Balakin [9] and, using delicate moment calculations, Blömer, Karp and Welzl [12] dealt with the rank of such matrices over finite fields.

Moreover, using expansion arguments, Costello and Vu [29, 30] studied the real rank of random symmetric matrices of a similar density.

The difficulty of the rank problem for dense random matrices strongly depends on the distribution of the matrix entries.

The single prior contribution on the rational rank of sparse random matrices is due to Bordenave, Lelarge and Salez [15], who computed the rational rank of the (symmetric) adjacency matrix of a random graph with a given vertex degree distribution.

Specifically, the ‘replica symmetric’ version of the cavity method predicts that the rank of a random matrix over a finite field can be expressed analytically as the maximum of a variational problem.

A

It is worth to point out that if μ𝜇\muitalic_μ is a non-degenerate probability measure on ΓΓ\Gammaroman_Γ such that the μ𝜇\muitalic_μ-random walk on ΓΓ\Gammaroman_Γ is transient, then the Martin boundary of ΓΓ\Gammaroman_Γ with respect to μ𝜇\muitalic_μ consists of positive μ𝜇\muitalic_μ-harmonic functions, provided that μ𝜇\muitalic_μ has finite support, or more generally, superexponential moment (cf. for instance [GGPY, Lemma 7.1]). Therefore, Theorem 1.2 yields the existence of non-constant, positive μ𝜇\muitalic_μ-harmonic functions with respect to such a measure μ𝜇\muitalic_μ. This evidently extends the result of [AK]. It is also worth to mention that the recent (relatively to the current version of the paper) [MST, Corollary 1.10] is also a special case of Theorem 1.2 and its proof is verbatim the proof of Theorem 1.2 of the original version of the paper under the additional assumption that μ𝜇\muitalic_μ has superexponential moment. It should also be noticed that [MST, Corollary 1.10] does not seem to contribute to the study of covering spaces, since superexponential moment has not been established for measures arising from the Lyons-Sullivan discretization under any assumption (besides the trivial case where the covering space is compact), and as a matter of fact, there are examples illustrating that this does not hold in general.

To set the stage, let p:M→N:𝑝→𝑀𝑁p\colon M\to Nitalic_p : italic_M → italic_N be a normal Riemannian covering of a closed manifold, with deck transformation group ΓΓ\Gammaroman_Γ. Lyons and Sullivan [LS, Theorem 3] showed that if ΓΓ\Gammaroman_Γ is non-amenable, then there exist non-constant, bounded harmonic functions on M𝑀Mitalic_M. The converse does not hold even if M𝑀Mitalic_M is the universal covering space of N𝑁Nitalic_N (cf. [Erschler, Theorem 5.2]). However, there are some results in the converse direction. More precisely, Kaimanovich [K] proved that if ΓΓ\Gammaroman_Γ has subexponential growth, or ΓΓ\Gammaroman_Γ is polycyclic (that is, solvable and any subgroup of ΓΓ\Gammaroman_Γ is finitely generated), then any bounded harmonic function on M𝑀Mitalic_M is constant. About the strong Liouville property, Lyons and Sullivan [LS, Theorem 1] showed that if ΓΓ\Gammaroman_Γ is virtually nilpotent, then any positive harmonic function on M𝑀Mitalic_M is constant. It should be noticed that according to a celebrated result of Gromov [Gromov], a finitely generated group is virtually nilpotent if and only if it is of polynomial growth.

It is quite evident that Theorem 1.1 is more general than the result of [BBE, BE]. From [LS, Theorem 3], it remains to investigate the strong Liouville property on M𝑀Mitalic_M, in the case where ΓΓ\Gammaroman_Γ is amenable. According to [Gu], a linear amenable group is virtually polycyclic. Therefore, from [BBE, BE], we obtain a characterization for the strong Liouville property on M𝑀Mitalic_M, if ΓΓ\Gammaroman_Γ is virtually polycyclic. On the other hand, Theorem 1.1 yields the following more

the Riemannian metric plays a role. Following earlier work of Furstenberg, in [LS], they constructed a discretization of the Brownian motion on the covering space. Their method was modified and extended in [BL, BP]. In particular, according to [BP, Theorems A, C], the cones of positive harmonic functions, and the spaces of bounded harmonic functions, respectively, on the covering space and the group (with respect to a symmetric probability measure, whose support is the whole group) are isomorphic. Therefore, it suffices to study the validity of the Liouville and the strong Liouville property on groups. Although one would expect this problem to be simpler, it is quite complicated and these properties are far from being comprehended completely. However, the Lyons-Sullivan discretization turned out to be quite fruitful.

In this paper, we focus on the triviality of the Martin boundary rather than the validity of the strong Liouville property on the deck transformation group. This is natural for our purposes, keeping in mind that if the base manifold is recurrent, then non-triviality of the Martin boundary of the deck transformation group (independently from the validity of the strong Liouville property on it) implies the existence of non-constant, positive harmonic functions on the covering space. Our main result in the group-theoretic setting is the following:

B

Let P⁢V𝑃𝑉PVitalic_P italic_V be a function from ℱ×ℱ0ℱsubscriptℱ0\mathcal{F}\times\mathcal{F}_{0}caligraphic_F × caligraphic_F start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT to

ℝ≥0subscriptℝabsent0\mathbb{R}_{\geq 0}blackboard_R start_POSTSUBSCRIPT ≥ 0 end_POSTSUBSCRIPT that satisfies definition 13.3.

ℝ≥0subscriptℝabsent0\mathbb{R}_{\geq 0}blackboard_R start_POSTSUBSCRIPT ≥ 0 end_POSTSUBSCRIPT that satisfies definition 13.3.

ℝ≥0subscriptℝabsent0\mathbb{R}_{\geq 0}blackboard_R start_POSTSUBSCRIPT ≥ 0 end_POSTSUBSCRIPT that satisfies definition 13.3.

ℝ≥0subscriptℝabsent0\mathbb{R}_{\geq 0}blackboard_R start_POSTSUBSCRIPT ≥ 0 end_POSTSUBSCRIPT that satisfies definition 13.3,

A

\mathbb{C}).italic_α ( italic_x × italic_y ) = [ italic_α ( italic_x ) , italic_α ( italic_y ) ] = italic_α ( italic_α ( italic_x ) italic_y ) = italic_y italic_x start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT - italic_x italic_y start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT , italic_α ( italic_R italic_x ) = italic_R italic_α ( italic_x ) italic_R start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT , italic_x , italic_y ∈ blackboard_C start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT , italic_R ∈ bold_O start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ( blackboard_C ) .

Let x⊂ℂ3𝑥superscriptℂ3x\subset\mathbb{C}^{3}italic_x ⊂ blackboard_C start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT be a surface applicable (isometric) to a surface

Let (u,v)∈D𝑢𝑣𝐷(u,v)\in D( italic_u , italic_v ) ∈ italic_D with D𝐷Ditalic_D domain of ℝ2superscriptℝ2\mathbb{R}^{2}blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT or ℂ2superscriptℂ2\mathbb{C}^{2}blackboard_C start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT and x:D↦ℂ3:𝑥maps-to𝐷superscriptℂ3x:D\mapsto\mathbb{C}^{3}italic_x : italic_D ↦ blackboard_C start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT be

Consider Lie’s viewpoint: one can replace a surface x⊂ℂ3𝑥superscriptℂ3x\subset\mathbb{C}^{3}italic_x ⊂ blackboard_C start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT with a 2222-dimensional

x0⊂ℂ3subscript𝑥0superscriptℂ3x_{0}\subset\mathbb{C}^{3}italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ⊂ blackboard_C start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT:

A

\operatorname{diam}Y|over^ start_ARG italic_d end_ARG start_POSTSUBSCRIPT caligraphic_G caligraphic_H end_POSTSUBSCRIPT ( italic_X , italic_Y ) ≥ divide start_ARG 1 end_ARG start_ARG 2 end_ARG | roman_diam italic_X - roman_diam italic_Y |. Assuming without loss of generality diam⁡X≥diam⁡Ydiam𝑋diam𝑌\operatorname{diam}X\geq\operatorname{diam}Yroman_diam italic_X ≥ roman_diam italic_Y, the call to FindLargeK⁢(DX,diam⁡X)FindLargeKsuperscript𝐷𝑋diam𝑋\textsc{FindLarge$K$}(D^{X},\operatorname{diam}X)FindLarge italic_K ( italic_D start_POSTSUPERSCRIPT italic_X end_POSTSUPERSCRIPT , roman_diam italic_X ) from VerifyLowerBound⁢(DX,DY,diam⁡X)VerifyLowerBoundsuperscript𝐷𝑋superscript𝐷𝑌diam𝑋\textsc{VerifyLowerBound}(D^{X},D^{Y},\operatorname{diam}X)VerifyLowerBound ( italic_D start_POSTSUPERSCRIPT italic_X end_POSTSUPERSCRIPT , italic_D start_POSTSUPERSCRIPT italic_Y end_POSTSUPERSCRIPT , roman_diam italic_X ) will yield K𝐾Kitalic_K with non-zero entries that each are at least (diam⁡X−diam⁡Y)diam𝑋diam𝑌(\operatorname{diam}X-\operatorname{diam}Y)( roman_diam italic_X - roman_diam italic_Y ) away from all DYsuperscript𝐷𝑌D^{Y}italic_D start_POSTSUPERSCRIPT italic_Y end_POSTSUPERSCRIPT entries. Because the (diam⁡X)diam𝑋(\operatorname{diam}X)( roman_diam italic_X )-bounded K𝐾Kitalic_K is also (diam⁡X−diam⁡Y)diam𝑋diam𝑌(\operatorname{diam}X-\operatorname{diam}Y)( roman_diam italic_X - roman_diam italic_Y )-bounded, the modified CheckTheoremB⁢(K,DY,diam⁡X)CheckTheoremB𝐾superscript𝐷𝑌diam𝑋\textsc{CheckTheoremB}(K,D^{Y},\operatorname{diam}X)CheckTheoremB ( italic_K , italic_D start_POSTSUPERSCRIPT italic_Y end_POSTSUPERSCRIPT , roman_diam italic_X ) will output d∗≥diam⁡X−diam⁡Ysuperscript𝑑diam𝑋diam𝑌d^{*}\geq\operatorname{diam}X-\operatorname{diam}Yitalic_d start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ≥ roman_diam italic_X - roman_diam italic_Y, ensuring that the resulting lower bound is at least 12⁢|diam⁡X−diam⁡Y|12diam𝑋diam𝑌\frac{1}{2}|\operatorname{diam}X-\operatorname{diam}Y|divide start_ARG 1 end_ARG start_ARG 2 end_ARG | roman_diam italic_X - roman_diam italic_Y |.

This is in contrast with the original FindLowerBound, in which the only easy guarantee for proving the baseline lower bound via Theorem B is the presence of diam⁡Xdiam𝑋\operatorname{diam}Xroman_diam italic_X values in K𝐾Kitalic_K produced by FindLargeK⁢(DX,diam⁡X−diam⁡Y)FindLargeKsuperscript𝐷𝑋diam𝑋diam𝑌\textsc{FindLarge$K$}(D^{X},\\

\operatorname{diam}Y|over^ start_ARG italic_d end_ARG start_POSTSUBSCRIPT caligraphic_G caligraphic_H end_POSTSUBSCRIPT ( italic_X , italic_Y ) ≥ divide start_ARG 1 end_ARG start_ARG 2 end_ARG | roman_diam italic_X - roman_diam italic_Y |. Assuming without loss of generality diam⁡X≥diam⁡Ydiam𝑋diam𝑌\operatorname{diam}X\geq\operatorname{diam}Yroman_diam italic_X ≥ roman_diam italic_Y, the call to FindLargeK⁢(DX,diam⁡X)FindLargeKsuperscript𝐷𝑋diam𝑋\textsc{FindLarge$K$}(D^{X},\operatorname{diam}X)FindLarge italic_K ( italic_D start_POSTSUPERSCRIPT italic_X end_POSTSUPERSCRIPT , roman_diam italic_X ) from VerifyLowerBound⁢(DX,DY,diam⁡X)VerifyLowerBoundsuperscript𝐷𝑋superscript𝐷𝑌diam𝑋\textsc{VerifyLowerBound}(D^{X},D^{Y},\operatorname{diam}X)VerifyLowerBound ( italic_D start_POSTSUPERSCRIPT italic_X end_POSTSUPERSCRIPT , italic_D start_POSTSUPERSCRIPT italic_Y end_POSTSUPERSCRIPT , roman_diam italic_X ) will yield K𝐾Kitalic_K with non-zero entries that each are at least (diam⁡X−diam⁡Y)diam𝑋diam𝑌(\operatorname{diam}X-\operatorname{diam}Y)( roman_diam italic_X - roman_diam italic_Y ) away from all DYsuperscript𝐷𝑌D^{Y}italic_D start_POSTSUPERSCRIPT italic_Y end_POSTSUPERSCRIPT entries. Because the (diam⁡X)diam𝑋(\operatorname{diam}X)( roman_diam italic_X )-bounded K𝐾Kitalic_K is also (diam⁡X−diam⁡Y)diam𝑋diam𝑌(\operatorname{diam}X-\operatorname{diam}Y)( roman_diam italic_X - roman_diam italic_Y )-bounded, the modified CheckTheoremB⁢(K,DY,diam⁡X)CheckTheoremB𝐾superscript𝐷𝑌diam𝑋\textsc{CheckTheoremB}(K,D^{Y},\operatorname{diam}X)CheckTheoremB ( italic_K , italic_D start_POSTSUPERSCRIPT italic_Y end_POSTSUPERSCRIPT , roman_diam italic_X ) will output d∗≥diam⁡X−diam⁡Ysuperscript𝑑diam𝑋diam𝑌d^{*}\geq\operatorname{diam}X-\operatorname{diam}Yitalic_d start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ≥ roman_diam italic_X - roman_diam italic_Y, ensuring that the resulting lower bound is at least 12⁢|diam⁡X−diam⁡Y|12diam𝑋diam𝑌\frac{1}{2}|\operatorname{diam}X-\operatorname{diam}Y|divide start_ARG 1 end_ARG start_ARG 2 end_ARG | roman_diam italic_X - roman_diam italic_Y |.

\operatorname{diam}X-\operatorname{diam}Y)FindLarge italic_K ( italic_D start_POSTSUPERSCRIPT italic_X end_POSTSUPERSCRIPT , roman_diam italic_X - roman_diam italic_Y ). However, because FindLargeK𝐾Kitalic_K prioritizes the number of entries ≥diam⁡X−diam⁡Yabsentdiam𝑋diam𝑌\geq\operatorname{diam}X-\operatorname{diam}Y≥ roman_diam italic_X - roman_diam italic_Y in a row of DXsuperscript𝐷𝑋D^{X}italic_D start_POSTSUPERSCRIPT italic_X end_POSTSUPERSCRIPT over their magnitude, and because a point with a distance diam⁡Xdiam𝑋\operatorname{diam}Xroman_diam italic_X can have arbitrarily many distances <diam⁡X−diam⁡Yabsentdiam𝑋diam𝑌<\operatorname{diam}X-\operatorname{diam}Y< roman_diam italic_X - roman_diam italic_Y, establishing practical conditions for the output of FindLowerBound to be ≥12⁢(diam⁡X−diam⁡Y)absent12diam𝑋diam𝑌\geq\frac{1}{2}(\operatorname{diam}X-\operatorname{diam}Y)≥ divide start_ARG 1 end_ARG start_ARG 2 end_ARG ( roman_diam italic_X - roman_diam italic_Y ) is non-trivial.

Notice that the bounds of the GH distance from the inequalities 12⁢|diam⁡X−diam⁡Y|≤d𝒢⁢ℋ⁢(X,Y)≤12⁢dmax12diam𝑋diam𝑌subscript𝑑𝒢ℋ𝑋𝑌12subscript𝑑\frac{1}{2}|\operatorname{diam}X-\operatorname{diam}Y|\leq d_{\mathcal{GH}}(X,%

A

Let ℋℋ\mathcal{H}caligraphic_H be a Hilbert space, V0,…,Vn⊆ℋsubscript𝑉0…subscript𝑉𝑛ℋV_{0},...,V_{n}\subseteq\mathcal{H}italic_V start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , italic_V start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ⊆ caligraphic_H be closed subspaces, and A𝐴Aitalic_A be the cosine matrix of V0,…,Vnsubscript𝑉0…subscript𝑉𝑛V_{0},...,V_{n}italic_V start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , italic_V start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT. If A𝐴Aitalic_A is positive definite, then for every τ⊆{0,1,…,n}𝜏01…𝑛\tau\subseteq\{0,1,...,n\}italic_τ ⊆ { 0 , 1 , … , italic_n }, it holds that ℋτ=⨁η⊆τℋηsubscriptℋ𝜏subscriptdirect-sum𝜂𝜏superscriptℋ𝜂\mathcal{H}_{\tau}=\bigoplus_{\eta\subseteq\tau}\mathcal{H}^{\eta}caligraphic_H start_POSTSUBSCRIPT italic_τ end_POSTSUBSCRIPT = ⨁ start_POSTSUBSCRIPT italic_η ⊆ italic_τ end_POSTSUBSCRIPT caligraphic_H start_POSTSUPERSCRIPT italic_η end_POSTSUPERSCRIPT.

The proof of the equivalence between this definition and the one given above is straightforward and can be found in [6, Lemma 9.5].

The original proof of this Theorem published in [14] contained a mistake which is corrected in this version of the paper.

The following definitions of the core complex and the Davis Chamber appear in the paper of Dymara and Januszkiewicz [8]. The inspiration to in is attributed in [8] to a similar construction of M.W. Davis in the setting of Coxeter complexes.

While the Theorems stated above concern vanishing of group cohomology, the ideas of Dymara and Januszkiewicz [8] reduce this problem to showing a decomposition theorem in Hilbert spaces and the main tool that they used to prove such a decomposition was the idea of angle between subspaces. In the technical heart of this paper we use the results of Kassabov [15] regarding angles between subspaces to prove a general decomposition theorem in Hilbert spaces that is interesting by its own right. After doing this, we show how this decomposition can be applied to deducing vanishing of cohomology in the general framework of Dymara and Januszkiewicz and how to apply this result for BN-pair buildings.

B