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As mentioned in the introduction, when q(x)=(x−1)d𝑞𝑥superscript𝑥1𝑑q(x)=(x-1)^{d}italic_q ( italic_x ) = ( italic_x - 1 ) start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT, mn(q)=1subscript𝑚𝑛𝑞1m_{n}(q)=1italic_m start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_q ) = 1 for every n𝑛nitalic_n, so equation (17) becomes the moment-cumulant formula proven in [AP18]. In some sense, this means that the combinatorial theory of addition can be recovered from that of multiplication. On the other hand, observe that we could have proven Lemma 3.2 by combining equation (17) with the moment-cumulant formula from [AP18], but we have chosen to present our results in the above form to underscore that Lemma 3.2 has a simple direct proof and to provide an alternative (and shorter) proof of the existing moment-cumulant formula. | In this section we will prove Theorems 1.2 and 1.4 by analyzing the asymptotic behavior of the formulas (15) and (17) as d→∞→𝑑d\to\inftyitalic_d → ∞. The results appearing in this section articulate in a precise way that the finite free multiplicative convolution converges to the free multiplicative convolution. | Firstly, as the expert reader may predict, we can use our result to show that the finite free multiplicative convolution ⊠dsubscript⊠𝑑\boxtimes_{d}⊠ start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT converges to the free multiplicative convolution ⊠⊠\boxtimes⊠. The fact that the finite free multiplicative convolution is related in the limit to the free multiplicative convolution was discovered by Marcus in [Mar21], where he showed that a transform that linearizes ⊠dsubscript⊠𝑑\boxtimes_{d}⊠ start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT (i.e. the logarithm of the d𝑑ditalic_d-finite S𝑆Sitalic_S-transform) converges, as d𝑑ditalic_d goes to infinity, to the logarithm of Voiculescu’s S𝑆Sitalic_S-transform, which is known to linearize ⊠⊠\boxtimes⊠. In the present paper we give a statement in terms of convergence of measures. Note that we do not need to restrict to the case when both measures are supported on [0,∞)0[0,\infty)[ 0 , ∞ ) and we do not require the sequences of polynomials to have a uniformly bounded root distribution. | Of greatest relevance to this paper is the combinatorial approach, based on cumulants, for the finite free additive convolution that was introduced by Arizmendi and Perales [AP18]. These finite free cumulants converge to free cumulants as d𝑑ditalic_d goes to infinity and share many of their properties. In the present work we further this approach to also include the finite free multiplicative convolution in the combinatorial description and present applications of our results to the asymptotic theory of polynomials. Below we give a brief summary of our main results, deferring to Section 2 the precise definitions of some of the notation used in these statements. | Apart from this introductory section, the rest of the paper is organized in four other sections. In Section 2 we introduce some notation and survey some of the theory that will be needed throughout this paper. Section 3 is divided in three parts: first we prove Theorem 1.1, that provides a formula for the finite free cumulants of a product of polynomials; then we prove Theorems 1.2 and 1.4, that articulate in a precise way that the finite free multiplicative convolution converges to the free multiplicative convolution; and the last part retrieves the interesting relation between derivatives of a polynomial and free additive convolution powers. In Section 4 we prove Theorem 1.3 and its generalization, which gives a topological interpretation of the terms of order Θ(1/dk)Θ1superscript𝑑𝑘\Theta(1/d^{k})roman_Θ ( 1 / italic_d start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ) appearing in Theorem 1.1. The proof of Theorem 1.6 and study of the infinitesimal limiting distribution of certain sequences of polynomials is given in Section 5. | A |
Any element of ℜhsubscriptℜℎ\mathfrak{R}_{h}fraktur_R start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT can be written as a linear combination of standard monomials of 𝒥hsubscript𝒥ℎ\mathcal{J}_{h}caligraphic_J start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT with integer coefficients. | Any element of ℜhsubscriptℜℎ\mathfrak{R}_{h}fraktur_R start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT can be written as a linear combination of standard monomials of 𝒥hsubscript𝒥ℎ\mathcal{J}_{h}caligraphic_J start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT with integer coefficients. | there must be a smallest element J∈ℳ(𝒥)𝐽ℳ𝒥J\in\mathcal{M}(\mathcal{J})italic_J ∈ caligraphic_M ( caligraphic_J ), which cannot be written as a linear combination of elements of 𝒮ℳ(𝒥)𝒮ℳ𝒥\mathcal{S}\mathcal{M}(\mathcal{J})caligraphic_S caligraphic_M ( caligraphic_J ) with integer coefficients. So J𝐽Jitalic_J is not standard. By Lemma 2.8, | If J=J1⋯Jb∈ℳ(𝒥)𝐽subscript𝐽1⋯subscript𝐽𝑏ℳ𝒥J=J_{1}\cdots J_{b}\in\mathcal{M}(\mathcal{J})italic_J = italic_J start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ⋯ italic_J start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT ∈ caligraphic_M ( caligraphic_J ) is not standard, J𝐽Jitalic_J can be written as a linear combination of elements of ℳ(𝒥)ℳ𝒥\mathcal{M}(\mathcal{J})caligraphic_M ( caligraphic_J ) preceding J1⋯Jb−1subscript𝐽1⋯subscript𝐽𝑏1J_{1}\cdots J_{b-1}italic_J start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ⋯ italic_J start_POSTSUBSCRIPT italic_b - 1 end_POSTSUBSCRIPT, with integer coefficients. | We only need to show that any element of ℜℜ\mathfrak{R}fraktur_R can be written as a linear combination of standard monomials of 𝒥𝒥\mathcal{J}caligraphic_J with integer coefficients. If the lemma is not true, | D |
Finally, in the last section we show that Theorem 3 can be used to give a new proof of the result of Lück and Lott that the Singer conjecture holds for 3-manifolds. As in [LL95], this is proved assuming the Geometrization conjecture. | Acknowledgments. LFDC thanks M. Anderson, G. Besson, and S. Maillot for introducing him to the theory of geometric 3333-manifolds during his time as a graduate student. He also thanks M. Stern for a stimulating ongoing collaboration on Price inequalities for harmonic forms, and for pointing out a mistake in a previous version of this paper. MH thanks D. Groves, D. B. McReynolds, and E. Einstein for helpful conversations about separability and virtually special groups. MH also thanks the University of Florida, which he visited during the preparation of this work, for its hospitality and support. Finally, the authors thank C. Löh for pointing out the reference [FFL19], and the referees for pertinent comments on the paper. | In this section, we repackage the Price inequality estimates for harmonic forms proved by the first author and M. Stern in [DS22], [DS19], and [DS20]. The statements presented here are tailored to the study of higher graph manifolds, and they may slightly differ from the presentation in the original sources. | We now provide a proof of the Singer conjecture (and Equation (39)) for aspherical manifold with MinVol=0𝑀𝑖𝑛𝑉𝑜𝑙0MinVol=0italic_M italic_i italic_n italic_V italic_o italic_l = 0 and residually finite fundamental group. The proof is along the lines of the strategy employed in the proofs of Theorem 3 and Theorem 4. Even if this was already known thanks to the work of Gromov and Sauer, we think it is somewhat interesting to have yet another approach. | for all i≥0𝑖0i\geq 0italic_i ≥ 0. In [Gro99, Section 5.33] Gromov outlines a proof of this fact under the assumption that π1(M)subscript𝜋1𝑀\pi_{1}(M)italic_π start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_M ) is residually finite. See the paper by Sauer [Sau09] for a proof of Equation (39) without the residual finiteness assumption. In particular, the Singer conjecture holds true for aspherical manifolds with MinVol=0𝑀𝑖𝑛𝑉𝑜𝑙0MinVol=0italic_M italic_i italic_n italic_V italic_o italic_l = 0 (see Conjecture 2). Finally, we observe that if the fundamental group is residually finite then Fauser-Friedl-Löh prove an integral approximation theorem for the simplicial volume of higher graph manifolds without pure pieces, see Theorem 1.2 in [FFL19]. This interesting result can then be used to prove the vanishing of gradient invariants of higher graph manifolds without pure pieces, see Corollary 1.4 and Remark 1.5 in [FFL19]. Gradient invariants are defined in [FFL19] as the normalization over finite coverings and finite index subgroups of several topological quantities. In the case of Betti numbers, this normalization process recovers the usual L2superscript𝐿2L^{2}italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT-Betti numbers. For more details, see the Introduction in [FFL19]. | A |
Survival analysis, a branch of statistics that deals with the analysis of time-to-event data, is a fundamental tool across various disciplines [40, 37]. It has been used in various domains such as healthcare [57, 11], medical prognostic [60], and manufacturing [1]. In such scenarios, the feature vector tends to be high-dimensional, and the variables tend to have complex correlations. | Therefore, we can conclude that the graph-based regularization has satisfactory performance on the pbcseq dataset. However, its performance improvement is limited by the fact that the problem is not high-dimensional (p=17𝑝17p=17italic_p = 17), and the graphical structure among the variables is relatively simple. | Survival analysis, a branch of statistics that deals with the analysis of time-to-event data, is a fundamental tool across various disciplines [40, 37]. It has been used in various domains such as healthcare [57, 11], medical prognostic [60], and manufacturing [1]. In such scenarios, the feature vector tends to be high-dimensional, and the variables tend to have complex correlations. | This is a fundamental problem in predictive modeling: when the feature vector is high-dimensional, it is crucial to select a subset of significant variables for model interpretation and predictability [28]. | For example, in organ transplantation, we are interested in knowing which variable is useful in predicting the post-transplant survival time of the patient. In such cases, efficient identification of the key variables will be useful for better decision-making. However, the variables tend to be highly correlated, and their mutual correlation can be represented by an undirected graph known from prior knowledge or estimated from data. For instance, a real-data example of the correlation structures among the predicting variables for the organ transplant dataset is illustrated in Figure 1. When such prior structural knowledge is available, incorporating it in variable selection may yield more precise results [70]. Instead of selecting individual variables, the graph structure enables us to utilize the neighborhood information to estimate or select the variables jointly. | C |
In [17], Gómez-Tato, Halperin and Tanré constructed an algebraic rational homotopy theory for topological spaces | Thanks to Theorem 1.2, the homotopy classification problem of cofibrant diffeological spaces is equivalent to that of minimal local systems; see, for example, [21] for the general theory of model categories. We observe that all diffeological spaces are fibrant in 𝖣𝗂𝖿𝖿𝖣𝗂𝖿𝖿\mathsf{Diff}sansserif_Diff and hence in 𝖣𝗂𝖿𝖿∗subscript𝖣𝗂𝖿𝖿\mathsf{Diff}_{*}sansserif_Diff start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ; see Theorem 2.4 and [21, Proposition 1.1.8]. | In this manuscript, by relating the local systems over K(π,1)𝐾𝜋1K(\pi,1)italic_K ( italic_π , 1 )-spaces to the model structure of 𝖣𝗂𝖿𝖿𝖣𝗂𝖿𝖿\mathsf{Diff}sansserif_Diff, | of stratifolds to 𝖣𝗂𝖿𝖿𝖣𝗂𝖿𝖿\mathsf{Diff}sansserif_Diff considered in [1]. The main result of this section (Theorem A.3) asserts that the functor k𝑘kitalic_k assigns an adjunction space in 𝖣𝗂𝖿𝖿𝖣𝗂𝖿𝖿\mathsf{Diff}sansserif_Diff to a p-stratifold up to smooth homotopy. | Local systems over K(π,1)𝐾𝜋1K(\pi,1)italic_K ( italic_π , 1 )-spaces are certainly key tools to construct the fibrewise localizations in [17] and then in this manuscript. Moreover, since more general local systems are defined over arbitrary simplicial sets, we expect that they will provide further applications. By using machinery in [19, Chapters 12 and 14], we also discuss applications of such local systems in diffeology and topology. | B |
The approach relies on showing that the dual Selmer group associated with the deformation problem is trivial on average. These groups characterize obstructions to lifting Galois representations subject to local conditions. The strategy employed involves showing that this dual Selmer group vanishes provided the p𝑝pitalic_p-primary Bloch-Kato Selmer group vanishes, see Proposition 4.2. This argument is a purely Galois theoretic argument, which we expect should extend to more general Galois representations. The result of F. Diamond, M. Flach and L. Guo (see [14]) gives a criterion for the vanishing of these Bloch-Kato Selmer groups. The strategy used does not explicitly rely on R=𝕋R𝕋\operatorname{R}=\mathbb{T}roman_R = blackboard_T results. | The deformation rings studied in this paper are equipped with a local condition at p𝑝pitalic_p, which we shall refer to as geometric deformation rings. There is one such ring ℛE,psubscriptℛ𝐸𝑝\mathcal{R}_{E,p}caligraphic_R start_POSTSUBSCRIPT italic_E , italic_p end_POSTSUBSCRIPT associated to each pair (E,p)𝐸𝑝(E,p)( italic_E , italic_p ), where E𝐸Eitalic_E is a rational elliptic curve and p𝑝pitalic_p an odd prime at which E𝐸Eitalic_E has good reduction. We study the structure of these rings on average, more precisely, how often they are unobstructed. Since the determinant is fixed throughout, the geometric deformation ring is smooth if and only if it is isomorphic to ℤpsubscriptℤ𝑝\mathbb{Z}_{p}blackboard_Z start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT. In this case, there is a unique characteristic-zero lift. We study the following questions. | is as in the previous section. In this section, we consider Galois representations which are ordinary/crystalline when localized at p𝑝pitalic_p, and thus in particular, are deRham. Such representations will be geometric and under a mild additional hypothesis, are known to arise from Hecke eigencuspforms. Following Böckle [5], we discuss presentations for the associated Galois deformation rings. We do not impose any local conditions at the primes ℓ∈Sℓ𝑆\ell\in Sroman_ℓ ∈ italic_S. However, at p𝑝pitalic_p, we impose the condition 𝒞psubscript𝒞𝑝\mathcal{C}_{p}caligraphic_C start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT defined in the previous section. Recall that throughout, the determinant of all deformations considered is fixed and equal to χ𝜒\chiitalic_χ, the cyclotomic character. | It is part (2) of Question 1.1 that is of primary interest from the point of view of arithmetic statistics. This question has not been studied before in any related context, and the answers provided in the last section of the paper are based on results proven in the first four sections. The answer to this question is the main goal of the paper and results are proven specifically with this application to arithmetic statistics in mind. We restrict ourselves to elliptic curves with good ordinary reduction at p𝑝pitalic_p, squarefree conductor and for which the residual representation ρ¯E,psubscript¯𝜌𝐸𝑝\bar{\rho}_{E,p}over¯ start_ARG italic_ρ end_ARG start_POSTSUBSCRIPT italic_E , italic_p end_POSTSUBSCRIPT on E[p]𝐸delimited-[]𝑝E[p]italic_E [ italic_p ] is irreducible. Furthermore, we insist that all primes ℓℓ\ellroman_ℓ dividing the conductor of E𝐸Eitalic_E are not ±1modpmoduloplus-or-minus1𝑝\pm 1\mod{p}± 1 roman_mod italic_p, see Definition 5.1. For this set of elliptic curves, it is shown that the deformation ring ℛE,psubscriptℛ𝐸𝑝\mathcal{R}_{E,p}caligraphic_R start_POSTSUBSCRIPT italic_E , italic_p end_POSTSUBSCRIPT is unobstructed provided the degree of the modular parametrization X0(N)→E→subscript𝑋0𝑁𝐸X_{0}(N)\rightarrow Eitalic_X start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_N ) → italic_E is coprime to p𝑝pitalic_p. This condition has been studied in detail by M.Watkins in [31]. Cohen-Lenstra heuristics indicate that the probability that p𝑝pitalic_p divides the modular degree of an elliptic curve is given by the product | Including the introduction, the article consists of five sections. In section 2, we discuss the essential objects of study in the paper, namely the deformation rings associated to elliptic curves. In section 3, we discuss presentations for deformation rings and establish a criterion for unobstructedness, in the setting in which there is a local condition at p𝑝pitalic_p. In section 4, we study the first of the two aforementioned questions. In this section, it is shown that given a non-CM elliptic curve, the deformation rings considered are unobstructed for all but a finite explicit set of primes. In section 5, the second question is studied, when p≥5𝑝5p\geq 5italic_p ≥ 5 is a fixed prime and E𝐸Eitalic_E varies over a certain collection of elliptic curves. In this section the question of unobstructedness of the deformation ring is related to the modular degree. Cohen-Lenstra heuristics are supported by computations in this section. | D |
If A𝐴Aitalic_A is an F𝐹Fitalic_F-split ℤ⩾0subscriptℤabsent0\mathbb{Z}_{\geqslant 0}blackboard_Z start_POSTSUBSCRIPT ⩾ 0 end_POSTSUBSCRIPT-graded ring, then ai(A)⩽0subscript𝑎𝑖𝐴0a_{i}(A)\leqslant 0italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_A ) ⩽ 0 for all i∈ℤ𝑖ℤi\in\mathbb{Z}italic_i ∈ blackboard_Z [HRFpurity, Lemma 2.32.32.32.3]. | Let d1,…,dr>0subscript𝑑1…subscript𝑑𝑟0d_{1},\ldots,d_{r}>0italic_d start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_d start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT > 0 be the generating degrees of A𝐴Aitalic_A as an A0subscript𝐴0A_{0}italic_A start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT-algebra. Let M𝑀Mitalic_M be a finitely generated ℤℤ\mathbb{Z}blackboard_Z-graded A𝐴Aitalic_A-module. Then, βA(M)⩽reg(M)+∑i=1r(di−1)subscript𝛽𝐴𝑀reg𝑀superscriptsubscript𝑖1𝑟subscript𝑑𝑖1\beta_{A}(M)\leqslant\operatorname{reg}(M)+\sum_{i=1}^{r}(d_{i}-1)italic_β start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_M ) ⩽ roman_reg ( italic_M ) + ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_r end_POSTSUPERSCRIPT ( italic_d start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - 1 ). | Given a finitely generated ℤℤ\mathbb{Z}blackboard_Z-graded A𝐴Aitalic_A-module, the Castelnuovo-Mumford regularity of M𝑀Mitalic_M is defined as | Let A=⊕n⩾0An𝐴subscriptdirect-sum𝑛0subscript𝐴𝑛A=\oplus_{n\geqslant 0}A_{n}italic_A = ⊕ start_POSTSUBSCRIPT italic_n ⩾ 0 end_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT be a ℤ⩾0subscriptℤabsent0\mathbb{Z}_{\geqslant 0}blackboard_Z start_POSTSUBSCRIPT ⩾ 0 end_POSTSUBSCRIPT-graded Noetherian ring. Given a finitely generated graded A𝐴Aitalic_A-module M=⊕n∈ℤMn𝑀subscriptdirect-sum𝑛ℤsubscript𝑀𝑛M=\oplus_{n\in\mathbb{Z}}M_{n}italic_M = ⊕ start_POSTSUBSCRIPT italic_n ∈ blackboard_Z end_POSTSUBSCRIPT italic_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT, we let | In this subsection we study the asymptotic behavior the depth and Castelnuovo-Mumford regularity of F𝐹Fitalic_F-split filtrations. We assume Setup 4.1. In the graded case, by depth of a graded R𝑅Ritalic_R-module we mean its grade with respect to the maximal ideal 𝔪𝔪\mathfrak{m}fraktur_m, i.e., the length of a maximal regular sequence for M𝑀Mitalic_M inside 𝔪0+R+subscript𝔪0subscript𝑅\mathfrak{m}_{0}+{R_{+}}fraktur_m start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_R start_POSTSUBSCRIPT + end_POSTSUBSCRIPT. On the other hand, by Castelnuovo-Mumford regularity we mean the regularity computed with respect to the ideal R+subscript𝑅{R_{+}}italic_R start_POSTSUBSCRIPT + end_POSTSUBSCRIPT. | B |
\hookrightarrow L^{1}(\Omega)italic_W start_POSTSUPERSCRIPT 1 , 2 end_POSTSUPERSCRIPT ( roman_Ω ) ↪ ↪ italic_L start_POSTSUPERSCRIPT divide start_ARG 2 italic_q end_ARG start_ARG italic_q - 2 end_ARG end_POSTSUPERSCRIPT ( roman_Ω ) ↪ italic_L start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ( roman_Ω ), | 1}(\Omega)italic_W start_POSTSUPERSCRIPT 1 , 2 end_POSTSUPERSCRIPT ( roman_Ω ) ↪ ↪ italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( roman_Ω ) ↪ italic_L start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ( roman_Ω ), Ehrling’s lemma in combination with the mass boundedness property (2.2) allows us to further fix K2>0subscript𝐾20K_{2}>0italic_K start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT > 0 such that | the Gagliardo–Nirenberg inequality asserts that there is K2>0subscript𝐾20K_{2}>0italic_K start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT > 0 with the property that | Ehrling’s lemma combined with (2.2) allows us to fix K2>0subscript𝐾20K_{2}>0italic_K start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT > 0 such that | \hookrightarrow L^{2}(\Omega)italic_W start_POSTSUPERSCRIPT 1 , 2 end_POSTSUPERSCRIPT ( roman_Ω ) ↪ ↪ italic_L start_POSTSUPERSCRIPT 2 italic_p start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT ( roman_Ω ) ↪ italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( roman_Ω ) to fix K2>0subscript𝐾20K_{2}>0italic_K start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT > 0 such that | C |
The solver’s input is the curvature on different nodes inside a domain, and the trace of hℎhitalic_h (the values of hℎhitalic_h on points on the boundary of the domain). Error estimates and empirical tests for the solver can be found in [Froese2018]. | This work consists of two parts. The first part of this paper provides the mathematical analysis needed for the experimental part and serves as the theoretical foundation for future implementations. There we deal with finding surfaces in ℝ3superscriptℝ3\mathbb{R}^{3}blackboard_R start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT which are as close as possible to being flat and span a given contour such that the contour is a geodesic on the sought surface. We look for a surface which minimizes the total Gaussian curvature squared. We show that by a change of coordinates the curvature of the optimal surface is controlled by a fourth order PDE. Then we show how to compute accurately the Dirichlet boundary condition and the Neumann boundary condition on the boundary of the domain on which we parametrize the unknown surface. In applications this PDE is heuristically reduced to the biharmonic equation. We then state a system of PDEs for the function whose graph is the optimal surface. The analysis, where an arbitrary affine plane is considered for reference, is coordinate-free: the surface described by a solution to the PDE system will be independent of the choice of plane as long as the projection of the optimal surface on it is one-to-one. The geodesic constraint on the contour is expressed via the Neumann boundary condition which we show how to compute. | Note that in running the solver we inputted the curvature function found in Section 7 with no coordinate transformation. Thus we regarded the curvature as given in Cartesian coordinates. These are the same coordinates used for solving the biharominc equation as described in the previous section. This is not compatible with the theory in the first part, which is also mentioned in Section 6, but is used to compute an approximation of the target surface. The underlying assumption is that the isothermal chart is close to the projection chart near the contour based on how we related the two chart in the first part of this study. | This part includes relevant numerical experiments which were performed throughout the study. I also describe an algorithm to construct an approximately optimal smooth surface when given a net of curves satisfying the following condition at the intersections. The condition is that the acceleration vectors of each two intersecting curves should be parallel at the intersection point. This is a necessary condition for the curves to be geodesics on a surface as the acceleration vector should be parallel to the normal of the surface everywhere for geodesics curves. (The smoothness of the result is based on the assumption that there is one smooth surface minimizing the considered energy and spanning the given net of curves such that the curves are geodesics on in it.) For each cell, the algorithm finds an approximate solution for the PDE system presented in Eq. 5. This is done by considering a cell’s contour, heuristically fixing a plane such that the projection of the unknown minimal Gaussian curvature surface spanning the contour is with high probability one-to-one, solving a biharmonic equation for the conformal factor for the approximate optimal curvature by using a finite element method, and then solving the curvature Monge-Ampère equation for a function whose graph over the chosen affine plane has the prescribed curvature at each point and its trace equals the cell’s contour. Note that in the first part of this study we work in isothermal coordinates (coordinates for which the surface’s metric can be written as ds2=e2f(dx2+dy2)𝑑superscript𝑠2superscript𝑒2𝑓𝑑superscript𝑥2𝑑superscript𝑦2ds^{2}=e^{2f}(dx^{2}+dy^{2})italic_d italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = italic_e start_POSTSUPERSCRIPT 2 italic_f end_POSTSUPERSCRIPT ( italic_d italic_x start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_d italic_y start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ), where f𝑓fitalic_f is a smooth real function on the parameterization domain) and we do not know how to compute the coordinate change maps from isothermal coordinates to ”projection on a plane” coordinates. Inputting the approximate optimal curvature in Cartesian coordinates seems to work well and is based on the assumption that the isothermal chart is not too ”wild” near the bounding contour. In addition, we are able to evaluate the error between the computed function and the solution of Eq. 5. This can be done by applying a discrete Laplace-Beltrami operator to evaluate the result. This will potentially allow a refinement of the net of geodesics in regions where the Laplace-Beltrami operator applied to each coordinate function of the projection (on the plane), with respect to the computed surface, is significantly larger in absolute value from zero. In addition, we can check if near the given curves, at the seam of two cells, the error is low. We plan to to implement this in a subsequent work which will put the emphasis on the implementation side of the method. | In the previous section we showed how to obtain the approximate optimal curvature of the minimal Gauss curvature surface spanning a geodesic contour. We also remark that we input the curvature as if it were given in Cartesian coordinates, which may also lead to inaccuracies, however we can evaluate the result. We elaborate on this in the end of this section and in the next section. | B |
This section motivates the idea of an initial c𝑐citalic_c-grid whose edges are not uniformly labeled 1. The section provides definitions, a worked out example, and notes various patterns which will be formalized in the three main theorems of the paper. | An important point, is that the four edge-ratio functions can be used to describe the ratio of any two edges in the 3-grid. To illustrate this, we introduce the heuristic of travel along the edges of the c𝑐citalic_c-grid. Heuristically, the edge ratio functions x,y,rR,L,rB,L𝑥𝑦subscript𝑟𝑅𝐿subscript𝑟𝐵𝐿x,y,r_{R,L},r_{B,L}italic_x , italic_y , italic_r start_POSTSUBSCRIPT italic_R , italic_L end_POSTSUBSCRIPT , italic_r start_POSTSUBSCRIPT italic_B , italic_L end_POSTSUBSCRIPT are used to describe vertical, horizontal, and rotational travel, respectively. The next two examples clarify how this concept of travel will be used. | The c𝑐citalic_c grid considered in this section uses the four edge ratio functions to label edges. More specifically, given a c𝑐citalic_c-grid we use the following equations to define labels. | For an arbitrary c𝑐citalic_c grid possessing symmetry, if the four edge-ratio relationships are satisfied for edges of triangles in the upper left half, then the four edge-factor relationships are satisfied in the entire c𝑐citalic_c-grid. | The next four conjectures study patterns in the ratios of certain edge-labels in the c𝑐citalic_c grid resulting from applying n−c𝑛𝑐n-citalic_n - italic_c row reductions to an n𝑛nitalic_n grid whose edges are uniformly labeled 1. The conjectures should be read as defining four functions, rR,L,rB,L,x,ysubscript𝑟𝑅𝐿subscript𝑟𝐵𝐿𝑥𝑦r_{R,L},r_{B,L},x,yitalic_r start_POSTSUBSCRIPT italic_R , italic_L end_POSTSUBSCRIPT , italic_r start_POSTSUBSCRIPT italic_B , italic_L end_POSTSUBSCRIPT , italic_x , italic_y and then conjecturing that the ratio of the labels specified in each conjecture have the indicated value of these functions. In the sequel, we will refer to these four functions as the (four) edge-ratio functions. | B |
For an algorithm with optimal scaling as r→1→𝑟1r\to 1italic_r → 1, a log-log plot of γ2superscript𝛾2\gamma^{2}italic_γ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT vs. −1/logr1𝑟-1/\log r- 1 / roman_log italic_r is therefore a line of slope −11-1- 1 as r→1→𝑟1r\to 1italic_r → 1. | Figs. 3 to 5 show that as the rate r→1→𝑟1r\to 1italic_r → 1, the sensitivity γ→0→𝛾0\gamma\to 0italic_γ → 0. In the limit of arbitrarily slow convergence, we obtain the desirable behavior of complete noise attenuation. However, these trade-off plots give limited insight on the asymptotic convergence rate as r→1→𝑟1r\to 1italic_r → 1. | As shown in Fig. 6, our algorithm designs appear to have optimal or near-optimal scaling as r→1→𝑟1r\to 1italic_r → 1. | In other words, our algorithms can easily be adjusted to be made faster (and more sensitive to noise), or more robust to noise (and slower). When tuned to be as fast as possible, our algorithm designs recover algorithms that are known to converge with the optimal linear rate in the noise-free setting for the respective function classes (see Section 7). Our algorithm designs may be used to converge at a linear rate to a predetermined noise level, or the trade-off parameter may be adjusted over time to construct multi-stage variants444While we do not propose a restart mechanism or time-varying parameter schedule in this work, we illustrate such an approach in Section 7.4 through a hand-tuned schedule.. | For an algorithm with optimal scaling as r→1→𝑟1r\to 1italic_r → 1, a log-log plot of γ2superscript𝛾2\gamma^{2}italic_γ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT vs. −1/logr1𝑟-1/\log r- 1 / roman_log italic_r is therefore a line of slope −11-1- 1 as r→1→𝑟1r\to 1italic_r → 1. | B |
M)\}{ roman_diam start_POSTSUBSCRIPT roman_Hol start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT ( italic_g ) end_POSTSUBSCRIPT italic_H italic_o italic_l start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT ( italic_g ) | italic_M italic_n - roman_manifold , italic_g ∈ roman_Riem ( italic_M ) } is bounded. | Now we want to switch roles of H𝐻Hitalic_H and K𝐾Kitalic_K in the last step. To this aim, we first restrict ourselves to the case of G𝐺Gitalic_G carrying a bi-invariant metric, which is the case for G=Hol0𝐺superscriptHol0G={\rm Hol}^{0}italic_G = roman_Hol start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT, as the latter is a closed subgroup of SO(n)𝑆𝑂𝑛SO(n)italic_S italic_O ( italic_n ) and thus compact, so a bi-invariant metric on G𝐺Gitalic_G is given as the negative of the Killing form. Then Theorem 7 is implied by the following theorem: | Proof. In view of the last paragraph preceding Theorem 4, it suffices to show that such an inclusion holds w.r.t. the Hausdorff metric after intersecting both sides with compact subsets of TxMsubscript𝑇𝑥𝑀T_{x}Mitalic_T start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT italic_M, say B(0,R)𝐵0𝑅B(0,R)italic_B ( 0 , italic_R ) and B(0,R+1)𝐵0𝑅1B(0,R+1)italic_B ( 0 , italic_R + 1 ) (if we choose δ<1𝛿1\delta<1italic_δ < 1), and this is the content of the last theorem. □□\quad\Box\quad□ | As now by Th. 11 the convexity radius of G/H𝐺𝐻G/Hitalic_G / italic_H can be bound uniformly from 00 for all subgroups H𝐻Hitalic_H of intrinsic diameter ≤Labsent𝐿\leq L≤ italic_L, Th. 7 follows by swapping K𝐾Kitalic_K and H𝐻Hitalic_H in the proof of Th. 10. □□\quad\Box\quad□ | Proof. Go through Berger’s finite list and Cartan’s classification of symmetric spaces, and noting that there are, up to conjugation, finitely many splittings into invariant subspaces, and that the intrinsic diameter is invariant under conjugation. □□\quad\Box\quad□ | D |
Here, each σ∈Σ𝜎Σ\sigma\in\Sigmaitalic_σ ∈ roman_Σ is a K𝐾Kitalic_K-morphism σ:Y→Y:𝜎→𝑌𝑌\sigma:Y\to Yitalic_σ : italic_Y → italic_Y. Hence any | every GLsubscript𝐺𝐿G_{L}italic_G start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT-torsor over the L𝐿Litalic_L-scheme SLsubscript𝑆𝐿S_{L}italic_S start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT can be viewed as a G𝐺Gitalic_G-torsor | Let 𝒢𝒢\mathcal{G}caligraphic_G be a G𝐺Gitalic_G-torsor over the K𝐾Kitalic_K-scheme SLsubscript𝑆𝐿S_{L}italic_S start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT, as in | G𝐺Gitalic_G-torsor 𝒢𝒢\mathcal{G}caligraphic_G over Y𝑌Yitalic_Y can be pulled back along σ𝜎\sigmaitalic_σ, resulting in | another G𝐺Gitalic_G-torsor σ∗𝒢superscript𝜎𝒢\sigma^{*}\mathcal{G}italic_σ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT caligraphic_G over Y𝑌Yitalic_Y. | C |
The statements of the present article cover the reduced twisted crossed product C∗∗{}^{\ast}start_FLOATSUPERSCRIPT ∗ end_FLOATSUPERSCRIPT-algebras. | We emphasis that the theorem involves non-unital C∗∗{}^{\ast}start_FLOATSUPERSCRIPT ∗ end_FLOATSUPERSCRIPT-algebras | even if one is only interested in unital C∗∗{}^{\ast}start_FLOATSUPERSCRIPT ∗ end_FLOATSUPERSCRIPT-algebras. | of the unital reduced crossed product C∗∗{}^{\ast}start_FLOATSUPERSCRIPT ∗ end_FLOATSUPERSCRIPT-algebras | non-unital C∗∗{}^{\ast}start_FLOATSUPERSCRIPT ∗ end_FLOATSUPERSCRIPT-algebras (namely, the reduced group C∗∗{}^{\ast}start_FLOATSUPERSCRIPT ∗ end_FLOATSUPERSCRIPT-algebras of open normal subgroups). This illustrates importance | A |
FS2∘S1∼FS2∘FS1.similar-tosubscript𝐹subscript𝑆2subscript𝑆1subscript𝐹subscript𝑆2subscript𝐹subscript𝑆1F_{S_{2}\circ S_{1}}\sim F_{S_{2}}\circ F_{S_{1}}.italic_F start_POSTSUBSCRIPT italic_S start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ∘ italic_S start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ∼ italic_F start_POSTSUBSCRIPT italic_S start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ∘ italic_F start_POSTSUBSCRIPT italic_S start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT . | Recall that the cobordism maps in Fan’s unoriented link Floer TQFT are computed by decomposing a given cobordism into elementary disoriented link cobordisms and composing the maps corresponding to each of these pieces. Among all possible pieces, we only have to consider nonorientable saddle moves, since this Lemma is verified in all other cases in the proof of [Zem19a, Theorem 1.3] (for compatible decorated link cobordisms). Since the cobordism map for a saddle move is defined via a count of pseudoholomorphic triangles on certain simple Heegaard triple-diagrams – see the proof of [Fan19, Theorem 5.1.4] – they homotopy-commute with conjugation maps. | For oriented link cobordisms, the notion of disoriented cobordism is equivalent to the notion of decorated cobordism in [Juh16] and [Zem19b]. This equivalence is a little subtle. The decoration in a decorated cobordism with underlying surface ΣΣ\Sigmaroman_Σ is an embedded 1111-manifold in ΣΣ\Sigmaroman_Σ whose boundary is required to be disjoint from the basepoints on the link. On the other hand, we have that the decoration in a disoriented cobordism is required to coincide with the basepoints on the link. Nevertheless, there is a notion of compatibility between disoriented and decorated orientable link cobordisms [GM23, Remark 4.14] and it is known that the cobordism maps from Zemke’s oriented link Floer TQFT induce the cobordism maps from Fan’s unoriented link Floer TQFT under truncation for appropriate compatible cobordisms [GM23, Remark 4.15]. | Figure 4.2. Left, the cobordism 𝒮𝒮\mathcal{S}caligraphic_S. Right, the cobordism representing τK∘(S¯,−a)subscript𝜏𝐾¯𝑆𝑎\tau_{K}\circ(\bar{S},-a)italic_τ start_POSTSUBSCRIPT italic_K end_POSTSUBSCRIPT ∘ ( over¯ start_ARG italic_S end_ARG , - italic_a ). Note that the cobordism on the right is isotopic to the one on the left. | In this section, we will use not only (disoriented) cobordisms in S3×Isuperscript𝑆3𝐼S^{3}\times Iitalic_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT × italic_I, but also cobordisms in B4superscript𝐵4B^{4}italic_B start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT. They can be seen as disoriented cobodisms from a nontrivial link to the empty link, or from the empty link to a nontrivial link. To define cobordism maps for such cobordisms, given a disoriented cobordism (S,a)𝑆𝑎(S,a)( italic_S , italic_a ) in B4superscript𝐵4B^{4}italic_B start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT, we choose any p𝑝pitalic_p in the interior of a𝑎aitalic_a and its sufficiently small neighborhood N(p)𝑁𝑝N(p)italic_N ( italic_p ) in B4superscript𝐵4B^{4}italic_B start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT, so that S∩N(p)𝑆𝑁𝑝S\cap N(p)italic_S ∩ italic_N ( italic_p ) is a boundary-parallel disk in B4superscript𝐵4B^{4}italic_B start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT and a∩N(p)𝑎𝑁𝑝a\cap N(p)italic_a ∩ italic_N ( italic_p ) is a properly embedded arc on S∩N(p)𝑆𝑁𝑝S\cap N(p)italic_S ∩ italic_N ( italic_p ). We then consider (S\N(p),a\N(p))\𝑆𝑁𝑝\𝑎𝑁𝑝(S\backslash N(p),a\backslash N(p))( italic_S \ italic_N ( italic_p ) , italic_a \ italic_N ( italic_p ) ), which is now a disoriented cobordism in S3×Isuperscript𝑆3𝐼S^{3}\times Iitalic_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT × italic_I, from some disoriented link L𝐿Litalic_L to the unknot. Then the map F(S,a)subscript𝐹𝑆𝑎F_{(S,a)}italic_F start_POSTSUBSCRIPT ( italic_S , italic_a ) end_POSTSUBSCRIPT induced by (S,a)𝑆𝑎(S,a)( italic_S , italic_a ) can be naturally defined as the disoriented cobordism map F(S\N(p),a\N(p))subscript𝐹\𝑆𝑁𝑝\𝑎𝑁𝑝F_{(S\backslash N(p),a\backslash N(p))}italic_F start_POSTSUBSCRIPT ( italic_S \ italic_N ( italic_p ) , italic_a \ italic_N ( italic_p ) ) end_POSTSUBSCRIPT induced by the punctured cobordism (S\N(p),a\N(p))\𝑆𝑁𝑝\𝑎𝑁𝑝(S\backslash N(p),a\backslash N(p))( italic_S \ italic_N ( italic_p ) , italic_a \ italic_N ( italic_p ) ) by 4.4. We can similarly define the opposite-direction cobordism map F(−S,a)subscript𝐹𝑆𝑎F_{(-S,a)}italic_F start_POSTSUBSCRIPT ( - italic_S , italic_a ) end_POSTSUBSCRIPT. Note that this definition is compatible with the birth/death maps of unknots in Zemke’s oriented link Floer TQFT. | B |
_{\overline{K}},x),{\mathbb L}_{x})italic_H start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_alg end_POSTSUBSCRIPT ( italic_π start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT roman_pro - roman_alg end_POSTSUPERSCRIPT ( italic_X start_POSTSUBSCRIPT over¯ start_ARG italic_K end_ARG end_POSTSUBSCRIPT , italic_x ) , italic_L start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ) → italic_H start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_cont end_POSTSUBSCRIPT ( italic_π start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT over´ start_ARG roman_e end_ARG roman_t end_POSTSUPERSCRIPT ( italic_X start_POSTSUBSCRIPT over¯ start_ARG italic_K end_ARG end_POSTSUBSCRIPT , italic_x ) , italic_L start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ) is an isomorphism because both groups classify extensions of the trivial representation Qpsubscript𝑄𝑝{\mathbb Q}_{p}italic_Q start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT by Lxsubscript𝐿𝑥{\mathbb L}_{x}italic_L start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT and the categories of finite-dimensional representations of π1e´t(XK¯,x)superscriptsubscript𝜋1´etsubscript𝑋¯𝐾𝑥\pi_{1}^{\mathrm{\acute{e}t}}(X_{\overline{K}},x)italic_π start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT over´ start_ARG roman_e end_ARG roman_t end_POSTSUPERSCRIPT ( italic_X start_POSTSUBSCRIPT over¯ start_ARG italic_K end_ARG end_POSTSUBSCRIPT , italic_x ) and π1pro−alg(XK¯,x)superscriptsubscript𝜋1proalgsubscript𝑋¯𝐾𝑥\pi_{1}^{\mathrm{pro}-\mathrm{alg}}(X_{\overline{K}},x)italic_π start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT roman_pro - roman_alg end_POSTSUPERSCRIPT ( italic_X start_POSTSUBSCRIPT over¯ start_ARG italic_K end_ARG end_POSTSUBSCRIPT , italic_x ) are equivalent. | There is a Leray spectral sequence with E2i,j=He´ti(UF¯,Rjπ∗Qp)superscriptsubscript𝐸2𝑖𝑗subscriptsuperscript𝐻𝑖´etsubscript𝑈¯𝐹superscript𝑅𝑗subscript𝜋subscript𝑄𝑝E_{2}^{i,j}=H^{i}_{\mathrm{\acute{e}t}}(U_{\overline{F}},R^{j}\pi_{*}{\mathbb Q% | The proof of Theorem 1.2 now proceeds by induction on the dimension of X𝑋Xitalic_X. The base case dimX=0dimension𝑋0\dim X=0roman_dim italic_X = 0 is given by Proposition 1.1. Assuming that the theorem has been proven for all schemes of dimension <dimXabsentdimension𝑋<\dim X< roman_dim italic_X, using resolution of singularities and the Gysin sequence, we may freely replace X𝑋Xitalic_X by a birational variety. We can therefore assume that X𝑋Xitalic_X admits a smooth proper morphism to a (possibly open) curve. Applying Belyi’s theorem to this curve we may moreover assume that X𝑋Xitalic_X admits a smooth proper morphism f:X→PF1∖{0,1,∞}:𝑓→𝑋subscriptsuperscript𝑃1𝐹01f:X\to{\mathbb P}^{1}_{F}\setminus\{0,1,\infty\}italic_f : italic_X → italic_P start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT ∖ { 0 , 1 , ∞ } to the projective line with three punctures. | After the preparatory work of the previous sections, the main result will follow by induction on the dimension, exhibiting the relevant variety as a fibration over a curve and applying a Leray spectral sequence. | We will never work with the pro-algebraic completion in terms of its points but will rather analyze the ring of regular functions on it. Given a continuous representation ρ:Γ→GL(V):𝜌→Γ𝐺𝐿𝑉\rho:\Gamma\to GL(V)italic_ρ : roman_Γ → italic_G italic_L ( italic_V ) on a finite-dimensional E𝐸Eitalic_E-vector space V𝑉Vitalic_V, denote by ℱ(V)ℱ𝑉{\mathcal{F}}(V)caligraphic_F ( italic_V ) the E𝐸Eitalic_E-span of the image of the composition Γ→𝜌GL(V)⊂EndEV𝜌→Γ𝐺𝐿𝑉subscriptEnd𝐸𝑉\Gamma\xrightarrow{\rho}GL(V)\subset\operatorname{{End}}_{E}Vroman_Γ start_ARROW overitalic_ρ → end_ARROW italic_G italic_L ( italic_V ) ⊂ roman_End start_POSTSUBSCRIPT italic_E end_POSTSUBSCRIPT italic_V. The dual space ℱ(V)∨ℱsuperscript𝑉{\mathcal{F}}(V)^{\vee}caligraphic_F ( italic_V ) start_POSTSUPERSCRIPT ∨ end_POSTSUPERSCRIPT is sometimes referred to as the space of matrix coefficients of the representation V𝑉Vitalic_V. One might think of the space of functions on ΓEpro−algsubscriptsuperscriptΓproalg𝐸\Gamma^{\mathrm{pro}-\mathrm{alg}}_{E}roman_Γ start_POSTSUPERSCRIPT roman_pro - roman_alg end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_E end_POSTSUBSCRIPT as of the ring of matrix coefficients of all representations: | C |
[13] was to optimize the resources distribution. In the present paper, we consider similar questions | It remains to investigate the effect of the time frequency 1/T1𝑇1/T1 / italic_T. In the case of a scalar equation in a bounded domain | case z=0𝑧0z=0italic_z = 0, and will then deduce the general case as a consequence of this one. Also, since the statement of Theorem | Indeed, this follows from the uniqueness of the solutions of the above systems and from the existence of radial solutions by | (i.e. how to distribute resources in a domain) in the case of systems and we also investigate the question of optimal | D |
It was shown in [27] that τ(E)=τ𝒱(E)𝜏𝐸subscript𝜏𝒱𝐸\tau(E)=\tau_{\mathscr{V}}(E)italic_τ ( italic_E ) = italic_τ start_POSTSUBSCRIPT script_V end_POSTSUBSCRIPT ( italic_E ) for every ordered defining sequence 𝒱𝒱\mathscr{V}script_V for E𝐸Eitalic_E. | First we recall the thickness of a Cantor set in ℝℝ\mathbb{R}blackboard_R from Newhouse [19] (see [1] for some recent progress). | The following lower bound for the Hausdorff dimension of a Cantor set in ℝℝ\mathbb{R}blackboard_R in terms of thickness was proven by Newhouse [20] (see also [21, P. 77]). | Two Cantor sets in ℝℝ\mathbb{R}blackboard_R are called interleaved if neither set lies in the closure of a gap of the other. | The key ingredient in our proof of Theorem 1.2 is Proposition 3.1 (see below), which indicates that the local Hausdorff dimension of Λ(x)Λ𝑥\Lambda(x)roman_Λ ( italic_x ) at some λ0∈Λ(x)subscript𝜆0Λ𝑥\lambda_{0}\in\Lambda(x)italic_λ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ roman_Λ ( italic_x ) is equal to the Hausdorff dimension of the self-similar set Kλ0subscript𝐾subscript𝜆0K_{\lambda_{0}}italic_K start_POSTSUBSCRIPT italic_λ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT. This property on the interplay between the | B |
0: The digital beamformer 𝐃msubscript𝐃𝑚{\mathbf{D}}_{m}bold_D start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT and the analog beamformer | of the kthsuperscript𝑘thk^{\rm th}italic_k start_POSTSUPERSCRIPT roman_th end_POSTSUPERSCRIPT TTD element and the adjustable phase shift parameter of the | 4: Determine the phase shift of the kthsuperscript𝑘thk^{\rm th}italic_k start_POSTSUPERSCRIPT roman_th end_POSTSUPERSCRIPT sub-array: | Here, rk′superscriptsubscript𝑟𝑘′r_{k}^{\prime}italic_r start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT denotes the adjustable distance parameter of the kthsuperscript𝑘thk^{\rm th}italic_k start_POSTSUPERSCRIPT roman_th end_POSTSUPERSCRIPT | 3: Determine the distance parameter of the kthsuperscript𝑘thk^{\rm th}italic_k start_POSTSUPERSCRIPT roman_th end_POSTSUPERSCRIPT TTD element: | D |
=\|X\|.∥ over~ start_ARG italic_X end_ARG ∥ = start_UNDERACCENT over^ start_ARG italic_f end_ARG ∈ over^ start_ARG italic_H start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT end_ARG , ∥ over^ start_ARG italic_f end_ARG ∥ = 1 end_UNDERACCENT start_ARG roman_sup end_ARG ∥ over~ start_ARG italic_X end_ARG over^ start_ARG italic_f end_ARG ∥ = start_UNDERACCENT italic_f ∈ italic_H start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT , ∥ italic_f ∥ = 1 end_UNDERACCENT start_ARG roman_sup end_ARG ∥ italic_X italic_f ∥ = ∥ italic_X ∥ . | Note in particular that the set of norm attaining linear functionals on X𝑋Xitalic_X is dense in X∗superscript𝑋X^{*}italic_X start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT. Therefore, the Bishop-Phelps-Bollobás theorem recovers the classical Bishop and Phelps theorem (see Theorem 1.1). We refer the reader to Aron and Lomonosov [6] for a rapid but excellent survey on Bishop-Phelps-Bollobás theorem, its applications, and relevant connections (also see the survey by Acosta [1]). | Now we are ready to prove the main theorem of this paper. However, for the reader’s convenience, we restate Theorem 1.4 by incorporating the definition of Bishop-Phelps-Bollobás property (see Definition 1.3). | The main objective of this paper is to connect two classical concepts of independent interest: the Bishop-Phelps-Bollobás theorem and H∞superscript𝐻H^{\infty}italic_H start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT, the algebra of all bounded analytic functions on the open unit disc 𝔻𝔻\mathbb{D}blackboard_D in the complex plane ℂℂ\mathbb{C}blackboard_C. More specifically, we prove that the Banach space of bounded linear operators from H∞superscript𝐻H^{\infty}italic_H start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT into itself satisfies the Bishop-Phelps-Bollobás property (see Theorem 1.4). Here, our formulation of the Bishop-Phelps-Bollobás property follows the construction of Acosta, Aron, García, and Maestre [2] (see Definition 1.3). | The main contribution of this paper is to demonstrate that the concept of Bishop-Phelps-Bollobás property of simultaneous approximations (in the sense of Definition 1.3 and following the classical construction of Bishop, Phelps, and Bollobás) fits perfectly in the framework of bounded analytic functions. This also includes specific values of the functions β𝛽\betaitalic_β and γ𝛾\gammaitalic_γ in Definition 1.3. More specifically: | B |
An immersed disk Σ⊂XΣ𝑋\Sigma\subset Xroman_Σ ⊂ italic_X with n𝑛nitalic_n double points and dual G𝐺Gitalic_G, together with a collection α={α1,…,αn}𝛼subscript𝛼1…subscript𝛼𝑛\alpha=\{\alpha_{1},\dots,\alpha_{n}\}italic_α = { italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_α start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT } of (possibly immersed) tubing arcs in ΣΣ\Sigmaroman_Σ connecting each double point to the point Σ⋔Gproper-intersectionΣ𝐺\Sigma\pitchfork Groman_Σ ⋔ italic_G, determines an embedded tubed disk (Σ,α)Σα(\Sigma,\alpha)( roman_Σ , italic_α ) which also has dual G𝐺Gitalic_G, constructed by using the Norman trick [norman] to tube each double point of ΣΣ\Sigmaroman_Σ over G𝐺Gitalic_G along the tubing arc α𝛼\alphaitalic_α as in Figure 2. If the tubing arcs in α𝛼\alphaitalic_α are immersed, then the isotopy class of the resulting tubed disk (Σ,α)Σ𝛼(\Sigma,\alpha)( roman_Σ , italic_α ) is not unique until an ordering of the arcs (i.e. an overstrand and an understrand) is chosen at each crossing to determine the relative radius of the tubes running along those arcs in the normal disk bundle of ΣΣ\Sigmaroman_Σ, as shown in Figure 2. A diagram on the disk D2superscript𝐷2D^{2}italic_D start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT of the pre-image of the tubing arcs, together with their endpoints and crossing information, is called a tube diagram for the tubed disk. | Isotopies of tubed disks which only move the “tubes” are convenient to visualize, especially from the point of view of the pre-image: the pre-images of the tubing arcs in D2superscript𝐷2D^{2}italic_D start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT determine how the tubes run along the embedded tubed disk in X𝑋Xitalic_X. In particular, Gabai’s isotopies of tubed surfaces from [dave:LBT, Figure 5.6] will come in handy. | Any composition of the isotopies of tubed disks illustrated in Figure 3, as well as [dave:LBT, Figure 5.6], will be referred to as a tube isotopy. These isotopies are “local” in the sense that they move only one tube at a time and leave the rest of the disk fixed. The support of each tube isotopy is also completely contained in a tubular neighborhood of the original surface and its dual sphere. The isotopies in Figure 3 are drawn from the perspective of tube diagrams, whereas Figure 4 shows the 3333-dimensional cross-sections of the tubed disk during isotopies A𝐴Aitalic_A and B𝐵Bitalic_B. | An immersed disk Σ⊂XΣ𝑋\Sigma\subset Xroman_Σ ⊂ italic_X with n𝑛nitalic_n double points and dual G𝐺Gitalic_G, together with a collection α={α1,…,αn}𝛼subscript𝛼1…subscript𝛼𝑛\alpha=\{\alpha_{1},\dots,\alpha_{n}\}italic_α = { italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_α start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT } of (possibly immersed) tubing arcs in ΣΣ\Sigmaroman_Σ connecting each double point to the point Σ⋔Gproper-intersectionΣ𝐺\Sigma\pitchfork Groman_Σ ⋔ italic_G, determines an embedded tubed disk (Σ,α)Σα(\Sigma,\alpha)( roman_Σ , italic_α ) which also has dual G𝐺Gitalic_G, constructed by using the Norman trick [norman] to tube each double point of ΣΣ\Sigmaroman_Σ over G𝐺Gitalic_G along the tubing arc α𝛼\alphaitalic_α as in Figure 2. If the tubing arcs in α𝛼\alphaitalic_α are immersed, then the isotopy class of the resulting tubed disk (Σ,α)Σ𝛼(\Sigma,\alpha)( roman_Σ , italic_α ) is not unique until an ordering of the arcs (i.e. an overstrand and an understrand) is chosen at each crossing to determine the relative radius of the tubes running along those arcs in the normal disk bundle of ΣΣ\Sigmaroman_Σ, as shown in Figure 2. A diagram on the disk D2superscript𝐷2D^{2}italic_D start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT of the pre-image of the tubing arcs, together with their endpoints and crossing information, is called a tube diagram for the tubed disk. | Figure 3. Isotopies of a tubed disk which move only the tubes and leave the rest of the disk fixed, as in Definition 1.5. These isotopies, drawn from the perspective of the tube diagram, are identical to those defined in [dave:LBT, Figure 5.6]. Re-ordering is also described by Schneiderman-Teichner in Figures 14141414 and 15151515 of [st] | A |
(a) ϵ=10−5italic-ϵsuperscript105\epsilon=10^{-5}italic_ϵ = 10 start_POSTSUPERSCRIPT - 5 end_POSTSUPERSCRIPT | Table 2: Computation times and accuracy for regression with Matérn kernels with ν∈[0.5,3.5]𝜈0.53.5\nu\in[0.5,3.5]italic_ν ∈ [ 0.5 , 3.5 ], and ρ∈[0.1,0.5]𝜌0.10.5\rho\in[0.1,0.5]italic_ρ ∈ [ 0.1 , 0.5 ]. The nodes and weights used are reported in Table 4. | 000.50.50.50.511111.51.51.51.522222.52.52.52.533333.53.53.53.544444.54.54.54.555555.55.55.55.566666.56.56.56.577777.57.57.57.5ξ𝜉\xiitalic_ξ | 000.50.50.50.511111.51.51.51.522222.52.52.52.533333.53.53.53.544444.54.54.54.555555.55.55.55.566666.56.56.56.577777.57.57.57.5ξ𝜉\xiitalic_ξ | Figure 1: Location of the 86868686 nodes for GPs defined on [−1,1]11[-1,1][ - 1 , 1 ] with Matérn kernels with ν∈[1.5,3.5]𝜈1.53.5\nu\in[1.5,3.5]italic_ν ∈ [ 1.5 , 3.5 ], and ρ∈[0.1,0.5]𝜌0.10.5\rho\in[0.1,0.5]italic_ρ ∈ [ 0.1 , 0.5 ]. | B |
}\cot{a\,x}}{2}\right)^{q}divide start_ARG d start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ( roman_cot italic_a italic_x ) end_ARG start_ARG d italic_x start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT end_ARG = ( 2 bold_italic_i italic_a ) start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ( - bold_italic_i + roman_cot italic_a italic_x ) ∑ start_POSTSUBSCRIPT italic_q = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT italic_q ! { FRACOP start_ARG italic_k end_ARG start_ARG italic_q end_ARG } ( - divide start_ARG 1 - bold_italic_i roman_cot italic_a italic_x end_ARG start_ARG 2 end_ARG ) start_POSTSUPERSCRIPT italic_q end_POSTSUPERSCRIPT | 7) and see if it’s possible to rewrite it. An expression for a similar sum exists in the literature, | The big breakthrough is really in the following straightforward identity, which only works for the analytic continuation of the Lerch ΦΦ\Phiroman_Φ and the polylogarithm functions at the negative integers, and makes it possible to obtain the former as a sum of the latter, | It’s possible to express this formula as a simple function of the polylogarithm. First, we rewrite it as, | And now that we know how to differentiate the cosecant successively, it’s possible to produce an explicit formula from f(x)𝑓𝑥f(x)italic_f ( italic_x ), again through the Leibniz rule. However, to make this process simpler, we resort to two artifices. First, to get rid of the extra x𝑥xitalic_x factor in the integral, we divide f(x)𝑓𝑥f(x)italic_f ( italic_x ) by x𝑥xitalic_x and take the (k−1)𝑘1(k-1)( italic_k - 1 )-th derivative instead of the k𝑘kitalic_k-th. Second, to avoid the complications of differentiating the sine, we replace it with an equivalent sum of exponential functions. | C |
},\ y\geq 0\},{ italic_x + italic_y blackboard_S : italic_x + italic_y italic_I start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∈ italic_U start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_x + italic_y italic_I start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ∈ italic_U start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_y ∈ blackboard_R , italic_y ≥ 0 } , | V+∪VΔ.superscript𝑉superscript𝑉Δ\displaystyle V^{+}\cup V^{\Delta}.italic_V start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT ∪ italic_V start_POSTSUPERSCRIPT roman_Δ end_POSTSUPERSCRIPT . | UΔ∼=UΔ∗=Ω.superscriptsubscript𝑈Δsimilar-tosuperscriptsubscript𝑈ΔΩU_{\Delta}^{\sim}=U_{\Delta}^{*}=\Omega.italic_U start_POSTSUBSCRIPT roman_Δ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∼ end_POSTSUPERSCRIPT = italic_U start_POSTSUBSCRIPT roman_Δ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT = roman_Ω . | W⊂V+Δ,W∩V+≠∅,formulae-sequence𝑊superscript𝑉Δ𝑊superscript𝑉W\subset V^{+\Delta},\qquad W\cap V^{+}\neq\varnothing,italic_W ⊂ italic_V start_POSTSUPERSCRIPT + roman_Δ end_POSTSUPERSCRIPT , italic_W ∩ italic_V start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT ≠ ∅ , | V+Δsuperscript𝑉Δ\displaystyle V^{+\Delta}italic_V start_POSTSUPERSCRIPT + roman_Δ end_POSTSUPERSCRIPT | D |
Cλ(ℚ)tor=ℤ/2ℤsubscript𝐶𝜆subscriptℚtorℤ2ℤC_{\lambda}(\mathbb{Q})_{\mathrm{tor}}=\mathbb{Z}/2\mathbb{Z}italic_C start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ( blackboard_Q ) start_POSTSUBSCRIPT roman_tor end_POSTSUBSCRIPT = blackboard_Z / 2 blackboard_Z (note that in [22, X.10.19 (d)] there is a misprint that Cλ(ℚ)tor=ℤ/3ℤsubscript𝐶𝜆subscriptℚtorℤ3ℤC_{\lambda}(\mathbb{Q})_{\mathrm{tor}}=\mathbb{Z}/3\mathbb{Z}italic_C start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ( blackboard_Q ) start_POSTSUBSCRIPT roman_tor end_POSTSUBSCRIPT = blackboard_Z / 3 blackboard_Z). Otherwise, we have Cλ(ℚ)torsubscript𝐶𝜆subscriptℚtorC_{\lambda}(\mathbb{Q})_{\mathrm{tor}}italic_C start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ( blackboard_Q ) start_POSTSUBSCRIPT roman_tor end_POSTSUBSCRIPT is trivial for a cube-free λ𝜆\lambdaitalic_λ. | The p𝑝pitalic_p-part of Conjecture 1.1 was obtained by Rubin [18] using Iwasawa theory for p≠2𝑝2p\neq 2italic_p ≠ 2 or 3333. In this paper, we contribute to the case p=3𝑝3p=3italic_p = 3. | The root number of Cλ/ℚsubscript𝐶𝜆ℚC_{\lambda}/\mathbb{Q}italic_C start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT / blackboard_Q was computed already in, for example, [12, Proposition A.2] when (3,λ)=13𝜆1(3,\lambda)=1( 3 , italic_λ ) = 1. The following is obtained easily by combining the results of Rohrlich [16] and Várilly-Alvarado [28], and is included for the convenience of the reader. | where ℒ=Ω𝒪ℒΩ𝒪\mathcal{L}=\Omega\mathcal{O}caligraphic_L = roman_Ω caligraphic_O denotes the period lattice. We explicitly compute the 3333-adic valuation of expressions involving ℘(u,ℒ)Weierstrass-p𝑢ℒ\wp(u,\mathcal{L})℘ ( italic_u , caligraphic_L ) following the method of Stephens [24], where it was used to obtain integrality of L(alg)(Cλ,1)superscript𝐿algsubscript𝐶𝜆1L^{(\mathrm{alg})}(C_{\lambda},1)italic_L start_POSTSUPERSCRIPT ( roman_alg ) end_POSTSUPERSCRIPT ( italic_C start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT , 1 ), even when 3∣λconditional3𝜆3\mid\lambda3 ∣ italic_λ. Stephen’s method, in turn, is based on the idea of Birch and Swinnerton-Dyer used in their proof of integrality of the algebraic part of the central L𝐿Litalic_L-value of quadratic twists of the curve y2=4x3−4xsuperscript𝑦24superscript𝑥34𝑥y^{2}=4x^{3}-4xitalic_y start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = 4 italic_x start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT - 4 italic_x in [1]. In this paper, we show in addition to integrality of L(alg)(Cλ,1)superscript𝐿algsubscript𝐶𝜆1L^{\mathrm{(alg)}}(C_{\lambda},1)italic_L start_POSTSUPERSCRIPT ( roman_alg ) end_POSTSUPERSCRIPT ( italic_C start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT , 1 ) that it is also divisible by an arbitrarily large power of 3333 as one increases the number of distinct prime divisors of λ𝜆\lambdaitalic_λ. In order to do this, we introduce an “averaged L𝐿Litalic_L-value” ΦλsubscriptΦ𝜆\Phi_{\lambda}roman_Φ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT, summing over the L𝐿Litalic_L-values of all cubic twists of Cλsubscript𝐶𝜆C_{\lambda}italic_C start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT with the same (sub)set of prime divisors, while fixing the exponent e(λ)𝑒𝜆e(\lambda)italic_e ( italic_λ ) of 3333 dividing λ𝜆\lambdaitalic_λ. It is important that e(λ)𝑒𝜆e(\lambda)italic_e ( italic_λ ) is fixed, as it turns out that the 3333-adic valuation of L(alg)(Cλ,1)superscript𝐿algsubscript𝐶𝜆1L^{(\mathrm{alg})}(C_{\lambda},1)italic_L start_POSTSUPERSCRIPT ( roman_alg ) end_POSTSUPERSCRIPT ( italic_C start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT , 1 ) depends on e(λ)𝑒𝜆e(\lambda)italic_e ( italic_λ ) (see Theorem 2.4). | In [12], we studied the 3333-adic valuation of L(alg)(Cλ,1)superscript𝐿algsubscript𝐶𝜆1L^{\mathrm{(alg)}}(C_{\lambda},1)italic_L start_POSTSUPERSCRIPT ( roman_alg ) end_POSTSUPERSCRIPT ( italic_C start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT , 1 ) in the case when (λ,3)=1𝜆31(\lambda,3)=1( italic_λ , 3 ) = 1. We crucially used that 3∤λnot-divides3𝜆3\nmid\lambda3 ∤ italic_λ, because 3333 divides the conductor of the base curve E1subscript𝐸1E_{1}italic_E start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT which is (27)27(27)( 27 ). The aim of this paper is to extend the lower bounds for the 3333-adic valuation of L(alg)(Cλ,1)superscript𝐿algsubscript𝐶𝜆1L^{\mathrm{(alg)}}(C_{\lambda},1)italic_L start_POSTSUPERSCRIPT ( roman_alg ) end_POSTSUPERSCRIPT ( italic_C start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT , 1 ) obtained in [12] to the case when 3∣λconditional3𝜆3\mid\lambda3 ∣ italic_λ. This is achieved by working with the Weierstrass ℘Weierstrass-p\wp℘-function whose real period ΩΩ\Omegaroman_Ω is given by Ω3=Ω1Ω3subscriptΩ1\frac{\Omega}{\sqrt{3}}=\Omega_{1}divide start_ARG roman_Ω end_ARG start_ARG square-root start_ARG 3 end_ARG end_ARG = roman_Ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT, satisfying the classical Weierstrass equation | A |
Therefore, we deduce from Lemmas 4.10 and 4.9 that H𝐻Hitalic_H embeds into ℬp,qsubscriptℬ𝑝𝑞\mathcal{B}_{p,q}caligraphic_B start_POSTSUBSCRIPT italic_p , italic_q end_POSTSUBSCRIPT for some p,q≥0𝑝𝑞0p,q\geq 0italic_p , italic_q ≥ 0. We conclude from Lemma 5.2 that H𝐻Hitalic_H must be finite. | It is worth noticing that, in our previous work [GLU22], we already constructed contractible cube complexes on which asymptotically rigid mapping class groups act. However, it may or may not be CAT(0). For instance, it turns out to be CAT(0) for the braided Ptolemy-Thompson groups brTnbrsubscript𝑇𝑛\mathrm{br}T_{n}roman_br italic_T start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT, but it turns out not to be CAT(0) for the braided Houghton groups (as shown in [GLU22, Section 3.3]) or some braided Higman-Thompson groups. The main obstruction comes from the absence of polygons with no puncture in the rigid structure. In the particular case of the braided Houghton and Higman-Thompson groups, one can overcome the difficulty by modifying artificially the planar trees and adding a “ghost vertex” that does not contribute to the puncture of the planar surface. Then, the construction of [GLU22] can adapted in a straightforward way, leading to a CAT(0) cube complex. Unfortunately, it seems unlikely that such a trick can be applied in full generality. For instance, difficulties already appear with the tree obtained from the right tree given by Figure 12 by subdividing all the horizontal edges. This is why a new approach was needed, and why the formalism of braided strand diagrams is fundamental in our work. | Finite subgroups in asymptotically rigid mapping class groups have been classified in [GLU22], and Theorem 1.2 proves that finitely generated torsion subgroups must be finite. This is sufficient to show that, in some cases, torsion subgroups are always finite. | Following the lines of [GLU22, Theorem 4.3], one can exploit the CAT(0) cube complexes we constructed in order to show that finite subgroups in Chambord groups are necessarily cyclic and to classify the possible orders of finite-order elements (depending on the arboreal semigroup presentation and the baseword). Because this is already done for asymptotically rigid mapping class groups in [GLU22], we do not pursue this direction further and leave the details to the reader. In this article, we restrict ourselves to the following observation about pure subgroups, as defined at the end of Section 2.4: | The main problem addressed in this article is the structure of the subgroups in asymptotically rigid mapping class groups. Our main result in this direction is the following statement: | C |
We do not discuss here in details the relation between the intervals of a permutation and its decomposition tree. | We point out to the interested readers that the nodes of T(σ)𝑇𝜎T(\sigma)italic_T ( italic_σ ) are the so-called | The proof is by induction on the depth (counted in number of edges) of T(σ)𝑇𝜎T(\sigma)italic_T ( italic_σ ). | Under this correspondence, σ(i)𝜎𝑖\sigma(i)italic_σ ( italic_i ) corresponds to the i𝑖iitalic_i-th leaf of T(σ)𝑇𝜎T(\sigma)italic_T ( italic_σ ) | we can represent every permutation by a tree, called its decomposition tree, which we denote T(σ)𝑇𝜎T(\sigma)italic_T ( italic_σ ). | A |
It is of importance in both theory and applications to investigate sample path properties of stochastic processes. One way of studying the irregularity properties of the sample paths of stochastic processes is by analyzing the smoothness of their local times. S. Berman has initiated this approach in a series of papers [3, 4, 5] by using Fourier analytic methods to Gaussian processes. Furthermore, he has introduced the concept of local nondeterminism (LND) for Gaussian processes to investigate the existence of jointly continuous local times. | differential equations driven by fractional Brownian motions. In [9], a new condition, called α𝛼\alphaitalic_α-local nondeterminism (α𝛼\alphaitalic_α-LND for short), has been introduced to investigate the existence and joint regularity of the local times of the solutions to systems of non-linear stochastic heat equations—which are neither Gaussian nor stable processes. Although the α𝛼\alphaitalic_α-LND condition has been specifically designed for non-Gaussian processes, in a Gaussian setting, the α𝛼\alphaitalic_α-LND property can be seen as a weaker condition than the classical LND (see Remark 2.9(ii)). Generally, the proof of the α𝛼\alphaitalic_α-LND condition relies on the technique of integration by parts derived from the Malliavin calculus. | for fixed t𝑡titalic_t, of the local times L(x,t)𝐿𝑥𝑡L(x,t)italic_L ( italic_x , italic_t ) of some classical Gaussian and stable processes, e.g. [12, 11]. However, to the best of our knowledge, there is no work in the literature treating the Besov regularity of L(x,∙)𝐿𝑥∙L(x,\bullet)italic_L ( italic_x , ∙ ) even for the Gaussian or stable processes. | It is well-known that the concept of local nondeterminism in the Gaussian framework means that “the value of the process at a given time point is relatively unpredictable based on a finite set of observations from the immediate past". In the Gaussian context, Berman uses conditional variance to express this. But unfortunately, he can’t use the conditional variance outside the Gaussian case because in a general framework the conditional variance is not deterministic. So, Berman has introduced the local g𝑔gitalic_g-nondeterminism concept for general processes by replacing the incremental variance, which is a measure of local unpredictability, by a measure of local | Since then, there have been a wide variety of extensions of the notion of local nondeterminism for Gaussian and stable processes, e.g. [20, 13, 19]. In the Gaussian case, the exponential form of the characteristic function allows expressing the LND property in terms of a condition on the variance. Nevertheless, the unknown form of the characteristic functions of general processes leads to difficulties in extending the LND condition beyond the Gaussian framework. Consequently, the LND property used in the Gaussian context should be replaced, for general processes, by fine estimations on the characteristic function of the increments. | D |
A lattice is distributive if it satisfies a∨(b∧c)=(a∨b)∧(a∨c)𝑎𝑏𝑐𝑎𝑏𝑎𝑐a\vee(b\wedge c)=(a\vee b)\wedge(a\vee c)italic_a ∨ ( italic_b ∧ italic_c ) = ( italic_a ∨ italic_b ) ∧ ( italic_a ∨ italic_c ) for all a,b,c∈L𝑎𝑏𝑐𝐿a,b,c\in Litalic_a , italic_b , italic_c ∈ italic_L. | The RE degrees are dense: for any RE sets A<TBsubscript𝑇𝐴𝐵A<_{T}Bitalic_A < start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT italic_B, there is a RE set C𝐶Citalic_C such that A<TC<TBsubscript𝑇𝐴𝐶subscript𝑇𝐵A<_{T}C<_{T}Bitalic_A < start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT italic_C < start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT italic_B. | The infimum A⊕Bdirect-sum𝐴𝐵A\oplus Bitalic_A ⊕ italic_B is defined as follows: A⊕Bdirect-sum𝐴𝐵A\oplus Bitalic_A ⊕ italic_B is a theory in the | We know that the interpretation degree structure of RE theories with the operators ⊕direct-sum\oplus⊕ and ⊗tensor-product\otimes⊗ is a distributive lattice. | The supremum A⊗Btensor-product𝐴𝐵A\otimes Bitalic_A ⊗ italic_B is defined as follows: A⊗Btensor-product𝐴𝐵A\otimes Bitalic_A ⊗ italic_B is a theory in | D |
τ𝜏\tauitalic_τ, σ𝜎\sigmaitalic_σ: Γ→ℝ→Γℝ\Gamma\to\mathbb{R}roman_Γ → blackboard_R as the value of the jump conditions; ΩΩ\Omegaroman_Ω as the | in the solvent region shuAccurateGradientApproximation2014 ; zhongImplicitBoundaryIntegral2018 ; zhouVariationalImplicitSolvation2014 . | ϵitalic-ϵ\epsilonitalic_ϵ, which can take on values of around 1111 or 2222 in the solute region and 80808080 | The coefficient ϵitalic-ϵ\epsilonitalic_ϵ is the same as (37). The source term and the jump conditions are calculated accordingly. | problem, with ϵitalic-ϵ\epsilonitalic_ϵ as thermal conductivity, can be solved for the temperature u𝑢uitalic_u. | B |
In this paper we solve the finite-horizon OSP featuring the identity as the gain function and an OUB as the underlying process. The solution is provided in terms of a non-linear, Volterra-type integral equation. Similarly to Shepp, (1969), our methodology relies on a time-space change that casts the original problem into an infinite-horizon OSP with a Brownian motion as the underlying process. Due to the complexity of our resulting OSP, we use a direct approach to solve it rather than using the common candidate-verification scheme. We then show that one can either apply the inverse transformation to recover the solution of the original OSP or, equivalently, solve the Volterra integral equation reformulated back in terms of OUB. It is worth highlighting that the BB framework is included in our analysis as a limit case. | The next three remarks broaden the scope of applicability of the OUB as the underlying model in (3). In particular, the two first reveal that setting the terminal time to 1111 and the pulling level (coming from the asymptotic mean of the OU process underneath) to 00 does not take a toll on generality, while the last one shows that the OSP for the BB arises as a limit case when α→0→𝛼0\alpha\to 0italic_α → 0. | In this paper we solve the finite-horizon OSP featuring the identity as the gain function and an OUB as the underlying process. The solution is provided in terms of a non-linear, Volterra-type integral equation. Similarly to Shepp, (1969), our methodology relies on a time-space change that casts the original problem into an infinite-horizon OSP with a Brownian motion as the underlying process. Due to the complexity of our resulting OSP, we use a direct approach to solve it rather than using the common candidate-verification scheme. We then show that one can either apply the inverse transformation to recover the solution of the original OSP or, equivalently, solve the Volterra integral equation reformulated back in terms of OUB. It is worth highlighting that the BB framework is included in our analysis as a limit case. | Arguing as Shepp, (1969) for the BB, we worked out the OUB case by coming up with an equivalent OSP having a Brownian motion as the underlying process after time-space transforming the OUB. Contrary to Shepp, (1969), the complexity of our problem did not allow us to guess a candidate solution, and we directly characterized the value function and the OSB by means of the pricing formula and the free-boundary equation. However, the equivalence between both OSPs was used only to facilitate technicalities along the proofs, and it is not necessary to compute the solution, since both the pricing formula and the free-boundary equation are also provided in the original formulation. We discussed how to use a Picard iteration algorithm to numerically approximate the OSB and displayed some examples to illustrate how different sets of values for the OUB’s parameters rule the shape of the OSB. | The rest of the paper is structured as follows. Section 2 introduces the main problem and some useful notation. In Section 3 we derive the transformed OSP and establish its equivalence to the original one. The most technical part of the paper is relegated to Section 4, in which we derive the solution of the reformulated OSP. From it, we use the reverse transformation to get the solution back to the original OSP in Section 5, where we also remark that both a BB and an OUB with general pulling level and terminal time are immediate consequences of our results. An algorithm for numerical approximations of the solution is given in Section 6, along with a compendium of illustrative cases for different values of the OUB’s parameters. Concluding remarks are relegated to Section 7. | D |
The performance of such techniques depend on multiple features of NPDs, including the functions used to construct these dictionaries and the datalength L𝐿Litalic_L [6]. The authors in [6] established that L≥2Pmax𝐿2subscript𝑃L\geq 2P_{\max}italic_L ≥ 2 italic_P start_POSTSUBSCRIPT roman_max end_POSTSUBSCRIPT is both necessary and sufficient to recover the exact support set of the sparse vector of a periodic signal (single period) using the program in (3), however, as discussed in Section II this is an NP-hard problem and is of limited practical use. | In a preliminary version of this work [16], we established a new and improved condition for exact sparse recovery with NPDs in the noise-free setting, limited to the special case where m=1𝑚1m=1italic_m = 1, i.e., when the periodic signal consists of one single period. In this work, we extend our study along several fronts. First, we establish exact recovery conditions for periodic mixtures consisting of multiple hidden periods. The conditions rest on newly introduced notions of nested periodic coherence and restricted coherence that leverage structural properties of NPDs. | We turn our attention in this section to the model in (19) in which the observation signal 𝐲~~𝐲\tilde{\mathbf{y}}over~ start_ARG bold_y end_ARG is a periodic signal 𝐲𝐲\mathbf{y}bold_y (18) contaminated with noise 𝐰𝐰\mathbf{w}bold_w. We limit the NPDs to those with real values, i.e., 𝐊∈ℝL×N𝐊superscriptℝ𝐿𝑁\mathbf{K}\in\mathbb{R}^{L\times N}bold_K ∈ blackboard_R start_POSTSUPERSCRIPT italic_L × italic_N end_POSTSUPERSCRIPT. In the context of period estimation, it is important to recover the exact support (not a subset), otherwise we could end up with an incorrect estimate of the period. Following [30], we focus on the OMP algorithm to recover the exact support of the periodic signal, where we use 𝐫isubscript𝐫𝑖\mathbf{r}_{i}bold_r start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT to denote the residual at iteration i𝑖iitalic_i of the OMP algorithm. We consider two cases for the additive noise 𝐰𝐰\mathbf{w}bold_w: the ℓ2subscriptℓ2\ell_{2}roman_ℓ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT-norm bounded noise, i.e., ‖𝐰‖2≤ϵsubscriptnorm𝐰2italic-ϵ\|\mathbf{w}\|_{2}\leq\epsilon∥ bold_w ∥ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ≤ italic_ϵ, and Gaussian noise, i.e., 𝐰∼𝒩(𝟎,σ2𝐈L)similar-to𝐰𝒩0superscript𝜎2subscript𝐈𝐿\mathbf{w}\sim\mathcal{N}\left(\mathbf{0},\sigma^{2}\mathbf{I}_{L}\right)bold_w ∼ caligraphic_N ( bold_0 , italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT bold_I start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT ). | The periodic signal y(n)𝑦𝑛y\left(n\right)italic_y ( italic_n ) could be a periodic mixture that consists of multiple hidden periods. In [6], the authors defined two criteria to characterize the hidden periods. First, the signal can be expressed as a combination of periodic signals yi,i=1…,mformulae-sequencesubscript𝑦𝑖𝑖1…𝑚y_{i},i=1\ldots,mitalic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_i = 1 … , italic_m, | We start off with the noise free case where the discrete-periodic signal is modeled as in (18). In general, we consider y𝑦yitalic_y to be a periodic mixture that consists of m𝑚mitalic_m periodic signals as expressed in (2). First, we introduce some definitions before stating the main results. All proofs are deferred to the appendix. | D |
^{L}p_{s}(x,y)e^{2\mu(y-L)}w_{1}(y)^{2}dy\;ds}_{\textstyle=:T_{2}}.under⏟ start_ARG blackboard_E start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT [ ∑ start_POSTSUBSCRIPT italic_v ∈ caligraphic_N start_POSTSUBSCRIPT italic_u end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_L end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_z ( italic_X start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT ( italic_u ) ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ] end_ARG start_POSTSUBSCRIPT = : italic_T start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT + 2 italic_ρ under⏟ start_ARG ∫ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_u end_POSTSUPERSCRIPT ∫ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_L end_POSTSUPERSCRIPT italic_p start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_x , italic_y ) italic_e start_POSTSUPERSCRIPT 2 italic_μ ( italic_y - italic_L ) end_POSTSUPERSCRIPT italic_w start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_y ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_d italic_y italic_d italic_s end_ARG start_POSTSUBSCRIPT = : italic_T start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT . | Let us now bound the double integral T2subscript𝑇2T_{2}italic_T start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT. First, recall from Equation (68) that the integral of the density pssubscript𝑝𝑠p_{s}italic_p start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT with respect to s𝑠sitalic_s can be bounded thanks to the Green function | Applying the many-to-one lemma and interchanging the integrals, the quantity U~~𝑈\tilde{U}over~ start_ARG italic_U end_ARG can be written as | Let us first bound the expectation T1subscript𝑇1T_{1}italic_T start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT. By the many-to-one lemma, | The many-to-two lemma (see Lemma 1.4)) combined with the formula for the first moment of Z′superscript𝑍′Z^{\prime}italic_Z start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT calculated in Lemma 3.2 yields | C |
We will prove that the recursive presentation of 𝒢2subscript𝒢2\mathcal{G}_{2}caligraphic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT | There are exactly 2nsuperscript2𝑛2^{n}2 start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT possible initial sets of relations and | (G,S)𝐺𝑆(G,S)( italic_G , italic_S ) and (H,S′)𝐻superscript𝑆′(H,S^{\prime})( italic_H , italic_S start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) satisfy exactly the same relations, and thus | 2nsuperscript2𝑛2^{n}2 start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT possible initial sets, and deleting incoherent ones until | relations while (s1,…,sm′)subscript𝑠1…subscript𝑠superscript𝑚′(s_{1},...,s_{m^{\prime}})( italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_s start_POSTSUBSCRIPT italic_m start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ) are not relations. Sets of this | A |
A marked group (G,S)𝐺𝑆(G,S)( italic_G , italic_S ) is called residually 𝒞𝒞\mathcal{C}caligraphic_C when any | Let 𝒞𝒞\mathcal{C}caligraphic_C be a class of marked groups, and (G,S)𝐺𝑆(G,S)( italic_G , italic_S ) a marked | For (G,S)𝐺𝑆(G,S)( italic_G , italic_S ) a marked group and 𝒞𝒞\mathcal{C}caligraphic_C a set of marked groups, | A residually 𝒞𝒞\mathcal{C}caligraphic_C marked group (G,S)𝐺𝑆(G,S)( italic_G , italic_S ) has a finite | A marked group (G,S)𝐺𝑆(G,S)( italic_G , italic_S ) is called residually 𝒞𝒞\mathcal{C}caligraphic_C when any | B |
Это подмногообразие расширенного фазового пространства. По теореме 1, оно является инвариантным многообразием системы (3). | 𝒯x˙x˙subscript𝒯˙𝑥˙𝑥\mathscr{T}_{\dot{x}\dot{x}}script_T start_POSTSUBSCRIPT over˙ start_ARG italic_x end_ARG over˙ start_ARG italic_x end_ARG end_POSTSUBSCRIPT положительно определена. При этом переменные x𝑥xitalic_x можно счиать локальными координатамми на некотором гладком многообразии размерности m𝑚mitalic_m. | Уравнения (3), (6) определенные на расширенном фазовом пространстве приходят из физики физики. При этом, фактически нас интересует | Из теоремы 6 вытекает, что уравнения Лагранжа (17) разрешимы относительно старших производных, т.е. представимы в нормальной форме. | Рассмотрим гладкую функцию ℱ(t,x,x˙)ℱ𝑡𝑥˙𝑥\mathscr{F}(t,x,\dot{x})script_F ( italic_t , italic_x , over˙ start_ARG italic_x end_ARG ), определенную на расширенном фазовом пространстве. | B |
The proof uses (besides the truncated maps jetN(Exp)𝑗𝑒subscript𝑡𝑁𝐸𝑥𝑝jet_{N}(Exp)italic_j italic_e italic_t start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT ( italic_E italic_x italic_p ), jet(Ln)𝑗𝑒𝑡𝐿𝑛jet(Ln)italic_j italic_e italic_t ( italic_L italic_n )) the full | The 𝒜𝒜\mathscr{A}script_A-equivalence is much more complicated. (Recall that the 𝒦𝒦{\mathscr{K}}\!script_K-equivalence was initially introduced as an auxiliary, | As was explained in §1.6, the methods of that lemma are not directly applicable to 𝒜𝒜\mathscr{A}script_A-equivalence. | The 𝒜𝒜\mathscr{A}script_A-case (the most important for the study of maps) is essentially more complicated than the (auxiliary) ℛ,𝒦ℛ𝒦\mathscr{R},{\mathscr{K}}\!script_R , script_K-cases. | simpler version of 𝒜𝒜\mathscr{A}script_A-equivalence.) A difficulty of 𝒜𝒜\mathscr{A}script_A (besides those mentioned in §1.1) is that the classical | A |
\langle x|roman_Φ start_POSTSUBSCRIPT caligraphic_M end_POSTSUBSCRIPT : italic_ρ ↦ ∑ start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ( roman_Tr italic_ρ italic_M start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ) | italic_x ⟩ ⟨ italic_x |, where | positive semidefinite operators on ℋℋ\mathcal{H}caligraphic_H is denoted as 𝒫(ℋ)𝒫ℋ\mathcal{P}(\mathcal{H})caligraphic_P ( caligraphic_H ). | Let σ𝜎\sigmaitalic_σ be a self-adjoint operator on ℋℋ\mathcal{H}caligraphic_H with spectral projections | on ℋℋ\mathcal{H}caligraphic_H is denoted as 𝒰(ℋ)𝒰ℋ\mathcal{U}(\mathcal{H})caligraphic_U ( caligraphic_H ), and the set of | eigenvalues of σ𝜎\sigmaitalic_σ. The associated pinching map ℰσsubscriptℰ𝜎\mathcal{E}_{\sigma}caligraphic_E start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT is | B |
For every phase k≥0𝑘0k\geq 0italic_k ≥ 0, we have c~(k)∈Im(A⊤)+c−rsuperscript~𝑐𝑘Imsuperscript𝐴top𝑐𝑟\tilde{c}^{(k)}\in\operatorname{Im}(A^{\top})+c-rover~ start_ARG italic_c end_ARG start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT ∈ roman_Im ( italic_A start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT ) + italic_c - italic_r for some 𝟘≤r≤kδ𝟙double-struck-𝟘𝑟𝑘𝛿double-struck-𝟙\mathbb{0}\leq r\leq k\delta\mathbb{1}blackboard_𝟘 ≤ italic_r ≤ italic_k italic_δ blackboard_𝟙. | Next, suppose that the statement holds for k𝑘kitalic_k, and consider the inductive step k+1𝑘1k+1italic_k + 1. | Next, suppose that the statement holds for k𝑘kitalic_k, and consider the inductive step k+1𝑘1k+1italic_k + 1. | Let t𝑡titalic_t be the first iteration of a phase k≥1𝑘1k\geq 1italic_k ≥ 1. For any iteration r≥t𝑟𝑡r\geq titalic_r ≥ italic_t, | Next, suppose that the statement holds for ℓ−k=dℓ𝑘𝑑\ell-k=droman_ℓ - italic_k = italic_d, and consider the inductive step ℓ−k=d+1ℓ𝑘𝑑1\ell-k=d+1roman_ℓ - italic_k = italic_d + 1. | A |
(r2−r02)2=β4{t2+(2−r0)t+1}2.superscriptsuperscript𝑟2superscriptsubscript𝑟022superscript𝛽4superscriptsuperscript𝑡22subscript𝑟0𝑡12(r^{2}-r_{0}^{2})^{2}=\beta^{4}\{t^{2}+(2-r_{0})t+1\}^{2}.( italic_r start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - italic_r start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = italic_β start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT { italic_t start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + ( 2 - italic_r start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) italic_t + 1 } start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT . | Repeating this inductively, we thus have G(n)(t0)=0superscript𝐺𝑛subscript𝑡00{G}^{(n)}(t_{0})=0italic_G start_POSTSUPERSCRIPT ( italic_n ) end_POSTSUPERSCRIPT ( italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) = 0 for all n≥0𝑛0n\geq 0italic_n ≥ 0, where G(n)(t)superscript𝐺𝑛𝑡G^{(n)}(t)italic_G start_POSTSUPERSCRIPT ( italic_n ) end_POSTSUPERSCRIPT ( italic_t ) denotes the n𝑛nitalic_n-th derivative of G(t)𝐺𝑡G(t)italic_G ( italic_t ) with respect to t𝑡titalic_t. | The polynomial c2(r)subscript𝑐2𝑟c_{2}(r)italic_c start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_r ) is separable. | Thus, we have that G(t)𝐺𝑡G(t)italic_G ( italic_t ) is not separable by (3.9), and hence this is a contradiction. | Assume that G(t)𝐺𝑡G(t)italic_G ( italic_t ) has a multiple root t0subscript𝑡0t_{0}italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, then we have G(t0)=G′(t0)=0𝐺subscript𝑡0superscript𝐺′subscript𝑡00G(t_{0})={G}^{\prime}(t_{0})=0italic_G ( italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) = italic_G start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) = 0 and G′′(t0)=0superscript𝐺′′subscript𝑡00{G}^{\prime\prime}(t_{0})=0italic_G start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ( italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) = 0 by (3.13). | C |
2}]blackboard_E [ ∑ start_POSTSUBSCRIPT italic_m = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n - 1 end_POSTSUPERSCRIPT italic_Y start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ] = blackboard_E [ ( ∑ start_POSTSUBSCRIPT italic_m = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n - 1 end_POSTSUPERSCRIPT italic_Y start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ], which yields (3.13) by way of (3.11), (3.12), and (3.13). This completes the proof. ∎ | By the argument used in the proof of Lemma 3.1 of [30], we may assume without loss of generality that x(0)=x(1)=0𝑥0𝑥10x(0)=x(1)=0italic_x ( 0 ) = italic_x ( 1 ) = 0. | which then reproduces Theorem 2.4. To prepare for the proof of Theorem 9.2, we define for n∈ℕ𝑛ℕn\in\mathbb{N}italic_n ∈ blackboard_N, | Now we move toward the proof of Theorem 2.4. The following auxiliary lemma will be needed in that proof. | to a martingale that is hidden in the Faber–Schauder expansion of x𝑥xitalic_x. The corresponding theory is developed in Section 3, where also the proof of Theorem 2.4 is given. | C |
α(v)←min(ℕ>0∖{α(u):u∈N(v)})←𝛼𝑣superscriptℕabsent0conditional-set𝛼𝑢𝑢𝑁𝑣\alpha(v)\leftarrow\min(\mathbb{N}^{>0}\setminus\{\alpha(u)\colon u\in N(v)\})italic_α ( italic_v ) ← roman_min ( blackboard_N start_POSTSUPERSCRIPT > 0 end_POSTSUPERSCRIPT ∖ { italic_α ( italic_u ) : italic_u ∈ italic_N ( italic_v ) } ) | Let α𝛼\alphaitalic_α be a proper vertex coloring of an oriented graph G𝐺Gitalic_G and let <<< be a total ordering of α(V(G))𝛼𝑉𝐺\alpha(V(G))italic_α ( italic_V ( italic_G ) ). We say that (α,<)𝛼(\alpha,<)( italic_α , < ) is an out-tree coloring of G𝐺Gitalic_G if for every v∈V(G)𝑣𝑉𝐺v\in V(G)italic_v ∈ italic_V ( italic_G ), α(N+(v))⊇{i∈α(V(G)):i<α(v)}conditional-set𝑖𝛼𝑉𝐺𝑖𝛼𝑣𝛼superscript𝑁𝑣\alpha(N^{+}(v))\supseteq\{i\in\alpha(V(G))\colon i<\alpha(v)\}italic_α ( italic_N start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT ( italic_v ) ) ⊇ { italic_i ∈ italic_α ( italic_V ( italic_G ) ) : italic_i < italic_α ( italic_v ) }. When the ordering <<< on the colors is clear from the context, we shorten “(α,<)𝛼(\alpha,<)( italic_α , < ) is an out-tree coloring” to just “α𝛼\alphaitalic_α is an out-tree coloring”. | (i.e., α(v)𝛼𝑣\alpha(v)italic_α ( italic_v ) is set equal to the least positive integer that has not already been assigned to a neighbour of v𝑣vitalic_v). | An undirected edge between the vertices u𝑢uitalic_u and v𝑣vitalic_v of G𝐺Gitalic_G shall be denoted as uv𝑢𝑣uvitalic_u italic_v, whereas an edge directed from u𝑢uitalic_u to v𝑣vitalic_v shall be denoted as (u,v)𝑢𝑣(u,v)( italic_u , italic_v ). Two vertices are said to be adjacent in G𝐺Gitalic_G if there is a (directed or undirected) edge between them in G𝐺Gitalic_G. The set of all vertices adjacent to a vertex v𝑣vitalic_v in G𝐺Gitalic_G is called the neighbourhood of v𝑣vitalic_v in G𝐺Gitalic_G and is denoted by NG(v)subscript𝑁𝐺𝑣N_{G}(v)italic_N start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( italic_v ) (the subscript G𝐺Gitalic_G is omitted when the graph under consideration is clear). In a directed graph G𝐺Gitalic_G, if (u,v)∈E(G)𝑢𝑣𝐸𝐺(u,v)\in E(G)( italic_u , italic_v ) ∈ italic_E ( italic_G ), then v𝑣vitalic_v is said to be an out-neighbour of u𝑢uitalic_u, and u𝑢uitalic_u is said to be an in-neighbour of v𝑣vitalic_v. For a directed graph G𝐺Gitalic_G and a vertex v∈V(G)𝑣𝑉𝐺v\in V(G)italic_v ∈ italic_V ( italic_G ), the out-neighbourhood (resp. in-neighbourhood) of v𝑣vitalic_v, i.e. the set of out-neighbours (resp. in-neighbours) of v𝑣vitalic_v in G𝐺Gitalic_G, is denoted by NG+(v)subscriptsuperscript𝑁𝐺𝑣N^{+}_{G}(v)italic_N start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( italic_v ) (resp. NG−(v)subscriptsuperscript𝑁𝐺𝑣N^{-}_{G}(v)italic_N start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( italic_v )). (Again, the subscript is omitted when the graph under consideration is clear from the context.) Note that for every vertex v∈V(G)𝑣𝑉𝐺v\in V(G)italic_v ∈ italic_V ( italic_G ), N(v)=N+(v)∪N−(v)𝑁𝑣superscript𝑁𝑣superscript𝑁𝑣N(v)=N^{+}(v)\cup N^{-}(v)italic_N ( italic_v ) = italic_N start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT ( italic_v ) ∪ italic_N start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT ( italic_v ). | Assign a direction to each edge uv𝑢𝑣uvitalic_u italic_v of G𝐺Gitalic_G as follows: If α(u)<α(v)𝛼𝑢𝛼𝑣\alpha(u)<\alpha(v)italic_α ( italic_u ) < italic_α ( italic_v ), direct the edge from v𝑣vitalic_v to u𝑢uitalic_u; otherwise, | B |
2.3 Edges of G𝐺Gitalic_G in case of non-uniqueness of continuations of a prefix of a tropical recurrent sequence | Proof. From subsection 6.1 it follows that the number of vertices V𝑉Vitalic_V in G𝐺Gitalic_G is less than the amount of orders on an n𝑛nitalic_n-element set, hence it is less than nn.superscript𝑛𝑛n^{n}.italic_n start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT . Thus, we can put M=1𝑀1M=1italic_M = 1 in the bounds. The remaining part of the proof is literally as in the proofs of the mentioned statements. □□\Box□ | Now we study the case when |S|≥2𝑆2|S|\geq 2| italic_S | ≥ 2 and the inequality (31) is true. We claim that in this case (x1,…,xn)∈Pwsubscript𝑥1…subscript𝑥𝑛subscript𝑃𝑤(x_{1},\dots,x_{n})\in P_{w}( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_x start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) ∈ italic_P start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT where the edge (v,w)𝑣𝑤(v,w)( italic_v , italic_w ) of G𝐺Gitalic_G is constructed according to Definition 2.11. Recall that in this case Bw=(B∖{1})−1subscript𝐵𝑤𝐵11B_{w}=(B\setminus\{1\})-1italic_B start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT = ( italic_B ∖ { 1 } ) - 1, and 2≤t≤n2𝑡𝑛2\leq t\leq n2 ≤ italic_t ≤ italic_n is the minimal element of S∖{1}𝑆1S\setminus\{1\}italic_S ∖ { 1 }. The inequalities (29) are fulfilled according to the construction in Definition 2.11. The inequalities (30) follow from (8) taking into account that s≤sw+1𝑠subscript𝑠𝑤1s\leq s_{w}+1italic_s ≤ italic_s start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT + 1 due to Lemma 1.4 and that ys≤ysw+1=xswsubscript𝑦𝑠subscript𝑦subscript𝑠𝑤1subscript𝑥subscript𝑠𝑤y_{s}\leq y_{s_{w}+1}=x_{s_{w}}italic_y start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ≤ italic_y start_POSTSUBSCRIPT italic_s start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT + 1 end_POSTSUBSCRIPT = italic_x start_POSTSUBSCRIPT italic_s start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT end_POSTSUBSCRIPT (cf. the similar argument in the case |S|=1𝑆1|S|=1| italic_S | = 1 above). The inequalities (32) for 1≤j<n1𝑗𝑛1\leq j<n1 ≤ italic_j < italic_n are justified as above in case |S|=1𝑆1|S|=1| italic_S | = 1. The inequality (32) for j=n𝑗𝑛j=nitalic_j = italic_n coincides with (31). Thus, the point (x1,…,xn)subscript𝑥1…subscript𝑥𝑛(x_{1},\dots,x_{n})( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_x start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) belongs to the polyhedron Pwsubscript𝑃𝑤P_{w}italic_P start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT for an edge (v,w)𝑣𝑤(v,w)( italic_v , italic_w ) construced in Definition 2.11. This proves the claim when |S|≥2𝑆2|S|\geq 2| italic_S | ≥ 2 and (31) holds. | The constructed vertex w𝑤witalic_w is the unique one to which there is an edge in the graph G𝐺Gitalic_G from the vertex v𝑣vitalic_v such that the coordinate xnsubscript𝑥𝑛x_{n}italic_x start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT is unbounded. Still we assume that |S|≥2,t∈Sformulae-sequence𝑆2𝑡𝑆|S|\geq 2,t\in S| italic_S | ≥ 2 , italic_t ∈ italic_S with a minimal possible t>1𝑡1t>1italic_t > 1. Now we construct vertices w𝑤witalic_w with a bounded coordinate xnsubscript𝑥𝑛x_{n}italic_x start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT to which there are edges from v𝑣vitalic_v. | Now we study the case when the set S𝑆Sitalic_S (see Definition 2.8) consists of more than one element. Take a minimal t>1𝑡1t>1italic_t > 1 such that | D |
The first-named author was supported by the National Science Foundation grant DMS-1904342 and CAREER grant DMS-2142575, by PSC-CUNY Award # 62074-00 50, and Fapesp (2019/19891-9). The second-named author was supported by grants from CNPq and Fapesp (2022/14254-3, 2022/16097-2), Brazil. | The cohomogeneity of a 𝖦𝖦\mathsf{G}sansserif_G-invariant submanifold Σ⊂MΣ𝑀\Sigma\subset Mroman_Σ ⊂ italic_M is the codimension k𝑘kitalic_k that principal 𝖦𝖦\mathsf{G}sansserif_G-orbits have inside ΣΣ\Sigmaroman_Σ. | For instance, principal orbits are themselves 𝖦𝖦\mathsf{G}sansserif_G-invariant submanifolds of M𝑀Mitalic_M, of cohomogeneity k=0𝑘0k=0italic_k = 0. | Clearly, e.g., by Palais’ symmetric criticality principle, such a cohomogeneity 00 submanifold Σ⊂MΣ𝑀\Sigma\subset Mroman_Σ ⊂ italic_M is minimal if and only if Σ/𝖦=Π(Σ)∈Mpr/𝖦Σ𝖦ΠΣsubscript𝑀pr𝖦\Sigma/\mathsf{G}=\Pi(\Sigma)\in M_{{\rm{pr}}}/\mathsf{G}roman_Σ / sansserif_G = roman_Π ( roman_Σ ) ∈ italic_M start_POSTSUBSCRIPT roman_pr end_POSTSUBSCRIPT / sansserif_G is a critical point of (2.1). This characterization is naturally extended to submanifolds of cohomogeneity k≥1𝑘1k\geq 1italic_k ≥ 1 in the following well-known result [HL71, Thm. 2]. | For the convenience of the reader, we recall the symmetry reduction principle for 𝖦𝖦\mathsf{G}sansserif_G-invariant minimal submanifolds pioneered by Hsiang and Lawson [HL71], | D |
)}}\,.over^ start_ARG italic_A end_ARG start_POSTSUPERSCRIPT ( ∘ ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_C end_POSTSUBSCRIPT start_FLOATSUPERSCRIPT - 1 end_FLOATSUPERSCRIPT = italic_π 2 over^ start_ARG italic_K end_ARG start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT | start_POSTSUBSCRIPT caligraphic_H start_POSTSUPERSCRIPT ( ∘ ) end_POSTSUPERSCRIPT end_POSTSUBSCRIPT , over^ start_ARG italic_A end_ARG start_POSTSUPERSCRIPT ( ∙ ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_C end_POSTSUBSCRIPT start_FLOATSUPERSCRIPT - 1 end_FLOATSUPERSCRIPT = italic_π over^ start_ARG italic_K end_ARG start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT | start_POSTSUBSCRIPT caligraphic_H start_POSTSUPERSCRIPT ( ∙ ) end_POSTSUPERSCRIPT end_POSTSUBSCRIPT . | In addition, since their convolution kernels are Bochner-Riesz integral kernels [27], the operators A^C(∘)subscriptsuperscript^𝐴C{\hat{A}}^{(\circ)}_{\rm C}over^ start_ARG italic_A end_ARG start_POSTSUPERSCRIPT ( ∘ ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_C end_POSTSUBSCRIPT and A^C(∙)subscriptsuperscript^𝐴∙C{\hat{A}}^{(\bullet)}_{\rm C}over^ start_ARG italic_A end_ARG start_POSTSUPERSCRIPT ( ∙ ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_C end_POSTSUBSCRIPT are positive definite (despite the fact that the kernels are oscillatory functions), bounded and invertible. We can obtain a resolution of the identity operator using the theory of frames (see the Appendix in | [3] and references therein), using the concept of dual frame, and for that purpose the inverse of the resolution operator is required. | Although the resolution operator is not the identity operator in ℋℋ{\cal H}caligraphic_H, the fact that it is bounded, invertible and with a bounded inverse, allows to define a dual family of coherent states (the dual frame, see the Appendix in [3]). These are defined by: | The content of the paper is as follows. In Section 2 we review the study of an infinite set of equally spaced homogeneous waveguide arrays, obtaining the differential equations describing the amplitudes of light along the waveguides and computing the propagator. In Section 3 we recall that the symmetry of this system is the Euclidean group and we construct coherent states of the Perelomov type. In Section 4 a resolution of the identity for these coherent states is built using the fact that coherent states satisfy the Helmholtz equation. In Sec. 5 a physical realization of E(2) coherent states and of Cauchy Coherent states is provided in tilted waveguide arrays. | C |
Then, there exists a K𝒳subscript𝐾𝒳K_{\mathcal{X}}italic_K start_POSTSUBSCRIPT caligraphic_X end_POSTSUBSCRIPT-flop 𝒳⇢𝒳+⇢𝒳superscript𝒳\mathcal{X}\dashrightarrow\mathcal{X}^{+}caligraphic_X ⇢ caligraphic_X start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT over T𝑇Titalic_T extending 𝒳0⇢𝒳0+⇢subscript𝒳0superscriptsubscript𝒳0\mathcal{X}_{0}\dashrightarrow\mathcal{X}_{0}^{+}caligraphic_X start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ⇢ caligraphic_X start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT. | After the stratification, T𝑇Titalic_T is the disjoint union of finitely many irreducible components. | Finally, since T𝑇Titalic_T has finitely many irreducible components, restricting to one of these components we may also assume that T𝑇Titalic_T is irreducible, by Noetherian induction. | By iteration of this argument on all the finitely many components of T𝑇Titalic_T and by Noetherian induction on the closed subsets of T𝑇Titalic_T removed in the construction, this shows the claim. | Up to passing to a stratification and an étale base change of the original parameter space T𝑇Titalic_T, we may assume that Theorem 3.4 and Theorem 4.2 apply to the pull-back of the morphisms in (6.44) to each of the finitely many irreducible components of T𝑇Titalic_T. | A |
The proposed class of q𝑞qitalic_q-gradient estimators for the residual dependence index η>0𝜂0\eta>0italic_η > 0 is here applied to aid the study of extreme values arising in tropical rainfall. Our data consists of daily rainfall measurements collected over 68 years, between the years 1950 and 2017, at 591 irregularly spaced stations across Ghana. The data were collected, processed and quality controlled by the Ghana Meteorological Agency (Israelsson et al.,, 2020). | Importantly, since even during the monsoon season there are many dry days (i.e., daily rainfall amounts <1mmabsent1𝑚𝑚<1mm< 1 italic_m italic_m), we have chosen to conduct estimation of the residual dependence index by drawing only on rainy days with rainfall measurements above the 90%-empirical quantile. | The proposed class of q𝑞qitalic_q-gradient estimators for the residual dependence index η>0𝜂0\eta>0italic_η > 0 is here applied to aid the study of extreme values arising in tropical rainfall. Our data consists of daily rainfall measurements collected over 68 years, between the years 1950 and 2017, at 591 irregularly spaced stations across Ghana. The data were collected, processed and quality controlled by the Ghana Meteorological Agency (Israelsson et al.,, 2020). | Ghana has a strong seasonal rainfall cycle regulated by the West African monsoon. For the purposes of this illustrative analysis, we have singled out a pair of nearby gauging stations (within the range 5-15km𝑘𝑚kmitalic_k italic_m) and a pair of stations distancing 190-200km𝑘𝑚kmitalic_k italic_m from each other. The focus of our analysis is on daily rainfall measurements collected every June in the 68 years worth of available data because we wish to include the main rainy season, with the obvious advantage that it contains the largest proportion of rainy days and likely the highest frequency of extreme rainfall occurrences as well. | In order to ensure that the data are identically distributed, we will only consider the stations in the southern part of the country (south of 8∘superscript88^{\circ}8 start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT lat) since there are significant contrasts in rainfall regimes across the country. Namely, the north of Ghana is semi-arid, exhibiting a uni-model seasonal cycle, peaking in July/August; the south is more humid, with a bi-modal seasonal cycle, with peaks in June and October, and a break in August. None of the individual time series are without missing values, and the proportion of these was found to range between 5% to 95%. As an initial screening, we removed all stations with more than 3000 missing values, roughly equating to 12% of the full time series, which left us with 40 stations from which to select pairs. To minimise distributional variations due to systematic features in the spatial domain, one of the three chosen stations (ASU) is common for the two pairs and the two pairs are roughly aligned on the same bearing. Figure 4 highlights this with BRI located 5-10km𝑘𝑚kmitalic_k italic_m away from ASU, and MAM 190-200km𝑘𝑚kmitalic_k italic_m away. After pre-processing the data recorded at each station, in order to remove inconsistencies, we were left with 1830 bivariate observations validated for our analysis of whether asymptotic independence is present for extreme values in the data. In Israelsson et al., (2020), it is concluded that for moderate values of rainfall, stations located more than 150km𝑘𝑚kmitalic_k italic_m apart appear to no longer exhibit dependence in terms of simultaneous rainfall occurrences. However, some evidence was found of tenuous dependence in very heavy rainfall (i.e. of magnitude >50mmabsent50𝑚𝑚>50mm> 50 italic_m italic_m), during the month of June, even for pairs of stations at such long distance as 150km150𝑘𝑚150km150 italic_k italic_m apart. Building on these findings, we have settled with 190-200km𝑘𝑚kmitalic_k italic_m as a benchmark for the distance at which we expect to find some evidence of asymptotic independence in extreme tropical rainfall. | C |
Noticing that the de Finetti state ζRnAnsubscript𝜁subscript𝑅𝑛superscript𝐴𝑛\zeta_{R_{n}A^{n}}italic_ζ start_POSTSUBSCRIPT italic_R start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT italic_A start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT end_POSTSUBSCRIPT in Lemma 8 is identical to ζR~R¯nAnsubscript𝜁~𝑅superscript¯𝑅𝑛superscript𝐴𝑛\zeta_{\tilde{R}\bar{R}^{n}A^{n}}italic_ζ start_POSTSUBSCRIPT over~ start_ARG italic_R end_ARG over¯ start_ARG italic_R end_ARG start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT italic_A start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT end_POSTSUBSCRIPT by setting Rn≡R~R¯nsubscript𝑅𝑛~𝑅superscript¯𝑅𝑛R_{n}\equiv\tilde{R}\bar{R}^{n}italic_R start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ≡ over~ start_ARG italic_R end_ARG over¯ start_ARG italic_R end_ARG start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT, we complete the proof. | The following proposition is about properties of the lower and upper bounds of the reliability function. | Based on Theorem 9 and Theorem 10, we prove the following characterization of the reliability function. | In this paper, we study the reliability function of reverse Shannon simulation of a quantum channel 𝒩A→Bsubscript𝒩→𝐴𝐵\mathcal{N}_{A\rightarrow B}caligraphic_N start_POSTSUBSCRIPT italic_A → italic_B end_POSTSUBSCRIPT. We use purified distance as the measure of performance, which, being a function of the fidelity, is a natural distance measure in quantum information. Building on the preceding work [33] of the authors, we derive tight upper and lower bounds for the reliability function. We show that these two bounds coincide when the rate of classical communication cost is below a critical value Rcriticalsubscript𝑅criticalR_{\textrm{critical}}italic_R start_POSTSUBSCRIPT critical end_POSTSUBSCRIPT. Thus, in the low-rate case, we have determined the exact formula of the reliability function. In the derivation, we have obtained an explicit achievability bound for the performance of simulating 𝒩A→B⊗nsuperscriptsubscript𝒩→𝐴𝐵tensor-productabsent𝑛\mathcal{N}_{A\rightarrow B}^{\otimes n}caligraphic_N start_POSTSUBSCRIPT italic_A → italic_B end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊗ italic_n end_POSTSUPERSCRIPT for a finite blocklength n𝑛nitalic_n, which is of realistic significance. These results are given in terms of the sandwiched Rényi divergence [35, 36]. More specifically, the channel’s sandwiched Rényi information [37], defined as follows, plays a crucial role: | We have derived fundamental achievability and converse bounds for the performance of reverse Shannon simulation of a quantum channel. These bounds translate to the corresponding lower and upper bounds for the reliability function. By showing that the lower bound and upper bound coincide when the classical communication rate r𝑟ritalic_r is such that r≤Rcritical𝑟subscript𝑅criticalr\leq R_{\textrm{critical}}italic_r ≤ italic_R start_POSTSUBSCRIPT critical end_POSTSUBSCRIPT, we have obtained the exact formula of the reliability function for r𝑟ritalic_r falling in this range. Remarkably, our result has provided an operational interpretation to the channel’s sandwiched Rényi information Iα(𝒩)subscript𝐼𝛼𝒩I_{\alpha}(\mathcal{N})italic_I start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ( caligraphic_N ) of order α∈(1,2]𝛼12\alpha\in(1,2]italic_α ∈ ( 1 , 2 ], in characterizing the direct error exponent. We point out that in [57], an operational interpretation for Iα(𝒩)subscript𝐼𝛼𝒩I_{\alpha}(\mathcal{N})italic_I start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ( caligraphic_N ) with α∈(1,+∞)𝛼1\alpha\in(1,+\infty)italic_α ∈ ( 1 , + ∞ ) has been found by the authors, in characterizing the strong converse exponent of entanglement-assisted communication. | A |
However, due to the large number of variables to be estimated, the acquisition of accurate instantaneous CSI for (massive) MIMO leads to significant training overhead, and channel estimation is a challenging task, especially in the scenarios that wireless channels are fast time-varying. The implementation of grant-free MIMO-NOMA systems is even more challenging, but it has great intentional for massive machine type communications. | This paper deals with the design of grant-free MIMO-NOMA for machine type communications, where each device is equipped with a single antenna and the AP is equipped with a (massive) number of antennas. We assume that the data payload of the devices is small (e.g., a few bits), and the devices send information to an access point (AP) randomly. In addition, the data packet length of the devices is not fixed, so the devices are not synchronized in the packet level, i.e., any device is allowed to freely access the channel whenever it has information to transmit, and it becomes inactive when its packet is delivered. Due to the high flexibility of the transmissions and the consideration of high transmission efficiency, the realization of the system is very challenging, especially the implementation of the receiver at the AP, as it needs to identify the active devices and perform multi-device data detection. The conventional grant-free NOMA strategy is to first perform active user identification and channel estimation, followed by coherent data detection. However, due to a large number of antennas at the AP and the data packet of the devices is very short, the conventional strategy will render the transmissions highly inefficient due to the requirement of long training signals for active user identification and channel estimation. Even the large training overhead is not an issue, it is unknown how to use the conventional strategy to handle the asynchronous transmissions of the IoT devices, where the devices have different packet lengths and are allowed to access and leave the channel freely. | (1) We propose to solve the challenge by adopting differential modulation at the devices and designing non-coherent multi-device data detection at the AP. Such design can bypass channel estimation. The implementation of the transmitters is very simple, where each device is assigned a spreading sequence and transmits information with differential modulation. The receiver needs to first identify active users without the assistance of any training signals, and then perform multi-device data detection for the active users. | We assume an IoT communication system, where an access point (AP) equipped with N𝑁Nitalic_N antennas is used to serve U𝑈Uitalic_U IoT devices, each equipped with a single antenna. In particular, we consider uplink transmission, i.e., the IoT devices send information to the AP. As the transmissions of the IoT devices are sporadic, i.e., not all devices are active at the same time, we use K𝐾Kitalic_K to denote the number of active devices served by the AP. | Due to the small payload of the IoT devices, the requirement of low latency and (larger) number of antennas used at the AP, the use of training symbols for channel estimation and active device detection is not preferable. In this work, we design a non-coherent scheme and differential modulation is employed by the IoT devices, so that the use of training symbols can be avoided. | A |
To state the result, we use ∑s,t∗superscriptsubscript𝑠𝑡\sum_{s,t}^{*}∑ start_POSTSUBSCRIPT italic_s , italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT to denote the sum over s𝑠sitalic_s and t𝑡titalic_t where the term with s=0𝑠0s=0italic_s = 0 is weighted by 1212\frac{1}{2}divide start_ARG 1 end_ARG start_ARG 2 end_ARG. | We have222Here the notation ∑±subscriptplus-or-minus\displaystyle\sum_{\pm}∑ start_POSTSUBSCRIPT ± end_POSTSUBSCRIPT means that we sum over the choice +++ as well as the choice −--. | We say that the set of xEsubscript𝑥𝐸x_{E}italic_x start_POSTSUBSCRIPT italic_E end_POSTSUBSCRIPT for E∈{Eα,β:α,β∈𝔽p}𝐸conditional-setsubscript𝐸𝛼𝛽𝛼𝛽subscript𝔽𝑝E\in\{E_{\alpha,\beta}:\alpha,\beta\in\mathbb{F}_{p}\}italic_E ∈ { italic_E start_POSTSUBSCRIPT italic_α , italic_β end_POSTSUBSCRIPT : italic_α , italic_β ∈ blackboard_F start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT } becomes equidistributed with respect to the Sato–Tate measure as p→∞→𝑝p\to\inftyitalic_p → ∞. Birch’s result has been extended to the case where 𝔽psubscript𝔽𝑝\mathbb{F}_{p}blackboard_F start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT is replaced with an arbitrary finite field 𝔽prsubscript𝔽superscript𝑝𝑟\mathbb{F}_{p^{r}}blackboard_F start_POSTSUBSCRIPT italic_p start_POSTSUPERSCRIPT italic_r end_POSTSUPERSCRIPT end_POSTSUBSCRIPT (see the work of Brock and Granville [3] as well as [16]). Instead of considering the sample space defined by the coefficients of the cubic polynomial, we investigate subsets of the set ℰ(pr)ℰsuperscript𝑝𝑟\mathcal{E}(p^{r})caligraphic_E ( italic_p start_POSTSUPERSCRIPT italic_r end_POSTSUPERSCRIPT ) of isomorphism classes of elliptic curves over 𝔽prsubscript𝔽superscript𝑝𝑟\mathbb{F}_{p^{r}}blackboard_F start_POSTSUBSCRIPT italic_p start_POSTSUPERSCRIPT italic_r end_POSTSUPERSCRIPT end_POSTSUBSCRIPT. Specifically, for m∈ℤ𝑚ℤm\in\mathbb{Z}italic_m ∈ blackboard_Z and M∈ℕ:={1,2,3,…}𝑀ℕassign123…M\in\mathbb{N}:=\{1,2,3,\dots\}italic_M ∈ blackboard_N := { 1 , 2 , 3 , … }, we generalize Birch’s result by considering the sample space | ck,m,M(n)=−12∑±±∑∗s,t∈ℕ0t2−s2=nt≡±m(modM)∗Fk,t(s)+Ok,M(nk+1+ε).subscript𝑐𝑘𝑚𝑀𝑛plus-or-minus12subscriptplus-or-minussubscriptsuperscript𝑠𝑡subscriptℕ0superscript𝑡2superscript𝑠2𝑛𝑡plus-or-minus𝑚mod𝑀subscript𝐹𝑘𝑡𝑠subscript𝑂𝑘𝑀superscript𝑛𝑘1𝜀c_{k,m,M}(n)=-\frac{1}{2}\sum_{\pm}\pm\sideset{}{{}^{*}}{\sum}_{\begin{% | −12∑±±∑∗s,t∈ℕ0t2−s2=nt≡±m(modM)∗Fk,t(s)+Ok,M(nk+1+ε).plus-or-minus12subscriptplus-or-minussubscriptsuperscript𝑠𝑡subscriptℕ0superscript𝑡2superscript𝑠2𝑛𝑡plus-or-minus𝑚mod𝑀subscript𝐹𝑘𝑡𝑠subscript𝑂𝑘𝑀superscript𝑛𝑘1𝜀-\frac{1}{2}\sum_{\pm}\pm\sideset{}{{}^{*}}{\sum}_{\begin{subarray}{c}s,t\in% | A |
\Lambda)}+\mathcal{H}(\bar{x}_{t},\bar{u}_{t},p_{t},P_{t})\geq 0,italic_G start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT + ⟨ roman_Δ over¯ start_ARG italic_x end_ARG start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT , italic_p start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ⟩ start_POSTSUBSCRIPT italic_H start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( roman_Λ ) × italic_H start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( roman_Λ ) end_POSTSUBSCRIPT + caligraphic_H ( over¯ start_ARG italic_x end_ARG start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT , over¯ start_ARG italic_u end_ARG start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT , italic_p start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT , italic_P start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ) ≥ 0 , | The higher regularity assumptions p𝑝pitalic_p and P𝑃Pitalic_P in (113) are necessary due to the unbounded term in (115). Notice that the adjoint states given by (20) and (23), respectively, satisfy this higher regularity. In case p𝑝pitalic_p and P𝑃Pitalic_P are the adjoint states, and | The two main approaches to mathematical control theory are the dynamic programming approach introduced by Bellman, see [1], which studies the value function, and the approach via Pontryagin’s maximum principle, see [22], which evolves around the adjoint process. Under smoothness assumptions, Pontryagin already established in the deterministic case that the adjoint state is given by the derivative of the value function evaluated along an optimal trajectory. In the stochastic case, the adjoint state p𝑝pitalic_p has to be complemented by another process q𝑞qitalic_q, which arises in the construction of a solution for the backward stochastic differential equation when applying Itô’s representation theorem. Bismut extended Pontryagin’s result for p𝑝pitalic_p to this case, see [4], and Bensoussan identified q𝑞qitalic_q in terms of the second order derivative of the value function, see [2]. Since then, there have been various generalizations of these results, dispensing with the smoothness assumptions on the value function, both in the deterministic and stochastic setting, as well as in finite and infinite dimensions, see [5, 6, 26, 27]. However, the full generalization to the case of controlled stochastic partial differential equations (SPDEs) was still missing. The first part of this paper closes this gap using a recently obtained characterization of the second order adjoint state as the solution of a function-valued backward SPDE, see [23]. In the second part of this paper, we show how this relationship between the adjoint states and the (generalized) derivatives of the value function naturally leads us to a non-smooth version of the classical verification theorem that was obtained in the finite dimensional case in [28, 16, 17]. | It turns out that the necessary optimality conditions from Theorem 3.1 and Corollary 3.4 are closely related to a non-smooth version of the classical verification theorem, which is a sufficient optimality condition. This relationship was first observed in the finite dimensional case in [28], and proven in [16, 17]. We are going to show how to extend this result to the case of controlled SPDEs. The results we are going to prove are mathematically independent of Section 3 and in fact hold in a more general setting. In particular, we do not use the adjoint states and therefore we drop the assumption that the coefficients of the control problem are Nemytskii operators and we do not restrict to one-dimensional space domains. Let us therefore first formulate the more general setting. | This result together with our main result Theorem 3.1 extends the necessary condition in Pontryagin’s maximum principle by adjoint state inclusions. In the deterministic infinite dimensional case, the differential inclusion for the first order adjoint state was proven in [6]. In [27], Zhou proved the corresponding result for the case of controlled SPDEs. However, going only to first order does not fully reflect the stochastic nature of the problem. In the finite dimensional case, Zhou proved the differential inclusion for the second order adjoint state in [26]. The main obstacle in the generalization of this result to infinite dimensions was the missing representation of the second order adjoint state, which appears due to the Itô correction term arising in stochastic calculus. | A |
For simplicity, let us denote 𝒢:=𝒢(𝒪)assign𝒢𝒢𝒪\mathcal{G}:=\mathcal{G}(\mathcal{O})caligraphic_G := caligraphic_G ( caligraphic_O ). Catsigeras and Enrich [10, Theorems 1.3] have shown that 𝒪𝒪\mathcal{O}caligraphic_O is non-empty. They have also proved the following results: | 𝒪𝒪\mathcal{O}caligraphic_O is the smallest weak* compact set that contains, for Lebesgue almost all initial states, the limits of all convergent subsequences of (1.3). | For any point x∈X𝑥𝑋x\in Xitalic_x ∈ italic_X, we denote by V(x)𝑉𝑥V(x)italic_V ( italic_x ) the set of those Borel probabilities on X𝑋Xitalic_X that are limits in the weak∗ topology of convergent subsequences of the sequence | and if 𝒪𝒪\mathcal{O}caligraphic_O is countable infinite, then 𝒪𝒪\mathcal{O}caligraphic_O is the weak*closure of the set of Phy measures by Theorem 2.4. In both cases, the equality is clear. | has positive Lebesgue measure. We call Bϵ(μ)subscript𝐵italic-ϵ𝜇B_{\epsilon}(\mu)italic_B start_POSTSUBSCRIPT italic_ϵ end_POSTSUBSCRIPT ( italic_μ ) the basin of ε𝜀\varepsilonitalic_ε-attraction of μ𝜇\muitalic_μ. We denote by 𝒪Tsubscript𝒪𝑇\mathcal{O}_{T}caligraphic_O start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT the set of all observable measures. When the context is clear, we denote by 𝒪:=𝒪Tassign𝒪subscript𝒪𝑇\mathcal{O}:=\mathcal{O}_{T}caligraphic_O := caligraphic_O start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT the set of all observable measures. | A |
Any Lyapunov transformation between systems of this form acts as Lyapunov transformations on the individual blocks, i.e., L(t)=diag(L1(t),…,Lp(t))𝐿𝑡diagsubscript𝐿1𝑡…subscript𝐿𝑝𝑡L(t)=\textnormal{diag}(L_{1}(t),\dots,L_{p}(t))italic_L ( italic_t ) = diag ( italic_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_t ) , … , italic_L start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ( italic_t ) ). This implies that Eq. 8 (resp. stability of the Lyapunov exponents of the subsystems and Eq. 5) holds for one such system if and only if it holds for all of them. | In this subsection, we introduce strong fast invertibility, which is a weaker concept than the stability of Lyapunov exponents but still sufficient for their computation. Our main objectives are a characterization result that allows us to compare strong fast invertibility to the stability of Lyapunov exponents and perturbation results in preparation for the analysis of Benettin’s algorithm. | One of our two main goals is to provide a characterization of strong fast invertibility that allows a direct comparison to stability of Lyapunov exponents via the above theorem. To connect strong fast invertibility, a property that concerns the evolution of volumes, to Lyapunov exponents, we work with certain induced systems on spaces of exterior products. These systems allow us to link Lyapunov exponents to volume growth assuming the original system is regular. Our characterization theorem is the following: | When testing for strong fast invertibility (resp. for stability of Lyapunov exponents), it is enough to check the conditions of Theorem 3.33 (resp. Theorem 3.22) after transforming the original system to any block diagonal system of the form | An immediate consequence of Theorem 3.33 is that stability of Lyapunov exponents implies strong fast invertibility. | D |
The test performance are rather satisfactory even for moderate sample sizes, also compared to that of the test by Jiménez-Gamero and Alba-Fernández, (2021), specifically tailored for the Geometric distribution. | The test shows some criticalities when an inflation of zeroes occurs, which could be overcome introducing a family of test statistics, indexed by a parameter hℎhitalic_h, depending on the cumulative distribution function at hℎhitalic_h instead of on the probability in zero. Further research will be devoted to this last issue and to the use of alternative estimators involved in the implementation of the statistic, in order to improve the performance of the test. Finally, the generalization of the test to families of count distributions indexed by a k𝑘kitalic_k-variate parameter will be investigated. | A huge literature deals with testing continuous distributions, while a few proposals have been developed for testing the fit of discrete distributions. Among these, many are tailored to deal with particular distributions and so they are of limited applicability, while there is an emerging need of tests allowing to specify an extremely broad class of distributions under the null hypothesis. Undoubtedly, the chi-squared test is the most widely adopted, notwithstanding the arbitrariness of its implementation due to the requirement on the frequency minimum values. Similarly to the chi-squared test, the proposed test shows considerable flexibility as it allows to specify any count distribution with finite second moment under the null hypothesis and can be considered deriving from a L1superscript𝐿1L^{1}italic_L start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT distance for the class of alternative distributions for which Dμ(s)subscript𝐷𝜇𝑠D_{\mu}(s)italic_D start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT ( italic_s ) has constant sign. The resulting test statistic has a simple expression, depending on the empirical p.g.f. only through the probability of zero occurrence, and an asymptotic normal distribution. Moreover, it must be pointed out that the test can be adopted for a much broader class of alternative distributions. Indeed, the test is consistent for the alternatives in the class but also for all the alternatives for which the probability of zero is different from that under the null hypothesis. | The test statistic has a manageable expression and depends on the empirical p.g.f. solely through its value in zero, thus avoiding the complexities of handling the whole empirical functional. In addition, the test statistic is proven to have an asymptotic normal distribution, which allows for a straightforward implementation of the test, without demanding intensive resampling methods. Moreover, the test can be also adopted for the very general hypothesis system with alternatives not necessarily belonging to the class, even though in this case the consistency is ensured only if the probability of getting zero is different under the null and alternative hypothesis. | The test stands in the long tradition of testing procedures based on distances, such as the Pearson chi-squared test. In particular, the proposed test statistic is justified by the L1superscript𝐿1L^{1}italic_L start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT distance of a suitable function of the p.g.f. of the model under the null hypothesis and the p.g.f. of the random variable actually generating data, when the derivative of the ratio of the p.g.f.s has constant sign. Therefore, given the distribution specified under the null hypothesis, the natural class of alternative distributions contains those ensuring the derivative constant sign and, for the corresponding hypothesis system, the test is proven to be consistent. | A |
An alternative approach, proposed by Meintanis and Nikitin, (2008), is to consider tests with suitable asymptotic properties with respect to a fairly wide class of alternatives, which are also the most likely when dealing with the Poissonity assumption. | It is at once apparent that Zn,ksubscript𝑍𝑛𝑘Z_{n,k}italic_Z start_POSTSUBSCRIPT italic_n , italic_k end_POSTSUBSCRIPT actually constitutes a family of test statistics giving rise to consistent test for k=0𝑘0k=0italic_k = 0 and for all the other values of k𝑘kitalic_k for which there is a discrepancy between the cumulative distribution of the Poisson and of X𝑋Xitalic_X. Among this family of test statistics, only Zn,0subscript𝑍𝑛0Z_{n,0}italic_Z start_POSTSUBSCRIPT italic_n , 0 end_POSTSUBSCRIPT belongs to the family of the Poisson zero indexes (Weiß et al.,, 2019). It is particularly attractive owing to its simplicity but, as already pointed out, its finite-sample performance may deteriorate, especially if the sample size is small and μ𝜇\muitalic_μ is relatively large. Therefore, the selection of the parameter k𝑘kitalic_k ensuring consistency and good discriminatory capability is crucial and a data-driven selection criterion is proposed. | The performance of the proposed test has been assessed by means of an extensive Monte Carlo simulation. First of all, fixed the nominal level α=0.05𝛼0.05\alpha=0.05italic_α = 0.05, the significance level of the test is empirically evaluated, as the proportion of rejections of the null hypothesis, by independently generating 10000100001000010000 samples of size n=50𝑛50n=50italic_n = 50 from Poisson distributions with μ𝜇\muitalic_μ varying from 1111 to 16161616 by 0.50.5{0.5}0.5. As early mentioned, the family of test statistics Zn,ksubscript𝑍𝑛𝑘Z_{n,k}italic_Z start_POSTSUBSCRIPT italic_n , italic_k end_POSTSUBSCRIPT depends on the parameter k𝑘kitalic_k and therefore, for any μ𝜇\muitalic_μ, the empirical significance level is computed for k=0,1,2,3𝑘0123k=0,1,2,3italic_k = 0 , 1 , 2 , 3 and reported in Figure 1. Simulation results confirm that for large values of μ𝜇\muitalic_μ the empirical level is far from the nominal one even for a reasonably large sample size and that a data-driven procedure is needed to select k𝑘kitalic_k. Thus, the test statistic Wnsubscript𝑊𝑛W_{n}italic_W start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT is considered, and its performance is compared to those of two tests having known asymptotic distributions: the test by Meintanis and Nikitin, (2008), MNn𝑀subscript𝑁𝑛MN_{n}italic_M italic_N start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT, also recommended by Mijburgh and Visagie, (2020) to achieve good power against a large variety of deviations from the Poisson distribution, and the Fisher index of dispersion, IDn𝐼subscript𝐷𝑛ID_{n}italic_I italic_D start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT, which, owing to its simplicity, is often considered as a benchmark. | In this paper, by referring to the same class of alternative distributions and by using the characterization of the Poisson distribution based on its p.g.f., we propose a family of consistent and asymptotically normally distributed test statistics, based on the difference between the plug-in estimator of the Poisson cumulative distribution function (c.d.f.) and the empirical c.d.f., and a data-driven procedure for the choice of the parameter indexing the statistics. In particular, the test statistics not only have an intuitive interpretation but, being simple to compute, allow a straightforward implementation of the test and lead to test procedures with satisfactory performance also in presence of contiguous alternatives. | the proposed family of test statistics seems to be an appealing alternative in the absence of prior information regarding the type of deviation from Poissonity. In particular, the statistics are rather simple and easily interpretable and the test implementation does not require intensive computational effort. Moreover, the test is consistent against any fixed alternative when k𝑘kitalic_k is equal to 00 and when it is selected using the data-driven criterion, that is k=kn∗𝑘subscriptsuperscript𝑘𝑛k=k^{*}_{n}italic_k = italic_k start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT. For k=0𝑘0k=0italic_k = 0 the test statistic basically compares an estimator of P(X=0)𝑃𝑋0P(X=0)italic_P ( italic_X = 0 ) assuming that X𝑋Xitalic_X is Poisson with the relative frequency of 00 but the finite sample performance of the test may not be satisfactory, especially for small sample size and relatively large Poisson parameter. The performance improves for kn∗subscriptsuperscript𝑘𝑛k^{*}_{n}italic_k start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT, when the test juxtaposes the plug-in estimator of the cumulative distribution function of a Poisson r.v. and the empirical cumulative distribution function in kn∗subscriptsuperscript𝑘𝑛k^{*}_{n}italic_k start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT. Indeed, even if kn∗subscriptsuperscript𝑘𝑛k^{*}_{n}italic_k start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT converges a.s. to 00, the convergence rate may be very slow for large values of the Poisson parameter, and thus, even for large sample sizes, kn∗subscriptsuperscript𝑘𝑛k^{*}_{n}italic_k start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT can be rather larger than 00. | C |
,\,\pi(G_{1})=\pi(G_{2})=\ldots=\pi(G_{N_{1}})\right\}.caligraphic_H = { ( italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_G start_POSTSUBSCRIPT italic_N start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) : italic_G start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∈ caligraphic_G , italic_π ( italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) = italic_π ( italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) = … = italic_π ( italic_G start_POSTSUBSCRIPT italic_N start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) } . | we denote the fact that for any coloring of the k𝑘kitalic_k-subspaces of an N𝑁Nitalic_N-space V𝑉Vitalic_V with r𝑟ritalic_r colors, there exists a n𝑛nitalic_n-subspace U⊆V𝑈𝑉U\subseteq Vitalic_U ⊆ italic_V such that [Uk]matrix𝑈𝑘\begin{bmatrix}U\\ | n\end{bmatrix}italic_U ∈ [ start_ARG start_ROW start_CELL italic_X end_CELL end_ROW start_ROW start_CELL italic_n end_CELL end_ROW end_ARG ] with [Uk]∩ℋmatrix𝑈𝑘ℋ\begin{bmatrix}U\\ | k\end{bmatrix}caligraphic_H ⊆ [ start_ARG start_ROW start_CELL italic_X end_CELL end_ROW start_ROW start_CELL italic_k end_CELL end_ROW end_ARG ] of k𝑘kitalic_k-subspaces of X𝑋Xitalic_X such that for any coloring of ℋℋ\mathcal{H}caligraphic_H with r𝑟ritalic_r colors, there exists an n𝑛nitalic_n-space U∈[Xn]𝑈matrix𝑋𝑛U\in\begin{bmatrix}X\\ | We claim that for any r𝑟ritalic_r-coloring of ℋℋ\mathcal{H}caligraphic_H, there exists a n𝑛nitalic_n-space U𝑈Uitalic_U such that [Uk]∩ℋmatrix𝑈𝑘ℋ\begin{bmatrix}U\\ | D |
\prime\prime}}over̊ start_ARG italic_μ end_ARG start_POSTSUPERSCRIPT roman_ℏ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_Γ , roman_Γ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ∘ over̊ start_ARG italic_μ end_ARG start_POSTSUPERSCRIPT roman_ℏ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_Γ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , roman_Γ start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT = over̊ start_ARG italic_μ end_ARG start_POSTSUPERSCRIPT roman_ℏ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_Γ , roman_Γ start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT holds for each triple of seeds Γ,Γ′,Γ′′∈|Γ|ΓsuperscriptΓ′superscriptΓ′′Γ\Gamma,\Gamma^{\prime},\Gamma^{\prime\prime}\in|\Gamma|roman_Γ , roman_Γ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , roman_Γ start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ∈ | roman_Γ |. | satisfying some favorable properties, where 𝒜|Γ∨|(ℤt)subscript𝒜superscriptΓsuperscriptℤ𝑡\mathscr{A}_{|\Gamma^{\vee}|}(\mathbb{Z}^{t})script_A start_POSTSUBSCRIPT | roman_Γ start_POSTSUPERSCRIPT ∨ end_POSTSUPERSCRIPT | end_POSTSUBSCRIPT ( blackboard_Z start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ) stands for the set of tropical integer points of the cluster 𝒜𝒜\mathscr{A}script_A-variety 𝒜|Γ∨|subscript𝒜superscriptΓ\mathscr{A}_{|\Gamma^{\vee}|}script_A start_POSTSUBSCRIPT | roman_Γ start_POSTSUPERSCRIPT ∨ end_POSTSUPERSCRIPT | end_POSTSUBSCRIPT associated to the mutation-equivalence class |Γ∨|superscriptΓ|\Gamma^{\vee}|| roman_Γ start_POSTSUPERSCRIPT ∨ end_POSTSUPERSCRIPT | of the ‘cluster 𝒜𝒜\mathscr{A}script_A-seed’ Γ∨superscriptΓ\Gamma^{\vee}roman_Γ start_POSTSUPERSCRIPT ∨ end_POSTSUPERSCRIPT, whose underlying exchange matrix is same as that of ΓΓ\Gammaroman_Γ. One of the desired properties is that the image of 𝕀̊|Γ|subscript̊𝕀Γ\mathring{\mathbb{I}}_{|\Gamma|}over̊ start_ARG blackboard_I end_ARG start_POSTSUBSCRIPT | roman_Γ | end_POSTSUBSCRIPT forms a basis of the algebra of regular functions 𝒪(𝒳|Γ|)𝒪subscript𝒳Γ\mathscr{O}(\mathscr{X}_{|\Gamma|})script_O ( script_X start_POSTSUBSCRIPT | roman_Γ | end_POSTSUBSCRIPT ). The existence of such duality maps for general initial seed ΓΓ\Gammaroman_Γ has been expected by the Fock-Goncharov duality conjectures [26, 27], and was proved for a large class of seeds by Gross, Hacking, Keel and Kontsevich [34]: namely, for all seeds whose underlying exchange matrices satisfy some combinatorial condition, such as the existence of a maximal green sequence. The solution uses a tool called a consistent scattering diagram, and except for few examples, an explicit enough construction of a consistent scattering diagram is not known, although the existence is proved; this makes the solution of [34] not constructive in general. Davison and Mandel [17] later constructed a quantum version of that duality map | Another stipulation is the equivariance under the mapping class group MCG(S)MCG𝑆{\rm MCG}(S)roman_MCG ( italic_S ). For this, we also require 𝒜Tℏsubscriptsuperscript𝒜Planck-constant-over-2-pi𝑇\mathcal{A}^{\hbar}_{T}caligraphic_A start_POSTSUPERSCRIPT roman_ℏ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT, μT,T′ℏsubscriptsuperscript𝜇Planck-constant-over-2-pi𝑇superscript𝑇′\mu^{\hbar}_{T,T^{\prime}}italic_μ start_POSTSUPERSCRIPT roman_ℏ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_T , italic_T start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT and 𝐊T,T′subscript𝐊𝑇superscript𝑇′{\bf K}_{T,T^{\prime}}bold_K start_POSTSUBSCRIPT italic_T , italic_T start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT to be invariant under the action of MCG(S)MCG𝑆{\rm MCG}(S)roman_MCG ( italic_S ); so they must only depend on the underlying exchange matrices for the relevant ideal triangulations T,T′𝑇superscript𝑇′T,T^{\prime}italic_T , italic_T start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT. | Another stipulation is that the entire construction should be invariant (or equivariant) under the action of the cluster mapping class group, which consists of transformations of seeds that preserve the underlying exchange matrices. That is, we also ask for: | These intertwiners allow us to identify the representations π̊Γsubscript̊𝜋Γ\mathring{\pi}_{\Gamma}over̊ start_ARG italic_π end_ARG start_POSTSUBSCRIPT roman_Γ end_POSTSUBSCRIPT for different seeds ΓΓ\Gammaroman_Γ in a unitary and consistent manner, yielding the sought-for quantization map 𝐐ℏsuperscript𝐐Planck-constant-over-2-pi{\bf Q}^{\hbar}bold_Q start_POSTSUPERSCRIPT roman_ℏ end_POSTSUPERSCRIPT for the cluster 𝒳𝒳\mathscr{X}script_X-variety 𝒳|Γ|subscript𝒳Γ\mathscr{X}_{|\Gamma|}script_X start_POSTSUBSCRIPT | roman_Γ | end_POSTSUBSCRIPT. Even without relating to the quantization map, constructing such a system of intertwiners can be interpreted as constructing a projective unitary representation of the groupoid of seeds in |Γ|Γ|\Gamma|| roman_Γ |. A morphism in this groupoid can be written as a pair of seeds (Γ,Γ′)ΓsuperscriptΓ′(\Gamma,\Gamma^{\prime})( roman_Γ , roman_Γ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ), and for a fixed exchange matrix ε𝜀\varepsilonitalic_ε, the set of all morphisms (Γ,Γ′)ΓsuperscriptΓ′(\Gamma,\Gamma^{\prime})( roman_Γ , roman_Γ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) such that ε𝜀\varepsilonitalic_ε is the underlying exchange matrix for both ΓΓ\Gammaroman_Γ and Γ′superscriptΓ′\Gamma^{\prime}roman_Γ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT form a group called the cluster mapping class group. This recovers the usual mapping class group in case when ε𝜀\varepsilonitalic_ε comes from an ideal triangulation of a punctured surface. The above quantum representation of the groupoid of seeds then leads to representations of the cluster mapping class groups, which have been viewed as one of the main results of the theory of quantum cluster varieties and quantum Teichmüller theory. The construction of such a representation is indeed the main achievement of [25]; this also hints that it is one of the most difficult and central parts of the entire quantization problem. | C |
Since we have 𝔼|X|2≤1𝔼superscript𝑋21\mathbb{E}|X|^{2}\leq 1blackboard_E | italic_X | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≤ 1, the log|X|𝑋\log|X|roman_log | italic_X | term cannot scale with SNR. Thus, the I(θ;Y|X)𝐼𝜃conditional𝑌𝑋I(\theta;Y|X)italic_I ( italic_θ ; italic_Y | italic_X ) term also achieves a DoF of 1/2121/21 / 2. | For the AWGN channel without RIS, the DoF of 1 can be thought of as being comprised of 1/2121/21 / 2 DoF in magnitude of X𝑋Xitalic_X and 1/2121/21 / 2 DoF in phase of θ𝜃\thetaitalic_θ. But with RIS as part of the channel, there is a 1/2121/21 / 2 DoF overlap between | of X𝑋Xitalic_X and θ𝜃\thetaitalic_θ achieves the same DoF with contributions of 1/2121/21 / 2 DoF in magnitude of X𝑋Xitalic_X and 1/2121/21 / 2 DoF in phase of either X𝑋Xitalic_X or θ𝜃\thetaitalic_θ. | deterministic, there is no 1/2121/21 / 2 DoF overlap between 𝐗𝐗\mathbf{X}bold_X and 𝚯𝚯\mathbf{\Theta}bold_Θ, | the phase of the input X𝑋Xitalic_X and the phase of the RIS θ𝜃\thetaitalic_θ. Thus, the joint transmission | A |
\mathsf{T}}+\Sigma_{V_{2}})∼ sansserif_N ( italic_C start_POSTSUBSCRIPT 23 end_POSTSUBSCRIPT italic_B ( italic_θ ) italic_U start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT , italic_C start_POSTSUBSCRIPT 23 end_POSTSUBSCRIPT roman_Σ start_POSTSUBSCRIPT italic_W start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_C start_POSTSUBSCRIPT 23 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT sansserif_T end_POSTSUPERSCRIPT + roman_Σ start_POSTSUBSCRIPT italic_V start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) | where we recall that filter gain F1subscript𝐹1F_{1}italic_F start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT is characterized by | where F(θ)∈ℝd𝖷×d𝖸𝐹𝜃superscriptℝsubscript𝑑𝖷subscript𝑑𝖸F(\theta)\in\mathbb{R}^{d_{\mathsf{X}}\times d_{\mathsf{Y}}}italic_F ( italic_θ ) ∈ blackboard_R start_POSTSUPERSCRIPT italic_d start_POSTSUBSCRIPT sansserif_X end_POSTSUBSCRIPT × italic_d start_POSTSUBSCRIPT sansserif_Y end_POSTSUBSCRIPT end_POSTSUPERSCRIPT is given by (2.7) and is characterized below in terms of a Riccati equation (2.6), dual to (2.1) and (2.2). We further denote | Equipped with Equation 5.9, uninformativeness (recall Definition 3.1) is readily characterized as follows below. | where it can readily by computed that K1subscript𝐾1K_{1}italic_K start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT is as in (5.4). The result follows by invoking Lemma 2.1. | C |
Unlike the usual Hilbert-Samuel function, the structure of the Hilbert-Kunz function f(n)=λ(NJ[pn]N)𝑓𝑛𝜆𝑁superscript𝐽delimited-[]superscript𝑝𝑛𝑁f(n)=\lambda(\frac{N}{{J}^{[{p}^{n}]}N})italic_f ( italic_n ) = italic_λ ( divide start_ARG italic_N end_ARG start_ARG italic_J start_POSTSUPERSCRIPT [ italic_p start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ] end_POSTSUPERSCRIPT italic_N end_ARG ) is rather elusive. We refer interested readers to [SecCoef], [Teixeira], [Flag]. | We review results on graded Betti numbers which we use in Section 5. References for most of these results are [Serre], and [BH]. Recall that R𝑅Ritalic_R is a finitely generated ℕℕ\mathbb{N}blackboard_N-graded k𝑘kitalic_k-algebra (see 1.1). | By a finitely generated ℕℕ\mathbb{N}blackboard_N-graded k𝑘kitalic_k-algebra, we mean an ℕℕ\mathbb{N}blackboard_N-graded commutative ring whose degree zero piece is k𝑘kitalic_k and which is finitely generated over k𝑘kitalic_k. | The references for this subsection are [BH] and [Serre]. Throughout, R𝑅Ritalic_R is a finitely generated ℕℕ\mathbb{N}blackboard_N-graded algebra over a field k𝑘kitalic_k. Recall that the Hilbert series(also called the Hilbert-Poincaré series) of a finitely generated ℤℤ\mathbb{Z}blackboard_Z-graded R𝑅Ritalic_R-module M𝑀Mitalic_M is the formal Laurent series HM(t):=∑n∈ℤλ(Mn)tnassignsubscript𝐻𝑀𝑡𝑛ℤ𝜆subscript𝑀𝑛superscript𝑡𝑛H_{M}(t):=\underset{n\in\mathbb{Z}}{\sum}\lambda(M_{n})t^{n}italic_H start_POSTSUBSCRIPT italic_M end_POSTSUBSCRIPT ( italic_t ) := start_UNDERACCENT italic_n ∈ blackboard_Z end_UNDERACCENT start_ARG ∑ end_ARG italic_λ ( italic_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) italic_t start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT. | In this section, we define the Frobenius-Poincaré function associated to a given triple (M,R,I)𝑀𝑅𝐼(M,R,I)( italic_M , italic_R , italic_I ), where R𝑅Ritalic_R is a finitely generated ℕℕ\mathbb{N}blackboard_N-graded k𝑘kitalic_k-algebra – k𝑘kitalic_k has characteristic p>0𝑝0p>0italic_p > 0, (see 1.1), I𝐼Iitalic_I is a homogeneous ideal of finite co-length and M𝑀Mitalic_M is a finitely generated ℤℤ\mathbb{Z}blackboard_Z-graded R𝑅Ritalic_R-module. In Theorem 3.1 we prove that Frobenius-Poincaré functions are entire functions. | A |
Moreover, to ease the reconstruction of the proof, and avoid mistakes by associating a constraint to the wrong dual value, a user can name a constraint through the set_name() method and access it later on through the get_name() method. | Note it is the responsibility of the user to set a name to the constraints to be able to recover it later. | Moreover, to ease the reconstruction of the proof, and avoid mistakes by associating a constraint to the wrong dual value, a user can name a constraint through the set_name() method and access it later on through the get_name() method. | Note that, for some classes, the associated constraints might not be tight, meaning that the methodology might only be able to generate upper-bound on the worst-case behaviors. | Since PEPit deals with interpolation constraints without any intervention from the user side, a short description of each of those constraints is set by default, based on Points’ names and func’s name which can also be set through a set_name method. | A |
\end{aligned}\right.{ start_ROW start_CELL end_CELL start_CELL - 5 square-root start_ARG 2 end_ARG + 24 italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - 2 italic_λ italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + 3 square-root start_ARG 10 end_ARG italic_t start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT = 0 , end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL 9 square-root start_ARG 6 end_ARG + 24 italic_t start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT - 2 italic_λ italic_t start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT + square-root start_ARG 30 end_ARG italic_t start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT = 0 , end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL 6 square-root start_ARG 5 end_ARG + 3 square-root start_ARG 10 end_ARG italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + square-root start_ARG 30 end_ARG italic_t start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT - 10 italic_t start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT - 2 italic_λ italic_t start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT = 0 , end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_t start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_t start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - 1 = 0 . end_CELL end_ROW | One can check that the points correspond to these two solutions are not in the interior of ℰℰ\mathcal{E}caligraphic_E. | We can show that the boundary curve α𝛼\alphaitalic_α of ℰℰ\mathcal{E}caligraphic_E is embedded in ∂∞𝒞subscript𝒞\partial_{\infty}\mathcal{C}∂ start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT caligraphic_C by solving a system of equations. For example, the equations ∂∞(𝒞∩ℬ0¯)subscript𝒞subscriptℬ¯0\partial_{\infty}(\mathcal{C}\cap\mathcal{B}_{\overline{0}})∂ start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT ( caligraphic_C ∩ caligraphic_B start_POSTSUBSCRIPT over¯ start_ARG 0 end_ARG end_POSTSUBSCRIPT ), ∂∞(𝒞∩ℬ11)subscript𝒞subscriptℬ11\partial_{\infty}(\mathcal{C}\cap\mathcal{B}_{11})∂ start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT ( caligraphic_C ∩ caligraphic_B start_POSTSUBSCRIPT 11 end_POSTSUBSCRIPT ) and t12+t22+t32−1superscriptsubscript𝑡12superscriptsubscript𝑡22superscriptsubscript𝑡321t_{1}^{2}+t_{2}^{2}+t_{3}^{2}-1italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_t start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_t start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - 1 have two solutions | By using the techniques of Section 2, we can study the intersection of 𝒞𝒞\mathcal{C}caligraphic_C with each face of DSsubscript𝐷𝑆D_{S}italic_D start_POSTSUBSCRIPT italic_S end_POSTSUBSCRIPT. For | Solving the system by standard Groebner basis techniques, we get two solutions which are given approximately | D |
Let H=Tn⟨t,2t⟩𝐻subscript𝑇𝑛𝑡2𝑡H=T_{n}\langle t,2t\rangleitalic_H = italic_T start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ⟨ italic_t , 2 italic_t ⟩ and a𝑎aitalic_a be an integer such that a≡n(modt)𝑎annotated𝑛pmod𝑡a\equiv n\pmod{t}italic_a ≡ italic_n start_MODIFIER ( roman_mod start_ARG italic_t end_ARG ) end_MODIFIER with 1≤a≤t1𝑎𝑡1\leq a\leq t1 ≤ italic_a ≤ italic_t. | size where d=gcd(t1,t2)𝑑gcdsubscript𝑡1subscript𝑡2d={\rm gcd}(t_{1},t_{2})italic_d = roman_gcd ( italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_t start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ). Consequently, for n<t1+t2𝑛subscript𝑡1subscript𝑡2n<t_{1}+t_{2}italic_n < italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_t start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT, | Let G=Tn⟨t1,…,tk⟩𝐺subscript𝑇𝑛subscript𝑡1…subscript𝑡𝑘G=T_{n}\langle t_{1},\ldots,t_{k}\rangleitalic_G = italic_T start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ⟨ italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_t start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ⟩ with gcd(t1,…,tk)=dsubscript𝑡1…subscript𝑡𝑘𝑑\gcd(t_{1},\ldots,t_{k})=droman_gcd ( italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_t start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) = italic_d. | Since gcd(t,2t)=t𝑡2𝑡𝑡\gcd(t,2t)=troman_gcd ( italic_t , 2 italic_t ) = italic_t, it follows from Lemma 4.1 that | Since gcd(t,2t)=t𝑡2𝑡𝑡\gcd(t,2t)=troman_gcd ( italic_t , 2 italic_t ) = italic_t, it follows from Lemma 4.1 that | C |
\leq 8\pi\,t+C(r_{1}).∫ start_POSTSUBSCRIPT italic_L ( 1 , italic_t ) end_POSTSUBSCRIPT | ∇ italic_u | italic_S ≤ ∫ start_POSTSUBSCRIPT italic_L ( 1 , italic_ξ ) end_POSTSUBSCRIPT | ∇ italic_u | italic_S ≤ 8 italic_π italic_t + italic_C ( italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) . | for a sequence Ri→∞→subscript𝑅𝑖R_{i}\rightarrow\inftyitalic_R start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT → ∞, where C>0𝐶0C>0italic_C > 0 is a universal constant. | We assert that there exists C>0𝐶0C>0italic_C > 0 such that for any r0≤R<Tsubscript𝑟0𝑅𝑇r_{0}\leq R<Titalic_r start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ≤ italic_R < italic_T, | To prove the first part (2.12), by scaling, it suffices to show there exists a universal constant C>0𝐶0C>0italic_C > 0 so that | We now turn to the proof of Theorem 1.4. Since M𝑀Mitalic_M is assumed to have finite first Betti number and | C |
The most prominent spectral problem one can study is the case of the Laplace operator acting on functions | Let X:(Mn,g)→(ℝn+m,can):𝑋→superscript𝑀𝑛𝑔superscriptℝ𝑛𝑚canX:(M^{n},g)\to(\mathbb{R}^{n+m},{\rm can})italic_X : ( italic_M start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT , italic_g ) → ( blackboard_R start_POSTSUPERSCRIPT italic_n + italic_m end_POSTSUPERSCRIPT , roman_can ) be an isometric immersion. Assume that l1≤KM≤l2subscript𝑙1superscript𝐾𝑀subscript𝑙2l_{1}\leq K^{M}\leq l_{2}italic_l start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ≤ italic_K start_POSTSUPERSCRIPT italic_M end_POSTSUPERSCRIPT ≤ italic_l start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT for some l1,l2∈ℝsubscript𝑙1subscript𝑙2ℝl_{1},l_{2}\in\mathbb{R}italic_l start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_l start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ∈ blackboard_R. Let ΩΩ\Omegaroman_Ω a domain in M𝑀Mitalic_M such that ΩΩ\Omegaroman_Ω is contained in the complement of the cut locus of x0∈Ωsubscript𝑥0Ωx_{0}\in\Omegaitalic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ roman_Ω. Let f=aρx0𝑓𝑎subscript𝜌subscript𝑥0f=a\rho_{x_{0}}italic_f = italic_a italic_ρ start_POSTSUBSCRIPT italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT for some positive real number a>0𝑎0a>0italic_a > 0. For any p∈{0,…,n}𝑝0…𝑛p\in\{0,\ldots,n\}italic_p ∈ { 0 , … , italic_n }, the eigenvalues of the drifting Hodge Laplacian ΔfsubscriptΔ𝑓\Delta_{f}roman_Δ start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT acting on p𝑝pitalic_p-forms on a domain ΩΩ\Omegaroman_Ω of M𝑀Mitalic_M with Dirichlet boundary conditions satisfy for any k≥1𝑘1k\geq 1italic_k ≥ 1 | Let Ω⊂MΩ𝑀\Omega\subset Mroman_Ω ⊂ italic_M be a bounded domain of a Riemannian manifold M𝑀Mitalic_M with smooth boundary ∂ΩΩ\partial\Omega∂ roman_Ω | (Ω,g)Ω𝑔(\Omega,g)( roman_Ω , italic_g ) with boundary ∂ΩΩ\partial\Omega∂ roman_Ω the Dirichlet problem on differential p𝑝pitalic_p-forms is given by | [Pe] Let (Mn,g)superscript𝑀𝑛𝑔(M^{n},g)( italic_M start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT , italic_g ) be a complete Riemannian manifold. | B |
\mathcal{F}}under¯ start_ARG caligraphic_B end_ARG start_POSTSUBSCRIPT caligraphic_F end_POSTSUBSCRIPT / under¯ start_ARG caligraphic_X end_ARG start_POSTSUBSCRIPT caligraphic_F end_POSTSUBSCRIPT restricted to a connected open subset V𝑉Vitalic_V of an invariant irreducible component of ℰℱsubscriptℰℱ{\mathcal{E}}_{\mathcal{F}}caligraphic_E start_POSTSUBSCRIPT caligraphic_F end_POSTSUBSCRIPT. | Indeed if p𝑝pitalic_p is a regular point, by local triviality, the section is zero along the whole regular part of V𝑉Vitalic_V. The vanishing at the remaining singularities follows by analytic continuation. If p𝑝pitalic_p is a singular point, then the germ of X𝑋Xitalic_X at a regular point close to p𝑝pitalic_p is zero and we conclude similarly. | If the germ of X𝑋Xitalic_X at some point p𝑝pitalic_p of V𝑉Vitalic_V is zero, then X=0𝑋0X=0italic_X = 0. | if D⊂ℰℱ𝐷subscriptℰℱD\subset{\mathcal{E}}_{\mathcal{F}}italic_D ⊂ caligraphic_E start_POSTSUBSCRIPT caligraphic_F end_POSTSUBSCRIPT is a ℱ♯superscriptℱ♯{\mathcal{F}}^{\sharp}caligraphic_F start_POSTSUPERSCRIPT ♯ end_POSTSUPERSCRIPT-invariant component and p∈D𝑝𝐷p\in Ditalic_p ∈ italic_D is a singular point of ℱ♯superscriptℱ♯{\mathcal{F}}^{\sharp}caligraphic_F start_POSTSUPERSCRIPT ♯ end_POSTSUPERSCRIPT, then the Camacho-Sad indices of ℱ♯superscriptℱ♯{\mathcal{F}}^{\sharp}caligraphic_F start_POSTSUPERSCRIPT ♯ end_POSTSUPERSCRIPT at p𝑝pitalic_p along D𝐷Ditalic_D and of 𝒢♯superscript𝒢♯\mathcal{G}^{\sharp}caligraphic_G start_POSTSUPERSCRIPT ♯ end_POSTSUPERSCRIPT at φ(p)𝜑𝑝\varphi(p)italic_φ ( italic_p ) along φ(D)𝜑𝐷\varphi(D)italic_φ ( italic_D ) are equal, | The central point of the paper is Theorem 2.12. It states the 𝒞exsuperscript𝒞ex{\mathcal{C}}^{{{\mathrm{ex}}}}caligraphic_C start_POSTSUPERSCRIPT roman_ex end_POSTSUPERSCRIPT-universality of the germ at any point of the parameter space of the global family ℱ¯𝐔subscript¯ℱ𝐔{\underline{{\mathcal{F}}}}_{\mathbf{U}}under¯ start_ARG caligraphic_F end_ARG start_POSTSUBSCRIPT bold_U end_POSTSUBSCRIPT constructed in [9] that contains all the topological types in a fixed SL-class. This property will be proven by explicitly computing the Kodaira-Spencer map of this family at each point, that provides an infinitesimal characterization of 𝒞exsuperscript𝒞ex{\mathcal{C}}^{{{\mathrm{ex}}}}caligraphic_C start_POSTSUPERSCRIPT roman_ex end_POSTSUPERSCRIPT-universality. | B |
\ignorespaces}}}}\ignorespaces{\mathcal{M}}_{g}italic_D start_POSTSUPERSCRIPT italic_o end_POSTSUPERSCRIPT caligraphic_M start_POSTSUBSCRIPT italic_g end_POSTSUBSCRIPT provides this sub-sheaf. (Strictly speaking, this is only true after passing to a finite cover of Dosuperscript𝐷𝑜D^{o}italic_D start_POSTSUPERSCRIPT italic_o end_POSTSUPERSCRIPT which does not affect the argument.) | Once isotriviality for g=1𝑔1g=1italic_g = 1 and g>1𝑔1g>1italic_g > 1 has been established, one can use the | The isotriviality is less trivial for g>1𝑔1g>1italic_g > 1. It follows from the observation that the following results of Miyaoka–Mori and Hwang–Viehweg contradict each other. | There are two possibilities depending on the genus g𝑔gitalic_g of the general leaf: If g=1𝑔1g=1italic_g = 1, then the fibration has to be isotrivial, for otherwise one of the fibres of π𝜋\piitalic_π would be rational, in which case we are done already, cf. Theorem 3.2. | Since ωD=𝒪X(D)|Dsubscript𝜔𝐷evaluated-atsubscript𝒪𝑋𝐷𝐷\omega_{D}=\mathcal{O}_{X}(D)|_{D}italic_ω start_POSTSUBSCRIPT italic_D end_POSTSUBSCRIPT = caligraphic_O start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT ( italic_D ) | start_POSTSUBSCRIPT italic_D end_POSTSUBSCRIPT, we have ν(D,ωD)=ν(X,D)−1𝜈𝐷subscript𝜔𝐷𝜈𝑋𝐷1\nu(D,\omega_{D})=\nu(X,D)-1italic_ν ( italic_D , italic_ω start_POSTSUBSCRIPT italic_D end_POSTSUBSCRIPT ) = italic_ν ( italic_X , italic_D ) - 1. However, the numerical dimension of a nef divisor in a hyperkähler manifold can be 0,n0𝑛0,n0 , italic_n or 2n2𝑛2n2 italic_n. Since n>1𝑛1n>1italic_n > 1, the only possibility is that n=2𝑛2n=2italic_n = 2 and g>1𝑔1g>1italic_g > 1, which is excluded as follows: A fibre S𝑆Sitalic_S of the canonical map is equivalent as a cycle, up to a multiple, to D⋅D⋅𝐷𝐷D\cdot Ditalic_D ⋅ italic_D. This means that S𝑆Sitalic_S is Lagrangian, for ∫Sσσ¯=q(D)=0subscript𝑆𝜎¯𝜎𝑞𝐷0\int_{S}\sigma\bar{\sigma}=q(D)=0∫ start_POSTSUBSCRIPT italic_S end_POSTSUBSCRIPT italic_σ over¯ start_ARG italic_σ end_ARG = italic_q ( italic_D ) = 0. Hence, | A |
Their mathematical modeling is typically based on mixed-dimensional geometries representing the fractures as a network of codimension-one surfaces. The first ingredient of the model is a mixed-dimensional flow, typically coupling a Poiseuille flow along the network of fractures with the Darcy flow in the surrounding porous rock (called the matrix). This type of mixed-dimensional flow models has been introduced in the pioneering works [6, 14, 35, 44, 46], and has been the object of an intensive research over the last twenty years, both in terms of discretization and numerical analysis [49, 47, 7, 43, 52, 50, 23, 33, 22, 24, 4, 25, 20, 36, 26, 48, 13, 9, 3]. The second ingredient is based on poromechanics, coupling the rock deformation with the Darcy flow in the matrix domain, see [29] as a reference textbook on this topic. In this paper we assume small strains and porosity variations, as well as a poroelastic mechanical behavior of the porous rock. The third ingredient of the fully coupled model is related to the mechanical behavior of the fractures, given the matrix mechanical deformation and the matrix and fracture fluid pressures. This behavior is typically based on mechanics governing the contact and slip conditions [45, 55, 54]. | Few works are available in the literature and they are all based on simplifying assumptions, none addressing all the difficulties raised by such models. In [40, 42], the authors consider a simplified model assuming open fractures with no contact and a frozen fracture conductivity leading to a linear poromechanical coupling. The well-posedness is obtained based on a detailed analysis of weighted function spaces accounting for the degeneracy of the fracture conductivity at the tips. | The two references [16, 17] mentioned above concern two-phase flow models, but without contact. In this situation, the existence of a solution to the discrete or continuous model is not proved, and the convergence analysis can only be done under the assumption that discrete solutions exist and that the fracture aperture (and matrix porosity) remain bounded below by strictly positive quantities; this is a requirement as the continuous model itself does | Our convergence proof elaborates on previous works. In [18] we proved stability estimates and existence of a solution for the GD of a mixed-dimensional two-phase poromechanical model with Coulomb frictional contact. In [16, 17] we obtained compactness estimates for a mixed-dimensional two-phase poromechanical model with no contact. Combining these results we can establish the relative compactness of the fracture aperture, which is a key element to pass to the limit in the discrete variational formulation of the flow model. The main new difficulty addressed in this work is related to passing to the limit in the mechanical variational inequality, which involves the Lagrange multiplier and the normal jump of the displacement field at matrix–fracture interfaces. | In this work, we restrict the analysis to a simplified contact model based on Signorini complementarity conditions, thereby assuming frictionless contact. Note also that fractures are assumed to pre-exist and that fracture propagation is not addressed here. | D |
\end{matrix}\right\}italic_q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ≔ ⋁ { start_ARG start_ROW start_CELL ∧ { italic_γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = caligraphic_R , italic_γ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = caligraphic_T , italic_γ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT = caligraphic_F } , end_CELL end_ROW start_ROW start_CELL ∧ { italic_γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = caligraphic_R , italic_γ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = caligraphic_F , italic_γ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT = caligraphic_T } , end_CELL end_ROW start_ROW start_CELL ∧ { italic_γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = caligraphic_T , italic_γ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = caligraphic_F , italic_γ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT = caligraphic_R } , end_CELL end_ROW start_ROW start_CELL ∧ { italic_γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = caligraphic_F , italic_γ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = caligraphic_T , italic_γ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT = caligraphic_R } end_CELL end_ROW end_ARG } | We’ll also need to reason about ¬q15superscriptsubscript𝑞15\neg q_{1}^{5}¬ italic_q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 5 end_POSTSUPERSCRIPT. | ¬q1subscript𝑞1\neg q_{1}¬ italic_q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT holds. Adding the γ1=ℛsubscript𝛾1ℛ\gamma_{1}=\mathcal{R}italic_γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = caligraphic_R possibilities to ¬q1subscript𝑞1\neg q_{1}¬ italic_q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT results in | We’ll also need to reason about ¬qℛ¯subscript𝑞¯ℛ\neg q_{\bar{\mathcal{R}}}¬ italic_q start_POSTSUBSCRIPT over¯ start_ARG caligraphic_R end_ARG end_POSTSUBSCRIPT. | We’ll also need to reason about ¬q1subscript𝑞1\neg q_{1}¬ italic_q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT. | D |
∂∂tdΩg(t)=−ζR(t)dΩg(t).𝑡𝑑subscriptΩ𝑔𝑡𝜁𝑅𝑡𝑑subscriptΩ𝑔𝑡\frac{\partial}{\partial t}d\Omega_{g(t)}=-\zeta R(t)d\Omega_{g(t)}.divide start_ARG ∂ end_ARG start_ARG ∂ italic_t end_ARG italic_d roman_Ω start_POSTSUBSCRIPT italic_g ( italic_t ) end_POSTSUBSCRIPT = - italic_ζ italic_R ( italic_t ) italic_d roman_Ω start_POSTSUBSCRIPT italic_g ( italic_t ) end_POSTSUBSCRIPT . | Indeed, stochastic gravity could be simply regarded as a rigorous mathematical technique to work out the solution of the semi-classical Einstein’s field equation that is extracted from the localized (i.e approximated) version of the genuine formula of Wiener fractal gravity. In fact, as we established above, according to the original formulation of the Wiener fractal measure with the dynamics of the space geometry due to the Ricci flow, both the quantum field fluctuations and the gravitational features are basically non-local effects, admitting no definite Lagrangian density in a genuine path-integral formulation of the Wiener fractal gravity. However, as we showed in section III, an explicit action formula for the quantum field theory on a curved space-time continuum could be extracted at the first approximation of the Wiener fractal gravity. In other words, stochastic gravity is in principle one of the best-understood frameworks to work out the solutions of the approximated version of the Wiener fractal gravity described in (III.23). | The above equation lets us work out the explicit formulation of the Wiener fractal measure (III.1) along with the Ricci flow evolution of the space geometry. Initially, one should note that we have already considered a normalization condition as: | As we explained above, the Wiener Brownian motion is actually an entropic force whose origin stems from the second law of thermodynamics. On the other hand, according to Perelman perelman the second law of thermodynamics causes the geometric evolution of the underlying manifold via the Hamilton-Perelman Ricci flow. Therefore, to consider the whole features of the Wiener Brownian motion of the quantum states on M𝑀Mitalic_M one must incorporate the Ricci flow in the calculations to produce a dynamical background geometry. Actually, we should assume that the space metric would evolve with the Ricci flow. By definition the Ricci flow is: | Hence, the Wiener stochastic process of quantum states on the Riemannian manifold M𝑀Mitalic_M along with the Hamilton-Perelman Ricci flow automatically involves the Einstein-Hilbert action of general relativity via the approximations I, II, and III. In fact, in the above formulations, the emergence of general relativity is merely due to the evolution of the cosmos geometry, hence having nothing to do with possible background interactions of the included matter fields. However, one should note that the analytic structure of the spatial metric gijsubscript𝑔𝑖𝑗g_{ij}italic_g start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT fulfills Einstein’s field equation via the Euler-Lagrange equations. In other words, although 𝐠μνsubscript𝐠𝜇𝜈\mathbf{g}_{\mu\nu}bold_g start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT is affected by the energy-momentum tensor of the matter fields via the optimization of the total action, it must obey the Ricci flow as the imposed gauge fixing term that determines the intrinsic evolution of the space-time geometry. | B |
Homogeneous host distribution. We make the assumption that the host population is initially distributed evenly across the spatial domain. One could argue that the more accurate representation of the host population is one that is patchy in space, with clusters of the host existing at random points throughout the domain. However, we assume here that blossoms are dense enough in an orchard such that we take an average number of blossoms at each point in space without losing biological relevancy. This might be a reasonable assumption that approximates the situation in many industrial orchards well. | Ooze conversion. The pathogen can be transferred from living within the secreted ooze to living on the surface of flower. We assume that this liberation process is dependent on the rate at which ooze-carrying vectors visit flowers, the amount of ooze at that location, and the density of healthy flowers at that position. The underlying assumption here is that ooze-carrying vectors prefer to feed on the secreted ooze of infected flowers over the nectar of healthy flowers, and are more likely to deposit greater amounts of ooze to flowers (by pollinating for longer at that location) when there are a greater number of healthy flowers available. | Infection. The rate at which flowers transition from susceptible to infectious is dependent on the local floral pathogen population. We further assume that in order for infection to occur, there needs to be some threshold amount of the pathogen exceeded. | Pathogen multiplication on flowers. The growth of the pathogen within the floral cup is dependent on the current size of the bacterial population and the amount of resources available for consumption. We assume that the amount of resources available for consumption is dependent on the health of the flower, such that as the flower begins to die, the amount of resources available for consumption decline. We further assume that when all flowers are dead at a particular location, there can still exist some small, possibly dormant, population of the pathogen. | Death. We assume that the rate at which flowers die is a function of how strongly infected the population is, such that a greater number of infected flowers will weaken the local host population as a whole. This is to reflect the fact that these flowers are connected via a cluster, shoot or branch, and since the bacteria systematically invades the tree, the health of each host is in some sense dependent on the health of all the other hosts at the same location. We further assume that when only a small number of hosts are infected at a location, the average death rate of the hosts is close to zero, but when the infected population exceeds some threshold, the average death rate approaches some maximum value. | C |
By well known invariant theory, cj(M)subscript𝑐𝑗𝑀c_{j}(M)italic_c start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_M ) is an integral over M𝑀Mitalic_M of a universal polynomial in the covariant derivatives of the curvature of M𝑀Mitalic_M where the total degree is ≤jabsent𝑗\leq j≤ italic_j. The only possible invariants in degree 00 is the volume and in degree 2222 it is the integral of the scalar curvature. The universality implies that c0(M)=α0voln(M,g)subscript𝑐0𝑀subscript𝛼0subscriptvol𝑛𝑀𝑔c_{0}(M)=\alpha_{0}\mathrm{vol}_{n}(M,g)italic_c start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_M ) = italic_α start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT roman_vol start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_M , italic_g ) and c2(M)=α2∫MsdVsubscript𝑐2𝑀subscript𝛼2subscript𝑀𝑠differential-d𝑉c_{2}(M)=\alpha_{2}\int_{M}s\mathrm{d}Vitalic_c start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_M ) = italic_α start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ∫ start_POSTSUBSCRIPT italic_M end_POSTSUBSCRIPT italic_s roman_d italic_V for dimensional constants α0subscript𝛼0\alpha_{0}italic_α start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT and α2subscript𝛼2\alpha_{2}italic_α start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT. The values α0=1subscript𝛼01\alpha_{0}=1italic_α start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = 1 and α2=n+16subscript𝛼2𝑛16\alpha_{2}=\frac{n+1}{6}italic_α start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = divide start_ARG italic_n + 1 end_ARG start_ARG 6 end_ARG can be read of from works of Willerton [31, Theorem 11]. Since the Taylor expansion of the geodesic distance function near the boundary only contains even degree terms, cj(M)=0subscript𝑐𝑗𝑀0c_{j}(M)=0italic_c start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_M ) = 0 if j𝑗jitalic_j is odd. | Let X𝑋Xitalic_X be an n𝑛nitalic_n-dimensional manifold with boundary equipped with a distance function. We assume that X𝑋Xitalic_X is a domain with smooth boundary in a manifold M𝑀Mitalic_M. Using a variation of Theorem 4.6, we know that | Let X𝑋Xitalic_X be a compact n𝑛nitalic_n-dimensional manifold with boundary equipped with a distance function dd\mathrm{d}roman_d with property (MR)𝑀𝑅(MR)( italic_M italic_R ), e.g. if dd\mathrm{d}roman_d is regular at the diagonal and smooth off-diagonally. The magnitude function ℳXsubscriptℳ𝑋\mathcal{M}_{X}caligraphic_M start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT admits an asymptotic expansion | Let X𝑋Xitalic_X be a compact manifold with boundary and dd\mathrm{d}roman_d a distance function regular at the diagonal. The inverse of | Let X𝑋Xitalic_X be a compact manifold with boundary and dd\mathrm{d}roman_d a distance function regular at the diagonal. We have that | A |
(a) A 64-depth sparse ground truth test object (≈1.5%absentpercent1.5\approx 1.5\%≈ 1.5 % sparsity, or 1000 non-zero values) with wavelength 3.2μm3.2𝜇m3.2\mu\mathrm{m}3.2 italic_μ roman_m, and the depth channels inversely spaced from 1.65mm1.65mm1.65\mathrm{mm}1.65 roman_mm to 6.6mm6.6mm6.6\mathrm{mm}6.6 roman_mm. Patterns of varying structure are “hidden” in three of the 64 channels, plotted in grayscale. (b) Two images formed of the ground truth object by the amorphous and crystalline metasurface states of the optimized two-shot structure, with additive Gaussian noise (1%). | (c) CS reconstruction of the ground truth object for the optimized two-shot system (18% relative RMSE). | (d) CS reconstruction of the ground truth object for a two-shot system with a random metasurface (59% relative RMSE). (e) CS reconstruction of the ground truth object for an optimized single-shot system (74% relative RMSE). | We also note the significant improvement over single-shot designs where the GSS4T1 metasurface is kept in only its crystalline state. On the same ground truth object with 3%percent33\%3 % sparsity (Fig. 5), an end-to-end optimized single-shot design obtains a significantly worse reconstruction (74% relative RMSE, Fig. 5). This is due in large part to the delayed sparsity-error phase transition for the two-shot design (the rightward shift and slower incline of the sparsity-error relations in Fig. 5), emphasizing the crucial role of the material index contrast and the ability of our end-to-end optimization to exploit it. | We also perform a visual test of the system using a physical ground truth object with ≈1000absent1000\approx 1000≈ 1000 nonzero values (≈1.5%absentpercent1.5\approx 1.5\%≈ 1.5 % sparsity) where patterns of varying structure have been hidden at 3 of the 64 depths. This ground truth object differs substantially from the objects in the training distribution which have uniformly distributed nonzero values across all depths. Nevertheless, the optimized imaging system accurately recovers the support and forms a reconstruction with 18%percent1818\%18 % relative RMSE (Fig. 5), despite the image being corrupted with 2%percent22\%2 % noise, showing how the system generalizes to out-of-distribution sparse data. | A |
Let ℱ={z=x+iy:|x|≤1/2,|z|≥1}ℱconditional-set𝑧𝑥𝑖𝑦formulae-sequence𝑥12𝑧1\mathcal{F}=\{z=x+iy:|x|\leq 1/2,\;|z|\geq 1\}caligraphic_F = { italic_z = italic_x + italic_i italic_y : | italic_x | ≤ 1 / 2 , | italic_z | ≥ 1 } be the standard fundamental domain of Γ\ℍ\Γℍ\Gamma\backslash\mathbb{H}roman_Γ \ blackboard_H. | Let k𝑘kitalic_k be a large even integer and 1≤M≤logk1𝑀𝑘1\leq M\leq\log k1 ≤ italic_M ≤ roman_log italic_k. | Let 1≤M≤logk1𝑀𝑘1\leq M\leq\log k1 ≤ italic_M ≤ roman_log italic_k. Let 1≤|m|≤M(logk)ε1𝑚𝑀superscript𝑘𝜀1\leq|m|\leq M(\log k)^{\varepsilon}1 ≤ | italic_m | ≤ italic_M ( roman_log italic_k ) start_POSTSUPERSCRIPT italic_ε end_POSTSUPERSCRIPT. | Let k𝑘kitalic_k be a large even integer and J≤(logk)δ−ε𝐽superscript𝑘𝛿𝜀J\leq(\log k)^{\delta-\varepsilon}italic_J ≤ ( roman_log italic_k ) start_POSTSUPERSCRIPT italic_δ - italic_ε end_POSTSUPERSCRIPT be a positive integer. | for 1≤Y≤(logk)51𝑌superscript𝑘51\leq Y\leq(\log k)^{5}1 ≤ italic_Y ≤ ( roman_log italic_k ) start_POSTSUPERSCRIPT 5 end_POSTSUPERSCRIPT. | A |
In Section 2, we formally introduce the problem. In Section 3, we state our main convergence results for general convex and non-convex local functions. We also provide a lower bound to show the tightness of our convergence rate bounds for reasonably small step sizes. We justify our theoretical bounds through extensive numerical results in Section 4. Conclusions are given in Section 5. We defer all the proofs to Section A. | We consider the problem of n𝑛nitalic_n nodes [n]={1,2,…,n}delimited-[]𝑛12…𝑛[n]=\{1,2,\ldots,n\}[ italic_n ] = { 1 , 2 , … , italic_n } that collaboratively want to learn an over-parameterized model with decentralized data as the following distributed stochastic optimization problem: | In Local SGD, each node performs local gradient steps, and after every K𝐾Kitalic_K steps, sends the latest model to the central server. The server then computes the average of all nodes’ parameters and broadcasts the averaged model to all nodes. Let T𝑇Titalic_T be the total number of iterations in the algorithm. There is a set of communication times ℐ={0,K,2K,…,T=RK}ℐ0𝐾2𝐾…𝑇𝑅𝐾{\mathcal{I}}=\{0,K,2K,\ldots,T=RK\}caligraphic_I = { 0 , italic_K , 2 italic_K , … , italic_T = italic_R italic_K }333To simplify the analysis, we assume without loss of generality that T𝑇Titalic_T is divisible by K𝐾Kitalic_K, i.e., T=RK𝑇𝑅𝐾T=RKitalic_T = italic_R italic_K for some R∈ℕ𝑅ℕR\in\mathbb{N}italic_R ∈ blackboard_N., and in every iteration t𝑡titalic_t, Local SGD does the following: i) each node performs stochastic gradient updates locally based on ∇fi(𝐱,ξi)∇subscript𝑓𝑖𝐱subscript𝜉𝑖\nabla f_{i}(\mathbf{x},\xi_{i})∇ italic_f start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( bold_x , italic_ξ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ), which is an unbiased estimation of ∇fi(𝐱)∇subscript𝑓𝑖𝐱\nabla f_{i}(\mathbf{x})∇ italic_f start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( bold_x ), and ii) if t𝑡titalic_t is a communication time, i.e., t∈ℐ𝑡ℐt\in{\mathcal{I}}italic_t ∈ caligraphic_I, it sends the current model to the central server and receives the average of all nodes’ models. The pseudo-code for the Local SGD algorithm is provided in Algorithm 1. | For this set of experiments, we run the Local SGD algorithm for R=20000𝑅20000R=20000italic_R = 20000 communication rounds with a different number of local steps per communication round K=1,2,5,10,20𝐾1251020K=1,2,5,10,20italic_K = 1 , 2 , 5 , 10 , 20 and report the training error of the global model along the process. We do not report the test accuracy of the model, which is related to the generalization of the model and is beyond the scope of this work777Without data augmentation, the final test accuracy of the model in this set of experiments is around 80%percent8080\%80 %. . Following the work Hsieh et al. (2020), we also use Layer Normalization (Ba et al., 2016) instead of Batch Normalization in the architecture of ResNet18 while keeping everything else the same. | Distributed optimization methods have become increasingly popular in modern machine learning, owing to the data privacy/ownership issues and the scalability of learning models concerning massive datasets. The large datasets often make training the model and storing the data in a centralized way almost infeasible. That mandates the use of distributed optimization methods for training machine learning models. However, a critical challenge in distributed optimization is to reduce the communication cost among the local nodes, which has been reported as a major bottleneck in training many large-scale deep learning models (Zhang et al., 2017; Lin et al., 2017). | A |
[Mori, Cor. 1.7]) we also have κ(X~)=−∞𝜅~𝑋\kappa(\widetilde{X})=-\inftyitalic_κ ( over~ start_ARG italic_X end_ARG ) = - ∞. As promised, this proves | Note also that if κ(Y)≥0𝜅𝑌0\kappa(Y)\geq 0italic_κ ( italic_Y ) ≥ 0 and the complement of V𝑉Vitalic_V in Y𝑌Yitalic_Y has codimension | In case (1), since κ(L)≥1𝜅𝐿1\kappa(L)\geq 1italic_κ ( italic_L ) ≥ 1, the Campana-Peternell conjecture implies that κ(Y)≥1𝜅𝑌1\kappa(Y)\geq 1italic_κ ( italic_Y ) ≥ 1, contradicting our assumption that κ(Y)=0𝜅𝑌0\kappa(Y)=0italic_κ ( italic_Y ) = 0. | If κ(Y)=−∞𝜅𝑌\kappa(Y)=-\inftyitalic_κ ( italic_Y ) = - ∞ the result follows from the non-vanishing conjecture together with Proposition G, while if κ(Y)=0𝜅𝑌0\kappa(Y)=0italic_κ ( italic_Y ) = 0, it follows from Theorem H(i). If κ(Y)>0𝜅𝑌0\kappa(Y)>0italic_κ ( italic_Y ) > 0, we can reduce to the case | In this step we assume that κ(Y)≥0𝜅𝑌0\kappa(Y)\geq 0italic_κ ( italic_Y ) ≥ 0, and show that we can reduce to the case κ(Y)=0𝜅𝑌0\kappa(Y)=0italic_κ ( italic_Y ) = 0. | D |
A robust watermarking technique should allow the legitimate owner to correctly extract the watermark content (i.e., BER = 0) even in cases when functionality-preserving modifications are incurred on the model parameters. | FTAL using a different dataset is a fine-tuning technique that serves to re-purpose a model on a new similar task. | Figure 3. The effect of model fine-tuning on TATTOOED watermark using a different dataset than the one used for the training. In each case, after the fine-tuning procedure, the watermark BER was equal to 0. | Fine-tuning is a process that takes a model that has already been trained for one given task and then tunes or tweaks the model to make it perform better on the same task or perform a similar task. For our robustness evaluation, we consider all the possible fine-tuning techniques as follows: | Fine-tune all layer is a common fine-tuning technique to boost the performance of a pre-trained model on the task using new data. | C |
The general result follows from considerations of a sequence of finite stopping lines Ln↑L↑subscript𝐿𝑛𝐿L_{n}\uparrow Litalic_L start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ↑ italic_L, where Lnsubscript𝐿𝑛L_{n}italic_L start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT is monotonically increasing towards L𝐿Litalic_L, and utilizing the monotone convergence and backwards martingale convergence theorems applied to the conditional expectations given ℱLnsubscriptℱsubscript𝐿𝑛\mathcal{F}_{L_{n}}caligraphic_F start_POSTSUBSCRIPT italic_L start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT as ℱLn↓ℱL↓subscriptℱsubscript𝐿𝑛subscriptℱ𝐿\mathcal{F}_{L_{n}}\downarrow\mathcal{F}_{L}caligraphic_F start_POSTSUBSCRIPT italic_L start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT ↓ caligraphic_F start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT. | Note that the events {Zn>0}subscript𝑍𝑛0\{Z_{n}>0\}{ italic_Z start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT > 0 } and {ζn≥1}subscript𝜁𝑛1\{\zeta_{n}\geq 1\}{ italic_ζ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ≥ 1 } are equivalent. Moreover, conditioned on {Zn>0}subscript𝑍𝑛0\{Z_{n}>0\}{ italic_Z start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT > 0 }, there exists an individual at generation 1111 with alive descendants at generation n𝑛nitalic_n. Therefore, the random variable ηnsubscript𝜂𝑛\eta_{n}italic_η start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT can be interpreted as the number of individuals at generation one, having alive descendants at generation n𝑛nitalic_n which are different from that individual. Note that for every k≥0𝑘0k\geq 0italic_k ≥ 0, | We will establish in Proposition 3.1 that the sequence-valued process 𝐃:=(Di,i≥0)assign𝐃subscript𝐷𝑖𝑖0{\mathbf{D}}:=(D_{i},i\geq 0)bold_D := ( italic_D start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_i ≥ 0 ) possesses the Markov property. To provide the transitions probabilities, we begin by recalling that for each (m,i)∈ℤ−×ℕ𝑚𝑖subscriptℤℕ(m,i)\in\mathbb{Z}_{-}\times\mathbb{N}( italic_m , italic_i ) ∈ roman_ℤ start_POSTSUBSCRIPT - end_POSTSUBSCRIPT × roman_ℕ, the forward-in-time dynamics starting at individual (m,i)𝑚𝑖(m,i)( italic_m , italic_i ) are produced by a branching process in a varying environment. To be precise, let Z(m,i)(k)superscript𝑍𝑚𝑖𝑘Z^{(m,i)}(k)italic_Z start_POSTSUPERSCRIPT ( italic_m , italic_i ) end_POSTSUPERSCRIPT ( italic_k ) denote the number of descendants of individual (m,i)𝑚𝑖(m,i)( italic_m , italic_i ) at generation m+k𝑚𝑘m+kitalic_m + italic_k. Then, (Z(m,i)(k),k≥0)superscript𝑍𝑚𝑖𝑘𝑘0(Z^{(m,i)}(k),k\geq 0)( italic_Z start_POSTSUPERSCRIPT ( italic_m , italic_i ) end_POSTSUPERSCRIPT ( italic_k ) , italic_k ≥ 0 ) constitutes a GWVE process with environment 𝒮(m):=(em+1,em+2,…,e0,0,0,…)assignsubscript𝒮𝑚subscript𝑒𝑚1subscript𝑒𝑚2…subscript𝑒000…\mathcal{S}_{(m)}:=(e_{m+1},e_{m+2},\dots,e_{0},0,0,\dots)caligraphic_S start_POSTSUBSCRIPT ( italic_m ) end_POSTSUBSCRIPT := ( italic_e start_POSTSUBSCRIPT italic_m + 1 end_POSTSUBSCRIPT , italic_e start_POSTSUBSCRIPT italic_m + 2 end_POSTSUBSCRIPT , … , italic_e start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , 0 , 0 , … ), starting with one individual. Consequently, for every m∈ℤ−𝑚subscriptℤm\in\mathbb{Z}_{-}italic_m ∈ roman_ℤ start_POSTSUBSCRIPT - end_POSTSUBSCRIPT, the probability that an individual at generation m𝑚mitalic_m has alive descendants at the present generation is pm=1−fm,0(0)subscript𝑝𝑚1subscript𝑓𝑚00p_{m}=1-f_{m,0}(0)italic_p start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT = 1 - italic_f start_POSTSUBSCRIPT italic_m , 0 end_POSTSUBSCRIPT ( 0 ), see (3). Moreover, if we denote by ζ(m)superscript𝜁𝑚\zeta^{(m)}italic_ζ start_POSTSUPERSCRIPT ( italic_m ) end_POSTSUPERSCRIPT the number of its daughters with alive descendants at generation zero, as a consequence of (4), we have | We are also interested in the law of the children of the root conditioned to have alive descendants at a fixed generation. We note that the survival probability up to generation n𝑛nitalic_n is given by | The distribution of 𝐀𝐀\mathbf{A}bold_A is not easy to determine, and in general, it is not a Markov process, except for some special cases. Following the approach of [6], let us define an auxiliary Markov process that characterizes the genealogy. For any fixed individual i𝑖iitalic_i, we follow its ancestral line (referred to as the i𝑖iitalic_i-th spine) and consider the subtrees with roots in this spine. Note that these roots are labeled by {(−n,𝔞i(n)),n∈ℕ}𝑛subscript𝔞𝑖𝑛𝑛ℕ\{(-n,\mathfrak{a}_{i}(n)),n\in\mathbb{N}\}{ ( - italic_n , fraktur_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_n ) ) , italic_n ∈ roman_ℕ }. At every subtree, we count the number of daughters of the root at the right-hand side of the spine, whose descendants are alive at the present generation. To be precise, let | C |
Kumjian and Renault showed that C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebras associated with groupoids provide versatile models for large classes of C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebras [kum:diagonals, ren:cartan]. More recently, Li showed that every classifiable C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra admits a (twisted) groupoid model [li:cartan]. One of the biggest open questions in the field concerns the Universal Coefficient Theorem (UCT) and work of Barlak and Li [bali:CartanUCT] showed that the UCT problem can be translated to the question whether every nuclear C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra admits a groupoid model. | Kumjian and Renault showed that C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebras associated with groupoids provide versatile models for large classes of C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebras [kum:diagonals, ren:cartan]. More recently, Li showed that every classifiable C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra admits a (twisted) groupoid model [li:cartan]. One of the biggest open questions in the field concerns the Universal Coefficient Theorem (UCT) and work of Barlak and Li [bali:CartanUCT] showed that the UCT problem can be translated to the question whether every nuclear C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra admits a groupoid model. | In another direction, Matui’s works [matui:hk, matui:productSFT] have kickstarted a fruitful line of research in topological dynamics using étale groupoids at its heart (see also [li:orbitequivalence]). In this area it turns out that many invariants for topological dynamical systems can most naturally been defined in the framework of groupoid homology or the K𝐾Kitalic_K-theory of groupoid C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebras. | Consequently, there is a great deal of interest around the homology and K𝐾Kitalic_K-theory of étale groupoids and their interaction. Examples of recent research in this direction are the HK conjecture of Matui [matui:hk], or the relation between the homology theory of Smale spaces and the K𝐾Kitalic_K-theory of their corresponding C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebras [put:HoSmale]. In this latter example, a special case of the methods developed here (i.e., when the groupoid is torsion-free and ample) has already been applied with great success and lead to many interesting results in topological dynamics, as is demonstrated by the papers [bdgw:matui, valmak:groupoid, valmak:groupoidtwo, valmak:groupoidthree]. | Over the last decades étale groupoids and their homological and K𝐾Kitalic_K-theoretical invariants have played an increasingly important role in the fields of operator algebras, noncommutative geometry and topological dynamics. | B |
The paper is divided in the following way. First, the main results are stated precisely in Section 2 as well as their possible extensions, and the precise strategy of the proof is presented in Section 3. Then, the proof is further divided in two parts. The first part deals with the proof of the main theorems in the abstract settings. | The proof of Theorem 1.1 relies on our compactness/duality method, allowing to prove the existence of a Fredholm operator for the backstepping method as long as α>1𝛼1\alpha>1italic_α > 1 (and in particular below the critical case α=3/2𝛼32\alpha=3/2italic_α = 3 / 2). | The precise statement of our main results relies on the spectral properties of the operator A𝐴Aitalic_A and on the regularity properties of the control operator B𝐵Bitalic_B such that it is admissible (see Assumption 2), and (1.1) is controllable. In order to introduce the techniques used in the article and to motivate the new compactness/duality method, we begin by giving a simplified statement of our main results when the operator only has simple eigenvalues (the precise statements for the general case are found in Section 2, see Theorems 2.5 and 2.7) | The paper is divided in the following way. First, the main results are stated precisely in Section 2 as well as their possible extensions, and the precise strategy of the proof is presented in Section 3. Then, the proof is further divided in two parts. The first part deals with the proof of the main theorems in the abstract settings. | In particular, Section 4 begins with the statement of technical estimates used throughout the article as well as the proof of the Riesz basis (Step 1 of Section 1.4) using the compactness/duality argument. | D |
Let X𝑋Xitalic_X be a one-relator complex with negative immersions. Then every finitely generated subgroup of π1(X)subscript𝜋1𝑋\pi_{1}(X)italic_π start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_X ) is hyperbolic. | All one-relator splittings of one-relator groups with negative immersions are ℤℤ\mathbb{Z}blackboard_Z-stable. | In [Wis21], Wise shows that Magnus–Moldavanskii hierarchies of one-relator groups with torsion are quasi-convex. We show that all one-relator hierarchies of one-relator complexes X𝑋Xitalic_X satisfying either of the following are ℤℤ\mathbb{Z}blackboard_Z-stable: | Extending any of these results to all one-relator groups would require a better understanding of one-relator hierarchies that are not ℤℤ\mathbb{Z}blackboard_Z-stable. Therefore, we ask the following question. | Our next result establishes an equivalence between quasi-convex one-relator hierarchies and ℤℤ\mathbb{Z}blackboard_Z-stable one-relator hierarchies of hyperbolic one-relator groups. | C |
DRIVE[62] is a retinal vessel dataset with 40 RGB 565 ×\times× 584 images, and binary labels for 8.63% vessel pixels. We take 20 images for training and the other 20 images for testing. | CHASE [63] is a retinal vessel dataset with 28 RGB 999 ×\times× 960 images, and binary labels for 7.19% vessel pixels. We take 20 images for training and 8 images for testing. | Fig. LABEL:fig:dataset shows some images and labels from the above three datasets. During the training process, all images are randomly cropped into patches of a uniform size 64×64646464\times 6464 × 64. During the testing process, images will be processed with overlapped moving window of 64×64646464\times 6464 × 64. We use the standard flipping and cropping data augmentation from PyTorch. | DRIVE[62] is a retinal vessel dataset with 40 RGB 565 ×\times× 584 images, and binary labels for 8.63% vessel pixels. We take 20 images for training and the other 20 images for testing. | CRACK is a pavement crack dataset with 206 RGB 448×448448448448\times 448448 × 448 images, and binary labels for 0.32% cracks pixels. It consists of crack-centered cropped images of a subset of Cracktree[64]. We take 190 images for training and 16 images for testing. | A |
Let F:Mn→ℝ1n+1:𝐹→superscript𝑀𝑛subscriptsuperscriptℝ𝑛11F:M^{n}\to{\mathbb{R}}^{n+1}_{1}italic_F : italic_M start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT → blackboard_R start_POSTSUPERSCRIPT italic_n + 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT | be a (co-orientable) null Crsuperscript𝐶𝑟C^{r}italic_C start_POSTSUPERSCRIPT italic_r end_POSTSUPERSCRIPT-wave front. | ((((co-orientable)))) null Crsuperscript𝐶𝑟C^{r}italic_C start_POSTSUPERSCRIPT italic_r end_POSTSUPERSCRIPT-wave front | ((((co-orientable)))) null Crsuperscript𝐶𝑟C^{r}italic_C start_POSTSUPERSCRIPT italic_r end_POSTSUPERSCRIPT-wave front, and let | ((((co-orientable)))) Crsuperscript𝐶𝑟C^{r}italic_C start_POSTSUPERSCRIPT italic_r end_POSTSUPERSCRIPT-wave front. | B |
{jk}(t,s)+\lambda^{2}_{j}u_{jk}(t,s)+\lambda_{k}^{2}u_{jk}(t,s)|^{2}dtds.= ∫ start_POSTSUBSCRIPT blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ∑ start_POSTSUBSCRIPT italic_j , italic_k end_POSTSUBSCRIPT | - ( ∂ start_POSTSUBSCRIPT italic_t italic_t end_POSTSUBSCRIPT + ∂ start_POSTSUBSCRIPT italic_s italic_s end_POSTSUBSCRIPT ) italic_u start_POSTSUBSCRIPT italic_j italic_k end_POSTSUBSCRIPT ( italic_t , italic_s ) + italic_λ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT italic_u start_POSTSUBSCRIPT italic_j italic_k end_POSTSUBSCRIPT ( italic_t , italic_s ) + italic_λ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_u start_POSTSUBSCRIPT italic_j italic_k end_POSTSUBSCRIPT ( italic_t , italic_s ) | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_d italic_t italic_d italic_s . | In this case we use the diffeomorphism f2(t,x,s,y)=(s,x,t,y)subscript𝑓2𝑡𝑥𝑠𝑦𝑠𝑥𝑡𝑦f_{2}(t,x,s,y)=(s,x,t,y)italic_f start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_t , italic_x , italic_s , italic_y ) = ( italic_s , italic_x , italic_t , italic_y ) that has the set of normals Nf2={(s,x,t,y,0,0,0,0)∈T∗(M×M)}subscript𝑁subscript𝑓2𝑠𝑥𝑡𝑦0000superscript𝑇𝑀𝑀N_{f_{2}}=\{(s,x,t,y,0,0,0,0)\in T^{*}(M\times M)\}italic_N start_POSTSUBSCRIPT italic_f start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT = { ( italic_s , italic_x , italic_t , italic_y , 0 , 0 , 0 , 0 ) ∈ italic_T start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_M × italic_M ) } which has empty intersection with WF(KG)𝑊𝐹subscript𝐾𝐺WF(K_{G})italic_W italic_F ( italic_K start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ). Then [35, Theorem 8.2.3] and the invariance of the Sobolev wavefront set implies that | Applying the Fourier transform in (s,t)𝑠𝑡(s,t)( italic_s , italic_t ), Plancherel’s theorem shows that | where (ℱu)(ξ)=(2π)−n/2∫e−ixξu(x)dnxℱ𝑢𝜉superscript2𝜋𝑛2superscript𝑒𝑖𝑥𝜉𝑢𝑥superscript𝑑𝑛𝑥(\mathcal{F}u)(\xi)=(2\pi)^{-n/2}\int e^{-ix\xi}u(x)\,d^{n}x( caligraphic_F italic_u ) ( italic_ξ ) = ( 2 italic_π ) start_POSTSUPERSCRIPT - italic_n / 2 end_POSTSUPERSCRIPT ∫ italic_e start_POSTSUPERSCRIPT - italic_i italic_x italic_ξ end_POSTSUPERSCRIPT italic_u ( italic_x ) italic_d start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT italic_x is the Fourier transform. | The diffeomorphism f1(t,x,s,y)=(s,y,t,x)subscript𝑓1𝑡𝑥𝑠𝑦𝑠𝑦𝑡𝑥f_{1}(t,x,s,y)=(s,y,t,x)italic_f start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_t , italic_x , italic_s , italic_y ) = ( italic_s , italic_y , italic_t , italic_x ) has the set of normals Nf1={(s,y,t,x,0,0,0,0)∈T∗(M×M)}subscript𝑁subscript𝑓1𝑠𝑦𝑡𝑥0000superscript𝑇𝑀𝑀N_{f_{1}}=\{(s,y,t,x,0,0,0,0)\in T^{*}(M\times M)\}italic_N start_POSTSUBSCRIPT italic_f start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT = { ( italic_s , italic_y , italic_t , italic_x , 0 , 0 , 0 , 0 ) ∈ italic_T start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_M × italic_M ) } which has empty intersection with WF(KG)𝑊𝐹subscript𝐾𝐺WF(K_{G})italic_W italic_F ( italic_K start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ). Then [35, Theorem 8.2.3] and the invariance of the Sobolev wavefront set implies that | B |
It is well know that the classical Brunn-Minkowski-theory of convex bodies (i.e., compact, convex sets) in n𝑛nitalic_n-dimensional Euclidean spaces ℝnsuperscriptℝ𝑛\mathbb{R}^{n}blackboard_R start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT plays an important role in the study of convex geometric analysis and develops rapidly in recent years. | Thus, if the given Borel measure μ𝜇\muitalic_μ has a positive continuous density, the classical Minkowski problem can be seen as the problem of prescribing the Gauss curvature in differential geometry. | In modern convex geometry, the Lpsubscript𝐿𝑝L_{p}italic_L start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT Minkowski problem [32], Orlicz Minkowski problem [20] and their dual Minkowski problem [16, 17, 22, 34] generalize and dualize the classical Minkowski problem, and then studied by [3, 9, 23, 24, 27, 28, 33, 42, 43, 44, 45, 46] and the references therein. | As far as we know, the existence for p<1𝑝1p<1italic_p < 1 are unknown until now. An extreme case of Equation (1.3) is k=n−1𝑘𝑛1k=n-1italic_k = italic_n - 1, which corresponds to the Lpsubscript𝐿𝑝L_{p}italic_L start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT dual Minkowski problem introduced in [22], and then followed by [6, 7, 8, 34] and the references therein. Moreover, the case of k=n−1𝑘𝑛1k=n-1italic_k = italic_n - 1 and q=n𝑞𝑛q=nitalic_q = italic_n is the Lpsubscript𝐿𝑝L_{p}italic_L start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT Minkowski problem posed in [32]. | The classical Minkowski problem, introduced by Minkiwski [38], is one of the cornerstones of the classical Brunn-Minkowski theory | D |
Wn(polp(B))subscript𝑊𝑛subscriptpol𝑝𝐵{W_{n}(\operatorname{pol}_{p}(B))}italic_W start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( roman_pol start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ( italic_B ) )Wn+1(polp(B))subscript𝑊𝑛1subscriptpol𝑝𝐵{W_{n+1}(\operatorname{pol}_{p}(B))}italic_W start_POSTSUBSCRIPT italic_n + 1 end_POSTSUBSCRIPT ( roman_pol start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ( italic_B ) )Wn(B)subscript𝑊𝑛𝐵{W_{n}(B)}italic_W start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_B )Wn+1(B)subscript𝑊𝑛1𝐵{W_{n+1}(B)}italic_W start_POSTSUBSCRIPT italic_n + 1 end_POSTSUBSCRIPT ( italic_B )V𝑉\scriptstyle{V}italic_Vpolp(V)subscriptpol𝑝𝑉\scriptstyle{\operatorname{pol}_{p}(V)}roman_pol start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ( italic_V ) | D(Fru(A))=colimnHomAbSchk(Fr(A),Wn)=colimWn(A)=CWu(A).∎𝐷superscriptFr𝑢𝐴subscriptcolim𝑛subscriptHomsubscriptAbSch𝑘Fr𝐴subscript𝑊𝑛colimsubscript𝑊𝑛𝐴𝐶superscript𝑊𝑢𝐴D(\operatorname{Fr}^{u}(A))=\operatornamewithlimits{colim}_{n}\operatorname{% | CWu(R)=colim(W1(R)→𝑉W2(R)→𝑉⋯)𝐶superscript𝑊𝑢𝑅colim𝑉→subscript𝑊1𝑅subscript𝑊2𝑅𝑉→⋯CW^{u}(R)=\operatornamewithlimits{colim}(W_{1}(R)\xrightarrow{V}W_{2}(R)% | Thus CWu(pol(B))≅colimVCWn(pol(B))≅colimVCWn(B)≅CWu(B)𝐶superscript𝑊𝑢pol𝐵subscriptcolim𝑉𝐶subscript𝑊𝑛pol𝐵subscriptcolim𝑉𝐶subscript𝑊𝑛𝐵𝐶superscript𝑊𝑢𝐵CW^{u}(\operatorname{pol}(B))\cong\operatornamewithlimits{colim}_{V}CW_{n}(% | CWu(A)=colim(W0(A)→𝑉W1(A(−1))→𝑉W2(A(−2))→𝑉⋯),𝐶superscript𝑊𝑢𝐴colim𝑉→subscript𝑊0𝐴subscript𝑊1𝐴1𝑉→subscript𝑊2𝐴2𝑉→⋯CW^{u}(A)=\operatornamewithlimits{colim}(W_{0}(A)\xrightarrow{V}W_{1}(A(-1))% | C |
0,1)]^{({n_{y}}\times 1)};\;italic_A = [ Uniform ( 0 , 1 ) ] start_POSTSUPERSCRIPT ( italic_n start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT × 1 ) end_POSTSUPERSCRIPT ; italic_B = [ Uniform ( 0 , 1 ) ] start_POSTSUPERSCRIPT ( italic_n start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT × 1 ) end_POSTSUPERSCRIPT ; | E=[Normal(0,1)](nu×1);G=[Uniform(0,1)](ny×1);formulae-sequence𝐸superscriptdelimited-[]Normal01subscript𝑛𝑢1𝐺superscriptdelimited-[]Uniform01subscript𝑛𝑦1\displaystyle E=[\text{Normal}(0,1)]^{({n_{u}}\times 1)};\;G=[\text{Uniform}(0% | H=[Normal(0,1)](nx×nl);J=[Normal(0,1)](ny×nl);formulae-sequence𝐻superscriptdelimited-[]Normal01subscript𝑛𝑥subscript𝑛𝑙𝐽superscriptdelimited-[]Normal01subscript𝑛𝑦subscript𝑛𝑙\displaystyle H=[\text{Normal}(0,1)]^{({n_{x}}\times{n_{l}})};\;J=[\text{% | C=[Normal(0,1)](nx×nu);D=[0](ny×nu);formulae-sequence𝐶superscriptdelimited-[]Normal01subscript𝑛𝑥subscript𝑛𝑢𝐷superscriptdelimited-[]0subscript𝑛𝑦subscript𝑛𝑢\displaystyle C=[\text{Normal}(0,1)]^{({n_{x}}\times{n_{u}})};\;D=[0]^{({n_{y}% | N=[Normal(0,1)](nl×1);𝑁superscriptdelimited-[]Normal01subscript𝑛𝑙1\displaystyle N=[\text{Normal}(0,1)]^{({n_{l}}\times 1)};\;italic_N = [ Normal ( 0 , 1 ) ] start_POSTSUPERSCRIPT ( italic_n start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT × 1 ) end_POSTSUPERSCRIPT ; | C |
If l=6k−3𝑙6𝑘3l=6k-3italic_l = 6 italic_k - 3, then I𝐚≡−42k+13modmsubscript𝐼𝐚modulo42𝑘13𝑚I_{\mathbf{a}}\equiv-42k+13\mod mitalic_I start_POSTSUBSCRIPT bold_a end_POSTSUBSCRIPT ≡ - 42 italic_k + 13 roman_mod italic_m, while if l=6k−1𝑙6𝑘1l=6k-1italic_l = 6 italic_k - 1, then I𝐚≡−42k−1subscript𝐼𝐚42𝑘1I_{\mathbf{a}}\equiv-42k-1italic_I start_POSTSUBSCRIPT bold_a end_POSTSUBSCRIPT ≡ - 42 italic_k - 1. Hence, for a fixed m𝑚mitalic_m and k𝑘kitalic_k, we can always choose l=6k−3𝑙6𝑘3l=6k-3italic_l = 6 italic_k - 3 or l=6k−1𝑙6𝑘1l=6k-1italic_l = 6 italic_k - 1 so that m∤I𝐚not-divides𝑚subscript𝐼𝐚m\nmid I_{\mathbf{a}}italic_m ∤ italic_I start_POSTSUBSCRIPT bold_a end_POSTSUBSCRIPT. This implies that X=(Y(𝐚)∖{0})/ℂ∗𝑋𝑌𝐚0superscriptℂX=\left(Y(\mathbf{a})\setminus\{0\}\right)/\mathbb{C}^{*}italic_X = ( italic_Y ( bold_a ) ∖ { 0 } ) / blackboard_C start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT admits no holomorphic contact structure by Proposition 2.9 and so the metrics we found are pairwise non-isometric by Proposition 2.8. | We shall use the following propositions to distinguish the Sasaki-Einstein metrics on the links of weighted homogeneous hypersurface singularities. | As we saw earlier, the proof of Proposition 3.2 reduces to showing a special case of Conjecture 3.8. | The rest of the section is devoted to the proof of Proposition 3.2. We will repeatedly use the following lemma. | Li’s motivation of introducing the normalized volume was to study the local K-stability theory for klt singularities. We refer the readers to the surveys [LLX20, Zhu23] for more backgrounds and discussions. Here we will focus on the case of K-polystable Fano cones. | C |
For each end of the manifold V𝑉Vitalic_V, we also obtain an homomorphism rϵ:𝒟(V)→𝒟ϵ(V):subscript𝑟italic-ϵ→𝒟𝑉superscript𝒟italic-ϵ𝑉r_{\epsilon}:\mathcal{D}(V)\to\mathcal{D}^{\epsilon}(V)italic_r start_POSTSUBSCRIPT italic_ϵ end_POSTSUBSCRIPT : caligraphic_D ( italic_V ) → caligraphic_D start_POSTSUPERSCRIPT italic_ϵ end_POSTSUPERSCRIPT ( italic_V ) by sending diffeomorphisms to their corresponding equivalence class with respect to the end ϵitalic-ϵ\epsilonitalic_ϵ. | We will be able to achieve a similar statement; in the case of non-compact 4-manifolds, however, we shall restrict only to diffeomorphisms of the punctured ends since the other ends might have a wild behavior that we cannot detect with the tools we have developed so far. As an example, there might be an end that is not topologically collared by any M×ℝ𝑀ℝM\times\mathbb{R}italic_M × blackboard_R, where M𝑀Mitalic_M is a compact 3-manifold. | The following two remarks assume that the ends of V𝑉Vitalic_V are isolated, which is a consistent assumption in our setting. If we drop this condition, the constructions in remarks 2.10 and 2.11 might not be well defined. | We impose the assumption S⊊M̊𝑆̊𝑀S\subsetneq\mathring{M}italic_S ⊊ over̊ start_ARG italic_M end_ARG in order to avoid removing points on the (possibly non-empty) boundary ∂M𝑀\partial M∂ italic_M. Going through the proof of Lemma 3.1, we see that we made almost no use of the empty boundary assumption. However if p∈∂M𝑝𝑀p\in\partial Mitalic_p ∈ ∂ italic_M, then the initial idea, connect-summing with ℛℛ\mathcal{R}caligraphic_R, needs to be modified. We have to take the boundary sum with ℍ4superscriptℍ4\mathbb{H}^{4}blackboard_H start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT, the upper half 4-space. We do not know how the diffeotopy groups of ℍ4superscriptℍ4\mathbb{H}^{4}blackboard_H start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT behave (note that this is D4∖ptsuperscript𝐷4𝑝𝑡D^{4}{\smallsetminus}ptitalic_D start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT ∖ italic_p italic_t where the point lies in the boundary 𝕊3superscript𝕊3\mathbb{S}^{3}blackboard_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT, a case we will not cover) and so no direct conclusions can be made from this construction. | In the following discussion, if M𝑀Mitalic_M is orientable, then we have to restrict to the orientation-preserving diffeomorphism of the ends, otherwise the Cerf-Palais disk theorem [Cerf, Palais] (which we use in the proof of Lemma 4.2) does not hold. We will however denote the groups by the usual 𝒟𝒟\mathcal{D}caligraphic_D, suppressing the + subscript, to ease the notation. | B |
As a particular case of frequent interest, we have singular foliations which are locally real analytic. A singular foliation ℱℱ\mathcal{F}caligraphic_F is said real analytic if there exists an open cover of M𝑀Mitalic_M such that the generators of ℱℱ\mathcal{F}caligraphic_F only involve real analytic functions of the local coordinates associated to each chart. Then, for every x∈M𝑥𝑀x\in Mitalic_x ∈ italic_M there exists a universal Lie ∞\infty∞-algebroid of ℱℱ\mathcal{F}caligraphic_F of length at most n+1𝑛1n+1italic_n + 1 defined over some neighborhood of x𝑥xitalic_x. The following theorem is a result about the universal property of universal Lie ∞\infty∞-algebroids: | Let ℱℱ\mathcal{F}caligraphic_F be a singular foliation on a smooth manifold M𝑀Mitalic_M, and let (E,lk,ρ)𝐸subscript𝑙𝑘𝜌(E,l_{k},\rho)( italic_E , italic_l start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT , italic_ρ ) be a Lie ∞\infty∞-algebroid over M𝑀Mitalic_M covering ℱℱ\mathcal{F}caligraphic_F. We say that E𝐸Eitalic_E is a universal Lie ∞\infty∞-algebroid of ℱℱ\mathcal{F}caligraphic_F if its linear part is a geometric resolution of ℱℱ\mathcal{F}caligraphic_F. The set of all universal Lie ∞\infty∞-algebroids of ℱℱ\mathcal{F}caligraphic_F is denoted 𝔈ℱsubscript𝔈ℱ\mathfrak{E}_{\mathcal{F}}fraktur_E start_POSTSUBSCRIPT caligraphic_F end_POSTSUBSCRIPT. | Let ℱℱ\mathcal{F}caligraphic_F be a solvable singular foliation and let E𝐸Eitalic_E be a universal Lie ∞\infty∞-algebroid of ℱℱ\mathcal{F}caligraphic_F of finite length. Let θ𝔘ℱsuperscriptsubscript𝜃𝔘ℱ\theta_{\mathfrak{U}}^{\mathcal{F}}italic_θ start_POSTSUBSCRIPT fraktur_U end_POSTSUBSCRIPT start_POSTSUPERSCRIPT caligraphic_F end_POSTSUPERSCRIPT be the cohomology class induced by the modular class of E𝐸Eitalic_E in the universal foliated cohomology of ℱℱ\mathcal{F}caligraphic_F. | Let ℱℱ\mathcal{F}caligraphic_F be a singular foliation on a smooth manifold M𝑀Mitalic_M and let E𝐸Eitalic_E be a universal Lie ∞\infty∞-algebroid of ℱℱ\mathcal{F}caligraphic_F. | Let ℱℱ\mathcal{F}caligraphic_F be a singular foliation on M𝑀Mitalic_M, and let (E,lk,ρ)𝐸subscript𝑙𝑘𝜌(E,l_{k},\rho)( italic_E , italic_l start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT , italic_ρ ) be a Lie ∞\infty∞-algebroid covering ℱℱ\mathcal{F}caligraphic_F (i.e. not necessarily a resolution). | C |
σ(X,X∘)𝜎𝑋superscript𝑋\sigma(X,X^{\circ})italic_σ ( italic_X , italic_X start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT )-(T(t))t≥0subscript𝑇𝑡𝑡0(T(t))_{t\geq 0}( italic_T ( italic_t ) ) start_POSTSUBSCRIPT italic_t ≥ 0 end_POSTSUBSCRIPT-equicontinuous which is easily seen as well. | Let (X,∥⋅∥,τ)(X,\|\cdot\|,\tau)( italic_X , ∥ ⋅ ∥ , italic_τ ) be a sequentially complete d-consistent Mazur–Saks space | Let (X,∥⋅∥,τ)(X,\|\cdot\|,\tau)( italic_X , ∥ ⋅ ∥ , italic_τ ) be a sequentially complete d-consistent Mazur–Saks space | Let (X,∥⋅∥,τ)(X,\|\cdot\|,\tau)( italic_X , ∥ ⋅ ∥ , italic_τ ) be a sequentially complete d-consistent Mazur–Saks space | Let (X,∥⋅∥,τ)(X,\|\cdot\|,\tau)( italic_X , ∥ ⋅ ∥ , italic_τ ) be a sequentially complete d-consistent Mazur–Saks space | A |
\tau(\xi^{2}).≤ ∑ start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT divide start_ARG italic_τ ( italic_d ) end_ARG start_ARG italic_d start_POSTSUPERSCRIPT 1 / 2 end_POSTSUPERSCRIPT square-root start_ARG italic_d start_POSTSUBSCRIPT italic_ξ end_POSTSUBSCRIPT end_ARG start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT end_ARG ≤ ∑ start_POSTSUBSCRIPT italic_η ∣ italic_ξ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_POSTSUBSCRIPT divide start_ARG italic_τ ( italic_η ) end_ARG start_ARG italic_η start_POSTSUPERSCRIPT 1 / 2 end_POSTSUPERSCRIPT end_ARG ∑ start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT divide start_ARG italic_τ ( italic_d ) end_ARG start_ARG italic_d start_POSTSUPERSCRIPT 1 / 2 end_POSTSUPERSCRIPT square-root start_ARG italic_d end_ARG start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT end_ARG ≪ italic_τ ( italic_ξ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) . | Combining this with (7.9), (7.6), the first part of Proposition 6.3 and choosing Ω=min(A,C,Z)1/50\Omega=\min(A,C,Z)^{1/50}roman_Ω = roman_min ( italic_A , italic_C , italic_Z ) start_POSTSUPERSCRIPT 1 / 50 end_POSTSUPERSCRIPT, we have shown | On the other hand, choosing Ω=3Ω3\Omega=3roman_Ω = 3, we also invoke (6.13) to obtain our second important bound | Combining this with the off-diagonal contribution (6.5) and choosing Ω=3Ω3\Omega=3roman_Ω = 3, we have shown | Combining this with (7.6) and the second part of Proposition 6.3 and choosing Ω=min(Y,W,A)1/50\Omega=\min(Y,W,A)^{1/50}roman_Ω = roman_min ( italic_Y , italic_W , italic_A ) start_POSTSUPERSCRIPT 1 / 50 end_POSTSUPERSCRIPT, we have shown | C |
α(xi)=αd(xixj)=αd(xixjxi)𝛼subscript𝑥𝑖subscript𝛼𝑑subscript𝑥𝑖subscript𝑥𝑗subscript𝛼𝑑subscript𝑥𝑖subscript𝑥𝑗subscript𝑥𝑖\displaystyle\alpha(x_{i})=\alpha_{d}(x_{i}x_{j})=\alpha_{d}(x_{i}x_{j}x_{i})italic_α ( italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) = italic_α start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_x start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ) = italic_α start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_x start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) | \quad i\vartriangleright j\coloneqq 2i-j\,(3);italic_c ( italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⊗ italic_x start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ) = - italic_x start_POSTSUBSCRIPT italic_i ⊳ italic_j end_POSTSUBSCRIPT ⊗ italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_i ⊳ italic_j ≔ 2 italic_i - italic_j ( 3 ) ; | ≔∑i≠j∈Xξij.≔absentsubscript𝑖𝑗𝑋superscriptsubscript𝜉𝑖𝑗\displaystyle\coloneqq\sum_{i\neq j\in X}\xi_{i}^{j}.≔ ∑ start_POSTSUBSCRIPT italic_i ≠ italic_j ∈ italic_X end_POSTSUBSCRIPT italic_ξ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT . | ,j\in X.≔ 1 ⊗ italic_x start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ⊗ 1 + 1 ⊗ 1 ⊗ italic_x start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT , italic_i , italic_j ∈ italic_X . | =0,i≠j∈X,formulae-sequenceabsent0𝑖𝑗𝑋\displaystyle=0,\ i\neq j\in X,= 0 , italic_i ≠ italic_j ∈ italic_X , | D |
For instance when the three types of balls are represented by 1111, 2222 and 3333 and when there are 3333 boxes, the configurations 221,212,122221212122221,212,122221 , 212 , 122 represent the same one, and the different configurations are | For instance when the three types of balls are represented by 1111, 2222 and 3333 and when there are 3333 boxes, the configurations 221,212,122221212122221,212,122221 , 212 , 122 represent the same one, and the different configurations are | ⩽2(x+1)2absent2superscript𝑥12\displaystyle\leqslant 2(x+1)^{2}⩽ 2 ( italic_x + 1 ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT | 111,222,333,123,112,122,113,133,223,233111222333123112122113133223233\displaystyle 111,222,333,123,112,122,113,133,223,233111 , 222 , 333 , 123 , 112 , 122 , 113 , 133 , 223 , 233 | 𝒥βsuperscript𝒥𝛽\displaystyle\mathcal{J}^{\beta}caligraphic_J start_POSTSUPERSCRIPT italic_β end_POSTSUPERSCRIPT | C |
To address the research gap outlined above, a multi-access coded caching model is considered in this paper wherein an arbitrary user is capable of accessing any set of r𝑟ritalic_r caches instead of a distinct set of r𝑟ritalic_r consecutive caches, as outlined in [21]. Further, each set of r𝑟ritalic_r caches is considered to serve L𝐿Litalic_L users in contrary to the model outlined in [13], where each cache subset serves only one user. Here-forth, in this paper, such caching system is referred to as decentralized multi-access coded caching system with cache access degree r𝑟ritalic_r and user association L𝐿Litalic_L. This framework is further extended to a more general setting, where each of these subset of r𝑟ritalic_r caches is accessed by any arbitrary number of users. This generalization removes all constraints on the number of users in the network. Such caching system is referred to as decentralized multi-access coded caching system with cache access degree r𝑟ritalic_r. | A linear delivery scheme is proposed for the decentralized multi-access coded caching system with cache access degree r𝑟ritalic_r. The optimality of the proposed delivery scheme for this general setting is proved for certain system parameters. | Motivated by [22, 23, 24], using techniques from index coding, the optimality of the proposed linear delivery scheme is proved among all linear delivery schemes for the decentralized multi-access coded caching setup with cache access degree r𝑟ritalic_r and user association L𝐿Litalic_L. | A linear delivery scheme is proposed for the decentralized multi-access coded caching system with cache access degree r𝑟ritalic_r and user association L𝐿Litalic_L, and a closed-form expression for the delivery rate of the proposed scheme is derived. | This paper introduces a system model for a decentralized multi-access coded caching network with cache access degree r𝑟ritalic_r and user association L𝐿Litalic_L. A novel linear delivery scheme is presented for this model and a closed-form expression of per user transmission rate is computed. Using index coding techniques, the proposed scheme is shown to be optimal. Furthermore, it demonstrates that existing frameworks [18], [2], and [22] are special cases of the proposed model. Additionally, a more general multi-access coded caching setting is proposed, eliminating all constraints on the number of users and optimal delivery scheme is proposed for the user-to-cache association profiles satisfying L1≥L2≥…≥L(cr)subscript𝐿1subscript𝐿2…subscript𝐿binomial𝑐𝑟{L}_{1}\geq{L}_{2}\geq\ldots\geq{L}_{\binom{c}{r}}italic_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ≥ italic_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ≥ … ≥ italic_L start_POSTSUBSCRIPT ( FRACOP start_ARG italic_c end_ARG start_ARG italic_r end_ARG ) end_POSTSUBSCRIPT. Future work may focus on devising an optimal delivery scheme for the generalized decentralized multi-access coded caching problem with arbitrary user-to-cache associations. | C |
𝐋ℱ(S)≥β|S|.subscriptsuperscript𝐋𝑆ℱsuperscript𝛽𝑆\mathbf{L}^{(S)}_{\mathcal{F}}\geq\beta^{|S|}\,.bold_L start_POSTSUPERSCRIPT ( italic_S ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT caligraphic_F end_POSTSUBSCRIPT ≥ italic_β start_POSTSUPERSCRIPT | italic_S | end_POSTSUPERSCRIPT . | The negation of the existence of a Følner sequence of G𝐺Gitalic_G implies that there exists a finite subset S𝑆Sitalic_S of G𝐺Gitalic_G and ε>0𝜀0\varepsilon>0italic_ε > 0 such that for all F𝐹Fitalic_F, there exists s∈S𝑠𝑆s\in Sitalic_s ∈ italic_S, | We showed that adding an element to the support multiplies the number of locally admissible patterns by β𝛽\betaitalic_β. Moreover, the empty pattern is always locally admissible, so we deduce the following corollary. | Since the conditions given in [13] are more general than the ones given in Lemma 18, we could also deduce a more general version of Corollary 19. Let us also mention that the result of Miller has identical conditions, but the conclusion only implies the non-emptiness of the subshift [11]. The ideas behind our proof and the proof from [11, 14] share some similarities, which explains the similarities of the conditions. The main difference lies in the fact that they consider the number of possible extensions of a word instead of looking at the suffixes of the words (intuitively, they look ahead at what could go wrong when one tries to extend the word further, while we look at the past to see what could have gone wrong when building the current word). | In particular, the previous Corollary implies that, for any finite support, there exists at least one locally admissible configuration. The usual compactness argument immediately implies the following Corollary. | D |
Then C(D)𝐶𝐷C(D)italic_C ( italic_D ) is isomorphic to the structure U(D)𝑈𝐷U(D)italic_U ( italic_D ) introduced in Lemma 3. | For every finite digraph D𝐷Ditalic_D there exists an (up to isomorphism unique) countable digraph C(D)𝐶𝐷C(D)italic_C ( italic_D ) with a homomorphism g:C(D)→D:𝑔→𝐶𝐷𝐷g\colon C(D)\to Ditalic_g : italic_C ( italic_D ) → italic_D such that | A finite digraph embeds into C(D)𝐶𝐷C(D)italic_C ( italic_D ) if and only if it maps homomorphically to D𝐷Ditalic_D (by Lemma 10) if | and only if it embeds into UDsubscript𝑈𝐷U_{D}italic_U start_POSTSUBSCRIPT italic_D end_POSTSUBSCRIPT. Thus, the theory T𝑇Titalic_T of C(D)𝐶𝐷C(D)italic_C ( italic_D ) is a companion of the theory of U(D)𝑈𝐷U(D)italic_U ( italic_D ), since | is universal in the sense has the property that a countable digraph (or graph) C𝐶Citalic_C has an embedding into U(D)𝑈𝐷U(D)italic_U ( italic_D ) if and only if C𝐶Citalic_C maps homomorphically to D𝐷Ditalic_D. | B |
To the best of our knowledge, every coregular representation appearing in the literature on arithmetic statistics of algebraic curves arises from Vinberg theory (that is, from a ℤ/mℤℤ𝑚ℤ\mathbb{Z}/m\mathbb{Z}blackboard_Z / italic_m blackboard_Z-grading on a semisimple Lie algebra), except for one family of notable examples: the representation of SLnsubscriptSL𝑛\operatorname{SL}_{n}roman_SL start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT acting on pairs of symmetric matrices Sym2(n)⊕Sym2(n)direct-sumsuperscriptSym2𝑛superscriptSym2𝑛\operatorname{Sym}^{2}(n)\oplus\operatorname{Sym}^{2}(n)roman_Sym start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_n ) ⊕ roman_Sym start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_n ). | For example, elements of Sel2EsubscriptSel2𝐸\operatorname{Sel}_{2}Eroman_Sel start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_E for an elliptic curve E/ℚ𝐸ℚE/\mathbb{Q}italic_E / blackboard_Q correspond to locally soluble genus-1111 curves C𝐶Citalic_C that are double covers of ℙℚ1subscriptsuperscriptℙ1ℚ\mathbb{P}^{1}_{\mathbb{Q}}blackboard_P start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT blackboard_Q end_POSTSUBSCRIPT [CremonaFisherStoll, §1.3], so give rise to PGL2(ℚ)subscriptPGL2ℚ\operatorname{PGL}_{2}(\mathbb{Q})roman_PGL start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( blackboard_Q )-orbits of binary quartic forms [BS-2selmerellcurves, Theorem 3.5]. | This representation is used in [BhargavaGrossWang-positiveproportionnopoints] to show that a positive proportion of locally soluble hyperelliptic curves over ℚℚ\mathbb{Q}blackboard_Q of fixed genus have no points over any odd degree extension. | So far we have considered properties of the pair (G,V)𝐺𝑉(G,V)( italic_G , italic_V ) over ℚℚ\mathbb{Q}blackboard_Q. In this subsection we define these objects over ℤℤ\mathbb{Z}blackboard_Z. | For example, if 𝖣=A2g𝖣subscript𝐴2𝑔\mathsf{D}=A_{2g}sansserif_D = italic_A start_POSTSUBSCRIPT 2 italic_g end_POSTSUBSCRIPT, then B=Specℚ[p2,…,p2g+1]𝐵Specℚsubscript𝑝2…subscript𝑝2𝑔1B=\operatorname{Spec}\mathbb{Q}[p_{2},\dots,p_{2g+1}]italic_B = roman_Spec blackboard_Q [ italic_p start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , … , italic_p start_POSTSUBSCRIPT 2 italic_g + 1 end_POSTSUBSCRIPT ] and C→B→𝐶𝐵C\rightarrow Bitalic_C → italic_B is the family of all monic odd hyperelliptic curves of genus g𝑔gitalic_g. | B |
If a functor becomes a homotopy equivalence when applying B𝐵Bitalic_B, then we say the functor is a homotopy equivalence. | The following proposition is proved in [Qui73, §1] and used in the special case that all spaces involved are discrete: | The constructions work for internal categories in any finitely complete ambient category (in particular for topological categories), but are also given here for categories in Set to clarify the nature of the objects and the arrows. | The following proposition is stated in [Qui73, §1, Proposition 2], for example, and holds for categories replaced by topological categories. | We need this formulation of NDR-pair in the slice category, so that the following product lemma holds. | C |
\operatorname{\mathrm{Hom}}(\fourIdx{t}{u}{r}{s}{A},\mathbb{C}).italic_t italic_r italic_u italic_s roman_Hom ( italic_A , blackboard_C ) ≅ roman_Hom ( italic_t italic_u italic_r italic_s italic_A , blackboard_C ) . | It is easily verified that the above definition of a regular partial Hopf algebra coincides with the one in [DCT15, Definition 1.11]. We also have the following. | By Proposition 3.7, we have a one-to-one correspondence between right A𝐴Aitalic_A-comodules and left A𝐴Aitalic_A-modules. | Then, because we are considering the opposite comultiplication, we have one extra flip around the horizontal axis. | Note that the ordering of the grading can be remembered as follows: when simply dualizing the algebra and co-algebra structure, the horizontal and vertical grading interchange by a flip around the anti-diagonal, so | C |
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