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The spectral gap conjecture of M. van den Berg [2, formula (65)] asserts that λ2 − λ1 ≥ 3π for all convex euclidean domains of diameter 1, where λ1 and λ2 denote the first two eigenvalues of the Dirichlet Laplacian. Notice that equality holds for the 1-dimensional unit interval, which can be regarded also as a degenerate n-dimensional rectangular box. The gap estimate is conjectured to hold more generally for Schrodinger operators with convex potentials, under Dirichlet boundary conditions; see the work of S.-T. Yau and collaborators [9, 11]. This Schrodinger gap conjecture was proved some time ago in 1 dimension by R. Lavine [8], and more recently in all dimensions by B. Andrews and J. Clutterbuck [1]. The proof in this journal by H. Donnelly [3] of the original gap conjecture in 2 dimensions (for the Dirichlet Laplacian with zero potential) is not correct. The Editors of Mathematische Zeitschrift have asked us to describe the flaws in the proof, in order to clarify the state of the literature. Donnelly’s approach to the problem is a natural one: first perform a shape optimization to rule out a non-degenerate minimizing domain, and then analyze the spectral gap for a sequence of domains degenerating to an interval, with the help of results by D. Jerison [5]. (For some history on this approach, and on the gap conjecture more generally, see the report on the AIM meeting “Low Eigenvalues of Laplace and Schrodinger Operators” [10], especially page 12 of the open problems list.) The error lies in the proof of the shape optimization step, as we now explain. Donnelly wishes to prove that no minimizing domain can exist for

the density of the winding process in the complex projective line is worked out; International audience; In the first part of this paper, we derive explicit expressions of the semi-group densities of generalized stochastic areas arising from the Anti-de Sitter and the Hopf fibrations. Motivated by the number-theoretical connection between the Heisenberg group and Dirichlet series, we express the Mellin transform of the generalized stochastic area corresponding to the one-dimensional Anti de Sitter fibration as a series of Riemann Zeta function evaluated at integers. In the second part of the paper, we focus on winding processes around the origin in the Poincar\'e disc and in the complex projective line. More pricesely, we derive the fixed-time marginal density of the former process while we give a ultraspherical series expansion of the characteristic function of the latter.

In a Hilbert space setting ${\mathcal{H}}$, in order to minimize by fast methods a general convex lower semicontinuous and proper function ${\Phi }: {\mathcal{H}} \rightarrow \mathbb {R} \cup \{+\infty \}$, we analyze the convergence rate of the inertial proximal algorithms. These algorithms involve both extrapolation coefficients (including Nesterov acceleration method) and proximal coefficients in a general form. They can be interpreted as the discrete time version of inertial continuous gradient systems with general damping and time scale coefficients. Based on the proper setting of these parameters, we show the fast convergence of values and the convergence of iterates. In doing so, we provide an overview of this class of algorithms. Our study complements the previous Attouch–Cabot paper (SIOPT, 2018) by introducing into the algorithm time scaling aspects, and sheds new light on the Guler seminal papers on the convergence rate of the accelerated proximal methods for convex optimization.

In this paper, we establish compactness results of some class of conformally compact Einstein 4-manifolds. In the first part of the paper, we improve the earlier results obtained by Chang-Ge. In the second part of the paper, as applications, we derive some compactness results under perturbation conditions when the L^2-norm of the Weyl curvature is small. We also derive the global uniqueness of conformally compact Einstein metrics on the 4-Ball constructed in the earlier work of Graham-Lee., Comment: 28 pages

International audience; The construction of anisotropic triangulations is desirable for various applications, such as the numerical solving of partial differential equations and the representation of surfaces in graphics. To solve this notoriously difficult problem in a practical way, we introduce the discrete Rieman-nian Voronoi diagram, a discrete structure that approximates the Riemannian Voronoi diagram. This structure has been implemented and was shown to lead to good triangulations in R 2 and on surfaces embedded in R 3 as detailed in our experimental companion paper. In this paper, we study theoretical aspects of our structure. Given a finite set of points P in a domain Ω equipped with a Riemannian metric, we compare the discrete Riemannian Voronoi diagram of P to its Riemannian Voronoi diagram. Both diagrams have dual structures called the discrete Riemannian Delaunay and the Riemannian Delaunay complex. We provide conditions that guarantee that these dual structures are identical. It then follows from previous results that the discrete Riemannian Delaunay complex can be embedded in Ω under sufficient conditions, leading to an anisotropic triangulation with curved simplices. Furthermore, we show that, under similar conditions, the simplices of this triangulation can be straightened. 1998 ACM Subject Classification Computational Geometry and Object Modeling 1 Introduction Anisotropic triangulations are triangulations whose elements are elongated along prescribed directions. Anisotropic triangulations are known to be well suited when solving PDE's [10, 19, 24]. They can also significantly enhance the accuracy of a surface representation if the anisotropy of the triangulation conforms to the curvature of the surface [15]. Many methods to generate anisotropic triangulations are based on the notion of Rieman-nian metric and create triangulations whose elements adapt locally to the size and anisotropy prescribed by the local geometry. The numerous theoretical and practical results [1] of the Euclidean Voronoi diagram and its dual structure, the Delaunay triangulation, have pushed authors to try and extend these well-established concepts to the anisotropic setting. La-belle and Shewchuk [17] and Du and Wang [12] independently introduced two anisotropic Voronoi diagrams whose anisotropic distances are based on a discrete approximation of the Riemannian metric field. Contrary to their Euclidean counterpart, the fact that the dual of these anisotropic Voronoi diagrams is an embedded triangulation is not immediate, and, despite their strong theoretical foundations, the anisotropic Voronoi diagrams of Labelle and

This paper is centred on the spectral study of a Random Fourier matrix, that is an $n\times n$ matrix $A$ whose $(j, k)$ entries are $\exp(2i\pi m X_jY_k)$, with $X_j$ and $Y_k$ two i.i.d sequences of random variables and $1\leq m\leq n$ is a real number. When they are uniformly distributed on a symmetric interval, this may be seen as a random discretization of the Finite Fourier transform, whose spectrum has been extensively studied in relation with band-limited functions. Our study is two-fold. Firstly, by pushing forward concentration inequalities, we find an accurate comparison in $\ell^2$- norm between the spectrum of $A^*A$ and the one of an integral operator that can be defined in terms of the two probability laws chosen for the rows and the columns. Our study includes the one of stationary Hermitian kernel matrices and can be generalized to non stationary ones, for which the same kind of comparison with an integral operator is possible. Because of possible applications in the data science area, these last matrices have been largely studied in the literature and our results are compared with previous ones. Secondly we concentrate on uniform distributions for the laws of $X_j$'s and $Y_k$'s, for which the integral operator is the well-known Sinc-kernel operator with parameter $m.$ Our previous study allows to translate to random Fourier matrices the knowledge that we have on the spectrum of this operator. We have for them asymptotic results for $m, n$ and $n/m$ tending to $\infty$, as well as non asymptotic bounds in the spirit of recent work on the integral operators. As an application, we give fairly good approximations of the number of degrees of freedom and the capacity of a MIMO wireless communication network approximation model. Finally, we provide the reader with some numerical examples that illustrate the theoretical results of this paper.

In this paper, we give a geometric expression for the multiplicities of the equivariant index of a spin-c Dirac operator., Comment: The authors have decided to divide the preprint "Multiplicities of equivariant Spin-c Dirac operators" into two separate publications. The present paper is the first of them. The other part is the preprint arXiv:1512.02367 entitled "Admissible orbits for compact Lie group"

27 pages. This is the third and final part of the series of 3 publications stemming from the preprint arXiv:1312.3161; International audience; In the third paper of the series we complete the proof of our main result: a description of the ergodic decomposition of infinite Pickrell measures. We first prove that the scaling limit of the determinantal measures corresponding to the radial parts of Pickrell measures is precisely the infinite Bessel process introduced in the first paper of the series. We prove that the `Gaussian parameter' for ergodic components vanishes almost surely. To do this, we associate a finite measure with each configuration and establish convergence to the scaling limit in the space of finite measures on the space of finite measures. We finally prove that the Pickrell measures corresponding to different values of the parameter are mutually singular.

In the present paper we advocate the Howland-Evans approach to solution of the abstract non-autonomous Cauchy problem (non-ACP) in a separable Banach space X. The main idea is to reformulate this problem as an autonomous Cauchy problem (ACP) in a new Banach space L^p(I,X), consisting of X-valued functions on the time-interval I. The fundamental observation is a one-to-one correspondence between solution operators (propagators) for a non-ACP and the corresponding evolution semigroups for ACP in L^p(I,X). We show that the latter also allows to apply a full power of the operator-theoretical methods to scrutinise the non-ACP including the proof of the Trotter product approximation formulae with operator-norm estimate of the rate of convergence. The paper extends and improves some recent results in this direction in particular for Hilbert spaces., Comment: 40 pages

International audience; In this paper, I study Richard Dedekind and Heinrich Weber's 1882 Theorie der algebraischen Funktionen einer Veränderlichen, with a focus on the inherently arithmetical aspects of their work. I show that their paper provides an arithmetical rewriting of Riemannian function theory, i.e. a rewriting built on elementary arithmetical notions such as divisibility. I start with contextual elements concerning what is “arithmetical”, to put Dedekind and Weber's works into perspective from that viewpoint. Then, through a detailed analysis of the 1882 paper and using elements of their correspondence, I suggest that Dedekind and Weber deploy a strategy of rewriting parts of mathematics using arithmetic, and that this strategy is essentially related to Dedekind's specific conception of numbers and arithmetic as intrinsically linked to the human mind.

We show that the Teichmuller space of a surface without boundary and with punctures, equipped with the Thurston metric, is the limit in an appropriate sense of Teichmuller spaces of surfaces with boundary, equipped with their arc metrics, when the boundary lengths tend to zero. We use this to obtain a result on the translation distances of mapping classes for their actions on Teichmuller spaces equipped with the Thurston metric. In this paper, we show that the arc metrics on the Teichmuller space of surfaces with boundary limit to the Thurston metric on the Teichmuller space of a surface without boundary, by making the boundary lengths tend to zero. We use this to prove a result on the translation distances for mapping classes. We introduce some notation before stating precisely the results. In all this paper, S = Sg,p,n is a connected orientable surface of finite type, of genus g with p punctures and n boundary components. We assume that S has negative Euler characteristic, i.e., χ(S) = 2 − 2g − p − n 0, we denote by ∂S the boundary of S. A hyperbolic structure on S is a complete metric of constant curvature −1 such that (i) each puncture has a neighborhood isometric to a cusp, i.e., to the quotient {z = x + iy ∈ H 2 | y > a}/hz 7→z + 1i

In this paper, we revisit the analyses of Antonie Stern (1925) and Hans Lewy (1977) devoted to the construction of spherical harmonics with two or three nodal domains. Our method yields sharp quantitative results and a better understanding of the occurrence of bifurcations in the families of nodal sets.This paper is a natural continuation of our critical reading of A. Stern's results for Dirichlet eigenfunctions in the square, see arXiv:14026054., Comment: Accepted for publication in "Monatshefte f{\"u}r Mathematik"

International audience; The purpose of this paper is to push forward the theory of operator-valued Riemann-Hilbert problems and demonstrate their effectiveness in respect to the implementation of a non-linear steepest descent method a la Deift-Zhou. In this paper, we demonstrate that the operator-valued Riemann-Hilbert problem arising in the characterization of so-called c-shifted integrable integral operators allows one to extract the large-x asymptotics of the Fredholm determinant associated with such operators.

International audience; In [Séries Gevrey de type arithmétique I Théorémes de pureté et de dualité, Annals of Math. 151 (2000), 705--740], André has introduced E-operators, a class of differential operators intimately related to E-functions, and constructed local bases of solutions for these operators. In this paper we investigate the arithmetical nature of connexion constants of E-operators at finite distance, and of Stokes constants at infinity. We prove that they involve values at algebraic points of E-functions in the former case, and in the latter one, values of G-functions and of derivatives of the Gamma function at rational points in a very precise way. As an application, we define and study a class of numbers having certain algebraic approximations defined in terms of E-functions. These types of approximations are motivated by the convergents to the number e, as well as by recent constructions of approximations to Euler's constant and values of the Gamma function. Our results and methods are completely different from those in our paper [On the values of G-functions, Commentarii Math. Helv., to appear], where we have studied similar questions for G-functions.

The second part of the paper mainly deals with convergence of infinite determinantal measures, understood as the convergence of the approximating finite determinantal measures. In addition to the usual weak topology on the space of probability measures on the space of configurations, we also consider the weak topology on the space of finite measures on the space of finite measures on the half-line, used via the natural immersion, well-defined almost surely with respect to the infinite Bessel point process, of the space of configurations into the space of finite measures on the half-line. The main results of the second part are sufficient conditions for tightness of families of determinantal measures, for convergence of sequences of induced proceses, as well as for sequences of finite-dimensional perturbations of determinantal processes., Comment: 20 pages. This is the second paper in the series of three stemming from the preprint arXiv:1312.3161

Dans un r\'ecent article, Lagarias et Soundararajan \'etudient les solutions friables \`a l'\'equation a+b=c. Sous l'hypoth\`ese de Riemann g\'en\'eralis\'ees aux fonctions L de Dirichlet, ils obtiennent une estimation pour le nombre de solutions pond\'er\'ees par un poids lisse et une minoration pour le nombre de solutions non pond\'er\'ees. Le but de cet article est de pr\'esenter des arguments qui permettent d'une part de pr\'eciser les termes d'erreur et d'\'etendre les domaines de validit\'e de ces estimations afin de faire le lien avec un travail de la Bret\`eche et Granville, d'autre part d'obtenir le comportement asymptotique exact du nombre de solutions non pond\'er\'ees. In a recent paper, Lagarias and Soundararajan study the y-smooth solutions to the equation a+b=c. Under the Generalised Riemann Hypothesis, they obtain an estimate for the number of those solutions weighted by a compactly supported smooth function, as well as a lower bound for the number of bounded unweighted solutions. In this paper, we aim to prove a more precise estimate for the number of weighted solutions that is valid when y is relatively large with respect to x, so as to connect our estimate with the one obtained by La Bret\`eche and Granville in a recent work. We also prove the conjectured upper bound for the number of bounded unweighted solutions, thus obtaining its exact asymptotic behaviour., Comment: 20 pages

This paper is devoted to the study of the low Mach number limit for the isentropic Euler system with axisymmetric initial data without swirl. In the first part of the paper we analyze the problem corresponding to the subcritical regularities, that is $H^s$ {with $s>\frac52$.} Taking advantage of the Strichartz estimates and using the special structure of the vorticity we show that the {lifespan $T_\epsilon$} of the solutions is bounded below by $\log\log\log\frac1\epsilon$, where $\epsilon$ denotes the Mach number. Moreover, we prove that the incompressible parts converge to the solution of the incompressible Euler system, when the parameter $\epsilon$ goes to zero. In the second part of the paper we address the same problem but for the Besov critical regularity $B_{2,1}^{\frac52}$. This case turns out to be more subtle at least due to two facts. The first one is related to the Beale-Kato-Majda criterion which is not known to be valid for rough regularities. The second one concerns the critical aspect of the Strichartz estimate $L^1_TL^\infty$ for the acoustic parts $(\nabla\Delta^{-1}\diver\vepsilon,\cepsilon)$: it scales in the space variables like the space of the initial data., Comment: 47 pages

This is the second paper of the authors in a series concerned with the development of a deterministic model for the transfer matrix of a MIMO system. Starting from the Maxwell equations, we have described in \cite{BCFM} the generic structure of such a deterministic transfer matrix. In the current paper we apply the results of \cite{BCFM} in order to study the (Shannon-Foschini) capacity behavior of a MIMO system as a function of the deterministic spread function of the environment, and the number of transmitting and receiving antennas. The antennas are assumed to fill in a given, fixed volume. Under some generic assumptions, we prove that the capacity grows much more slowly than linearly with the number of antennas. These results reinforce previous heuristic results obtained from statistical models of the transfer matrix, which also predict a sublinear behavior., Comment: 12 pages, to appear in Math. Meth. Appl. Sci

The present paper introduces an efficient and accurate numerical scheme for the solution of a highly anisotropic elliptic equation, the anisotropy direction being given by a variable vector field. This scheme is based on an asymptotic preserving reformulation of the original system, permitting an accurate resolution independently of the anisotropy strength and without the need of a mesh adapted to this anisotropy. The counterpart of this original procedure is the larger system size, enlarged by adding auxiliary variables and Lagrange multipliers. This Asymptotic-Preserving method generalizes the method investigated in a previous paper [arXiv:0903.4984v2] to the case of an arbitrary anisotropy direction field.

We prove that in the Wasserstein space built over R the subset of measures that does not charge the non-differentiability set of convex functions is not displacement convex. This completes the study of Gigli on the geometric structure of measures meeting the sharp hypothesis of the refined version of Brenier’s Theorem. Optimal transport is nowadays a central tool in many fields of analysis, differential geometry and probability theory (see e.g. the books by Villani [11, 12]). It takes its origin in the problem of Monge, asking for the shortest way to displace an amount of soil from one place of the Euclidean space to a heap of soil at another place. This problem induces a very natural distance between probability measures – the Wasserstein distance – where the measures represent the heaps of soil. Wasserstein distance is somewhat complementary with the Lebesgue L norms because it corresponds to an “horizontal” displacement: in first approximation, for two localized probability measures on R, the Wasserstein distance is the distance between their barycenters. Brenier’s Theorem [4] on monotone rearrangement of maps of R has become the very core of the theory of optimal transport. It gives a representation of the optimal transport map in term of gradient of convex functions. A very enlightening heuristic on (P2(R ),W2) is proposed in [7] where it appears with an infinite differential structure and the Wasserstein distance is seen as a Riemannian-like distance. This point of view has raised a lot of developments, among which the gradient flow theory presented in [2]. In [6], Gigli explores the Riemannian-like structure and proposes to think of the measures meeting the hypothesis of a refined version of Brenier’s Theorem as the regular points of the Wasserstein space. In this paper we show that the set of those transport-regular measures is not geodesically convex (Theorems 2.2 and 2.5). It is quite surprising because it is well-known that the subset made of absolutely continuous measures (the most notorious transport-regular measures) is geodesically convex. It answers a question suggested by Gigli [6, Remark 2.12]. The paper is built as follows: we first recall some main results on the quadratic Monge(-Kantorovich) problem. The key idea of our result can be found in Lemma 1.6 while Proposition 1.8 gives a way, together with Proposition 1.7, to characterize transport-regular measures. In the second (and last) part, we prove the main theorem (Theorem 2.2) and its generalization (Theorem 2.5) thanks to explicit constructions. 2010 Mathematics Subject Classification. Primary 28A75.

We characterize the polynomial decay of orbits of Hilbert space $C_0$-semigroups in resolvent terms. We also show that results of the same type for general Banach space semigroups and functions obtained recently in the paper by C.J.K.Batty and T.Duyckaerts, Non-uniform stability for bounded semi-groups on Banach spaces (J. Evol. Eq. 2008), are sharp. This settles a conjecture posed in the paper by C.J.K.Batty and T.Duyckaerts., 21 pages, to appear in Matematische Annalen

In this paper we consider the following Timoshenko system: in (0, L) × ℝ + {ρ 1 ϕ tt -k 1 (ϕ x +ψ) x =0 ρ 2 ψ tt - k 2 ψ xx + ∫ t 0 g(t - τ)(a(x)ψ x (τ)) x dτ+k 1 (ϕ x + ψ) + b(x)h(ψ t ) =0 in (0, L) × ℝ + with Dirichlet boundary conditions and initial data where a, b, g and h are specific functions and ρ 1 , ρ 2 , κ 1 , κ 2 and L are given positive constants. We establish a general stability estimate using the multiplier method and some properties of convex functions. Without imposing any growth condition on h at the origin, we show that the energy of the system is bounded above by a quantity, depending on g and h, which tends to zero as time goes to infinity. Our estimate allows us to consider a large class of functions h with general growth at the origin. We use some examples (known in the case of wave equations and Maxwell system) to show how to derive from our general estimate the polynomial, exponential or logarithmic decay. The results of this paper improve and generalize some existing results in the literature and generate some interesting open problems.

In this paper, we deal with controllability properties of linear and nonlinear Korteweg-de Vries equations in a bounded interval. The main part of this paper is a result of uniform controllability of a linear KdV equation in the limit of zero-dispersion. Moreover, we establish a result of null controllability for the linear equation via the left Dirichlet boundary condition, and of exact controllability via both Dirichlet boundary conditions. As a consequence, we obtain some local exact controllability results for the nonlinear KdV equation.

National audience; This paper is devoted to the study of the internal stabilization of the Bernoulli-Euler plate equation in a square. The continuous and the space semi-discretizated problems are successively considered and analyzed using a frequency domain approach. For the infinite dimensional problem, we provide a new proof of the exponential stability result, based on a two dimensional Ingham's type result. In the second and main part of the paper, we propose a finite difference space semi-discretization scheme and we prove that this scheme yields a uniform exponential decay rate (with respect to the mesh size).

International audience; In this paper we study the pseudo-spectra of the rotated harmonic oscillator. The pseudo-spectra of an operator are subsets in the complex plane which describe where the resolvent is large in norm. The study of such subsets allows us to understand the stability of the spectrum under perturbations and the possible calculation of 'false eigenvalues' far from the spectrum by algorithms for computing eigenvalues. The rotated harmonic oscillator is the simplest classic non-self-adjoint quadratic Hamiltonian, which has already been studied by Davies and Boulton. In one of his works, Boulton states a conjecture about the pseudo-spectra of this operator, which describes the instabilities for high energies. We can deduce this conjecture from a study of Dencker, Sjostrand and Zworski, which gives bounds on the resolvent for a semi-classical pseudo-differential operator in a very general setting. In the present paper, we give a more elementary proof of this result using only some non-trivial localization scheme in the frequency variable.

International audience; In this paper, the mathematical tools, which are required to solve the axisymmetric Maxwell equations, are presented. An in-depth study of the problems posed in the meridian half-plane, numerical algorithms, as well as numerical experiments, based on the implementation of the theory described hereafter, shall be presented in forthcoming papers. In the present paper, the attention is focused on the (orthogonal) splitting of the electromagnetic field in a regular part and a singular part, the former being in the Sobolev space H-1 component-wise. It is proven that the singular fields are related to singularities of Laplace-like operators, and, as a consequence, that the space of singular fields is finite dimensional. Copyright (C) 2002 John Wiley Sons, Ltd.

In this paper, we extend the non-cooperative one-period game of Dutang et al. (Journal of Operational Research 231(3):702–711, 2013) to model a non-life insurance market over several periods by considering the repeated (one-period) game. Using Markov chain methodology, we derive general properties of insurer portfolio sizes given a price vector. In the case of a regulated market (identical premium), we are able to obtain convergence measures of long run market shares. We also investigate the consequences of the deviation of one player from this regulated market. Finally, we provide some insights of long-term patterns of the repeated game as well as numerical illustrations of leadership and ruin probabilities.

Article dans revue scientifique avec comité de lecture.; International audience; A considerable number of asymptotic distributions arising in random combinatorics and analysis of algorithms are of the exponential-quadratic type, that is, Gaussian. We exhibit a class of ``universal'' phenomena that are of the exponential-cubic type, corresponding to distributions that involve the Airy function. In this paper, such Airy phenomena are related to the coalescence of saddle points and the confluence of singularities of generating functions. For about a dozen types of random planar maps, a common Airy distribution (equivalently, a stable law of exponent $3/2$) describes the sizes of cores and of largest (multi)connected components. Consequences include the analysis and fine optimization of random generation algorithms for multiply connected planar graphs. Based on an extension of the singularity analysis framework suggested by the Airy case, the paper also presents a general classification of compositional schemas in analytic combinatorics.

The present paper is devoted to clustering geometric graphs. While the standard spectral clustering is often not effective for geometric graphs, we present an effective generalization, which we call higher-order spectral clustering. It resembles in concept the classical spectral clustering method but uses for partitioning the eigenvector associated with a higher-order eigenvalue. We establish the weak consistency of this algorithm for a wide class of geometric graphs which we call Soft Geometric Block Model. A small adjustment of the algorithm provides strong consistency. We also show that our method is effective in numerical experiments even for graphs of modest size., Comment: 23 pages, 6 figures

We prove an asymptotic formula for the mean-square average of L-functions associated with subgroups of characters of sufficiently large size. Our proof relies on the study of certain character sums ${\mathcal{A}}(p,d)$ recently introduced by E. Elma, where p ≥ 3 is prime and d ≥ 1 is any odd divisor of p − 1. We obtain an asymptotic formula for ${\mathcal{A}}(p,d),$ which holds true for any odd divisor d of p − 1, thus removing E. Elma’s restrictions on the size of d. This answers a question raised in Elma’s paper. Our proof relies on both estimates on the frequency of large character sums and techniques from the theory of uniform distribution. As an application, in the range $1\leq d\leq\frac{\log p}{3\log\log p}$, we obtain a significant improvement $h_{p,d}^- \leq 2(\frac{(1+o(1))p}{24})^{m/4}$ over the trivial bound $h_{p,d}^- \ll (\frac{dp}{24} )^{m/4}$ on the relative class numbers of the imaginary number fields of conductor $p\equiv 1\mod{2d}$ and degree $m=(p-1)/d$, where d ≥ 1 is odd.

The main result of this paper is that one cannot hear orientability of a surface with boundary. More precisely, we construct two isospectral flat surfaces with boundary with the same Neumann spectrum, one orientable, the other non-orientable. For this purpose, we apply Sunada's and Buser's methods in the framework of orbifolds. Choosing a symmetric tile in our construction, and adapting a folklore argument of Fefferman, we also show that the surfaces have different Dirichlet spectra. These results were announced in the {\it C. R. Acad. Sci. Paris S\'er. I Math.}, volume 320 in 1995, but the full proofs so far have only circulated in preprint form., Comment: Minor changes. Accepted for publication in Mathematische Zeitschrift

In this paper we prove a regularity and rigidity result for displacements in$$GSBD^p$$GSBDp, for every$$p>1$$p>1and any dimension$$n\ge 2$$n≥2. We show that a displacement in$$GSBD^p$$GSBDpwith a small jump set coincides with a$$W^{1,p}$$W1,pfunction, up to a small set whose perimeter and volume are controlled by the size of the jump. This generalises to higher dimension a result of Conti, Focardi and Iurlano. A consequence of this is that such displacements satisfy, up to a small set, Poincaré-Korn and Korn inequalities. As an application, we deduce an approximation result which implies the existence of the approximate gradient for displacements in$$GSBD^p$$GSBDp.

We consider the energy supercritical defocusing nonlinear Schrödinger equation $$\begin{aligned} i\partial _tu+\Delta u-u|u|^{p-1}=0 \end{aligned}$$ i ∂ t u + Δ u - u | u | p - 1 = 0 in dimension $$d\ge 5$$ d ≥ 5 . In a suitable range of energy supercritical parameters (d, p), we prove the existence of $${\mathcal {C}}^\infty $$ C ∞ well localized spherically symmetric initial data such that the corresponding unique strong solution blows up in finite time. Unlike other known blow up mechanisms, the singularity formation does not occur by concentration of a soliton or through a self similar solution, which are unknown in the defocusing case, but via a front mechanism. Blow up is achieved by compression for the associated hydrodynamical flow which in turn produces a highly oscillatory singularity. The front blow up profile is chosen among the countable family of $${\mathcal {C}}^\infty $$ C ∞ spherically symmetric self similar solutions to the compressible Euler equation whose existence and properties in a suitable range of parameters are established in the companion paper (Merle et al. in Preprint (2019)) under a non degeneracy condition which is checked numerically.

In this paper we study the conforming Galerkin approximation of the problem: find $u\in{{\mathcal{U}}}$ such that $a(u,v) = \langle L, v \rangle $ for all $v\in{{\mathcal{V}}}$, where ${{\mathcal{U}}}$ and ${{\mathcal{V}}}$ are Hilbert or Banach spaces, $a$ is a continuous bilinear or sesquilinear form and $L\in{{\mathcal{V}}}^{\prime}$ a given data. The approximate solution is sought in a finite-dimensional subspace of ${{\mathcal{U}}}$, and test functions are taken in a finite-dimensional subspace of ${{\mathcal{V}}}$. We provide a necessary and sufficient condition on the form $a$ for convergence of the Galerkin approximation, which is also equivalent to convergence of the Galerkin approximation for the adjoint problem. We also characterize the fact that ${{\mathcal{U}}}$ has a finite-dimensional Schauder decomposition in terms of properties related to the Galerkin approximation. In the case of Hilbert spaces we prove that the only bilinear or sesquilinear forms for which any Galerkin approximation converges (this property is called the universal Galerkin property) are the essentially coercive forms. In this case a generalization of the Aubin–Nitsche Theorem leads to optimal a priori estimates in terms of regularity properties of the right-hand side $L$, as shown by several applications. Finally, a section entitled ‘Supplement’ provides some consequences of our results for the approximation of saddle point problems.

The existence of weak solutions to the Navier–Stokes–Fourier system describing the stationary states of a compressible, viscous, and heat conducting fluid in bounded 2 D -domains is shown under fairly general and physically relevant constitutive relations. The equation of state of a real fluid is considered, where the admissible range of density is confined to a bounded interval (hard sphere model). The transport coefficients depend on the temperature in a general way including both gases and liquids behavior. The heart of the paper are new a priori bounds resulting from Trudinger–Moser inequality.

In this paper, we revisit the proof of the large deviations principle of Wiener chaoses partially given by Borel, and then by Ledoux in its full form. We show that some heavy-tail phenomena observed in large deviations can be explained by the same mechanism as for the Wiener chaoses, meaning that the deviations are created, in a sense, by translations. More precisely, we prove a general large deviations principle for a certain class of functionals $f_n : \mathbb{R}^n \to \mathcal{X}$, where $\mathcal{X}$ is some metric space, under the $n$-fold probability measure $\nu_{\alpha}^n$, where $\nu_{\alpha} =Y_{\alpha}^{-1}e^{-|x|^{\alpha}}dx$, $\alpha \in (0,2]$, for which the large deviations are due to translations. We retrieve, as an application, the large deviations principles known for the Wigner matrices without Gaussian tails, of the empirical spectral measure by Bordenave and Caputo, the largest eigenvalue and traces of polynomials by the author. We also apply our large deviations result to the last-passage time, which yields a large deviations principle when the weights have the density $Z_{\alpha}^{-1} e^{-x^{\alpha}}$ with respect to Lebesgue measure on $\mathbb{R}_+$, with $\alpha \in (0,1)$.

In this paper we discuss the generalizations of the concept of Chebyshev's bias from two perspectives. First we give a general framework for the study of prime number races and Chebyshev's bias attached to general $L$-functions satisfying natural analytic hypotheses. This extends the cases previously considered by several authors and involving, among others, Dirichlet $L$-functions and Hasse--Weil $L$-functions of elliptic curves over $\mathbf{Q}$. This also apply to new Chebyshev's bias phenomena that were beyond the reach of the previously known cases. In addition we weaken the required hypotheses such as GRH or linear independence properties of zeros of $L$-functions. In particular we establish the existence of the logarithmic density of the set $\lbrace x\geq 2 : \sum_{p\leq x} \lambda_{f}(p) \geq 0 \rbrace$ for coefficients $(\lambda_{f}(p))$ of general $L$-functions conditionally on a much weaker hypothesis than was previously known., Comment: 8 figures, exposition improved including more details on Kronecker--Weyl Equidistribution Theorem

International audience; Stability criteria have been derived and investigated in the last decades for many kinds of periodic traveling wave solutions to Hamiltonian PDEs. They turned out to depend in a crucial way on the negative signature - or Morse index - of the Hessian matrix of action integrals associated with those waves. In a previous paper (published in Nonlinearity in 2016), the authors addressed the characterization of stability of periodic waves for a rather large class of Hamiltonian partial differential equations that includes quasilinear generalizations of the Korteweg–de Vries equation and dispersive perturbations of the Euler equations for compressible fluids, either in Lagrangian or in Eulerian coordinates. They derived a sufficient condition for orbital stability with respect to co-periodic perturbations, and a necessary condition for spectral stability, both in terms of the negative signature of the Hessian matrix of the action integral. Here the asymptotic behavior of this matrix is investigated in two asymptotic regimes, namely for small amplitude waves and for waves approaching a solitary wave as their wavelength goes to infinity. The special structure of the matrices involved in the expansions makes possible to actually compute the negative signature of the action Hessian both in the harmonic limit and in the soliton limit. As a consequence, it is found that nondegenerate small amplitude waves are orbitally stable with respect to co-periodic perturbations in this framework. For waves of long wavelength, the negative signature of the action Hessian is found to be exactly governed by M''(c), the second derivative with respect to the wave speed of the Boussinesq momentum associated with the limiting solitary wave. This gives an alternate proof of Gardner's result [J. Reine Angew. Math. 1997] according to which the instability of the limiting solitary wave, when M''(c) is negative, implies the instability of nearby periodic waves. Interestingly enough, it is found here that in the inverse situation, when M''(c) is positive, nearby periodic waves are orbitally stable with respect to co-periodic perturbations.

In recent work, the authors derived a tropical interpretation of monotone and strictly monotone double Hurwitz numbers. In this paper, we apply the technique of tropical flows to this interpretation in order to provide a new proof of the piecewise polynomiality of these enumerative invariants. Moreover, we derive new types of wall-crossing formulae., 18 pages

International audience; This paper considers the problem of restricting the short-time Fourier transform to domains of nonzero measure in the plane and studies sampling bounds of such systems. In particular, we give a quantitative estimate for the lower sampling bound in the case of Hermite windows and derive a sufficient condition for a large class of windows in terms of a certain planar density. On the way, we prove a Remez-type inequality for polyanalytic functions. MSC2010: 42C40, 46E15, 46E20, 42C15Revised according to referee's recomandation

This paper addresses enforcing non-vanishing constraints for solutions to a second order elliptic partial differential equation by appropriate choices of boundary conditions. We show that, in dimension $d\geq2$, under suitable regularity assumptions, the family of $2d$ solutions such that their Jacobian has maximal rank in the domain is both open and dense. The case of less regular coefficients is also addressed, together with other constraints, which are relevant for applications to recent hybrid imaging modalities. Our approach is based on the combination of the Runge approximation property and the Whitney projection argument [Greene and Wu, Ann. Inst. Fourier (Grenoble), 25(1, vii):215-235, 1975]. The method is very general, and can be used in other settings., 13 pages

For the random interval partition of $[0,1]$ generated by the uniform stick-breaking scheme known as GEM$(1)$, let $u_k$ be the probability that the first $k$ intervals created by the stick-breaking scheme are also the first $k$ intervals to be discovered in a process of uniform random sampling of points from $[0,1]$. Then $u_k$ is a renewal sequence. We prove that $u_k$ is a rational linear combination of the real numbers $1, \zeta(2), \ldots, \zeta(k)$ where $\zeta$ is the Riemann zeta function, and show that $u_k$ has limit $1/3$ as $k \to \infty$. Related results provide probabilistic interpretations of some multiple zeta values in terms of a Markov chain derived from the interval partition. This Markov chain has the structure of a weak record chain. Similar results are given for the GEM$(\theta)$ model, with beta$(1,\theta)$ instead of uniform stick-breaking factors, and for another more algebraic derivation of renewal sequences from the Riemann zeta function., Comment: 25 pages. This paper is published by https://www.ams.org/journals/tran/2019-371-08/S0002-9947-2018-07516-X/

We prove the linear stability of slowly rotating Kerr black holes as solutions of the Einstein vacuum equations: linearized perturbations of a Kerr metric decay at an inverse polynomial rate to a linearized Kerr metric plus a pure gauge term. We work in a natural wave map/DeTurck gauge and show that the pure gauge term can be taken to lie in a fixed 7-dimensional space with a simple geometric interpretation. Our proof rests on a robust general framework, based on recent advances in microlocal analysis and non-elliptic Fredholm theory, for the analysis of resolvents of operators on asymptotically flat spaces. With the mode stability of the Schwarzschild metric as well as of certain scalar and 1-form wave operators on the Schwarzschild spacetime as an input, we establish the linear stability of slowly rotating Kerr black holes using perturbative arguments; in particular, our proof does not make any use of special algebraic properties of the Kerr metric. The heart of the paper is a detailed description of the resolvent of the linearization of a suitable hyperbolic gauge-fixed Einstein operator at low energies. As in previous work by the second and third authors on the nonlinear stability of cosmological black holes, constraint damping plays an important role. Here, it eliminates certain pathological generalized zero energy states; it also ensures that solutions of our hyperbolic formulation of the linearized Einstein equations have the stated asymptotics and decay for general initial data and forcing terms, which is a useful feature in nonlinear and numerical applications., National Science Foundation (U.S.) (Grant DMS-1955614)

International audience; Probability constraints are often employed to intuitively define safety of given decisions in optimization problems. They simply express that a given system of inequalities depending on a decision vector and a random vector is satisfied with high enough probability. It is known that, even if this system is convex in the decision vector, the associated probability constraint is not convex in general. In this paper, we show that some degree of convexity is still preserved, for the large class of elliptical random vectors, encompassing for example Gaussian or Student random vectors. More precisely, our main result establishes that, under mild assumptions, eventual convexity holds, i.e. the probability constraint is convex when the safety level is large enough. We also provide tools to compute a concrete convexity certificate from nominal problem data. Our results are illustrated on several examples, including the situation of polyhedral systems with random technology matrices and arbitrary covariance structure.

This paper examines the convergence of no-regret learning in games with continuous action sets. For concreteness, we focus on learning via "dual averaging", a widely used class of no-regret learning schemes where players take small steps along their individual payoff gradients and then "mirror" the output back to their action sets. In terms of feedback, we assume that players can only estimate their payoff gradients up to a zero-mean error with bounded variance. To study the convergence of the induced sequence of play, we introduce the notion of variational stability, and we show that stable equilibria are locally attracting with high probability whereas globally stable equilibria are globally attracting with probability 1. We also discuss some applications to mixed-strategy learning in finite games, and we provide explicit estimates of the method's convergence speed., Comment: 36 pages, 2 figures; completely reworked structure of first version and dropped individual concavity assumptions

The Euler numbers occur in the Taylor expansion of tan ⁡ ( x ) + sec ⁡ ( x ) \tan (x)+\sec (x) . Since Stieltjes, continued fractions and Hankel determinants of the even Euler numbers, on the one hand, of the odd Euler numbers, on the other hand, have been widely studied separately. However, no Hankel determinants of the (mixed) Euler numbers have been obtained. The reason for that is that some Hankel determinants of the Euler numbers are null. This implies that the Jacobi continued fraction of the Euler numbers does not exist. In the present paper, this obstacle is bypassed by using the Hankel continued fraction, instead of the J J -fraction. Consequently, an explicit formula for the Hankel determinants of the Euler numbers is being derived, as well as a full list of Hankel continued fractions and Hankel determinants involving Euler numbers. Finally, a new q q -analog of the Euler numbers E n ( q ) E_n(q) based on our continued fraction is proposed. We obtain an explicit formula for E n ( − 1 ) E_n(-1) and prove a conjecture by R. J. Mathar on these numbers.

International audience; Our goal in this paper is to present a generalization of the spectral zeta regularization for general Feynman amplitudes on Riemannian manifolds. Our method uses complex powers of elliptic operators but involves several complex parameters in the spirit of the analytic renormalization by Speer, to build mathematical foundations for the renormalization of perturbative interacting quantum field theories. Our main result shows that spectrally regularized Feynman amplitudes admit an analytic continuation as meromorphic germs with linear poles in the sense of the works of Guo-Paycha and the second author. We also give an explicit determination of the affine hyper-planes supporting the poles. Our proof relies on suitable resolution of singularities of products of heat kernels to make them smooth. As an application of the analytic continuation result, we use a universal projection from mero-morphic germs with linear poles on holomorphic germs to construct renormalization maps which subtract singularities of Feynman amplitudes of Euclidean fields. Our renormalization maps are shown to satisfy consistency conditions previously introduced in the work of Nikolov-Todorov-Stora in the case of flat space-times.

In geophysics, the shallow water model is a good approximation of the incompressible Navier-Stokes system with free surface and it is widely used for its mathematical structure and its computational efficiency. However, applications of this model are restricted by two approximations under which it was derived, namely the hydrostatic pressure and the vertical averaging. Each approximation has been addressed separately in the literature: the first one was overcome by taking into account the hydrodynamic pressure (e.g. the non-hydrostatic or the Green-Naghdi models); the second one by proposing a multilayer version of the shallow water model. In the present paper, a hierarchy of new models is derived with a layerwise approach incorporating non-hydrostatic effects to approximate the Euler equations. To assess these models, we use a rigorous derivation process based on a Galerkin-type approximation along the vertical axis of the velocity field and the pressure, it is also proven that all of them satisfy an energy equality. In addition, we analyse the linear dispersion relation of these models and prove that the latter relations converge to the dispersion relation for the Euler equations when the number of layers goes to infinity. Ministerio de Economía y Competitividad MTM2015-70490-C2-2-R

In this paper we construct a family of cohomology classes on the moduli space of stable curves generalizing Witten's $r$-spin classes. They are parameterized by a phase space which has one extra dimension and in genus $0$ they correspond to the extended $r$-spin classes appearing in the computation of intersection numbers on the moduli space of open Riemann surfaces, while when restricted to the usual smaller phase space, they give in all genera the product of the top Hodge class by the $r$-spin class. They do not form a cohomological field theory, but a more general object which we call F-CohFT, since in genus $0$ it corresponds to a flat F-manifold. For $r=2$ we prove that the partition function of such F-CohFT gives a solution of the discrete KdV hierarchy. Moreover the same integrable system also appears as its double ramification hierarchy., Comment: v3: minor changes, 34 pages

International audience; In this paper we consider the transmission eigenvalue problem for Maxwell's equations corresponding to nonmagnetic inhomogeneities with contrast in electric permittivity that has fixed sign (only) in a neighborhood of the boundary. Following the analysis made by Robbiano in the scalar case we study this problem in the framework of semiclassical analysis and relate the transmission eigenvalues to the spectrum of a Hilbert-Schmidt operator. Under the additional assumption that the contrast is constant in a neighborhood of the boundary, we prove that the set of transmission eigenvalues is discrete, infinite and without finite accumulation points. A notion of generalized eigenfunctions is introduced and a denseness result is obtained in an appropriate solution space.; Nous considérons le problème de valeurs propres de transmission pour les équations de Maxwell où le contraste sur la permittivité électrique admet un signe fixe seulement sur un voisinage du bord du domaine. En suivant l’approche développée par Robbiano dans le cas scalaire, nous formulons le problème aux valeurs propres comme un problème spectral pour un opérateur d’Hilbert-Schmidt. Sous l’hypothèse supplémentaire que le contraste est constant au voisinage de la frontière nous montrons que l’ensemble des valeurs propres de transmission est discret sans points d’accumulation finis. Nous définissons une famille de fonctions propres généralisées pour laquelle un résultat de densité est obtenu dans un espace de solutions bien choisi.