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International audience; In a previous paper, we have described the construction of an automaton from a rational expression which has the property that the automaton built from an expression which is itself computed from a co-deterministic automaton by the state elimination method is co-deterministic. It turned out that the definition on which the construction is based was inappropriate, and thus the proof of the property was flawed. We give here the correct definition of the broken derived terms of an expression which allow to define the automaton and the detailed full proof of the property.

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

In this short paper, we construct a unipotent nearby cycle functor and show a p-adic analogue of Beilinson’s equivalence comparing two derived categories: the derived category of holonomic arithmetic $${\mathcal {D}}$$ -modules and the derived category of arithmetic $${\mathcal {D}}$$ -modules whose cohomologies are holonomic.

International audience; In this short paper we prove that the almost Schur theorem, introduced by De Lellis and Topping, is true on 4-dimensional Riemannian manifolds of nonnegative scalar curvature and discuss some related problems on other dimensional manifolds.

International audience; Recently, Iwahori-Hecke algebras were associated to Kac-Moody groups over non-Archimedean local fields. In a previous paper, we introduced principal series representations for these algebras and partially generalized Kato's irreducibility criterion. In this paper, we study how some of these representations decompose when they are reducible and deduce information on the irreducible representations of these algebras.

In this paper we study the geodesic flow for a particular class of Riemannian non-compact manifolds with variable pinched negative sectional curvature. For a sequence of invariant measures we are able to prove results relating the loss of mass and bounds on the measure entropies. We compute the entropy contribution of the cusps. We develop and study the corresponding thermodynamic formalism. We obtain certain regularity results for the pressure of a class of potentials. We prove that the pressure is real analytic until it undergoes a phase transition, after which it becomes constant. Our techniques are based on the one side on symbolic methods and Markov partitions and on the other on geometric techniques and approximation properties at level of groups., 34 pages, 4 figures. In this new version we have improved the organization of the paper and the clarity of some statements

The paper is devoted to generic translation flows corresponding to Abelian differentials on flat surfaces of arbitrary genus $g\ge 2$. These flows are weakly mixing by the Avila-Forni theorem. In genus 2, the Hölder property for the spectral measures of these flows was established in our papers [10,12]. Recently Forni [17], motivated by [10], obtained Hölder estimates for spectral measures in the case of surfaces of arbitrary genus. Here we combine Forni's idea with the symbolic approach of [10] and prove Hölder regularity for spectral measures of flows on random Markov compacta, in particular, for translation flows in all genera., Corrects the formulation of Proposition 5.4 in the published version (reversed order of quantifiers); the results and proofs are unchanged, with a minor rewording

In this paper, we study the algebraic relations satisfied by the solutions of q-difference equations and their transforms with respect to an auxiliary operator. Our main tools are the parametrized Galois theories developed in Hardouin and Singer (Math Ann 342(2):333–377, 2008) and Ovchinnikov and Wibmer (Int Math Res Not 12:3962–4018, 2015). The first part of this paper is concerned with the case where the auxiliary operator is a derivation, whereas the second part deals with a $$\mathbf {q}$$ -difference operator. In both cases, we give criteria to guarantee the algebraic independence of a series, solution of a q-difference equation, with either its successive derivatives or its $$\mathbf {q}$$ -transforms. We apply our results to q-hypergeometric series.

We present a solution of the algebraic version of Birkhoff Conjecture on integrable billiards. Namely we show that every polynomially integrable real bounded convex planar billiard with smooth boundary is an ellipse. We extend this result to billiards with piecewise-smooth and not necessarily convex boundary on arbitrary two-dimensional surface of constant curvature: plane, sphere, Lobachevsky (hyperbolic) plane; each of them being modeled as a plane or a (pseudo-) sphere in $\mathbb R^3$ equipped with appropriate quadratic form. Namely, we show that a billiard is polynomially integrable, if and only if its boundary is a union of confocal conical arcs and appropriate geodesic segments. We also present a complexification of these results. These are joint results of Mikhail Bialy, Andrey Mironov and the author. The proof is split into two parts. The first part is given by Bialy and Mironov in their two joint papers. They considered the tautological projection of the boundary to $\mathbb{RP}^2$ and studied its orthogonal-polar dual curve, which is piecewise algebraic, by S.V.Bolotin's theorem. By their arguments and another Bolotin's theorem, it suffices to show that each non-linear complex irreducible component of the dual curve is a conic. They have proved that all its singularities and inflection points (if any) lie in the projectivized zero locus of the corresponding quadratic form on $\mathbb C^3$. The present paper provides the second part of the proof: we show that each above irreducible component is a conic and finish the solution of the Algebraic Birkhoff Conjecture in constant curvature., To appear in the Journal of the European Mathematical Society (JEMS), 69 pages, 2 figures. A shorter proof of Theorem 4.24. Minor precisions and misprint corrections

We prove a long exact sequence in KK-theory for both full and reduced amalgamated free products in the presence of conditional expectations. In the course of the proof, we established the KK-equivalence between the full amalgamated free product of two unital C*-algebras and a newly defined reduced amalgamated free product that is valid even for non GNS-faithful conditional expectations. Our results unify, simplify and generalize all the previous results obtained before by Cuntz, Germain and Thomsen., V.3, the paper has been splitted into two papers, this is the first part on amalgamated free products

International audience; This paper is devoted to the numerical analysis of a numerical scheme dedicated to the simulation of front advection (see https://hal.archives-ouvertes.fr/hal-02940407v1 for a preprint version presenting this scheme and some numerical results). The latter has been recently proposed and it is based on the ideas used for the Glimm's scheme. It relies on a two-step approach: a convection step is followed by a projection step which is based on a random choice. The main advantage of this scheme is that it applies to multi-dimensional problems. In the present paper a convergence result for this scheme is provided for a particular class of multi-dimensional problems. This work has been accepted for publication in IMA Journal of Numerical Analysis: https://doi.org/10.1093/imanum/drab053.

This paper is devoted to the abstract study of operational second-order differential equations of elliptic type with nonregular coefficient-operator boundary conditions in a non-commutative framework. The study is performed when the second member f belongs to $$L^{p}(0,1;X)$$, with general $$p\in ]1,+\infty [$$, X being a UMD Banach space. We give some new results by using semigroups and interpolation theory. Existence, uniqueness and optimal regularity of the classical and semi-classical solution are proved. This paper improves naturally the ones studied in the commutative case by Hammou et al. (Mediterr J Math, 1669–1683, 2015).

In this paper we are interested in the large time behavior of linear kinetic equations with heavy-tailed local equilibria. Our main contribution concerns the kinetic Lévy-Fokker-Planck equation, for which we adapt hypocoercivity techniques in order to show that solutions converge exponentially fast to the global equilibrium. Compared to the classical kinetic Fokker-Planck equation, the issues here concern the lack of symmetry of the non-local Lévy-Fokker-Planck operator and the understanding of its regularization properties. As a complementary related result, we also treat the case of the heavy-tailed BGK equation.; In this paper we are interested in the large time behavior of linear kinetic equations with heavy-tailed local equilib-ria. Our main contribution concerns the kinetic Lévy-Fokker-Planck equation, for which we adapt hypocoercivity techniques in order to show that solutions converge exponentially fast to the global equilibrium. Compared to the classical kinetic Fokker-Planck equation, the issues here concern the lack of symmetry of the non-local Lévy-Fokker-Planck operator and the understanding of its regularization properties. As a complementary related result, we also treat the case of the heavy-tailed BGK equation. Résumé Une note sur l'hypocoercivité pour leséquations cinétiques avecéquilibresà queue lourde. Dans cet article, on s'intéresse au comportement en temps long d'équations cinétiques linéaires dont leséquilibres locaux sontà queue lourde. Notre contribution principale concerne l'équation de Lévy-Fokker-Planck cinétique, pour laquelle nous adaptons des techniques d'hypocoercivité afin de démontrer la convergence exponentielle des solutions vers unéquilibre global. En comparant au cas de l'équation de Fokker-Planck cinétique classique, les enjeux ici sont liés au manque de symétrie de l'opérateur non-local de Lévy-Fokker-Planck età la compréhension de ses propriétés de régularisation. En complément de notre analyse, nous traitonségalement le cas de l'équation de BGKà queue lourde.

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.

International audience; In general, the group decompositions are formulated by employing automorphisms and semidirect products to determine continuity and compactification properties. This paper proposes a set of constructions of novel topological decompositions of groups and analyzes the behaviour of group actions under the topological decompositions. The proposed topological decompositions arise in two varieties, such as decomposition based on topological fibers without projections and decomposition in the presence of translated projections in topological spaces. The first variety of decomposition introduces the concepts of topological fibers, locality of group operation and the partitioned local homeomorphism resulting in formulation of transitions and symmetric surjection within the topologically decomposed groups. The reformation of kernel under decomposed homeomorphism and the stability of group action with the existence of a fixed point are analyzed. The first variety of decomposition does not require commutativity maintaining generality. The second variety of projective topological decomposition is formulated considering commutative as well as noncommutative projections in spaces. The effects of finite translations of topologically decomposed groups under projections are analyzed. Moreover, the embedding of a decomposed group in normal topological spaces is formulated in this paper. It is shown that Schoenflies homeomorphic embeddings preserve group homeomorphism in the decomposed embeddings within normal topological spaces. This paper illustrates that decomposed group embedding in normal topological spaces is separable. The applications aspects as well as parametric comparison of group decompositions based on topology, direct product and semidirect product are included in the paper.

Any germ of a complex analytic space is equipped with two natural metrics: the {\it outer metric} induced by the hermitian metric of the ambient space and the {\it inner metric}, which is the associated riemannian metric on the germ. We show that minimal surface singularities are Lipschitz normally embedded (LNE), i.e., the identity map is a bilipschitz homeomorphism between outer and inner metrics, and that they are the only rational surface singularities with this property., This paper is a major revision of the 2015 version. It now builds on the paper arXiv:1806.11240 by the same authors which gives a general characterization of Lipschitz normally embedded surface singularities

International audience; We show that an order antimorphism on a finite-dimensional cone having no one-dimensional factors is homogeneous of degree −1. A consequence is that the existence of an order antimorphism on a finite-dimensional cone implies that the cone is a symmetric cone.

In this paper, we give explicit rates in the central limit theorem and in the almost sure invariance principle for general R d {\mathbb R}^d -valued cocycles that appear in the study of the left random walk on linear groups. Our method of proof lies on a suitable martingale approximation and on a careful estimation of some coupling coefficients linked with the underlying Markov structure. Concerning the martingale part, the available results in the literature are not accurate enough to give almost optimal rates either in the central limit theorem for the Wasserstein distance, or in the strong approximation. A part of this paper is devoted to circumvent this issue. We then exhibit near optimal rates both in the central limit theorem in terms of the Wasserstein distance and in the almost sure invariance principle for R d {\mathbb R}^d -valued martingales with stationary increments having moments of order p ∈ ( 2 , 3 ] p \in (2, 3] (the case of sequences of reversed martingale differences is also considered). Note also that, as an application of our results for general R d {\mathbb R}^d -valued cocycles, a special attention is paid to the Iwasawa cocycle and the Cartan projection for reductive Lie groups (like for instance G L d ( R ) {\mathrm {GL}}_d(\mathbb {R}) ).

International audience; The aim of this paper is to present a new approach, based on singular volume integral equations and a perturbative approach in order to compute the size dependency of plasmonic resonances of metallic nanoparticles when the quasistatic regime is no longer valid. The paper also provides rigorous derivations of the extinction and absorption cross sections for elliptical particles.; Cet article présente une nouvelle méthode pour l'étude de la dépendance des fréquences de résonances plasmoniques des nano-particules métalliques dans un régime où l'approximation quasistatique n'est plus valide. Ces travaux reposent sur l'étude du spectre d'un opérateur integral singulier ainsi que sur l'application d'une méthode perturbative.

Topological classification of even the simplest Morse-Smale diffeomorphisms on 3-manifolds does not fit into the concept of singling out a skeleton consisting of stable and unstable manifolds of periodic orbits. The reason for this lies primarily in the possible "wild" behaviour of separatrices of saddle points. Another difference between Morse-Smale diffeomorphisms in dimension 3 from their surface analogues lies in the variety of heteroclinic intersections: a connected component of such an intersection may be not only a point as in the two-dimensional case, but also a curve, compact or non-compact. The problem of a topological classification of Morse-Smale cascades on 3-manifolds either without heteroclinic points (gradient-like cascades) or without heteroclinic curves was solved in a series of papers from 2000 to 2006 by Ch. Bonatti, V. Grines, F. Laudenbach, V. Medvedev, E. Pecou, O. Pochinka. The present paper is devoted to a complete topological classification of the set $MS(M^3)$ of orientation preserving Morse-Smale diffeomorphisms $f$ given on smooth closed orientable 3-manifolds $M^3$. A complete topological invariant for a diffeomorphism $f\in MS(M^3)$ is an equivalent class of its scheme $S_f$, which contains an information on a periodic date and a topology of embedding of two-dimensional invariant manifolds of the saddle periodic points of $f$ into the ambient manifold., Comment: 48 pages, 15 figures

In this paper, we define, a priori, a natural two-dimensional model for a time-dependent flexural shell. As expected, this model takes the form of a set of hyperbolic variational equations posed over the space of admissible linearized inextensional displacements, and a set of initial conditions. Using a classical argument, we prove that the model under consideration admits a unique strong solution. However, the latter strategy makes use of function spaces, which are not amenable for numerically approximating the solution. We thus provide an alternate formulation of the studied problem using a suitable penalty scheme, which is more suitable in the context of numerical approximations. For the sake of completeness, in the final part of the paper, we also provide an existence and uniqueness theorem for the case where the linearly elastic shell under consideration is an elliptic membrane shell.

The main subject of this paper is equilibrium problems on an unbounded conductor $\Sigma$ of the complex plane in the presence of a weakly admissible external field. An admissible external field $Q$ on $\Sigma$ satisfies, along with other mild conditions, the following growth property at infinity: $$\lim_{|x| \rightarrow \infty}(Q(x) - \log |x|) = +\infty.$$ This condition guarantees the existence and uniqueness of the equilibrium measure in the presence of $Q$, and the compactness of its support. In the last 10-15 years, several papers have dealt with weakly admissible external fields, in the sense that $Q$ satisfies a weaker condition at infinity, namely, $$\exists M\in(-\infty,\infty],\quad\liminf_{|x| \rightarrow \infty}(Q(x) - \log |x|) = M.$$ Under this last assumption, there still exists a unique equilibrium measure in the external field $Q$, but the support need not be a compact subset of $\Sigma$ anymore. In most examples considered in the literature the support is indeed unbounded. Our main goal in this paper is to illustrate this topic by means of a simple class of external fields on the real axis created by a pair of attractive and repellent charges in the complex plane, and to study the dynamics of the associated equilibrium measures as the strength of the charges evolves. As one of our findings, we exhibit configurations where the support of the equilibrium measure in a weakly admissible external field is a compact subset of the real axis. To achieve our goal, we extend some results from potential theory, known for admissible external fields, to the weakly admissible case. These new results may be of independent interest. Finally, the so--called signed equilibrium measure is an important tool in our analysis. Its relationship with the (positive) equilibrium measure is also explored., Comment: To appear in Journal of Approximation Theory

Many papers are devoted to study logarithmic sheaves associated to reduced divisors, in particular logarithmic bundles associated to plane curves since forty years in differential and algebraic topology or geometry. An interesting family of these curves are the so-called free ones for which the associated logarithmic sheaf is the direct sum of two line bundles. When the curve is a finite set of distinct lines (i.e. a line arrangement), Terao conjectured thirty years ago that its freeness depends only on its combinatorics. A lot of efforts were done to prove it but at this time it is only proved up to 12 lines. If one wants to find a counter example to this conjecture a new family of curves arises naturally: the nearly free curves introduced by Dimca and Sticlaru. We prove here that the logarithmic bundle associated to a nearly free curve possesses a minimal non zero section that vanishes on one single point $P$, called jumping point, and that characterizes the bundle. Then we give a precise description of the behaviour of $P$. In particular we show, based on detailed examples, that the position of $P$ relatively to its corresponding nearly free arrangement of lines may or may not be a combinatorial invariant, depending on the chosen combinatorics., 24 pages, Comments are welcome. We improve the exposition of the paper with more details

International audience; In my paper at Inventiones 2009, we determine the cohomology of Lubin-Tate spaces globally using the comparison theorem of Berkovich by computing the fibers at supersingular points of the perverse sheaf of vanishing cycle Ψ of some Shimura variety of Kottwitz-Harris-Taylor type. The most difficult argument deals with the control of maps of the spectral sequences computing the sheaf cohomology of both Harris-Taylor perverse sheaves and those of Ψ. In this paper, we bypass these difficulties using the classical theory of representations of the mirabolic group and a simple geometric argument.; Dans [2], on détermine les groupes de cohomologie des espaces de Lubin-Tate par voie globale en calculant les fibres des faisceaux de cohomologie du faisceau pervers des cyclesévanescentscyclesévanescents Ψ d'une variété de Shimura de type Kottwitz-Harris-Taylor. L'ingrédient le plus complexe consistè a contrôler lesfì eches de deux suites spectrales calculant l'une les faisceaux de cohomologie des faisceaux pervers d'Harris-Taylor, et l'autre ceux de Ψ. Dans cet article, nous contournons ces difficultés en utilisant la théorie classique des représentations du groupe mirabolique ainsi qu'un argument géométrique simple. Abstract (Mirabolic group, ramified Newton stratification and cohomology of Lubin-Tate spaces) In [2], we determine the cohomology of Lubin-Tate spaces globally using the comparison theorem of Berkovich by computing the fibers at supersingular points of the perverse sheaf of vanishing cycle Ψ of some Shimura variety of Kottwitz-Harris-Taylor type. The most difficult argument deals with the control of maps of the spectral sequences computing the sheaf coho-mology of both Harris-Taylor perverse sheaves and those of Ψ. In this paper, we bypass these difficulties using the classical theory of representations of the mirabolic group and a simple geometric argument.

DOI: https://doi.org/10.1090/memo/1235; International audience; A wandering domain for a diffeomorphism is an open connected set whose iterates are pairwise disjoint. We endow A^n = T^n x R^n with its usual exact symplectic structure. An integrable diffeomorphism Φ^h, i.e. the time-one map of a Hamiltonian h which depends only on the action variables, has no nonempty wandering domains. The aim of this paper is to estimate the size (measure and Gromov capacity) of wandering domains in the case of an exact symplectic perturbation of Φ^h , in the analytic or Gevrey category. Upper estimates are related to Nekhoroshev theory, lower estimates are related to examples of Arnold diffusion. This is a contribution to the "quantitative Hamiltonian perturbation theory" initiated in previous works on the optimality of long term stability estimates and diffusion times; our emphasis here is on discrete systems because this is the natural setting to study wandering domains. We first prove that the measure (or the capacity) of these wandering domains is exponentially small, with an upper bound of the form exp(-c ε^(-1/2nα)), where ε is the size of the perturbation, α is the Gevrey exponent (α is at least 1, α=1 for analytic systems) and c is some positive constant depending mildly on h. This is obtained as a consequence of an exponential stability theorem for near-integrable exact symplectic maps, in the analytic or Gevrey category, for which we give a complete proof based on the most recent improvements of Nekhoroshev theory for Hamiltonian flows, and which requires the development of specific Gevrey suspension techniques. The second part of the paper is devoted to the construction of near-integrable Gevrey systems possessing wandering domains, for which the capacity (and thus the measure) can be estimated from below. We suppose that n is at least 2, essentially because KAM theory precludes Arnold diffusion in too low a dimension. For any α>1, we produce examples with lower bounds of the form exp(-c ε^(-1/2(n-1)(α-1))). This is done by means of a "coupling" technique, involving rescaled standard maps possessing wandering discs in A and near-integrable systems possessing periodic domains of arbitrarily large periods in A^(n-1). The most difficult part of the construction consists in obtaining a perturbed pendulum-like system on A with periodic islands of arbitrarily large periods, whose areas are explicitly estimated from below. Our proof is based on a version due to Herman of the translated curve theorem.

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

The class A of countable groups that admit a faithful, transitive, amenable –in the sense that there is an invariant mean– action on a set has been widely investigated in the past. In this paper, we no longer require the action to be transitive, but we ask for it to preserve a locally finite metric (and still to be faithful and amenable). The groups having such actions are those that embed into a totally disconnected amenable locally compact group. Then we focus on the subclass A 1 of groups for which the actions are moreover transitive. This class is strictly contained into A and includes non-amenable groups. An important particular case of actions preserving a locally finite metric is given by actions by automorphisms of locally finite connected graphs. We take this opportunity, in our partly expository paper, to review some nice results about amenable actions in this setting.

51 pages, in French, v2: extended introduction, journal version; International audience; The aim of this paper is to prove a generalization of the famous Theorem A of Quillen for strict $\infty$-categories. This result is central to the homotopy theory of strict $\infty$-categories developed by the authors. The proof presented here is of a simplicial nature and uses Steiner's theory of augmented directed complexes. In a subsequent paper, we will prove the same result by purely $\infty$-categorical methods.

International audience; We deal with a severe ill posed problem, namely the reconstruction process of an image during tomography acquisition with (very) few views. We present different methods that we have been investigated during the past decade. They are based on variational analysis. This is a survey paper and we refer to the quoted papers for more details. Mathematics Subject Classification (2010). 49K40, 45Q05,65M32.

International audience; The last years have seen a growing interest from mathematicians in Mahler functions. This class of functions includes the generating series of the automatic sequences. The present paper is concerned with the following problem, which is rather frequently encountered in combinatorics: a set of Mahler functions u 1 , ..., un being given, are u 1 , ..., un and their successive derivatives algebraically independent? In this paper, we give general criteria ensuring an affirmative answer to this question. We apply our main results to the generating series attached to the so-called Baum-Sweet and Rudin-Shapiro automatic sequences. In particular, we show that these series are hyperalge-braically independent, i.e., that these series and their successive derivatives are algebraically independent. Our approach relies of the parametrized difference Galois theory (in this context, the algebro-differential relations between the solutions of a given Mahler equation are reflected by a linear differential algebraic group).

This is a continuation of a previous paper by the same authors. In the former paper, it was proved that in order to obtain local uniformization for valuations centered on local domains, it is enough to prove it for rank one valuations. In this paper, we extend this result to the case of valuations centered on rings which are not necessarily integral domains and may even contain nilpotents.

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.

This paper deals with a non-local parabolic equation of Lotka-Volterra type that describes the evolution of phenotypically structured populations. Nonlinearities appear in these systems to model interactions and competition phenomena leading to selection. In this paper, the equation on the structured population is coupled with a differential equation on the nutrient concentration that changes as the total population varies. Different methods aimed at showing the convergence of the solutions to a moving Dirac mass are reviewed. Using either weak or strong regularity assumptions, the authors study the concentration of the solution. To this end, BV estimates in time on appropriate quantities are stated, and a constrained Hamilton-Jacobi equation to identify where the solutions concentrates as Dirac masses is derived.

In this paper we consider the optimal mass transport problem for relativistic cost functions, introduced in [12] as a generalization of the relativistic heat cost. A typical example of such a cost function is c t ( x , y ) = h ( y − x t ) , h being a strictly convex function when the variable lies on a given ball, and infinite otherwise. It has been already proved that, for every t larger than some critical time T > 0 , existence and uniqueness of optimal maps hold; nonetheless, the existence of a Kantorovich potential is known only under quite restrictive assumptions. Moreover, the total cost corresponding to time t has been only proved to be a decreasing right-continuous function of t. In this paper, we extend the existence of Kantorovich potentials to a much broader setting, and we show that the total cost is a continuous function. To obtain both results the two main crucial steps are a refined “chain lemma” and the result that, for t > T , the points moving at maximal distance are negligible for the optimal plan.

International audience; In this paper, we consider a vibrating system of Timoshenko-type in a bounded one-dimensional domain with discrete time delay and complementary frictional damping and infinite memory controls all acting on the transversal displacement. We show that the system is well-posed in the sens of semigroup and that, under appropriate assumptions on the weights of the delay and the history data, the stability of the system holds in case of the equal-speed propagation as well as in the opposite case in spite of the presence of a discrete time delay, where the decay rate of solutions is given in terms of the smoothness of the initial data and the growth of the relaxation kernel at infinity. The results of this paper extend the ones obtained by the present author and Messaoudi in (Acta Math Sci 36:1–33, 2016) to the case of presence of discrete delay.

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"

In 1967, Schmidt wrote a seminal paper (Schmidt in Ann Math 85:430–472, 1967) on heights of subspaces of $$\mathbb {R}^n$$ or $$\mathbb {C}^n$$ defined over a number field K, and diophantine approximation problems. The going-down Theorem— one of the main theorems he proved in his paper—remains valid in two cases depending on whether the embedding of K in the complex field $$\mathbb {C}$$ is a real or a complex non-real embedding. For the latter, and more generally as soon as K is not totally real, at some point of the proof, the arguments in Schmidt (Ann Math 85:430–472, 1967) do not exactly work as announced. In this note, Schmidt’s ideas are worked out in details and his proof of the complex case is presented, solving the aforementioned problem. Some definitions of Schmidt are reformulated in terms of multilinear algebra and wedge product, following the approaches of Laurent (Analytic number theory, essays in honour of Klaus Roth. Cambridge University Press, Cambridge, pp 306–314, 2009), Bugeaud and Laurent (Mathematische Zeitschrift 265(2):249–262, 2010) and Roy (Journal de theorie des nombres de Bordeaux 27:591–603, 2015), Roy (Mathematische Zeitschrift 283(1–2):143–155, 2016). In Laurent (Analytic number theory, essays in honour of Klaus Roth. Cambridge University Press, Cambridge, pp 306–314, 2009) Laurent introduces in the case $$K=\mathbb {Q}$$ a family of exponents and he gives a series of inequalities relating them. In Sect. 5 these exponents are defined for an arbitrary number field K. Using the going-up and the going-down Theorems Laurent’s inequalities are generalized to this setting.

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.

In this paper we refine the main result of a previous paper of the author with Grimeland on derived invariants of surface algebras. We restrict to the case where the surface is a torus with one boundary component and give an easily computable derived invariant for such surface algebras. This result permits to give answers to open questions on gentle algebras: it provides examples of gentle algebras with the same AG-invariant (in the sense of Avella-Alaminos and Geiss) that are not derived equivalent and gives a partial positive answer to a conjecture due to Bobi\'nski and Malicki on gentle $2$-cycles algebras., Comment: 22 pages, a mistake concerning the computation of the mapping class group has been fixed, version 3: 25 pages, to appear in Algebras and Representation Theory

This is an expository paper on the theory of gradient flows, and in particular of those PDEs which can be interpreted as gradient flows for the Wasserstein metric on the space of probability measures (a distance induced by optimal transport). The starting point is the Euclidean theory, and then its generalization to metric spaces, according to the work of Ambrosio, Gigli and Savar{\'e}. Then comes an independent exposition of the Wasserstein theory, with a short introduction to the optimal transport tools that are needed and to the notion of geodesic convexity, followed by a precise desciption of the Jordan-Kinderleher-Otto scheme, with proof of convergence in the easiest case: the linear Fokker-Planck equation. A discussion of other gradient flows PDEs and of numerical methods based on these ideas is also provided. The paper ends with a new, theoretical, development, due to Ambrosio, Gigli, Savar{\'e}, Kuwada and Ohta: the study of the heat flow in metric measure spaces.

In 1980, Gizatullin classified rational surfaces endowed with an automorphism whose action on the Neron-Severi group is parabolic: these surfaces are endowed with an elliptic fibration invariant by the automorphism. The aim of this expository paper is to present for non-experts the details of Gizatullin's original proof, and to provide an introduction to a recent paper by Cantat and Dolgachev., Minor corrections

This paper is chiefly concerned with qualitative properties of some reaction–diffusion fronts. The recently defined notions of transition fronts generalize the standard notions of traveling fronts. In this paper, we show the existence and the uniqueness of the global mean speed of bistable transition fronts in R N . This speed is proved to be independent of the shape of the level sets of the fronts. The planar fronts are also characterized in the more general class of almost-planar fronts with any number of transition layers. These qualitative properties show the robustness of the notions of transition fronts. But we also prove the existence of new types of transition fronts in R N that are not standard traveling fronts, thus showing that the notions of transition fronts are broad enough to include other relevant propagating solutions.

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.

International audience; This paper is about surfaces of infinite topological type. Unlike the case of surfaces of finite type, there are several deformation spaces associated with a surface of infinite topological type. Such spaces depend on the choice of a basepoint (that is, the choice of a fixed conformal structure or hyperbolic structure on the surface) and they also depend on the choice of a distance on the set of equivalence classes of marked hyperbolic structures. We address the question of the comparison between two deformation spaces, namely, the quasiconformal Teichmüller space and the length-spectrum Teichmüller space. There is a natural inclusion map of the quasiconformal space into the length-spectrum space, which is not always surjective.We work under the hypothesis that the basepoint (a hyperbolic surface) satisfies a condition we call ``upper-boundedness''. This means that this surface admits a pants decomposition defined by curves whose lengths are bounded above. The theory under this upper-boundedness hypothesis shows a dichotomy. On the one hand there are surfaces satisfying what we call Shiga's condition, i.e. they admit a pants decomposition defined by curves whose lengths are bounded above and below. If the base point satisfies Shiga's condition, then the inclusion of the quasiconformal space into the length-spectrum space is surjective, and it is a homeomorphism. In this paper we concentrate on the other kind of upper-bounded surfaces, which we call ``upper-bounded with short interior curves''. This means that the corresponding hyperbolic surface admits a pants decomposition defined by curves whose lengths are bounded above, and such that the lenghts of some interior curves approach zero. We show that in this case the behavior is completely different. Under this hypothesis, the image of the inclusion between the two Teichmüller spaces is nowhere dense in the length-spectrum space. As a corollary of the methods used, we obtain an explicit parametrization of the length-spectrum Teichmüller space in terms of Fenchel-Nielsen coordinates and we prove that the length-spectrum Teichmüller space is path-connected.

In this paper, we obtain sharp Strichartz estimates for solutions of the wave equation $\square_\gg\phi=0$ where $\gg$ is a rough Lorentzian metric on a 4 dimensional space-time $\MM$. This is the last step of the proof of the bounded $L^2$ curvature conjecture proposed in [3], and solved by S. Klainerman, I. Rodnianski and the author in [8], which also relies on the sequence of papers [16][17][18][19]. Obtaining such estimates is at the core of the low regularity well-posedness theory for quasilinear wave equations. The difficulty is intimately connected to the regularity of the Eikonal equation $\gg^{\a\b}\pr_\a u\pr_\b u=0$ for a rough metric $\gg$. In order to be consistent with the final goal of proving the bounded $L^2$ curvature conjecture, we prove Strichartz estimates for all admissible Strichartz pairs under minimal regularity assumptions on the solutions of the Eikonal equation., Comment: 30 pages, 5 figures