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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.

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

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.

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.

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.

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

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.

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 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.

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 a previous paper, the authors have shown that Eilenberg's variety theorem can be extended to more general structures, called formations. In this paper, we give a general method to describe the languages corresponding to saturated formations of groups, which are widely studied in group theory. We recover in this way a number of known results about the languages corresponding to the classes of nilpotent groups, soluble groups and supersoluble groups. Our method also applies to new examples, like the class of groups having a Sylow tower.; Dans un article précédent, les auteurs avaient montré comment étendre le théorème des variétés d'Eilenberg à des structures plus générales, les formations. Dans cet article, nous donnons une méthode générale pour décrire les langages correspondant à des formations saturées de groupe, qui sont beaucoup étudiées en théorie des groupes. Nous retrouvons de cette façon les résultats connus sur les langages correspondant aux classes des groupes nilpotents, résolubles, et superrésolubles. Notre méthode s'applique aussi à de nouveaux exemples, telles que la classe des groupes ayant une tour de Sylow.

This paper studies the space of $BV^2$ planar curves endowed with the $BV^2$ Finsler metric over its tangent space of displacement vector fields. Such a space is of interest for applications in image processing and computer vision because it enables piecewise regular curves that undergo piecewise regular deformations, such as articulations. The main contribution of this paper is the proof of the existence of a shortest path between any two $BV^2$ curves for this Finsler metric. % The method of proof relies on the construction of a martingale on a space satisfying the Radon-Nikodym property and on the invariance under reparametrization of the Finsler metric. This method applies more generally to similar cases such as the space of curves with $H^k$ metrics for $k\geq 2$ integer. When $k \geq 2$ is integer, this space has a strong Riemannian structure and is geodesically complete. Thus, our result shows that the exponential map is surjective, which is complementary to geodesic completeness in infinite dimensions. We propose a finite element discretization of the minimal geodesic problem, and use a gradient descent method to compute a stationary point of a regularized energy. Numerical illustrations shows the qualitative difference between $BV^2$ and $H^2$ geodesics.

20 pages; International audience; In recent papers models of the human locomotion by means of an optimal control problem have been proposed. In this paradigm, the trajectories are assumed to be solutions of an optimal control problem whose cost has to be determined. The purpose of the present paper is to analyze the class of optimal control problems defined in this way. We prove strong convergence result for their solutions on the one hand for perturbations of the initial and final points (stability), and on the other hand for perturbations of the cost (robustness).

In a paper with Jean-Paul Dufour in 1999 \cite{DufourZung-Nambu1999}, we gave a classification of linear Nambu structures, and obtained linearization results for Nambu structures with a nondegenerate linear part. There was a case left open in \cite{DufourZung-Nambu1999}, namely the case of smooth linearization of Nambu structures with a Type 1 hyperbolic linear part which satisfies a natural signature condition. In this paper, we will show that such hyperbolic Nambu structures are also smoothly linearizable. We will also give a strong version of the analytic linearization theorem in the analytic case, improving a result obtained in \cite{DufourZung-Nambu1999}., 1st version, 12 pages

International audience; New investigations in the problems of polynomial stabilization are presented in this paper, where some relaxed results related to homogeneity theory leading to this polynomial stability with optimal decay rate are developed. To achieve our analysis, several physical examples are presented showing how we can construct stabilizing feedback laws making these closed loop systems polynomially stable with optimal decay rates. This allows the redesign of (a) homogeneous feedbacks stabilizing polynomially the Heisenberg system in weak sense, (b) and the polynomial observer for the angular momentum satellite with one control input.

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

In a recent paper, the authors proved that, under natural assumptions on the first marginal, the Monge problem in $ {\mathbb{R}^d} $ for the cost given by a general norm admits a solution. Although the basic idea of the proof is simple, it involves some complex technical results. Here we will give a proof of the result in the simpler case of a uniformly convex norm, and we will also use very recent results by Ahmad, Kim, and McCann. This allows us to reduce the technical burdens while still giving the main ideas of the general proof. The proof of the density of the transport set in the particular case considered in this paper is original. Bibliography: 22 titles.

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.

This paper explains the recent developments on the symplectic theory of Hamiltonian completely integrable systems on symplectic 4-manifolds, compact or not. One fundamental ingredient of these developments has been the understanding of singular affine structures. These developments make use of results obtained by many authors in the second half of the twentieth century, notably Arnold, Duistermaat and Eliasson, of which we also give a concise survey. As a motivation, we present a collection of remarkable results proven in the early and mid 1980s in the theory of Hamiltonian Lie group actions by Atiyah, Guillemin-Sternberg and Delzant among others, and which inspired many people, including the authors, to work on more general Hamiltonian systems. The paper concludes discussing a spectral conjecture for quantum integrable systems., 40 pages

International audience; The extended Lüroth's Theorem says that if the transcendence degree of $\KK(\mathsf{f}_1,\dots,\mathsf{f}_m)/\KK$ is 1 then there exists $f \in \KK(\underline{X})$ such that $\KK(\mathsf{f}_1,\dots,\mathsf{f}_m)$ is equal to $\KK(f)$. In this paper we show how to compute $f$ with a probabilistic algorithm. We also describe a probabilistic and a deterministic algorithm for the decomposition of multivariate rational functions. The probabilistic algorithms proposed in this paper are softly optimal when $n$ is fixed and $d$ tends to infinity. We also give an indecomposability test based on gcd computations and Newton's polytope. In the last section, we show that we get a polynomial time algorithm, with a minor modification in the exponential time decomposition algorithm proposed by Gutierez-Rubio-Sevilla in 2001.

In this paper we consider generalized eigenvalue problems for a family of operators with a quadratic dependence on a complex parameter. Our model is L ( λ ) = − △ + ( P ( x ) − λ ) 2 in L 2 ( R d ) where P is a positive elliptic polynomial in R d of degree m ⩾ 2 . It is known that for d even, or d = 1 , or d = 3 and m ⩾ 6 , there exist λ ∈ C and u ∈ L 2 ( R d ) , u ≠ 0 , such that L ( λ ) u = 0 . In this paper, we give a method to prove existence of non-trivial solutions for the equation L ( λ ) u = 0 , valid in every dimension d ⩾ 1 . This is a partial answer to a conjecture in Helffer, Robert and Xue Ping Wang (2004) [13] .

In this paper, we study the maximal stable domains on minimal and constant mean curvature 1 1 catenoids in hyperbolic space. In particular we investigate whether half-vertical catenoids are maximal stable domains (Lindelöf’s property). Our motivation comes from Lindelöf’s 1870 paper on catenoids in Euclidean space.

We consider the MEMS equation with fringing field ―Δu=λ(1+δ|∇u| 2 )(1―u) ―2 in Ω, u = 0 on ∂Ω, where λ, δ > 0 and Ω C ℝ n is a smooth and bounded domain. We show that when the fringing field exists (i.e. δ > 0), given any μ > 0, we have a uniform upper bound of classical solutions u away from the rupture level 1 for all λ ≥ μ. Moreover, there exists λ * δ > 0 such that there are at least two solutions when λ ∈ (0,λ * δ ); a unique solution exists when λ = λ * δ ; and there is no solution when λ λ * δ . This represents a dramatic change of behavior with respect to the zero fringing field case (i.e., δ = 0) and confirms the simulations in a paper by Pelesko and Driscoll as well as a paper by Lindsay and Ward.

55 pages; International audience; In this paper and its follow-up, we study the deformation theory of morphisms of properads and props thereby extending Quillen's deformation theory for commutative rings to a non-linear framework. The associated chain complex is endowed with an L_infinity-algebra structure. Its Maurer-Cartan elements correspond to deformed structures, which allows us to give a geometric interpretation of these results. To do so, we endow the category of prop(erad)s with a model category structure. We provide a complete study of models for prop(erad)s. A new effective method to make minimal models explicit, that extends the Koszul duality theory, is introduced and the associated notion is called homotopy Koszul. As a corollary, we obtain the (co)homology theories of (al)gebras over a prop(erad) and of homotopy (al)gebras as well. Their underlying chain complex is endowed with an L_infinity-algebra structure in general and a Lie algebra structure only in the Koszul case. In particular, we make the deformation complex of morphisms from the properad of associative bialgebras explicit. For any minimal model of this properad, the boundary map of this chain complex is shown to be the one defined by Gerstenhaber and Schack. As a corollary, this paper provides a complete proof of the existence of an L_infinity-algebra structure on the Gerstenhaber-Schack bicomplex associated to the deformations of associative bialgebras.

International audience; Interpolation inequalities play an essential role in Analysis with fundamental consequences in Mathematical Physics, Nonlinear Partial Differential Equations (PDEs), Markov Processes, etc., and have a wide range of applications in various other areas of Science. Research interests have evolved over the years: while mathematicians were originally focussed on abstract properties (for instance appropriate notions of functional spaces for the existence of weak solutions in PDEs), more qualitative questions (for instance, bifurcation diagrams, multiplicity of the solutions in PDEs and their qualitative behaviour) progressively emerged. The use of entropy methods in nonlinear PDEs is a typical example: in some cases, the optimal constant in the inequality can be interpreted as an optimal rate of decay of an entropy for an associated evolution equation. Much more has been learned by adopting this point of view. This paper aims at illustrating some of these recent aspect of entropyentropy production inequalities, with applications to stability in Gagliardo-Nirenberg-Sobolev inequalities and symmetry results in Caffarelli-Kohn-Nirenberg inequalities. Entropy methods provide a framework which relates nonlinear regimes with their linearized counterparts. This framework allows to prove optimality results, symmetry results and stability estimates. Some emphasis will be put on the hidden structure which explain such properties. Related open problems will be listed.

The paper introduces the notion of skew-evolution semiflows and presents the concept of pointwise trichotomy in the case of skew-evolution semiflows on a Banach space. The connection with the classic notion of trichotomy presented by us in a previous paper in 2006 for evolution operators, is also emphasized, as well as some characterizations. The approach of the theory is from uniform point of view. The study can also be extended to systems with control whose state evolution can be described by skew-evolution semiflows.

Isometric subgraphs of hypercubes are known as partial cubes. These graphs have first been investigated by Graham and Pollak [R.L Graham, H.Pollak On the addressing problem for loop switching, Bell System Technol. J. 50 (1971) 2495–2519] and Djokovic [D. Djokovic, Distance preserving subgraphs of the hypercubes, J. Combin. Theory, Ser B41 (1973), 263–267]. Several papers followed with various characterizations of partial cubes. In this paper, we determine all subdivisions of a given configuration which can be embedded isometrically in the hypercube. More specially, we deal with the case where this configuration is a connected graph of order 4 on one hand and the case where the configuration is a fan F k ( k ⩾ 3 ) on the other hand. Finally, we conjecture that a subdivision of a complete graph of order n ( n ⩾ 5 ) is a partial cube if and only if this one is isomorphic to S ( K n ) or there exists n − 1 edges of K n adjacent to a common vertex in the subdivision and the other edges of K n contain odd added vertices. This proposition is true when the order n ∈ { 4 , 5 , 6 } .

The attached file may be somewhat different from the published version; International audience; The aim of this paper is to provide the conditions necessary to reduce the complexity of state filtering for finite stochastic systems (FSSs). A concept of lumpability for FSSs is introduced. This paper asserts that the unnormalised filter for a lumped FSS has linear dynamics. Two sufficient conditions for such a lumpability property to hold are discussed. It is shown that the first condition is also necessary for the lumped FSS to have a linear dynamics. Next, it is proven that the second condition allows the filter of the original FSS to be directly obtained from the filter for the lumped FSS. Finally, the paper generalises an earlier published result for the approximation of a general FSS by a lumpable one.

The paper is devoted to initial boundary value problems for transport equations with non-divergence-free external field. The crucial role is played by integration along characteristics and associated Green's formula for which we provide a new proof which generalizes and clarifies previous versions. The paper concludes with an application of general theory to the Spencer–Lewis equation. Copyright © 2007 John Wiley & Sons, Ltd.

International audience; The purpose of this paper is to build sequences of suitably smooth approximate solutions to the Saint-Venant model that preserve the mathematical structure discovered in [D. Bresch, B. Desjardins, Comm. Math. Phys. 238 (1–2) (2003) 211–223]. The stability arguments in this paper then apply to such sequences of approximate solutions, which leads to the global existence of weak solutions for this model. Extension of this mollifying procedure to the case of compressible Navier–Stokes equations is also provided. Using the recent paper written by the authors, this provides global existence results of weak solutions for the barotropic Navier–Stokes equations and for compressible Navier–Stokes equations with heat conduction using a particular cold pressure term close to vacuum.

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).

We review the recent proof of the N.Takahashi's conjecture on genus $0$ Gromov-Witten invariants of $(\mathbb{P}^2, E)$, where $E$ is a smooth cubic curve in the complex projective plane $\mathbb{P}^2$. The main idea is the use of the algebraic notion of scattering diagram as a bridge between the world of Gromov-Witten invariants of $(\mathbb{P}^2, E)$ and the world of moduli spaces of coherent sheaves on $\mathbb{P}^2$. Using this bridge, the N.Takahashi's conjecture can be translated into a manageable question about moduli spaces of coherent sheaves on $\mathbb{P}^2$. This survey is based on a three hours lecture series given as part of the Beijing-Zurich moduli workshop in Beijing, 9-12 September 2019., Comment: Expository paper. 18 pages, 1 figure

In this paper we establish an unbounded version of the integral representation theorem by Buttazzo and Dal Maso [see Buttazzo Dal Maso (Nonlinear Anal 9(6):515–532, 1985) and also Bouchitte et al. (Arch Ration Mech Anal 165(3), 187–242, 2002)]. More precisely, we prove an integral representation theorem (with a formula for the integrand) for functionals defined on $$W^{1,p}$$ with $$p>N$$ (N being the dimension) that do not satisfy a standard p-growth condition from above and can take infinite values. Applications to $$\Gamma $$ -convergence, relaxation and homogenization are also developed.

In this paper, we provide asymptotic upper bounds on the complexity in two (closely related) situations. We confirm for the total doubling coverings and not only for the chains the expected bounds of the form [Formula: see text] This is done in a rather general setting, i.e. for the [Formula: see text]-complement of a polynomial zero-level hypersurface [Formula: see text] and for the regular level hypersurfaces [Formula: see text] themselves with no assumptions on the singularities of [Formula: see text]. The coefficient [Formula: see text] is the ambient dimension [Formula: see text] in the first case and [Formula: see text] in the second case. However, the question of a uniform behavior of the coefficient [Formula: see text] remains open. As a second theme, we confirm in arbitrary dimension the upper bound for the number of a-charts covering a real semi-algebraic set [Formula: see text] of dimension [Formula: see text] away from the [Formula: see text]-neighborhood of a lower dimensional set [Formula: see text], with bound of the form [Formula: see text] holding uniformly in the complexity of [Formula: see text]. We also show an analogue for level sets with parameter away from the [Formula: see text]-neighborhood of a low dimensional set. More generally, the bounds are obtained also for real subanalytic and real power-subanalytic sets.

In this paper, we consider hashing with linear probing for a hashing table with m places, n items (n > m), and l = m - n empty places. For a noncomputer science-minded reader, we shall use the metaphore of n cars parking on m places: each car ci chooses a place pi at random, and if pi is occupied, ci tries successively pi + 1, pi + 2, until it finds an empty place. Pittel [42] proves that when l/m goes to some positive limit β > 1, the size B1m,l1 of the largest block of consecutive cars satisfies 2(β - 1 - log β)B1m,l - 3 log log m + Ξm, where Ξm converges weakly to an extreme-value distribution. In this paper we examine at which level for n a phase transition occurs between B1m,l = o(m) and m - B1m,l = o(m). The intermediate case reveals an interesting behavior of sizes of blocks, related to the standard additive coalescent in the same way as the sizes of connected components of the random graph are related to the multiplicative coalescent.

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 goal of this paper is the study of simple rank 2 parabolic vector bundles over a 2 2 -punctured elliptic curve C C . We show that the moduli space of these bundles is a non-separated gluing of two charts isomorphic to P 1 × P 1 \mathbb {P}^1 \times \mathbb {P}^1 . We also showcase a special curve Γ \Gamma isomorphic to C C embedded in this space, and this way we prove a Torelli theorem. This moduli space is related to the moduli space of semistable parabolic bundles over P 1 \mathbb {P}^1 via a modular map which turns out to be the 2:1 cover ramified in Γ \Gamma . We recover the geometry of del Pezzo surfaces of degree 4 and we reconstruct all their automorphisms via elementary transformations of parabolic vector bundles.

After Hölder proved his classical theorem about the Gamma function, there has been a whole bunch of results showing that solutions to linear difference equations tend to be hypertranscendental (i.e., they cannot be solution to an algebraic differential equation). In this paper, we obtain the first complete results for solutions to general linear difference equations associated with the shift operator x ↦ x + h x\mapsto x+h ( h ∈ C ∗ h\in \mathbb {C}^* ), the q q -difference operator x ↦ q x x\mapsto qx ( q ∈ C ∗ q\in \mathbb {C}^* not a root of unity), and the Mahler operator x ↦ x p x\mapsto x^p ( p ≥ 2 p\geq 2 integer). The only restriction is that we constrain our solutions to be expressed as (possibly ramified) Laurent series in the variable x x with complex coefficients (or in the variable 1 / x 1/x in some special case associated with the shift operator). Our proof is based on the parametrized difference Galois theory initiated by Hardouin and Singer. We also deduce from our main result a general statement about algebraic independence of values of Mahler functions and their derivatives at algebraic points.

International audience; The main result of this paper, Theorem 1.4, establishes a conjecture of Lyons and Peres: for a determinantal point process governed by a self-adjoint reproducing kernel, the system of kernels sampled at the points of a random configuration is complete in the range of the kernel. A key step in the proof, Lemma 1.9, states that conditioning on the configuration in a subset preserves the determinantal property, and the main Lemma 1.10 is a new local property for kernels of conditional point processes. In Theorem 1.6 we prove the triviality of the tail σ-algebra for determinantal point processes governed by self-adjoint kernels.

In this paper we prove a convergence result for a discretization of the three-dimensional stationary compressible Navier–Stokes equations assuming an ideal gas pressure law $p(\rho )=a \rho ^{\gamma }$ with $\gamma> \frac{3}{2}$. It is the first convergence result for a numerical method with adiabatic exponents $\gamma $ less than $3$ when the space dimension is 3. The considered numerical scheme combines finite volume techniques for the convection with the Crouzeix–Raviart finite element for the diffusion. A linearized version of the scheme is implemented in the industrial software CALIF3S developed by the French Institut de Radioprotection et de Sûreté Nucléaire.

International audience; In this paper we present a derivation of a back-scatter rotational Large Eddy Simulation model, which is the extension of the Baldwin and Lomax model to non-equilibrium problems. The model is particularly designed to mathematically describe a fluid filling a domain with solid walls and consequently the differential operators appearing in the smoothing terms are degenerate at the boundary. After the derivation of the model, we prove some of the mathematical properties coming from the weighted energy estimates and which allow to prove existence and uniqueness of a class of regular weak solutions.

International audience; In this paper, we study topological properties of the right action by translation of the Weyl Chamber flow on the space of Weyl chambers. We obtain a necessary and sufficient condition for topological mixing. (1)

In this paper, we give a very general criterion for elimination of imaginaries using an abstract independent relation. We also study germs of definable functions at certain well-behaved invariant types. Finally we apply these tools to the elimination of imaginaries in bounded pseudo-p-adically closed fields., Comment: Minor corrections. Acknowledgement of support by ValCoMo (ANR-13-BS01-0006) added

We study a single-valued integration pairing between differential forms and dual differential forms which subsumes some classical constructions in mathematics and physics. It can be interpreted as a $p$-adic period pairing at the infinite prime. The single-valued integration pairing is defined by transporting the action of complex conjugation from singular to de Rham cohomology via the comparison isomorphism. We show how quite general families of period integrals admit canonical single-valued versions and prove some general formulas for them. This implies an elementary 'double copy' formula expressing certain singular volume integrals over the complex points of a smooth projective variety as a quadratic expression in ordinary period integrals of half the dimension. We provide several examples, including non-holomorphic modular forms, archimedean N\'{e}ron-Tate heights on curves, single-valued multiple zeta values and polylogarithms. In a sequel to this paper we apply this formalism to the moduli space of curves of genus zero with marked points, to deduce a recent conjecture due to Stieberger in string perturbation theory, which states that closed string amplitudes are the single-valued projections of open string amplitudes., Comment: The results are unchanged. We made some minor corrections and added several remarks and clarifications

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.

International audience; This paper is devoted to the decomposition of vectors into sampled complex exponentials; or, equivalently, to the information over discrete measures captured in a finite sequence of their Fourier coefficients. We study existence, uniqueness, and cardinality properties, as well as computational aspects of estimation using convex semidefinite programs. We then explore optimal transport between measures, of which only a finite sequence of Fourier coefficients is known.

We are interested in the large-time behaviour of solutions to finite volume discretizations of convection–diffusion equations or systems endowed with nonhomogeneous Dirichlet- and Neumann-type boundary conditions. Our results concern various linear and nonlinear models such as Fokker–Planck equations, porous media equations or drift–diffusion systems for semiconductors. For all of these models, some relative entropy principle is satisfied and implies exponential decay to the stationary state. In this paper we show that in the framework of finite volume schemes on orthogonal meshes, a large class of two-point monotone fluxes preserves this exponential decay of the discrete solution to the discrete steady state of the scheme. This includes for instance upwind and centred convections or Scharfetter–Gummel discretizations. We illustrate our theoretical results on several numerical test cases.